Je remercie les membres du jury non seulement d'avoir accepté d'en faire partie, mais également pour leur vif intérêt vis à vis de mon projet de thèse, leur bienveillance au cours de ma soutenance et tout particulièrement pour les discussions constructives en découlant.

Un grand merci à Nicolas Holzschuch et Pauline Martinetto, mes directeur et co-directrice de thèse. Merci pour votre précieux soutien sans faille tout au long de ces trois (et quelques) années. Merci de m'avoir fait confiance pour mener à bien ce projet et de m'avoir aidé notamment au cours de nos déjà regrettées réunions hebdomadaires. Je vous suis profondément reconnaissante de m'avoir tant appris, mais également pour votre sollicitude dans les moments potentiellement difficiles. Travailler à vos côtés fut si agréable et enrichissant que je ne pourrais qu'espèrer être amenée à collaborer à nouveau avec vous deux.

Je me dois de nommer celles et ceux avec qui j'ai eu la chance de collaborer. Merci à Alain Ibanez (Institut Néel) et Simon Giraud pour avoir synthétisé cette si belle palette de poudre d'hématite sur laquelle repose toute mon étude, à Corinne Félix (Institut Néel) pour ses conseils et son aide concernant la conception du banc de mesure de la rétro-diffusion, ainsi qu'à Bernard Schmitt (IPAG) pour l'accès à SHADOWS et son expertise quant à l'interprétation des résultats. Ce fut non seulement une opportunité de travailler avec vous, une nécessité sous peine de ne pas donner à mes recherches la direction souhaitée, mais également un véritable plaisir.

Plus généralement je tiens à remercier les membres des deux laboratoires m'ayant accueilli en leur sein. J'en profite pour également remercier les membres de Patrimalp, ce fut un véritable plaisir d'apporter ma pierre à ce magnifique projet, et d'échanger avec vous sur mes avancées mais également sur les vôtres.

Je souhaite également remercier les quelques compagnons de galère que j'ai pu rencontrer d'un labo à l'autre. Ceux de PLUM ou presque : Laurie, Marco, Alexandra et Juan pour les moments rafraîchissants à vos côtés. Et ceux de Maverick : mes colocataires de bureau Vincent et Nolan, Ronak, et Sunrise pour une quantité de choses que je n'aurais la place de lister.

Un grand merci à mes amis de Grenoble et d'ailleurs m'ayant supporté tout au long de cette thèse. Parmis eux, Cyrielle et Ayoub pour des pauses cafés non réglementées, et Clément pour un final inoubliable.

Les "vrais" sauront se reconnaître : Un énorme merci à Jovanie et Pierre, premiers partenaires du meilleur comité jamais créé, Hippolyte pour les sorties ciné et quelques débats enflammés. Et, bien sûr, évidemment, merci Clothilde pour ton inestimable soutien de presque tous les jours.

Enfin, merci à ma famille, mes parents et mes soeurs, pour son soutien inconditionnel depuis toujours.

7

Abstract

Parietal paintings are subject to natural and anthropogenic degradations. Although they may find themselves weakened to the point of becoming almost illegible, their broadcasting to the general public remains of historical, societal and cultural importance. Thus, the chemical and structural characterization, at the microscopic level, of the materials that were used to make these paintings, is necessary to reproduce their visual appearance. A digital reproduction through photorealistic images synthesis will validate their observed appearance, and promote their dissemination.

If the coloring matters used on parietal paintings were applied with a binder, there is no trace of it left nowadays which allows us to describe these paintings as a dry pigment layer on top of a rock substrate. Numerous shades of red ochre implying the presence of hematite α-Fe 2 O 3 are observed in parietal paintings. In order to limit the study to this chromophore, we chose to synthesize several samples through routes allowing to control the nucleation and growth of hematite crystals in solution under standard or hydrothermal conditions. These were chosen from those described in the literature so as to obtain colored powders ranging from orange to purple, comparable in terms of their morphology and/or their colors to those observed on parietal paintings.

We used several characterization methods (X-ray powder diffraction, optical microscopy and scanning electron microscopy) in order to study the chemical composition of synthetic hematite crystalline powders, as well as the morphological and microstructural parameters of their grains. The color of the different hematite powders is characterized by reflectance, measured by spectroscopy. The colorimetric analysis of these results makes it possible to deduce a correlation between the color and the size of the grains composing the powder.

Computer graphics is involved in order to accurately capture the appearance of objects through photo-realistic images rendering. As a first approximation, a paint is considered to be a thin, semitransparent uniform layer on a substrate. Thus we developed a Bidirectional Reflectance Distribution function (BRDF) model that takes into account multiple reflections and attenuation of light within the layer and implemented it in Mitsuba renderer. This model has been experimentally validated, but requires to be complexified in order to take into account the light scattering introduced by the presence of grains randomly dispersed within a more realistic paint layer.

The BRDF measurement of dry and optically thick pigment layers using spectro-goniophotometry allows a finer characterization of their appearance. We identified three scattering phenomena occurring at the surface or in the volume of the sample as responsible for their appearance : specular scattering, back-scattering and volume scattering. Several analytical models are defined so as to simulate each of these contributions as a function of the morphology of the grains of the studied powders.

If the modelizations carried out are efficient in the case of a single pigment, a more realistic approach will be to study the case of mixtures, ochres and natural samples.

Résumé

Les peintures pariétales sont sujettes à des dégradations naturelles ou anthropiques. Bien qu'elles s'en trouvent fragilisées au point de devenir presque illisibles, leur diffusion auprès du grand public reste d'une importance à la fois historique, sociétale et culturelle. Ainsi, la caractérisation chimique et structurale, au niveau microscopique, des matériaux ayant permis la réalisation de ces peintures, est nécessaire pour restituer leur apparence visuelle. Une restitution numérique par la synthèse d'images photo-réalistes permettra de valider l'apparence observée de ces peintures, et d'en favoriser la diffusion.

Si les matières colorantes utilisées en art pariétal furent appliquées à l'aide d'un éventuel liant, il n'en reste aucune trace aujourd'hui ce qui permet de décrire une telle oeuvre comme une couche pigmentaire sèche sur un substrat rocheux. De nombreuses teintes d'ocres rouges impliquant la présence d'hématite α-Fe 2 O 3 sont observées sur les peintures rupestres. Afin de limiter l'étude à ce chromophore, plusieurs poudres d'hématite ont été synthétisées par des voies de synthèses permettant de contrôler la nucléation et la croissance des cristaux en solution dans des conditions standards ou hydrothermales. Celles-ci ont été choisies parmi celles décrites dans la littérature de manière à obtenir une palette de poudres colorées allant de l'orangé au pourpre, parmi lesquelles certaines sont comparables de part leur morphologie et/ou leur teinte à celles observées sur des peintures pariétales.

Plusieurs méthodes de caractérisations sont employées (diffraction de rayons X sur poudre, microscopie optique et microscopie électronique à balayage) afin d'étudier la composition chimique des poudres cristallines d'hématite synthétique, ainsi que les paramètres morphologiques et microstructuraux de leurs grains. La couleur des différentes poudres d'hématite est caractérisée par la réflectance, mesurée par spectroscopie. Une analyse colorimétrique de ces résultats permet de mettre en évidence une corrélation entre la teinte et la taille des grains composant la poudre.

L'informatique graphique entre en jeu dans le but de fidèlement rendre compte de l'apparence d'objets par le biais de la synthèse d'images photo-réalistes. En première approximation, une peinture est considérée comme une couche mince uniforme semi-transparente sur un substrat. Ainsi un modèle de Bidirectional Reflectance Distribution function (BRDF) prenant en compte les réflexions multiples et l'atténuation de la lumière au sein de la couche a été développé et implémenté sous Mitsuba. Ce modèle est validé expérimentalement, mais nécessite d'être complexifié afin de prendre en compte les phénomènes de diffusion de la lumière introduits par la présence de grains aléatoirement dispersés au sein d'une couche de peinture plus réaliste.

La mesure de la BRDF par spectro-gonio-photométrie sur des couches pigmentaires sèches et optiquement épaisses permet de mieux en caractériser l'apparence. Trois phénomènes de diffusion intervenant à la surface ou dans le volume de l'échantillon sont identifiés comme responsables de leur apparence : diffusion spéculaire, rétro-diffusion et diffusion volumique. Plusieurs modèles analytiques sont définis de manière à simuler chacune de ces contributions en fonction de la morphologie des grains des poudres étudiées.

Si les modélisations effectuées sont efficaces dans le cas d'un unique pigment, une approche plus réaliste consistera à étudier le cas de mélanges, d'ocres et d'échantillons naturels.

Introduction

Parietal paintings are objects from our cultural heritage that have been degraded over the years by both environmental and anthropogenic causes, so that some of them are nowadays almost illegible. This can be characterized by the alterations occurring on the rock substrate, including its disaggregation, or the detachment of small pieces of the wall, but also by the alterations of the paint layer. Indeed, time, weathering and human activities are responsible for the peeling off, the erasure and possible color changing of the original paintings. In the objective to preserve such paintings and to give access to the general public, its reproduction through photo-realistic image renderings would be a real asset. It might also allow to predict the original appearance of these paintings at the time they have been made, and to estimate the alterations' effects through time. In order to proceed to such a restitution, it is necessary to know about the physico-chemical properties of the materials involved in the composition of the paintings, and to understand their impact on the physical phenomena at stake in the overall appearance.

Only a limited range of colors are observed on parietal paintings. Apart from black that will not be studied in this work, we can mainly distinguish yellowish and reddish hues. The coloring matters that are identified on the parietal paintings are powders of ferruginous rocks that have been obtained and altered through numerous processes such as mineralization, sedimentary, weathering or volcanic processes [START_REF] Chalmin | How to distinguish red coloring matter used in prehistoric time ? the contribution of visible near-infrared diffuse reflectance spectroscopy[END_REF]. Ochres formed after weathering with mixtures of clays and various minerals are among the most commonly found. Most of its constituents are colorless or white (quartz, calcite, dolomite...) but it includes iron oxides or hydroxides responsible for the coloration of the powder. Depending on the chemical composition of this chromophore, different colors are observed : the yellow hues are consequences of the presence of goethite FeO(OH) while the red hues going from light red to purple involve hematite α-Fe 2 O 3 , on which we will focus in this work. We choose to focus on hematite because it shows completely different appearance between a bulk crystal and a powder. Not only the surface appearance which goes from smooth and shiny for an hematite crystal to matte and quite uniform when milled into powder, but also the drastic color changes from dark grey to several reddish hues. Thus, the appearance and therefore the color also depend on the morphology of the grains composing the powder. When making a painting, the coloring matter may have been dispersed into a binder such as water to ease its application on the rock wall. Traces of organic binder such as animal fat or eggs are identified in parietal paintings from the Upper Paleolithic to the Neolithic [START_REF] Casoli | Research on the organic binders in archaeological wall paintings[END_REF][START_REF] Rampazzi | Characterisation of wall paintings in the sos furrighesos necropolis (anela, italy)[END_REF]. Yet it is not always noticeable, and it is not the main component of the painting responsible for its color. Hence the appearance of a parietal painting boils down to the appearance a dry pigment deposited as a thin layer on top of a rock substrate.

There are multiple non destructive physico-chemical characterization methods to fully investigate, from microscopic scale to macroscopic scale, the chemical composition, the structural properties and the appearance of powder materials. However some of them can only be performed in laboratories which requires taking a significant sample from the artwork. We have hematite crystalline powders, synthesized so as to obtain several shades of red similar to those observed on the paintings. This limitation to only one crystalline phase also allows a better appreciation of the color variations that a single mineral can reach. The appearance of dry powders is complex to describe because it involves multiple physical phe-CONTENTS nomena that depend on the nature, the size and the morphology of the grains. In a bulk material of known chemical composition, absorption is the main phenomenon that is responsible for perceived color. Light scattering at the surface of the material may also contributes to its appearance. But for powders, because it is not a homogeneous crystal but a collection of grains that are randomly organized and quite close to each other, several scattering phenomena also contribute to the coloration of the material. This may happen at the surface of the powder layer accounting for light diffraction by the surface asperities, and in the volume of the layer accounting for the interactions between the grains. In this work we are aiming to connect the appearance of dry powders to the chemical composition, the morphological and microstructural parameters of their grains. This includes the measurement of the Bidirectionnal Reflectance Distribution function (BRDF) describing the appearance of a material, the identification of the physical phenomena it involves, and their analytical modelization as functions of the powders' physico-chemical properties. Although we did it by studying the particular case of pure hematite α-Fe 2 O 3 crystalline powders, a similar and relevant approach could be followed on any other powder material. Dry powders are very commonly found in our world. They not only appear as pigments in paintings, but also in cosmetics (eyeshadows, blush), in food (spices), or in soils (dusts). Computer graphics would benefit from this modelization to accurately reproduce the appearance of fragile parietal paintings and therefore ease their dissemination to the general public. In a more general way, this would give the opportunity to render photo-realistic images of powder materials in other contexts than cultural heritage.

In this manuscript we report the steps we carried out in order to fulfill these objectives as follows : In chapter I we introduce the physical concepts that are necessary to the description and the modelization of material appearance. As our work is focused on the appearance of dry inorganic pigments, we highlight the phenomena responsible for the coloration of minerals, especially the one that are at stake in hematite α-Fe 2 O 3 . The hematite crystalline powder samples on which I performed this study are also introduced. They are either natural or synthesized in such a way that different sizes of grains and shades of red going from light red to purple, mimicking the colors observed on parietal paintings are obtained.

In chapter II, a first approximation of a paint layer as an absorptive homogeneous and smooth thin film on top of a substrate is investigated. The appearance of such a material involves the attenuation of light as it travels across the layer, a contribution of the substrate seen by transparency, and interference phenomena caused by multiple reflections between the film's interfaces. I propose a modelization that can be used to reproduce the appearance of these semi-transparent thin films through photo realistic rendering. It is validated by comparing the simulation results with experiments on comparable real materials.

The chapter III focuses on the case of the one specific pigment responsible for the various shades of red observed on parietal paintings : hematite α-Fe 2 O 3 . Many hematite crystalline powders, either synthetic or natural, are characterized using non destructive analytical techniques in order to study their chemical composition, their color, as well as the morphological and micro structural parameters of their grains. This study leads to the determination of a correlation between the color variation and the size of the grains composing the powders, that cannot be modelized using traditional Mie or Rayleigh scattering theories.

We investigate in more details the appearance of dry powders in the chapter IV. We describe both the measurement of the BRDF and its modelization. Multiple scattering phenomena, occurring both at the surface of the sample and inside its volume are identified from these measurements. They are first modelized using existing appearance models, modified to be representative of powders. I designed a new model taht produces better results. As it takes into account morphological parameters of the grains composing the powder, it is able to reproduce their appearance with a good accuracy.

In chapter V we discuss and illustrate the limitations of our model to simulate the appearance of parietal paintings observed on actual archaeological sites with preliminary results on more complex powders, mixing several mineral phases such as ochres.

According to these results, other parameters than the mineral composition or the size of the powder's grains would need to be further investigated. We conclude this manuscript with suggestions for future works that would be of interest for a better understanding and a more realistic modelization of the parietal paintings' appearance.

Chapter I

State of the Art

The appearance of a material is first described using perceptual concepts, characterizing its color, the matte or glossy look of its surface, as well as its transparency or opacity. Physically, it is the result of the optical phenomena occurring as light is reflected or transmitted by its surface. It mainly corresponds to scattering, absorption or combination of these phenomena. The analytical modelization of material's appearance is widely studied in computer graphics to create photorealistic images used in many domains such as video games or visual effects. In this chapter we both introduce some basic appearance concepts and their modelizations that will be essential for the rest of our study.

Since we are mainly interested in pigments or colored dry powders, it is also necessary to focus our investigation on the apperance of these specific materials. This requires the full understanding of the coloring processes involved, and will be discussed in the particular case of hematite.

Finally, our study is experimentally based in order for our appearance modelizations to closely match to measurements performed on real samples. For this reason we introduce numerous optical characterization methods that can be applied on powders to infer their physical and chemical properties.

I.1 Appearance of materials

The perceived appearance of an object is described using various attributes such as its shape, its color, its shininess, its surface texture, its transparency... These parameters depend not only on the properties of the material the object is made of, but also on the environment under which it is illuminated and observed ( [START_REF] Dorsey | Digital modeling of material appearance[END_REF] [START_REF] Glassner | Principles of digital image synthesis[END_REF]). For example the same object seen under two different illuminants can show two very different colors. In the same way the perceived appearance may changed with the observer experience or color blindness.

Here, we focus on the material's appearance, involving its intrinsic physical and chemical properties. As light interacts with a material it undergoes physical phenomena either in reflection, transmission or absorption, that are responsible for its appearance. The fraction of incident flux that is reflected, transmitted or absorbed by a given element is then respectively introduce as the Reflectance R, the Transmittance T and the Absorbance A. These quantities are linked to the material chemical composition and physical properties and may depend on both the incoming light direction and the observation direction. They are dimensionless, and for energy conservation they satisfy the following condition : R + T + A = 1 (I.1)

We will investigate each one of them in further details in the next sections.

As some perceptual insights are mandatory for the perceived color description, we will also introduce main colorimetry concepts.

I.1.1 Scattering

Let's first discuss in more details the physical phenomena that can be accounted in the reflectance R. We define as θ i the direction of the incoming ray of light on a material.

In the case of a perfect mirror, light is reflected in exactly one specific direction known as the specular direction (-θ i ). The reflectance is then maximum in this direction and null elsewhere. For more common materials, the light is mainly reflected towards the favored specular direction but also in a set of different directions all around it. Light is said to be scattered and can then be described in any direction by an angular reflectance function. For a perfect diffuser, also called a lambertian material, the scattered flux is the same in every direction of the observation hemisphere, for whatever incident direction. The visual appearance of these different kind of scattering is illustrated on the Figure I.1.

We distinguish surface scattering from volume scattering. The former is occurring at the rough interface between two materials and is responsible for its matte or shiny aspect : the rougher the surface of the material is, the matter it appears to be. This correlation has been highlighted and first modelized by Torrance & Sparrow [START_REF] Torrance | Off-specular peaks in the directional distribution of reflected thermal radiation[END_REF][START_REF] Torrance | Polarization, directional distribution, and off-specular peak phenomena in light reflected from roughened surfaces[END_REF]106]. Other modelizations are introduced in the Sections I.2.2 and I.2.3. Surface scattering can also impact the transmittance of a material, which results in the blurring of the image seen through the object, as seen on the Figure I.2. FIGURE I.1 -Surface scattering in reflexion : the flower pots shows different scattering depending on the material it is made of. On the left : flower pot made of paper. The light is scattered homogeneously in every direction giving the uniform look. In the middle : flower pot made of aluminium foil (reverse side). The light is reflected in a large range of directions, but not uniformly. On the right : flower pot made of aluminium foil. The light is only reflected in a sharp direction.

Scattering effects can also occur inside a material. This happens in participating media, media that are hosting a collection of small scattering elements named scatterers, such as droplets in clouds, proteins in milk, particles in smoke, or pigments in a binder for paintings. When interacting with such materials the light undergoes multiple reflections on different scatterers before it goes out of the medium. It is a phenomenon that impacts both the reflectance and the transmittance, and that is responsible for the translucency of materials such as marble, skin, or wax for example.

I.1.2 Transparency or opacity

Both the transmittance T and the absorbance A permits us to define a material as transparent or opaque. These two quantities are highly connected to intrinsic optical properties of materials defined by their optical constant η. For material representation, we usually separate between transparent and opaque materials in the visible range [START_REF] Haines | Real-time rendering[END_REF]. Dielectric materials are fully transparent to light in the visible range and have a real optical index η = n, also called refractive index, corresponding to the ratio between the speed of light inside the material and the speed of light in the vacuum (Figure I.3a). Only the transmittance is at stake in such cases (A = 0).

On the other hand, conductors have a complex optical constant with a non-null imaginary part [START_REF] Born | Principles of optics : electromagnetic theory of propagation, interference and diffraction of light[END_REF] : η = n + ik (Figure I.3b). The optical constant depends on the wavelength and this variation is responsible for the color of the material. Physically, the imaginary part of the optical constant is called the extinction coefficient and is connected to the attenuation of electromagnetic waves inside the material through an exponential falloff (Section I. 1.3). Conductors then involve a non null absorbance, and for most practical uses these materials can be treated as opaque (T = 0). This binary distinction is incorrect since, because of the wavelength dependency of the optical constants, there are also materials that are both transparent and absorptive (T = 0 and A = 0). This is the case for dielectric materials colored using dyes, or for semi-conductors whose extinction coefficient may be null over part of the visible spectrum as shown for hematite on the Figure I.3c.

Let's enumerate some physical phenomena that can be involved in the transparent appearance of a material. First there is refraction, which is responsible for the distortion of an image seen through a transparent object [START_REF] Fleming | Visual perception of thick transparent materials[END_REF]. For objects with complex shape, a combination of refraction and reflection phenomena at the interfaces can results in the creation of caustics [START_REF] Lynch | Color and light in nature[END_REF]. Colored phenomena can also be observed for more particular shapes such as light dispersion with prisms or interferences with thin layers [START_REF] Hirayama | Visualization of optical phenomena caused by multilayer films based on wave optics[END_REF], [START_REF] Belcour | A practical extension to microfacet theory for the modeling of varying iridescence[END_REF]. The later are quite common objects : windows, treatments on lenses, varnish coating on paintings... The potential substrate on top of which it is applied then contributes to the appearance of such an object. All of these phenomena can be described using geometric optics, as it will be shown later in this work (Sections I.2.1.a, I.2.1.b, II.2).

I.1.3 Absorption

The absorbance accounts for the light that is attenuated through an exponential falloff as it travels inside the object according to the extinction coefficient : the larger the imaginary part of the optical constant is, the faster the light intensity decreases. This is known as the absorption phenomenon and is described by the Beer Lambert's law [50].

Assuming that light goes along an optical path length x through a medium containing a uniform concentration c of one chemical species, it is absorbed with a fraction A = xc where is the absorptivity of the absorbing species. A is the absorbance of the medium, and is closely related to the Transmittance T by the relationship T = 10 -A . In the case of a particle suspension with a density N , the law involves the attenuation coefficient which is linked to the extinction coefficient by the relationship α = 4πk λ . Beer Lambert's law can then be written : A = N α ln [START_REF] Belcour | Bidirectional reflectance distribution function measurements and analysis of retroreflective materials[END_REF] x and T = e -N αx (I.2)

Note that for an absorptive homogeneous bulk of thickness x, the transmittance is T = e -αx . The law can be extended to multiple chemical species, by summing the respective absorbance of each one of them.

I.1.4 Color

The color of a material is a perceptual concept consequence of the scattering, transparency and absorption effects mentionned earlier. It depends on the illuminant, and on the observer experience [118]. In this study we will consider a normal observer as defined by the International Commission on Illumination (CIE) for whom only light with wavelengths in the visible spectrum from 380 nm to 780 nm can be perceived by a normal human eye [START_REF] Colorimetry | [END_REF]. Color can be quantified using different representations.

I.1.4.a Spectrum representation :

The more accurate representation of a color is its spectrum, that is to say its decomposition in a set of monochromatic wavelengths. It pictures ranges of wavelengths, known as absorption bands, that are consequence of electronic exchanges in the atomic structure of the material and for which the light therefore tends to be more absorbed. Thereby, the spectrum acts as a signature of the material under study.

However, such a representation is too complex to actually be representative of the human vision. The human eye only has three types of cones sensitive to specific stimulus in order to per-ceive color. Only three parameters should be enough to describe a color that we are able to perceive.

I.1.4.b Trichromatic representation :

A much simpler representation consists of reproducing a spectral color using three light stimuli called primaries. Two main sets of primaries are defined : Red, Blue, Green (RGB) which is used for display devices, and Cyan, Yellow, Magenta (CYM) for printing systems.

Not all of the colors a human eye perceive can be described using this representation [START_REF] Colorimetry | [END_REF]118]. The set of reproducible colors of a display or a printing device is called the color gamut and is represented on Figure I.4. A fourth primary, black (K) is often added to the CYM representation in order to extend its gamut.

There is one tristimulus representation that accounts for any color that a normal observer can see, using imaginary primaries XYZ. Let's consider a material lightened by a spectral stimuli S(λ) whose perceived color is given by the spectrum R(λ). Then the coordinates in the XYZ colors space are : X, Y, Z = 1 K x(λ), ȳ(λ), z(λ)R(λ)S(λ)dλ where K = ȳ(λ)S(λ)dλ (I.3) with x, ȳ and z the color matching functions empirically determined for a standard 10 • observer [START_REF] Colorimetry | [END_REF]. ȳ is chosen so that it is equal to the human eye spectral sensitivity.

I.1.4.c Color appearance model : CIE L*a*b* color space :

A color appearance model is a mathematical model that intends to represent colors as they are perceived by a human observer in a given context. We will focus on the CIE L*a*b* because it is the color model that accounts the most faithfully for the human eye's perception [START_REF] Colorimetry | [END_REF]118].

The color is described by three coordinates : the lightness L * describes if the color is either clear (L * = 100) or dark (L * = 0). The position on a green-red axis is given by the coordinate a * , and the position of a blue-yellow axis is given by the coordinate b * . [X n , Y n , Z n ] are the coordinates of the illuminant in the virtual color space XYZ. We choose the D65 illuminant with spectrum S(λ) and XYZ coordinates [0.95, 1.0, 1.09] to reproduce the observation conditions in a natural environment [START_REF] Colorimetry | [END_REF].

       L * = 116 f (Y/Y n ) -16 a * = 500[ f (X/X n ) -f (Y/Y n )] b * = 200[ f (Y/Y n ) -f (Z/Z n )] with f (t ) =      t 1/3 , if t > 6
Other colorimetric parameters are determined from color coordinates in that color space :

• The hue, describing the angle position on the color diagram. For example, this parameter permits to distinguish between a reddish color and a greenish one. It is defined as follows : All the reachable colors are contained in the corresponding polygon. The colors outside of the polygons can be perceived by a human eye but cannot be reproduced using these color representations. The CMYK gamut is obtained for Fogra39 characterization data set [3].

h * ab = arctan
In order to compare a perceived color to another one, metrics ∆E have been developed and reviewed to better take into account the non uniform perception of a human eye. If the difference ∆E is smaller than a threshold called Just Noticeable Difference, then the two compared colors cannot be perceptually differentiate. The simplest definition is the euclidean distance between two colors' coordinates :

∆E 76 = (L * 1 -L * 2 ) 2 + (a * 1 -a * 2 ) 2 + (b * 1 -b * 2 ) 2 (I.7)
with a just noticeable difference of 2.3. More accurate definitions accounting for perceptual nonuniformities ∆E 94 and ∆E 00 have a just noticeable difference of 1.0.

Color appearance models reduces the color analysis to only three coordinates. It is a much simpler but less accurate representation than the spectrum. We should especially pay attention to metamerism cases where two different spectral stimuli have the same coordinates in the color space. An example from Akbarinia et al. [START_REF] Akbarinia | Color metamerism and the structure of illuminant space[END_REF] is given on the Figure I.6 : Two yellow flowers with different reflectance spectra appear to be of similar color when seen under a street lamp lighting. The use of a different illuminant highlights that the colors are actually different from one another.

The representation using a color appearance model can mislead the color interpretation in case of metamerism, but it remains a powerful and straightforward model that we will use in this work to quantify the color of the studied materials.

I.1.4.d Optical constant :

The optical constant of a material is an intrisic complex parameter, which is essential for the study of its appearance, since it permits to describe the transparency and the color of the material. It is written as :

η = n + i k (I.8)
Its real part corresponds to the refractive index n and its imaginary part k describes the absorption power of the material.

Usually, at least in computer graphics, it is assumed to be a well known parameter that is often referenced in large databases such as RefractiveIndex.info. However we will be confronted to uncertainties that may be introduced by a poor knowledge of the optical constant at multiple times in this work.

Let's introduce some techniques that are used to compute or to measure the optical constant of a material.

There some theoretical models that permit to determined the optical constant of a material, based on the study of its electronic structure : the Lorentz-Drude model [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF] as well as its improvement known as the Brendel-Bordmann model [START_REF] Brendel | An infrared dielectric function model for amorphous solids[END_REF], that approximates each electron as a harmonic oscillator. Another model, the Density functional theory (DFT) allows through quantum considerations to determine the electronic structure of a material and so its optical constants [START_REF] Werner | Optical constants and inelastic electron-scattering data for 17 elemental metals[END_REF].

Despite these models, the actual measurement of the optical constant of the material under study is necessarily more reliable. As far as we know, two main experimental methods can be applied to measure the optical constants of a bulk crystal [START_REF] Callet | Couleur et apparence visuelle-l'aspect métallique[END_REF].

• By ellipsometry :

This optical technique is based on the change of light polarization after its reflection on a planar surface. In practice, we measure the ratio ρ = r p /r s , where r p and r s are the Fresnel's reflection coefficients of light respectively polarized parallel and perpendicularly to the incident plane (see Section I.2.1.a). Let's denote θ i the incident angle on the material, η i the complex optical constant of the host medium (air) and η the one of the material under study. The later can then be expressed as :

η = η i sin θ i 1 + 1 -ρ 1 + ρ 2 tan 2 θ o (I.9)
Both the refractive index n and the absorption coefficient k are deduced by the identification of the real and imaginary part of such expression.

• By spectroscopy :

The real and imaginary parts of a material's optical constant are actually linked to one another [START_REF] Toll | Causality and the dispersion relation : logical foundations[END_REF]. Another measurement method is to deduce one of them from spectroscopy, and then deduce the second one using mathematical relationships.

A first method which only requires the measure of the reflectance was proposed by Königsberger (1913) : the reflectances R a and R h are measured on one crystal immersed in two different host media with respective refractive indices n a and n h . The real and the imaginary parts of the optical constant can be calculated using the following equations :

               n = (n a /2)(n 2 h -n a ) n h ( n a +R h n a -R h ) - n a +R a n a -R a k = 2 n n a + R a n a -R a - n 2 + n a n 2 (I.10a) (I.10b)
This method was the first method proposed to measure the optical constants of material and is quite convenient for absorptive materials since the transmittance is not needed. Another more recent, reliable and common method is the Kramers-Kronig's analysis [START_REF] Kronig | On the theory of dispersion of x-rays[END_REF][START_REF] Kramers | La diffusion de la lumiere par les atomes[END_REF]. One of the two components (n or k) of the complex optical index is retrieved from a Reflectance or Transmittance measured spectrum by inversion of the Fresnel's formula I.2.1.a, while the other component (k or n) is computed using the Kramers-Kronig's relationships :

         n(λ) ∝ ∞ 0 k(λ ) λ -λ d λ k(λ) ∝ ∞ 0 n(λ ) λ -λ d λ (I.11a) (I.11b)
In theory, it requires the knowledge of the reflectance (or transmittance) over the whole electromagnetic spectrum. In practice this can not be measured and may introduce some errors in the optical constant estimation.

The previous methods are used on homogeneous bulk crystals, and not on powder samples. Because the optical constant is supposed to be intrinsic to the material, its values for a powder should be the same as its values measured for a bulk. However, some study assert that the same material as a bulk and as powder does not necessarily have the same optical constants [START_REF] Marra | Fine hematite particles of martian interest : absorption spectra and optical constants[END_REF][START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]. Some absorption band might be affected by the presence of surface modes which are highly dependent on the size and shape of the grains composing the powder.

Jarrigue et al. [START_REF] Jarrige | Determination of the complex optical index of red pigments, vermillon[END_REF] have introduced a method for the measurement of the optical constant of a pigment in the particular case of vermilion : a small concentration of pigment is dispersed into a polyester matrix. By neglecting the absorption, the refractive index of the ensemble ma-trix+pigment is retrieved from the specular reflectance measurement using the Fresnel's coefficients I.2.1.a. As for the absorption coefficient, it is deduced from transmittance measurements. The main drawback of this experiment is that it is not exactly the optical constant of the pigment that is measured, but the one of the matrix containing the pigment.

Although the optical constants of materials are supposed to be well known, it appears that their modelization or measurement raise some accuracy issues that can lead to errors in the results. Thereby, for each material that we use in this study we will confront different optical constant data to ensure a better reliability.

I.2 Photo-realistic rendering : Reflectance & BRDF

One of our main objective is to reproduce the appearance of a material through photo-realistic rendering. To do so, we need to modelize the physical phenomena described earlier such as scattering using analytical expression of the reflectance and the transmittance. This requires a fine understanding of the interactions between light and matter.

At the interface between two media 0 and 1 of respective optical indices η 0 and η 1 , the incoming light undergoes two main interactions : it is partially or totally reflected towards the incident medium and can also be transmitted to propagates inside the medium 1.

We must distinguish between specular and scattering interactions. For specular materials, the light is reflected (or transmitted) in only one specific direction which is described by the Reflectance (or Transmittance) in this direction. For diffusing materials the incoming light is reflected (or transmitted) towards every direction of the hemisphere. To describe the amount of light resulting in any direction, we should consider the angular variations of the reflectance (and transmittance). This is equivalent to study the so called Bidirectionnal Reflectance Distribution Function (BRDF) (or the Bidirectionnal Transmittance Distribution Function (BTDF)).

It is defined as the ratio between the scattered radiance L d and the incoming irradiance E i . We make the assumption that this function only depends on the incident θ i and observation (θ o , ϕ) directions as well as the wavelength λ.

BRDF(θ

i , θ o , ϕ, λ) = L r (θ i , θ o , ϕ, λ) E i (θ i , θ o , ϕ, λ) (I.12)

I.2.1 Specular interactions

The specular behavior of a material is quite simple and is responsible for the shiny look of a metal or the transparency of glass. For these materials, the light coming from the direction θ i is reflected and transmitted towards only two specific directions θ r and θ t given by the Snell-Descartes' laws [START_REF] Born | Principles of optics : electromagnetic theory of propagation, interference and diffraction of light[END_REF] : • The ray that is reflected at the interface between two media is reflected in the incident plane, with an angle θ r = -θ i to the surface's normal (see Figure I.7a).

• The ray that is transmitted at the interface between two media is refracted in the incident plane with an angle θ t satisfying the condition (see Figure I.7b) :

η 1 sin θ t = η 0 sin θ i (I.13)
Total reflection may occur for an incident direction larger than θ i , max such as sin θ i ,max = η 1 η 0 . For a strongly absorptive material, since the optical constant η 1 is complex, so is the refraction angle. This case has been studied leading to new expressions of the refracted direction. For example, by using geometric optics, Born and Wolf [START_REF] Born | Principles of optics : electromagnetic theory of propagation, interference and diffraction of light[END_REF] proposed a new expression. A more reliable approach is the one of Kovalenko et al. [START_REF] Kovalenko | Descartes-snell law of refraction with absorption[END_REF] who used the wave definition of light to give a new definition of the refracted direction such as :

sin θ t = 2 sin θ i n 2 1 -k 2 1 + sin 2 θ i + (n 2 1 -k 2 1 -sin 2 θ i ) 2 + 4n 2 1 k 2 1 (I.14)
The fraction of light being reflected and transmitted by a smooth surface between two uniform materials in these directions (Figure I.7) is characterized by the Fresnel's coefficients.

I.2.1.a Fresnel's coefficients :

After reaching the interface between the materials 0 and 1, the light is decomposed into two polarized contributions. The former is parallel to the incident plane (s polarized) while the latter is perpendicular to it (p polarized) [START_REF] Born | Principles of optics : electromagnetic theory of propagation, interference and diffraction of light[END_REF]. In reflection, corresponding Fresnel's coefficients are written :

r 01,s = η 0 cosθ i -η 1 cosθ t η 0 cosθ i + η 1 cosθ t r 01,p = η 0 cosθ t -η 1 cosθ i η 0 cosθ t + η 1 cosθ i (I.15)
In transmission, they become :

t 01,s = 2η 0 cosθ i η 0 cosθ i + η 1 cosθ t t 01,p = 2η 0 cosθ i η 0 cosθ t + η 1 cosθ i (I.16)
The reflectance and the transmittance are defined as the intensity of the Fresnel's coefficients : R 01,i = |r 01,i | 2 et T 01,i = η 1 cosθ t η 0 cosθ i |t 01,i | 2 , with i the polarization of interest. The η 1 cosθ t η 0 cosθ i factor involved in the transmittance definition permits to take into account the change of solid angle occurring with the refraction phenomenon between the two media.

The reflected and transmitted intensities for non polarized light is the average of the polarized contributions : R 01 = 1 2 (R 01,s + R 01,p ) (I.17)

T 01 = 1 2 (T 01,s + T 01,p ) (I.18)
with energy conservation, R 01 + T 01 = 1.

The BRDF (or BTDF) of specularly reflective materials can then be deduced as equal to R (or T) in the specular direction, and null eslewhere.

In practice, because the Fresnel's coefficients apply to one interface it can be used to modelized the reflectance and transmittance of a semi-infinite transparent medium or the reflectance of a completely opaque material.

For finite transparent object, every interface the light reaches should be taken into account. In particular cases, it can induce physical coloration effects such as interferences in thin films.

I.2.1.b Specular thin films :

Let's consider a thin film made out of a homogeneous material with smooth interfaces and constant thickness. As not only one but multiple reflection and transmission events occur, the reflectance and transmittance cannot be described by directly using the Fresnel's coefficients defined in Section I.2.1.a. Indeed when light enters such a transparent thin layer, multiple reflections inside the layer produce subtle interference effects, due to difference in optical path length between the outgoing rays. The case of non-absorbing layer has already been widely studied. Hirayama et al. [START_REF] Hirayama | An accurate illumination model for objects coated with multilayer films[END_REF][START_REF] Hirayama | Visualization of optical phenomena caused by multilayer films based on wave optics[END_REF] have presented an algorithm for rendering these effects with multiple transparent layers. The paper also contains pictures with thin metallic layers but only gives formulas for dielectrics.

Belcour et al. [START_REF] Belcour | A practical extension to microfacet theory for the modeling of varying iridescence[END_REF] provide a fast method for rendering these interference effects in thin transparent layers. Their key idea is to project the effect onto three well chosen basis functions, providing real-time rendering. The method is efficient, but limited to transparent materials, with a refractive index that does not depend on the wavelength.

In these references, any material with a non-null extinction coefficient is considered to be completely opaque. Therefore it only requires the reflectance to be fully described.

I.2.2 Semi-specular interactions

The specular interaction is an ideal modelization suitable for perfectly smooth materials. A more realistic approach is to consider that light is mainly reflected in the specular direction but that some of the incident flux is reflected in slightly different directions. It results in a scattered lobe around the specular direction, responsible for the glossy look of some materials such as polished metals or glass. Because light is scattered in many direction, it cannot be described using the reflectance in only one direction as it was done for specular interactions. Its complete description requires to study the BRDF.

A first modelization can be performed using geometric optics through the micro-facets models. The surface of the material is assumed to be composed of small perfectly specular micro facets whose orientations distribution is known. This was first introduced by Cook and Torrance [START_REF] Cook | A reflectance model for computer graphics[END_REF]106] in the case of reflection, and has been extended to transmission by Walter & al. [START_REF] Walter | Microfacet models for refraction through rough surfaces[END_REF]. The roughness of the surface is controlled through an empirical parameter, with which the scattering lobe size increases.

Another model that predicts the reflectance of a rough surface while taking into account the diffraction effects caused by fine details is known as the generalized Harvey-Shack theory [62].

The rough surface is characterized by two parameters : the standard deviation σ that describes the height variations of the surface, and the correlation length l c that quantifies the mean distance for which the surface roughness of two points of the surface are completely independent. The surface topography is then modelized by a gaussian power spread function as follows :

PSD(f) = πσ 2 l 2 c exp - π 2 l 2 c λ 4 f 2 (I.19)
where f = f 2 x + f 2 y is the norm of the projection of the difference vector between the specular direction -θ i and the observation direction According to the Generalized Harvey Shack theory, the BRDF for any vector f is then given by :

BRDF GHS = 16π 2 λ 4 cos θ i cos θ o PSD(f) (I.20)
This model has been experimentally validated by Choi et al. [START_REF] Choi | Numerical validation of the generalized harvey-shack surface scatter theory[END_REF], and appears to be a good approximation for smooth to quite rough surfaces, even at grazing angles.

Holzschuch and Pacanowski [START_REF] Holzschuch | A two-scale microfacet reflectance model combining reflection and diffraction[END_REF] proposed a variation of the Generalized-Harvey Shack model to account for diffraction phenomena that may occur at a fine scale of details of the surface. It uses the K-correlation model (or abc-model) to characterize the height distribution of a rough surface [START_REF] Hoenders | K correlations and facet models in diffuse scattering[END_REF][START_REF] Church | Statistical and signal processing concepts in surface metrology[END_REF]. It involves the roughness parameter from the micro-facet theory, as well as three parameters a,b and c to describe the geometry of the surface at the nanometric scale.

This model allows a nice control of the scattering lobe that fits experimental results. However, for materials whose specular reflection is not very intense, four editing parameters appear to be too complex relatively to the importance of the phenomenon.

I.2.3 Diffusing interactions

I.2.3.a Light scattering by a rough interface :

For very diffuse materials, the semi-specular models may reach their limitations. Indeed, the specular contribution is so spread out that it is not well noticeable anymore. This happens for materials such as paper, powders, velvet... A perfect diffuser, also called a Lambertian surface, reflects the light uniformly over the whole hemisphere, whatever the incident geometry is. In that case, its BRDF is given by :

BRDF L (θ i , θ o , ϕ, λ) = ρ π (I.21)
where ρ is the material albedo, a parameter that is intrinsic to the material and characterizes its reflective properties. Lambertian material is an ideal modelization. In reality, the material that is the closest to such a case, known as a Spectralon, also shows some uniformities in its BRDF [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF].

I.2.3.b Light scattering by particles :

Until now, only scattering interactions occuring at the surface of the materials have been introduced. Yet, light can also be scattered inside the medium's volume because of scattering inclusions : leaves in a canopy, proteins in milk, grains in a powder... [START_REF] Hulst | Light scattering by small particles[END_REF].

Let's consider a region 0 free of scatterers with wavenumber k 0 = 2π/λ, and a medium 1 characterized by an effective wavenumber K = K + i K containing N inclusions per unit volume responsible for the light scattering. All scatterers have the same scattering properties (Figure I.9).

The incoming light undergoes multiple scattering events before being observed : it penetrates the medium without former scattering until it encounters one scatterer. Then it either propagates towards the observer (simple scattering), or continues inside the medium to reach other scatterers (multiple scattering) until it is finally observed.

θ i θ o ⃗ K i ⃗ K o r 1 r 2 r 3 r N -2 r N -1 r N 0, k 0 1, K ψ(r i ) G 11 (r n , r m ) G 01 (r n , r o )
x z y FIGURE I.9 -Light scattering by a medium containing diffusing inclusions.

The light either propagates without being deviated or is scattered by a particle it encounters. Let's introduce the formalism we will use to describe each of these interactions.

Foldy's approximation : Firstly, we make the assumption that the effective wavenumber K of region 1 can be expressed using the Foldy's approximation [START_REF] Foldy | The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers[END_REF] : under some statistical considerations and for a sufficiently low density of scatterers, it is given by :

K ≈ k 0 + i 2πN k 0 f (I.22)
where the parameter f is the complex scattering amplitude, describing the scattering properties of the inclusions.

Light propagation :

In this section we only consider the propagation of light as it travels from region 0 to the medium 1, between two particles in region 1 or from the medium 1 to region 0. The propagation of light inside a medium is described by a wave function that is impacted by the properties of the medium in which light is travelling. The different wave functions that we introduce are represented with different line style on the Figure I.9.

The light that is coming from the light source in region 0 without being scattered by any inclusion is described by the incident field ψ m :

ψ m (r i ) = exp(i K i • r i ) (I.23)
where K i is of amplitude K given by the Foldy's approximation I.2.3.b, and oriented towards the incident direction θ i .

After it has been scattered by an encountered particle, the light travels in the region 1 towards another scatterer, as if it was in a space free of scatterer. Under this hypothesis, the propagation of light from a position r towards r is given by the Green's function G(r, r ), that satisfies the Helmholtz equation :

(∆ + k 2 )G(r, r ) = δ(r -r ) (I.24)
By averaging over all of the scatterers in the medium, the mean Green's function that refers to the propagation of light from a inclusion to another one is given as follows :

G 11 (r n , r m ) = exp(i K • |r n -r m |) 4π|r n -r m | (I.25)
where r m and n are the positions of two distinct scatterers in region 1.

In the same way, the mean Green's function

G 01 (r n , r o ) = exp(i k 0 r o ) 4πr o exp(-i K o • r n ) (I.26)
describes the propagation from a scatterer at position r n in region 1 towards the camera at r o in region 0. The vector K o is of amplitude K and is oriented towards the camera.

Light scattering : When the light reaches a particles in the medium 1 it is scattered : the direction of propagation is modified according to the scattering properties of the inclusion. We identify single scattering which describes the interaction of one ray of light with only one scatterer before it propagates towards the observer, and multiple scattering which involves numerous scatterers.

• Light scattering by one particle : single scattering Part of the incoming light on a medium containing scatterers is only scattered once before being observed. This phenomenon is known as simple scattering. After the light reaches the diffusing inclusion, only the ray scattered into the camera direction in region 0 is collected. Any other ray scattered in another direction is neglected. For strongly absorptive material, the restriction of scattering phenomenon to single scattering is a good approximation.

• Light scattering by more than one particle : multiple scattering Light that is not scattered towards the observation direction after its first interaction with a scatterer can either be scattered in another direction towards the incident medium or propagates towards other scatterers in the medium 1 before going out to the observer. This is multiple scattering. This contribution cannot be neglected in the case of weakly absorptive materials or dense media. To take into account the multiple scattering is equivalent to sum up the contributions of all possible paths along which a light beam would undergo N scattering events, with N = 2 ; ∞ . As there is no analytical solution for this scattering problem, numerical approximations need to be done.

Various models have been developed and used in that sense. In the following, we propose to study some of them that are relevant for our work.

• Scattering phase function

As it interacts with a scattering particle, one incoming ray of light is divided into multiple outgoing rays with different directions. The intensity of light scattered by one scatterer in any direction Θ is given by the scattering phase function p. Different phase functions are existing, depending on morphological parameters of the scatterers. In the following we introduce some of these functions, and compare them along with each other on the Figure I.11.

To ensure energy conservation, they have to be normalized such as :

Isotropic scattering :

The simplest case is the one of isotropic scatterers. The light is uniformly scattered in every direction, and is therefore described by the following normalized constant phase function :

p = 1 4π (I.28)
Obviously, in most cases the particles are not isotropic and then scatter the light with an intensity that depends on the direction.

Mie scattering : Let's consider a collection of spherical and homogeneous scatterers, that are larger than the wavelength (r λ 0 ) and in suspension in a host medium with a relatively low concentration. The full description of the scattering problem in such a material according to radiative transfer theory (see section I.2.3.c) would lead to the Mie scattering theory [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]. It is among other things commonly used to describe light scattering of water droplets in clouds, of proteins in milk, of pigments in a binder... Even though the solution expressions are quite complex, important morphological and intrinsic properties of the scatterers are taken into account : their size r , the effective optical index η 1 of the medium 1 accounting for the actual indices of both the material and the host medium.

The Mie theory can be extended to non-spherical scatterers. In that case, the radius of a sphere equivalent in volume or in surface to the real scatterer shape is used as the size parameter.

Rayleigh scattering :

For non absorptive scatterers, with a size much smaller than the wavelength (4πr λ 0 ), the Mie theory can be approximated as the Rayleigh theory [START_REF] Rayleigh | on the electromagnetic theory of light[END_REF] : light scattering is comparable to the scattering of an electric dipole. This simplification permits to reduce the computing time and cost of the scattering parameters calculations.

For an unpolarized light, the Rayleigh phase function can be written as :

p(Θ) = 3 16π (1 -cos 2 Θ) (I.29)
with Θ the scattering angle defined on Figure I.10. It appears that the intensity of light scattered by any of the small spherical particle is proportional to the wavelength λ according to the relationship : I ∝ I 0 λ p(Θ), with I o the intensity of the incoming light. The blue light is therefore more scattered than larger wavelength which explains the blue color of the sky in daylight as a consequence of Rayleigh scattering from molecules that are in the atmosphere [START_REF] Rayleigh | on the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky[END_REF].

Even thought the mathematical simplifications that can be done with this theory are quite useful, it is at the expense of the dependency on the morphological parameter r of the scatterers.

Henyey Greenstein scattering :

This phase function was introduced by Henyey [START_REF] Henyey | Diffuse radiation in the galaxy[END_REF] with the objective of simplifying the Mie theory into an accurate and easy to use expression. It depends on the scattering angle Θ and an anisotropy parameter g that takes its values between -1 and 1. If light is mainly scattered towards the specular direction, then g ∼ 1. Conversely, if back-scattering is favored then g ∼ -1. For a perfectly isotropic scattering, g = 0, which is consistent with the phase function introduced in Section I.2.3.b.

p(Θ, g ) = 1 4π 1 -g 2 (1 + g 2 -2g cos Θ) 3/2 (I.30)
It is used in various domains such as astrophysics to model the scattering of light by interstellar dust clouds, but also to describe the light scattering in biological tissues [START_REF] Van De Hulst | Multiple light scattering[END_REF][START_REF] Wang | Mcml-monte carlo modeling of light transport in multi-layered tissues[END_REF]. Improvements have been made to more accurately reproduce measurement results in both atmospheric and oceanic optics [START_REF] Kattawar | A three-parameter analytic phase function for multiple scattering calculations[END_REF][START_REF] Haltrin | One-parameter two-term henyey-greenstein phase function for light scattering in seawater[END_REF].

Plant stand scattering : Another phase function that we will consider in this work has been developped by Ross [START_REF] Ross | The radiation regime and architecture of plant stands[END_REF] for a plant stand with a uniform leaf orientations distribution. Its expression is given by :

P(Θ) = 1 3π 2 ((π -2Θ) cos Θ + 2si nΘ) (I.31)
It is used in teledetection to simulate the scattering inside a canopy, as we will explain in Section I.2.3.d, and can be extended in the case of bare soils.

I.2.3.c Radiative transfer theory :

The scattering phase functions cited earlier only describe the scattering of light by a single particle. The solving of the scattering problem by a collection of particles is done using the Radiative transfer theory, developped by Chandrasekhar [START_REF] Chandrasekhar | Radiative transfer[END_REF] and Ishimaru [START_REF] Ishimaru | Wave propagation and scattering in random media[END_REF], that describes the propagation of an electromagnetic wave in a medium that includes a random distribution of spherical scattering inclusions. It allows to account for single scattering as well as multiple scattering, requiring the following parameters : the scattering phase function p, the absorption σ a , scattering σ s and extinction σ t = σ a + σ s cross sections. Instead of considering the contributions of each individual scatterer, an averaging is made over their ensemble. It leads to a macroscopic description of the participating medium composed of microscopic entities. In order to ease that averaging an important hypothesis is made : all the scattered fields following a path inside the medium are supposed to be incoherent with each other. The light propagation is then described by the radiative transfer equation :

(k o • ∇)L(r o , k o ) = -σ t L(r o , k o ) + σ s 4π p(θ , θ o )L(r o , k )d ω (I.32)
where L(r, k) is the radiance at the position r in the k direction. The first term characterizes the absorption and possible emission from the studied material, while the scattering is modelled by the integral term. This equation does not have an analytical solution so it requires numerical methods to be solved.

Radiative transfer theory can be applied to solids, liquids or gazes, and is used in multiple scientific domains such as astrophysics for atmospheric studies (clouds) and teledetection. Here we are most interested in photo-realistic image rendering through computer graphics. Equation I.32 can be rewritten into another simpler expression known as the rendering equation that we will prefer in computer graphics :

L o (ω o ) = L e (ω o ) + Ω L i (ω i ) f (ω i , ω o ) cos θ i d ω i (I.33)
with ω o the vector from a point to the camera, of magnitude r o and direction k o . The first term represents the possible emission of the studied material. The second one involves the BRDF f describing its macroscopic appearance. It takes into account scattering through the scattering phase function which is implicitly involved, and the absorption phenomena.

In the following, we introduce some BRDF models from literature that we used throughout our study.

I.2.3.d RTLSR model :

A first model we used is the Ross-Thick Li-Sparse (RTLSR) model proposed by Lucht et al. [START_REF] Lucht | An algorithm for the retrieval of albedo from space using semiempirical brdf models[END_REF] and that has been developed to simulate the BRDF of canopies mesured by teledetection. Other references on this subject [START_REF] Nilson | A reflectance model for the homogeneous plant canopy and its inversion[END_REF][START_REF] Ni | An analytical hybrid gort model for bidirectional reflectance over discontinuous plant canopies[END_REF][START_REF] Li | Geometric-optical bidirectional reflectance modeling of a conifer forest canopy[END_REF] agree that it can be extended to the study of bare soils, but do not explicitly explain the changes that should be done in such a case. In this section, we will introduce the original model for canopies, and we will show in section IV.3.2 that adaptations are required to make the model valid in the case of dry pigment powders.

The RTLSR model is a semi-empirical model, linear combination of a volumetric contribution accounting for volume scattering inside the medium and a geometric contribution accounting for surface scattering. These two terms are weighted with a empirical parameter α :

BRDF=αR vol + (1 -α)R g eo
At first it had been developed for thick and sparse foliage. Improvements have been brought up by Wanner et al. [START_REF] Wanner | On the derivation of kernels for kernel-driven models of bidirectional reflectance[END_REF] so that the model can be extended to other configurations, especially for thick and dense foliage which is the configuration we will use.

Volumic contribution :

The volumetric contribution is deduced from radiative transfer theory and can take into account single and multiple scattering. In order to give a simple analytical expression, some hypotheses are made :

-The incoming light that penetrates into the medium undergoes only one scattering event before being observed. It is equivalent to only consider the single scattering and neglect the multiple scattering. This hypothesis is valid for a sparse medium or a strongly absorptive material.

-The lambertian reflectance s and transmittance t of the leaves are equal. These two quantities are indeed quite similar for leaves [START_REF] Jacquemoud | Compréhension et modélisation de la couleur des feuilles[END_REF].

-The interface between air and the diffusing medium can be described by micro-facets oriented with an isotropic normal distribution.

Using both these hypotheses and the radiative transfer theory approximation from Ross [START_REF] Ross | The radiation regime and architecture of plant stands[END_REF], Roujean [START_REF] Roujean | A bidirectional reflectance model of the earth's surface for the correction of remote sensing data[END_REF] gave an explicit analytical expression of the volumetric contribution. Let's denote µ i = cos θ i and µ o = cos θ o , then : The first term corresponds to the scattering and the attenuation of light by the leaves encountered in the canopy medium and is represented by the solid line on the Figure I.12. The second term, in dotted line, is the contribution of a light beam reaching the lambertian soil of albedo ρ o before being observed which has not been scattered but attenuated by the scatterers on its path. The exponential law is an adaptation of the Beer-Lambert law [50]. From references, the phase function introduced in Section I.2.3.b is usually used.

R vol = s 2 p(Θ) µ i + µ o (1 -e -LAI µ i +µo 2µ i µo ) + ρ 0 e -LAI µ i +µo 2µ i µo (I.

Geometric contribution :

The contribution of surface scattering is deduced from geometrical considerations and requires the knowledge of the media's surface geometry. It accounts for shading effects of leaves on each other. Although it is not physical but statistical, it is very interesting since it does involve the scatterers' size.

The total geometric contribution R g eo can be decomposed in four terms :

• A proportion K C of illuminated material, with reflectance C.

• A proportion K T of shadowed material, with reflectance T.

• A proportion K G of illuminated ground, with reflectance G.

• A proportion K Z of shadowed ground, with reflectance Z.

Supposing that the shadows are perfectly black, then T = Z = 0 which leads to the following expression of the complete contribution :

R g eo = K C C + K G G (I.35)
In order to fully express the parameters at stake, trees are considered by Roujean [START_REF] Roujean | A bidirectional reflectance model of the earth's surface for the correction of remote sensing data[END_REF] as parallelepipedic protuberances with heigth h, length b and width r . To better modelize a tree shape, and in our case the shape of the grains in a powder (Section IV. According to a Boolean model, the spacing probability between the trees is given by an exponential law e -DA(θ,ϕ) , where A(θ, ϕ) is the projected surface of a tree on the ground of total surface A, and D = N/A is the density of trees in the canopy. This probability represents the proportion of background, either illuminated or shaded, that is seen from the camera's viewpoint (θ o , ϕ). The proportion 1e -DA(θ o ,ϕ) then corresponds to the area within the field of view that is covered by trees.

K Z + K G = e -DA(θ o ,ϕ o ) K T + K C = 1 -e -DA(θ o ,ϕ o ) (I.36a) (I.36b)
These proportions take into account both illuminated and shaded areas. As we have considered that the shadows are perfectly black, only sunlit areas are involved in the observed reflectance. We need to deduce the proportions of the background and of the trees in the scene that are both lightened and in the camera's field of view.

• K G definition : FIGURE I.14 -Projected shadows of a sphere. Two shadows are defined, in the source light and in the camera point of view : A i is the absence of light caused by an obstacle between the light source and the ground. A o is the area that cannot be seen from the camera point of view. Parts of the latter can actually be lightened. The overlap is given by O(θ i , θ o , ϕ).

Two projected areas of a tree can be defined : a first one from the light source point of view and a second one from the camera's point of view (Figure I.14). The former is denoted as A i (θ i ) and corresponds to the lack of light on the ground caused by an obstacle (tree) between the light source and the ground. The latter is the observation shadow, A o (θ o , ϕ), which is the area that can be sunlit but that is not in the camera field of view. The proportion of the background that is both illuminated and viewed in the scene can then be written :

K G = e -D(A i (θ i )+A o (θ o ,ϕ)-O(θ i ,θ o ,ϕ)) (I.37)
where O is the overlap between the two projected shadows (see Figure I.14. Again, using the shape of the scattering objects and the illumination/observation geometry, it can be written :

O(θ i , θ o ) = 1 π (t -sin t cos t )(sec θ i + sec θ o ) (I.38)
with t depending on the dimensions of the objects so that :

cos t = h b D 2 + (tan θ i tan θ o sin ϕ) 2 sec θ i + sec θ o (I.39)
The parameters θ and D involved in the previous equations have been introduced for readability reasons. Their full expressions are given by the two following equations :

D = tan 2 θ i + tan 2 θ o -2 tan θ i tan θ o cos ϕ (I.40) θ = tan -1 b r tan θ (I.41)
• K C definition : Some knowledge about the shape of the objects is necessary to express this parameter. With the ellipsoid representation, the relative proportion of a tree that is illuminated can be written as :

K C K C + K T = 1 2 (1 + cos Θ ) (I.42)
where Θ is the scattering angle, defined as the difference between the adapted incoming and outgoing angles, respectively θ i and θ o . Thus, using the equation I.36 :

K C = 1 2 (1 + cos Θ )(1 -e -DA o (θ o ,ϕ) ) (I.43)
This expression I.43 is usable for a sparse canopy. The model has been adapted by Wanner et al. [START_REF] Wanner | On the derivation of kernels for kernel-driven models of bidirectional reflectance[END_REF] to account for dense canopies where the shadowing of a tree on another is significant. In that case, K C = F(1-K G ) with K G given by Equation I.37, and F the ratio between the area of objects that is both lightened by the source light and observed by the camera, and the shadow on the ground.

F = 1/2(1 + cos Θ ) sec θ o sec θ i + sec θ o -O(θ i , θ o , ϕ) (I.44)
Finally, for a sparse canopy the geometrical contribution would be :

R g eo = CDπr 2 O(θ i , θ o , ϕ) -sec θ i -sec θ o + sec θ o 1 + cos Θ 2 + C (I.45)
with D defined earlier (see Equation I.40, while for a dense canopy it would be :

R g eo = C 2 (1 + cos Θ ) sec θ o sec θ o + sec θ i -O(θ i , θ o ) -2 + C (I.46)
The final model is obtained by replacing R vol I.34 and R g eo I.46 by their respective expressions into the model's equation I.2.3.d.

Conclusions and Limitations :

Our interest for this model relies on its physical approach. Indeed it is partially deduced from radiative transfer theory, and involves the reflectance C which is intrinsic to the studied material and can be approximated as the diffuse reflectance or the single scattering albedo of the scattering elements. It is also of great interest because of its simplicity. Only one empirical parameter α is used to modify the amount of the volumetric contribution over the geometric one.

As a consequence it is not a very flexible model, and it requires more fundamental modifications to be adapted for other media than canopies. Originally deduced for thick and sparse canopies, some adaptations have been proposed to describe thin or dense ones [START_REF] Wanner | On the derivation of kernels for kernel-driven models of bidirectional reflectance[END_REF] as well as other materials such as snow [START_REF] Jiao | Development of a snow kernel to better model the anisotropic reflectance of pure snow in a kernel-driven brdf model framework[END_REF]. As far as we know, although it is said to be adaptable for bare soils, it had not been done.

Nevertheless part of the model is only deduced from geometric considerations. This contribution account for the back-scattering phenomenon but not for any specular contribution. Another term should be added to simulate this scattering effect.

Finally, the RTLSR model was developed for teledetection purposes, and is used to describe the scattering of leaves with approximate size of 10 cm at a few kilometers from the observer. Let's define the ratio between the scatterer size and the distance at which it is observed. For a canopy, is would be of magnitude 10 -6 . In the case of powders of much smaller scale (grains of 10 nm to 1 µm seen from a distance of around 1 m), it is of magnitude 10 -8 to 10 -6 . Such a difference might make the RTLSR model unsuitable for especially fine powders.

To conclude, multiple adaptations that will be explained in Section IV.3.2 should be made in order to use that model to describe the appearance of died powders.

I.2.3.e Spectralon's model :

Another model we are interested in has been developed by Lévesque et al. [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF] to reproduce the BRDF of a Spectralon, a reference white and almost lambertian surface used to calibrate reflectance and BRDF's measurements (see Figure I.15). This material is commercialized by Labsphere and composed of compressed transparent thermoplastic based upon poly-tetrafluoroethylene (PTFE) polymer particles. The main reason why this model is of great interest for our study is the similarity of the spectralon object with a powder material : both are made of a quite dense collection of particles uniform in size and shape. The scattering phenomena identified for the spectralon should then be identified in the powders. Four main physical phenomena have been identified : volume scattering at small (deep scattering, D) and large (Subsurface scattering, SS) observation's angles, and surface scattering in the specular (Specular scattering, S) and in the backward (Back scattering, B) direction. The total BRDF is the sum of these contributions :

BRDF = D(θ i , θ o ) + S S (θ i , θ o ) + B(θ i , θ o ) + S(θ i , θ o ) (I.47)
In the following we will go through the definition of each contribution, by order of importance in the overall appearance of the material.

Volumetric contribution :

Multiple scattering between the particles that are inside the medium is identified as the more important contribution to the BRDF : part of the incoming light on a participating medium penetrates the medium and undergoes multiple reflection or transmission events before being observed. For small incident angles, there is a higher probability for the light to penetrates deeply into the medium before it goes out with a direction close to the surface normal. In that case the entry and the exit points of the light are quite close. Such a phenomenon is called Deep scattering. For larger incidences, light penetrates much less deeply and propagates right underneath the medium surface to exit further away from the entry point. This is Subsurface scattering. These two kind of volume scattering are represented on • Deep scattering :

It is the major contribution of the four that have been identified. It permits to model the overall shape of the BRDF. The mathematical formula that is used is inspired from an empirical model developped by Walthall [START_REF] Walthall | Simple equation to approximate the bidirectional reflectance from vegetative canopies and bare soil surfaces[END_REF] to simulate the BRDF of canopies seen in teledetection.

D(θ i , θ o ) = d 0 + d 1 (θ 2 i + θ 2 i ) + d 2 (θ 4 i + θ 4 o ) + d 3 θ i θ o + d 4 θ 2 i θ 2 o (I.48)
where the parameters d j with j = 0, 1, 2, 3, 4 are intrinsic to the material, but whose physical meaning remain unexplained.

It results in a large scattering lobe (see Figure I.18) that fits the general shape of the measured BRDF.

• Subsurface scattering :

The second volume scattering phenomena is mainly visible for large to grazing incident angles. Two empirical expressions are given by the authors [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF]. The first and simpler one is a good approximation for small enough angles :

SS(θ i , θ o ) = ss 0 exp(ss 1 |θ o + θ i |) (I.49)
where ss 0 and ss 1 depends on the material under study. At grazing angles (empirically defined as larger than 70 • ), this first model does not permit to nicely fit the BRDF measurements. Beyond these angular conditions, a more accurate but complex formula is recommended : This second model involves not less than 11 material related parameters ss j , without explicit physical meaning. Depending on the importance of this phenomenon, some materials would not need such a complexity.

SS(θ i , θ o ) = g SS a (θ i )exp - 1 2 θ o -g SS c (θ i ) g SS w (θ i ) (I.

Surface contribution :

The light that doesn't penetrate into the medium is scattered at its interface with the air. This surface scattering is responsible for finer details of the measured BRDF : two reflectance peaks are observed in two privileged directions. The first one that is identified around the forward direction is commonly named Specular direction (θ o = -θ i ), and the second one around the backward direction (θ o = θ i ) leads to the so-called Back scattering. These two kind of surface scattering phenomena are represented on Figure I.17, and are detailed in the following.

• Back scattering :

Among the two surface contribution, this is the major one. A large amount of the incoming light is scattered back into the incident direction (θ o = θ i ) in a more or less narrow scattering lobe. This is the back scattering phenomenon and it can be empirically modelled using either a Gaussian or a Lorentzian function. According to Levesque et al. [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF], the Lorentzian matches more faithfully the measurements which is why only this modelization is introduced here : with b 0 is the amplitude and b 1 the width of the scattered peak. These two parameters should depend on the studied material and more precisely on the morphology of the grains. No precision is given about the physical meaning of such variables.

B(θ i , θ o ) = b 0 b 2 1 (θ o + θ i ) 2 + b 2 1 (I.54)
• Specular scattering :

Finally, some light is scattered around the specular direction (θ o = -θ i ) according to the Snell Descartes' law detailed in Section I.2.1. A Gaussian function is used to model the scattering lobe of this phenomenon.

S(θ

i , θ o ) = a f (θ i )exp - 1 2 θ o -θ i s 1 2 (I.55) with a f (θ i ) = s 0 [1 + cos(s 2 θ i )] (I.56)
The three parameters s j allow to adjust the amplitude and the width of such a peak, but once again, their physical signification is not investigated.

Conclusion & Limitations :

The identification of four physical phenomena occurring in a participating media and their individual modelization is of great interest to simulate the BRDF of powder materials other than Spectralon. The large number of parameters at stake allows the model to be quite flexible and easy to adapt.

However, these parameters are too many (12 for the simpler modelization or 21 for the more complex one) to have an actual physical meaning. Indeed each model used for simulating any of the physical phenomenon is purely empirical. Moreover, the model has been developed in the incident plane. To model the complete BRDF of a material some extension is necessary to account for the different azimuth angles.

I.2.3.f Enhanced Back-scattering :

The last BRDF model we are interested in is the Enhanced Back-Scattering (EBS) model, also called Coherent back-scattering, which offers a physical description of the back scattering phenomenon observed in quite dense media. It is an extension of radiative transfer (RT) from section I.2.3.c : the so called ladder terms for single scattering BRDF bs,L 1 and multiple scattering BRDF bs,L 2+ take into account the contribution of uncorrelated scattered fields already described by the RT theory. This model also accounts for coherent fields that induce interference effects as light is scattered multiple times in the medium. It is described by a cyclical term BRDF bs,C and results in a sharp peak in the back-scattering direction.

The BRDF is then composed of three contributions : The model has been developed for thick layers of a medium composed of isotropic spherical scatterers of radius r with a constant phase function p(Θ) = p embedded in a host medium [START_REF] Van Der | Light scattering in strongly scattering media : multiple scattering and weak localization[END_REF][START_REF] Tsang | Scattering of electromagnetic waves : advanced topics[END_REF][START_REF] Bret | Multiple light scattering in porous gallium phosphide[END_REF][START_REF] L^nn | Utilizing enhanced backscattering for determination of scattering properties in turbid media[END_REF]. We note N the number of scatterers per unit volume. To ease the calculations, they are supposed to act as point scatterers : the scattering event does not occur on the surface but at the center point of the scatterer (Figure I. [START_REF] Cervelle | Détermination de l'ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. application à la pyrrhotite hexagonale[END_REF]).

BRDF bs = BRDF bs,L 1 + BRDF bs,L 2+ + BRDF bs,C (I.57)
Among the studies on enhanced back scattering, we should note that in the work of Bret [START_REF] Bret | Multiple light scattering in porous gallium phosphide[END_REF], only materials with high albedos are considered which leads to simplifications of the model we will not introduce here. An interesting extension of the model to the case thin films is proposed by [START_REF] Van Der | Light scattering in strongly scattering media : multiple scattering and weak localization[END_REF] but has not been investigated in our work. Let's explain in more details the physical determination of such a model for optically thick layers. The propagation of light without being scattered in both region 0 and medium 1 is modelized using the Green's functions introduced earlier in Section I.2.3.b. As for the scattering events, the particles are supposed to be isotropic and therefore involves a constant phase function.

Scattering by one scatterer :

The scattering events caused by the encountering of light with the n-th scatterer is described using the transition operator T n that we introduced as :

T n (r n ) = 4π f (Θ n )δ(r i n -r on ) = 4π f (Θ n ) (I.58)
where Θ n = θ on -θ i n is the scattering angle between the incident direction on the n-th scatterer θ i n and the outgoing direction θ on shown on the Figure I.19. Because of isotropic scattering, the complex scattering amplitude f (Θ n ) = f is constant and linked to the isotropic phase function p = 1 4π using the relationship :

ρp = N κ | f | 2 (I.59)
with ρ the single scattering albedo, and κ is called the extinction factor.

Scattering with two scatterers -Comparison with RT theory : Before studying the general expression for multiple scattering, let's first consider the case of only two scatterers P 1 and P 2 randomly arranged at the respective positions r 1 and r 2 in the host medium. Light can either be scattered first by P 1 then by P 2 , or first by P 2 then by P 1 (Figure I.20). The respective scattered field of these contributions will be denoted as E [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]

∆Φ = (k i + k o ) • (r 1 -r 2 ) (I.61)
Two specific cases can arrise :

k i +k o = 0 : If k o is not in the exact back-scattering direction, then the phase difference is not null and depends on the random positions of the scatterers at stake. It follows that ∆φ is random.

By averaging over the possible positions, the term 〈E 12 E * 21 〉 is null which, with the reciprocity E 12 = E 21 it leads to :

I = 2I 1 (I.62)
This result is identical to the one obtain considering uncorrelated scattered fields. That is to say that in other directions than the back-scattering one, the intensity of the total scattered field can accurately be computed from the radiative transfer theory I.2.3.c.

k i + k o = 0 : If k o is in the exact back-scattering direction, then the phase difference is null whatever the position of the scatterers is. It follows 〈E 12 E * 21 〉 = 0, and using the reciprocity relationship E 12 = E 21 it leads to the following result :

I = 4I 1 (I.63)
The scattered intensity is twice as important in the back-scattering direction as in any other direction : the fields are coherent and produce constructive interferences in this specific direction. This result is not predicted by the radiative transfer theory, and is known as coherent scattering.

Scattering by multiple scatterers : Now that we have seen that both incoherent and coherent scattering should be taken into account in the case of only two scatterers, we can extend the result for a large number of scatterers in the medium 1. Incoherent scattering is composed of single and multiple scattering in the hypothesis of uncorrelated scattered fields and will be modelized by the so called ladder terms. Under the assumption of correlated scattered fields, a third term called cyclical term should be accounted for the coherent scattering.

• First order ladder term L 1 :

The main phenomenon that is involved in the BRDF of a medium containing scatterers with a density N is the single scattering I.2.3.b. Light only undergoes one scattering event before it is observed. In this specific case, the scattering angle is given by Θ n = Θ = θ o -θ i . By listing the propagation and scattering events of a light beam as it travels through the medium encountering one scatterer, the observed intensity can be written :

L 1,n = |G 01m (r o , r n )| 2 |T j (r n )| 2 |ψ(r n )| 2 (I.64)
As single scattering can occur on any particle in the medium, the complete single scattering contribution is obtained by summing over all the scatterers :

L 1 = N | f (Θ)| 2 (4π) 2 |G 01m (r , r n )| 2 |ψ m (r n )| 2 dr n (I.65)
This can be analytically solved using I.59 and the expressions of G 11 and G 01 respectively given by the equations I.25 and I.26. The BRDF is then deduced as follows :

BRDF bs,L 1 = 4πr 2 L 1 Aµ i µ o (I.66)
with µ i = cos θ i and µ o = cos θ o . The final result is summarized in the equation I.75.

• Multiple order ladder term L 2+ :

For quite dense materials, the multiple scattering I.2.3.b cannot be neglected. If we consider the scattered fields as uncorrelated, the incoherent contribution of multiple scattering can be described using radiative transfer theory I.2.3.c. We call n-th order scattered field, the ensemble of fields from a light ray that undergoes n scattering events as it travels through the medium. Its intensity L 2+ is obtained by summing the n-th order intensities L n over n with n = 2 ; ∞ .

L 2+ = ∞ n=2 L n (I.67)
As for the n-th order intensity L n , it is the sum of every possible path that encounters n scatterers before reaching the observer. The scattered intensity resulting from a light beam following a path j going from the scatterer P 1 to the scatterer P n can be visualized in the Figure I.9 and has the following expression :

L n, j = |G 01m (r , r n, j )| 2 |T n (r n, j )| 2 |G m11 (r n, j , r n-1, j )| 2 ...|G m11 (r 2, j , r 1, j )| 2 |T 1 (r 1, j )| 2 |ψ m (r 1, j )| 2
(I.68) After the summation over all possible paths, and using the expression of the scattering transition operator T n I.58 introduced earlier, it follows :

L n = (4π) n-2 (ρpκ) n r 2 |G m11 (r n , r n-1 )| 2 ...|G m11 (r 2 , r 1 )| 2 exp κ z 1 µ o + z n µ i dr 1 ...dr n (I.69)
The complete multiple order ladder term is finally obtained by replacing the expression of L n I.69 into the Equation I.67. After some developments it can written in a compact recursive way [START_REF] Van Der | Light scattering in strongly scattering media : multiple scattering and weak localization[END_REF][START_REF] Tsang | Scattering of electromagnetic waves : advanced topics[END_REF] : There is no analytical solution for this multiple scattering contribution. It will be approximated to a simpler formulation in the Section I.2.3.f.

L 2+ = exp(κ( z i µ o + z o µ i )A(r i ,
• Cyclical term C :

θ i θ i θ o θ o ⃗ K i ⃗ K i ⃗ K o ⃗ K o r 1 r 2 r 3 r N -2 r N -1 r N 0, k 0 1, K ψ(r i ) G 11 (r n , r m ) G 01 (r n , r o )
x z y FIGURE I.21 -Light scattering by a medium containing diffusing inclusions : the incoming light encounters n particles before it is observed. It can either propagates from the particle P 1 to the particle P n (dark lines) or the other way around (grey lines). Under the assumption of correlated fields, this creates interference that are constructive in the exact back scattering direction.

The assumption of coherent scattered fields that was not taken into account in the RT theory leads to another contribution resulting from multiple scattering. This assumption is valid for dense enough media. Because of such a correlation, the light ray propagating from P 1 to P n may interfere with the light ray going from P n to P 1 (see Figure I.21). Therefore another multiple scattering term needs to be accounted for.

As it has been done for the n-th order ladder term, and using the symmetry of Green's function G m11 ( r, r ) = G m11 ( r , r ), we can express the n-th order cyclical term along a path j .

C n, j = G 01m (r , r n, j )T n (r n, j )G m11 (r n, j , r n-1, j )...G m11 (r 2, j , r 1, j )T 1 (r 1, j )ψ m (r 1, j ) × G * 01m (r , r 1, j )T * 1 (r 1, j )G * m11 (r 1, j , r 2, j )...G * m11 (r n-1, j , r n, j )T * n (r n, j )ψ * m (r n, j ) C n, j = G 01m (r , r n, j )G * 01m (r , r 1, j )|T n (r n, j )| 2 |G m11 (r n, j , r n-1, j )| 2 ... × |G m11 (r 2, j , r 1, j )| 2 |T 1 (r 1, j )| 2 ψ m (r 1, j )| 2 ψ * m (r n, j ) (I.72)
The interactions between the light and the first and latest encountered scatterers, respectively described by ψ m , G m01 and their conjugates are different from the interactions identified for the ladder term. This is caused by the opposite arrangement of the scatterers along the rays that are interfering.

Considering any possible path j , the n-th order cyclical term is then deduced as :

C n = (4π) n-2 (ρpκ) n r 2 |G m11 (r n , r n-1 )| 2 ...|G m11 (r 2 , r 1 )| 2 × cos k(µ i -µ o )(z 1 -z n ) exp κ 2 1 µ o + 1 µ i (z 1 + z n ) dr 1 ...dr n (I.73)
The complete cyclical term C is obtained summing over the scattering orders. As for the multiple ladder term it can be written after some calculations in a recursive manner :

C = cos(2k(µ i -µ o )(z o -z i ))exp κ 2 z i µ o + z o µ i A(r i , r o )dr i dr o (I.74)
with A(r i , r o ) the factor from Equation I.71. Again, this contribution does not have an analytical solution. In order to be useable, it will be approximated to a simpler formulation in the next Section I.2.3.f. Diffusion approximation : Contrary to the first order ladder term, the multiple scattering terms in Equations I.70 and I.74 cannot be solved analytically. Mathematically, the difference between the coherent and incoherent multiple scattering only relies on the interaction of light with the first and the latest encountered scatterer. The path that light follows in between can be approximated using statistic considerations. This is the diffusion approximation, and is valid for optically thick media with large albedos. It permits to deduce analytical expressions of every contribution involved in the EBS theory I. [START_REF] Nassau | The physics and chemistry of color : the fifteen causes of color[END_REF].

                     BRDF bs,L 1 = 4πr 2 L 1 Aµ i µ o = ρp(Θ) µ o + µ i 1 -exp -κd 1 µ o + 1 µ i BRDF bs,L 2+ = 4πr 2 L 2+ Aµ i µ o = 3ρ 2 1 -µ 2 i c 2 2Q 1 -µ o c - µ 2 i µ i + µ o BRDF bs,C = 4πr 2 C Aµ i µ o = 3ρ 2 [c + v(1 -exp(-2cτ o ))] 2µ i µ o c v[(c + v) 2 + u 2 ] (I.75a) (I.75b) (I.75c)
The parameters u, v, c and Q in the previous Equations I. [START_REF] Nassau | The physics and chemistry of color : the fifteen causes of color[END_REF] vary with the isotropy and with the geometry under which the medium is illuminated and observed. It has been described by Van der Mark et al. [START_REF] Van Der | Light scattering in strongly scattering media : multiple scattering and weak localization[END_REF] for isotropic scatterers such as :

                               u = k 0 κ (µ i -µ o ) v = 1 2 1 µ i + 1 µ o α = k 0 κ f c = 3(1 -ρ)κ t r + α 2 Q = 1 + µ i c -(1 -µ i c)exp(-2cτ o ) /4c (I.76a) (I.76b) (I.76c) (I.76d) (I.76e)
where f is the norm of the vector f previously defined on the Figure I.8 which allows the BRDF modelization in any azimuthal plane.

Only the single scattering term involves the phase function in such a case. Tsang et Kong [START_REF] Tsang | Scattering of electromagnetic waves : advanced topics[END_REF] introduce the consideration of anisotropic scatterers but only with normal incidence. We will discuss the extension of the model for anisotropic scattering in the Section IV.3.3.

The last parameter involved in the BRDF expressions I.75 is the extinction ratio τ o that is defined as intrinsic to the scatterers' material. It accounts for internal reflections that may occur between the interface of the medium. A definition is given by Zhu et al. [START_REF] Zhu | Internal reflection of diffusive light in random media[END_REF].

τ o = 2 3 1 - R 1 + R (I.77) with R = 3C 2 + 2C 1 3C 2 -2C 1 + 2 and C i = π 2 0 R(θ) cos i θ sin θdθ (I.78)
In the previous equations, R(θ) is the Fresnel's angular reflection coefficient defined in Section I.2.1.a. The parameter τ o thereby depends on the optical constants of the material the particles are made of, and likewise, on the wavelength.

Conclusion & Limitations :

For sufficiently sparse media, the scattered fields can be considered as uncorrelated, and thus incoherent. This is well predicted by the radiative transfer theory. The Enhanced back scattering model permits to take into account the interference effects that are occurring with higher scatterer density.

In the diffusion approximation, it has been mainly developed for isotropic scatterers [START_REF] Van Der | Light scattering in strongly scattering media : multiple scattering and weak localization[END_REF][START_REF] Bret | Multiple light scattering in porous gallium phosphide[END_REF]. Tsang et Kong [START_REF] Tsang | Scattering of electromagnetic waves : advanced topics[END_REF] extended the model to anisotropic scatterers in a medium illuminated with a normal incidence.

It requires albedos close to 1 and an optically thick medium. Results for a thin layer are given by [START_REF] Van Der | Light scattering in strongly scattering media : multiple scattering and weak localization[END_REF]. Despite all of these assumptions, the results are analytical and involve only a few physical parameters that depend on the size and composition of the scatterers.

I.2.4 Photo realistic rendering

The Reflectance models, and more generally BRDF models, are required and used for the creation of photo-realistic images through computer graphics. These images reproduce with high fidelity the appearance of 3D scenes with a chosen illumination and observation geometry, and will be computed using the Mitsuba renderer [START_REF] Jakob | [END_REF] in our work.

I.2.4.a Creation of a virtual 3D scene :

First, the user defines a virtual scene by controlling the organization of the scene, its illumination, and its observation geometry. One or multiple 3D objects are placed in the scene, and observed through a camera in a chosen position.

Different kind of illumination configurations are available : we can use local light source such as a point light source in a given position, that can either be diffuse and illuminates in every directions, or collimated towards a specific direction. It is also possible to simulates more natural lighting conditions. To do so, we use an environment map which is an image used as the background of the scene, and whose each pixel is considered as a colored point light source.

Other light source can be defined, but only these two types of lighting will be of interest in our work.

I.2.4.b Rendering :

The renderings are made using ray tracing, which simulates the propagation of a ray of light through the scene. Physically, the light is coming from the light source, and is reflected or scattered as it encounters the surface of the objects, which is modeled by the BRDF of the material the object is made of. The rays that reach the camera can then be converted into pixel whose color is the result of the former light/object interactions, creating the result image. The probability for the ray to end up being reflected towards the camera is actually quite low, which increase the computation time.

To improve the calcultation cost, ray tracing works the other way around as shown on the Figure I.22 : instead of considering the rays coming from the light source and may be reflected out of the camera's field of view, we only account for the rays that are actually reaching the camera.

Let's consider a ray that reach one pixel of the camera. We can trace back its interaction with the object's surface and its propagation into the scene until it reaches a light source. Knowing the color of the light source and its alteration through the BRDF describing the object's material, the FIGURE I.22 -Principle of ray tracing in computer graphics. The color of on pixel is approximated by extending a ray from this pixel into the virtual scene until it reaches a light source. The color is computed by taking into account the interactions with the objects' surfaces present in the scene that the ray undergoes, through BRDF models. If the ray does not reach a light source, it contributes to the shadow of the objects.

color of the pixel is therefore approximated.

Because it has some powerful advantages, we have chosen the Mitsuba renderer [START_REF] Jakob | [END_REF] for our work. First of all, its interface is quite simple and easy to use. It already contains many BRDF models, including classical ones describing the appearance of transparent materials or metals, and more complex ones for varnish materials for example. Its flexibility allows to easily add and use new BRDF models.

More importantly, it is a spectral renderer which means that, contrary to many other renderers, it accounts for a spectral representation of light (between 360 to 830 nm) and not only a trichromatic represenation using RGB values I.1.4. It takes into account the wavelength dependency of the optical constants of the materials, and therefore leads to more physically reliable color reproductions.

I.3 Appearance of dry powders

In this work, we focus on the appearance of a specific kind of material which is dry powders, and more specifically pigments. Since we are aiming to reproduce their appearance using physically based BRDF models, we should understand what physical phenomena are at stake in their coloration. The particular case of hematite α-Fe 2 O 3 is our main example.

I.3.1 Dry powder description and occurancy

We refer as dry powder any collection of nanoscopic to microscopic grains in the absence of a binder. The grains can have various morphologies that is to say various sizes and shapes. If only one morphology is at stake, the powder is monodisperse.

Dry powders are commonly found materials in our everyday life environment. We can for example mention sandy or dusty soils, food such as spices and flour, eyeshadow and blush in cosmetics, and pigments used in painting.

I.3.2 Pigments

I.3.2.a Description and uses :

A pigment is an organic or inorganic powdered coloring matter whose origin is either natural or synthetic. The main ones have been listed in books such as [START_REF] Roy | Artists' pigments. a handbook of their history and characteristics[END_REF][START_REF] Riederer | Artists' pigments : a handbook of their history and characteristics[END_REF][START_REF] Feller | Artist's pigments : a handbook of their history and characteristics[END_REF][START_REF] Eastaugh | The Pigment Compendium : Optical Microscopy of Historical Pigments[END_REF]. They can be separated into two categories : organic and inogranic pigments.

Organic pigments are made out of carbon bonds and mostly come from plants (ex : indigo) or animals (ex : sepia, carmine). Examples are shown of the Figure I.23. They allow bright colours, but are not very resistant to light exposure and weathering. Because they are quite rare at the natural state, they are nowadays mainly synthesized, which extends the reachable color range.

Inorganic pigments like oxides, hydroxides and metallic salts are the results of milled minerals. Their colors are more dull but have a high covering power and show a nice resistance to change of light, weather and temperature. Some of them are derived from gemstones and are very pricy such as ultramarine that comes from Lapis-Lazuli (see Figure I.24). Even though they can easily be found at natural states, they also can be synthesized to obtain larger amount at a lower cost. Pigments can either be used as a suspension into a binder to ease its application on numerous substrate or dried. For example oil painting pigments are mainly mixed with linseed oil, while for watercolor gum arabic is preferred. Many other medium can be used depending on desired application : different kind of painting, cosmetics, plastic colouring...

In this work we focus on dry pigments that can be observed in parietal paintings from paleolithic to neolithic. The pigments identified come from ferruginous rocks obtained and altered through numerous processes such as mineralization, sedimentary, weathering or volcanic processes [START_REF] Chalmin | How to distinguish red coloring matter used in prehistoric time ? the contribution of visible near-infrared diffuse reflectance spectroscopy[END_REF]. We will only mention ochres, a mix of clays, sands and a coloring agent, consequences of weathering processes. Two main iron oxides or hydroxides are responsible for the colors of ochres : goethite FeO(OH) for the yellow hues and hematite α-Fe 2 O 3 for the red ones. If the pigments may have been applied using a binder such as water, there is no trace left of it nowadays. The appearance of rock painting artefacts then relies on the dry pigments appearance.

I.3.2.b Colors in minerals :

In order to understand how pigments get their colors, let's overview the different causes of the mineral's colors that have been explained by Nassau [START_REF] Nassau | The fifteen causes of color : the physics and chemistry of color[END_REF][START_REF] Nassau | The physics and chemistry of color : the fifteen causes of color[END_REF]. We may distinguish between the chemical colors and the physical colors. The former depends on the chemical structure of the material, at the microscopic scale. One specific material can have a number of color variations depending on its morphology : a powder does not have the same color as the mineral bulk it comes from (see Figure I.24). This is described by optical phenomena at a larger scale and leads to the physical colors.

The color of a mineral is closely related to light absorption caused by exchanges of electrons between its constituents.

A first type of exchange responsible for the color of many pigments including Cobalt blue Al 2 CoO 4 and malachite Cu 2 CO 3 (OH) 2 , is known as the ligand-field effect. The surrounding of one metal ion by ligand atoms results in a degeneration of its least filled orbitals levels (either the f or the d orbitals). Light is either absorbed or emitted at a specific wavelength to allow exchanges between the orbitals' energy levels, giving the materials its color. In some gemstones this effect happens for metal impurities. For example, the deep red color of ruby is due to ligand-field effects involving Cr impurities in the Al 2 O 3 structure. The transition-metal atom causing the coloration of these minerals is called chromophore.

Charge transfer can also happen between two different atoms. In some minerals this occurs between a metal and a ligand atom. For example, Lapis-Lazuli also known as ultramarine when milled into a pigment takes its blue color from charge transfer involving sulfide ions. In other minerals the exchanges of electrons are made between two separate metal atoms. They can be either identical as iron Fe in magnetite or different as iron Fe and titane Ti in sapphire.

In some materials, atomic orbitals can be combined to form molecular orbitals. Instead of the discrete energy levels of an isolated atom it results in energy bands with different widths depending on the studied material : this is known as the band theory. The incoming light must have a minimum amount of energy, known as the Fermi energy, to be absorbed by the material. Any other ray with a higher energy is also absorbed. This is responsible for the color of metals and semi-conductors such as cadmium sulfide CdS or vermilion HgS. It can be complicated in the case of semi-conductors with the presence of impurities in the crystal structure. This can create energy trapping levels, responsible for phosphorescent effect as it does for ZnS including some copper Cu.

There can also be a lack of ions in a crystalline structure, creating a color center. In that case an electron may occupy the created hole to stabilize the material. It gives its color to some gemstones such as amethyst or topaz. In that case, the color can be modified by heating or irradiation.

Physical phenomena may change the color of one material depending on its macroscopic morphology. Scattering and diffraction effects may occur once a mineral is milled into powders. As for thin and quite transparent gemstones, colors can be created through refraction and interference phenomena. These phenomena requires a knowledge of the size and shape of the crystals or the grains, as well as their chemical composition.

I.3.2.c A specific pigment : hematite :

Our study restricted to one particular pigment that is hematite α-Fe 2 O 3 . This iron oxide is a well known material for being responsible for the color of red ochres used in parietal paintings. As a bulk, hematite shows a shiny dark gray look while as a pigment it can cover a wide range of colors from light red to purple.

In this section, we introduce some information about the hematite crystalline structure as well as the chemical and physical processes responsible for its coloring.

Crystalline structure of hematite : Hematite belongs to the R 3c symmetry group, and does crystallize as a trigonal mesh. It is commonly converted into a hexagonal mesh with structural parameters given by Blake et al. [START_REF] Blake | Refinement of the hematite structure[END_REF] Color in hematite : When it comes to hematite α-Fe 2 O 3 , the color mechanisms at stake involve the chromophore ions Fe 3+ . Three main absorption bands respectively near 550 nm, 650 nm and 860 nm can be observed in the visible spectral domain (Figure I.26), which correspond to distinct electronic transitions. We found two interpretations of those electronic exchanges in the literature.

The first one that is proposed by Sherman [START_REF] Sherman | Electronic spectra of fe3+ oxides and oxide hydroxides in the near ir to near uv[END_REF] only implies transitions of electrons between the chromophore ions. They can either occur between the d-orbitals of two separated but adjacent Fe 3+ cations leading to the 550 nm band (charge transfer), or between the d-orbitals of one Fe 3+ leading to the 650 nm and 860 nm bands (ligand field transitions).

Nevertheless, Pailhé et al. [START_REF] Pailhé | Correlation between structural features and vis-nir spectra of α-fe2o3 hematite and afe2o4 spinel oxides (a= mg, zn)[END_REF] questioned this interpretation by raising some issues in the crystal-field study that have been made. Instead, the authors give a new explanation involving the six ligands O 2 -that are surrounding the chromophore Fe 3+ in the hematite crystal structure. According to their study, the first two bands at 550 nm and 650 nm are the results of charge transfers between the two degenerated d-orbitals of lower energies of the iron cation and the p-orbitals of the O 2 -anions, and are respectively called CT1 and CT2. As in the former studies, the third band at 860 nm is interpreted as a ligand transition of Fe 3+ , denoted as D.

Although these differences do not have any impact on our results, we will refer to the Pailhé's interpretation in the following, which is represented on the Figure I.26.

Depending on the morphology of the material, physical phenomena such as different kind of light scattering I.2.3, diffraction and interferences I.2.1.b may also occur and highly influence the perceived color and appearance of hematite powders.

Hematite and Hydrohematite :

Other absorption bands that do not play an important role in the coloration since they are in the infrared spectral domain, may be of great interest to precisely defined the sample under study. It permits to validate the purity of the sample and, in the case of hematite, to distinguish the possible presence of water H 2 O in the crystal structure responsible for an hydrated form of hematite known as hydro hematite [START_REF] Wolska | The structure of hydrohematite[END_REF]. According to Schwertmann [START_REF] Schwertmann | Iron oxides in the laboratory : preparation and characterization[END_REF], this particular form is most of the time obtained when hematite is synthesized in aqueous are characteristic to hematite. These two are also observed for the iron hydroxide goethite, the latter being even more important in such a case. An intense band at 3150 nm would therefore question the purity of the studied hematite sample.

Although the two spectra are representative for hematite samples, they also show some important dissimilarities.

First, two main groups of bands can be identified around 1450 nm and 1930 nm for only one of the samples (light blue spectrum on the Figure I.27). They are caused by multiple vibration modes of H 2 O molecules that are be trapped inside the crystal structure. Another major difference it the amplitude of the broad band between 2600 and 3200 nm. This is once again explained by the different amount of water that can be either trapped in the crystal structure (bands around 2650 -2800 nm and 3100 -3200 nm) or adsorbed at the surface of the material (bands around 2850 -2950 nm). The more hydrated the sample is, the more saturated these bands are. Among the two samples presented here, we can assert that one is made of hydrohematite (light blue spectrum) while the other one is purer (dark blue spectrum).

As a conclusion, two quite pure hematite samples may show different reflectance spectra in the infrared spectral domain : the absorption bands are measured at the same spectral positions but with very different amplitudes. The more or less saturated bands at 1930 and 3100 nm are in particular explained by the potential presence of water in the crystal structure.

Natural hematite : It is difficult to find pure hematite powder at a natural state. It is usually found in red ochers, where it is mixed with other minerals [START_REF] Chalmin | How to distinguish red coloring matter used in prehistoric time ? the contribution of visible near-infrared diffuse reflectance spectroscopy[END_REF][START_REF] Salomon | Les matières colorantes au début du paléolithique supérieur[END_REF][START_REF] Elias | The colour of ochres explained by their composition[END_REF] like quartz, kaolinite, calcite or goethite... The appearance of these coloring matters then may vary according to the mixture proportions. For their use on parietal paintings, they were either directly found in soils or obtained by annealing some yellow ochres containing goethite. Gialanella et al. [START_REF] Gialanella | Artificial or natural origin of hematite-based red pigments in archaeological contexts : the case of riparo dalmeri (trento, italy)[END_REF] and Pomiès [START_REF] Pomies | Xrd study of the goethite-hematite transformation : application to the identification of heated prehistoric pigments[END_REF] show that such a process was carried out to this end on some archaeological sites.

Purer samples can be obtained from the milling of bulk crystals, which is a more or less easy task depending on its friability. Moreover, the powder has to be quite fine in order to show the reddish hues used as pigments.

FIGURE I.28 -Natural hematite powder samples, obtained from milling of pure hematite bulks. The photographs are calibrated using an X-rite 24 patches Classic Colorchecker.

Four natural and quite pure bulk hematite samples have been acquired for our study. Among them, two referenced samples from the Pigmentothèque [START_REF] Salomon | Construction d'une Pigmentothèque : un outil pour comprendre l'approvisionnement en matériaux colorants durant la Préhistoire[END_REF] come from a single seam in the area of Lesquerde (66220, France) : one is smaller and more dense N Ls than the bigger one N Lb . A third sample, referred as N I in the following, is a lending from the "Collection géologique of OSUG". The geographic origin of the last sample remains unknown. It has been found in a flea market in Grenoble and will be denoted as N F . The resulting powders are illustrated on the Figure I.28.

Powders have been obtained from those rocks by manual grinding using a mortar and a pestle. Because some of them are quite hard, this grinding process is not efficient enough to reach uniform and fine powders : some bigger crystals are visible in the resulting powder. This can be seen on the Figure I.28, especially for the samples N Ls and N I . A planetary grinder from Institut de Planétologie et d'Astrophysique de Grenoble has been used on the sample N Ls to obtain more uniform grains of size smaller than 50 µm and thereby overcome this issue.

Hematite synthesis : Hematite synthesis is widely documented in the literature. We whose to use different routes that permit to control the nucleation and the growth of α-Fe 2 O 3 crystallites in solution under standard or hydrothermal conditions. It covers a wide range of morphologies and colors observed on actual parietal paintings from the paleolithic and the neolithic. For example, a synthesis has been chosen to reproduce the morphology and the color of the purple hematite powder used on the "blason de Lascaux" drawing (Figure I.29) FIGURE I.29 -Blason de Lascaux. The purple parts of the drawing have been identified as painted using hematite powders with platelet grains. We intend to reproduce such a pigment using a route under hydrothermal conditions [START_REF] Sugimoto | Preparation of monodisperse peanuttype α-fe2o3 particles from condensed ferric hydroxide gel[END_REF].

Let's overview the syntheses routes that we have chosen.

• Syntheses in acidic solution :

Syntheses of hematite by hydrolysis are described by the methods 1 to 4 from Schwertmann et al. [START_REF] Schwertmann | Iron oxides in the laboratory : preparation and characterization[END_REF] : the dissolution of ferric salts into an aqueous solution leads to the formation of a hematite precipitate. Its sedimentation is eased by centrifugation. The deposition is then collected after being washed and dried. Different ferric salts can be used such as ferric nitrate FeNO 3 (method 1 and 4), ferric perchlorate Fe(ClO 4 ) 3 (method 2), ferrous chloride FeCl 2 (method 1) or ferric chloride FeCl 3 (method 3).

The method 1 can be extended as methods 1 2 and 1 3 , respectively obtained using FeNO 3 {1} and FeCl 2 {2} precursors and adding an aging process of one month at 100 • C before drying.

1L HNO 3 (2.10 -3 M) + 8.08g Fe(NO 3 ) 3 • 9 H 2 O( 2.10 -2 M) 7d, 98 • C (1) 26d, 100 • C (1 2 ) Fe 2 O 3 {1} 1L HNO 3 (2.10 -3 M) + 3.975g FeCl 2 • 4 H 2 O( 2.10 -2 M) 38d, 100 • C Fe 2 O 3 {2}
Other ferric salts can be used : ferric perchlorate in the method 2 {3} and ferric chloride in the method 3. For the later, the route {4} proposed by Schwertmann [START_REF] Schwertmann | Iron oxides in the laboratory : preparation and characterization[END_REF] produces hematite powder composed of two size populations. A small amount a grains are five times larger than the other grains. In order to get a more monodisperse powder, we reduced the concentration of the hydrochloric acid to 1.10

-3 M.L -1 . 100ml H 2 O + 10g Fe(ClO 4 ) 3 • 9 H 2 O (2.10 -2 M) 7d, 98 • C Fe 2 O 3 {3} 1L HCl (2.10 -3 M ou 1.10 -3 M) + 5.4g FeCl 3 • 6 H 2 O (2.10 -2 M) 10d, 98 • C Fe 2 O 3 {4}
Finally, the solution in which the sample 4 is formed undergoes a step of purification through dialysis before its centrifugation {5}. This technique consists of pouring the solution into a microporous bag immersed into deionized water. To ensure the ionic equilibrium between the inside and the outside of the bag, nitrate ions go through the nanometric pores. Such a constraint imposed during the chemical reaction leads to the formation of particularly fine grains.

100ml H 2 O + 8ml Fe(NO 3 ) 3 • 9 H 2 O( 1M) 30min, dialysis Fe 2 O 3 {5}
• Syntheses in aqueous solution :

The two following routes are once again described by Schwertmann et al. [START_REF] Schwertmann | Iron oxides in the laboratory : preparation and characterization[END_REF]. They first result in a ferrihydrite precipitate which is then transformed into hematite before being collected. Nevertheless, because ferrihydrite solubility is quite weak at these pH, it is possible that not only hematite but also goethite is formed in the process. Indeed, without taking any precaution an amorphous phase has first been produced using the {7} route. To minimize such a side effect, the temperature conditions and the neutral pH (around 6.5) must be well controlled at every step of the protocol.

To do so, it is suggested to add some oxalic acid. We found the addition of 200 mg of citric acid more efficient.

30ml KOH (1M) + 50ml Fe(NO 3 ) 3 • 9 H 2 O 0.2M) + 5ml NaHCO 3 (1M) 48h,98 • C Fe 2 O 3 {6} 60ml KOH (1M) + 100ml Fe(NO 3 ) 3 • 9 H 2 O (0.2M) + 200mg C 6 H 8 O 7 • 2 H 2 O (2.10 -3 M) {7} 36h, 90 • C ------→ pH=6.5 Fe 2 O 3
• Annealing of goethite :

The last but not the least route used from Schwertmann et al. [START_REF] Schwertmann | Iron oxides in the laboratory : preparation and characterization[END_REF] is denoted as purple hematite.

Unlike the authors' statement, pure goethite has been obtained. We annealed that powder during 16 hours at 270 • C to transform this unwanted pigment into hematite {8}. This method would be denoted as AG in the following. Such a process resembles the one presumably used by Palaeolithic painter. As goethite is more abundant at a natural state, Gialanella et al. [START_REF] Gialanella | Artificial or natural origin of hematite-based red pigments in archaeological contexts : the case of riparo dalmeri (trento, italy)[END_REF] and Pomiès [START_REF] Pomies | Xrd study of the goethite-hematite transformation : application to the identification of heated prehistoric pigments[END_REF] show that on some archaeological sites goethite was actually annealed with the objective to reach red colors.

60ml NaOH (5M) + 100ml Fe(NO 3 )

3 • 9 H 2 O (0.2M) 8d,70 • C +16h,270 • C FeO(OH) {8}
• Synthesis under hydrothermal conditions :

Purple hematite can be found in parietal paintings from the Palaeolithic era, as it has been observed by Chalmin et al. [START_REF] Chalmin | Les blasons de lascaux[END_REF] in Lascaux, France. To mimic the morphology of such pigments, sample can be synthesized under hydrothermal conditions (high temperature and pressure), following the route by Sugimoto et al. [START_REF] Sugimoto | Characterization of hematite particles of different shapes[END_REF]. The reaction takes place in a autoclave at 180 • Syntheses in base solution :

More creative shapes of grains can be obtained under such conditions. Sugimoto et al. [START_REF] Sugimoto | Preparation of monodisperse peanuttype α-fe2o3 particles from condensed ferric hydroxide gel[END_REF] suggest the use of sulfate or phosphate in order to get peanut-like shaped particles. Two of our samples P 1 and P 2 come from this route.

30ml NaOH (6M)

+ 33ml FeCl 3 • 6 H 2 O (2M) + 3, 33ml Na 2 SO 4 ouNaH 2 PO 4 • 2 H 2 O (3.10 -2 M) {10} 8j, 100 • C ------→ Fe 2 O 3
By changing the concentrations of the chemical reagents, one can obtain shapes from ellipsoid to narrow peanuts.

• Commercial sample :

We also possess one commercial sample C coming from the Puratronic brand. It is a synthetic powder whose detailed production process is not known. As far as we know, its synthesis is based on a technique developed by Meng et al. [START_REF] Meng | Synthesis and application of carbon-iron oxide microspheres' black pigments in electrophoretic displays[END_REF]. For one hour, in a high temperature oven at 700 • C, very fine droplets of a FeCl 2 solution are formed using an ultrasonic technology and collected in a water container. A chemical reaction occurs once the droplets are in contact with the water, forming an hematite precipitate. The morphology of the grains depends on the FeCl 2 solution concentration : the higher it is, the bigger and rougher the grains are. Only the sample C comes from a commercial manufacturer. Its production process is not well known. The photographs are calibrated using an X-rite 24 patches Classic Colorchecker.

I.4 Physico-chemical characterization

Various characterization methods are used in this work to study the physico-chemical properties and the appearance of dry powders, at different scales. Non-contact and non-invasive techniques are favored in order to preserve the small available amounts of samples.

The crystal structure of the samples are studied by X-ray powder diffraction analysis. Their grains morphology is studied from the nanometric to mesoscopic scale using two microscopy methods : scanning electron and optical microscopy. The color and the appearance of the powders from the observer point of view is also investigated through reflectance and BRDF measurements.

In the following, we briefly describe the basic principles of these characterization techniques and the instruments that are used to perform such measurements and analyses.

I.4.1 X-ray powder diffraction (XRPD)

This technique is used to identify the crystalline phases that are present in a sample. It gives pieces of information about its purity and its crystallisation state. Further data permit to deduce micro-structural parameters. I.4.1.a Principle : X-ray powder diffraction (XRPD) is based on the diffraction of X-rays by a crystal lattice. As the wavelength at use is of same magnitude as the inter-atomic distance, incoming rays are scattered by each atom in the lattice, producing interferences. Thanks to the periodicity of the atoms' arrangement, constructive interferences only occur in discrete directions given by the Bragg's law. This phenomenon is known as crystal diffraction. Let's consider a periodic crystal lattice whose atoms are organized along a family of reticular planes hkl, illuminated by a coherent monochromatic X-ray beam of wavelength λ (see Figure I.31). The incoming wave scattered in the same direction by two different atoms will interfere. With straightforward geometric considerations, the optical path length difference between these two outgoing waves can be written :

δ = 2d sin θ (I.79)
where d is the inter-reticular spacing, which is intrinsic to the crystal structure and depends on the lattice dimensions and geometry.

Coherent scattering, that is to say constructive interferences, is ensured if and only if the path length difference satisfies the relationship :

2d sin θ B = kλ (I.80)
where k in an integer. It amounts to saying that it only happens in discrete directions θ B , known as Bragg's directions.

The scattered intensity distribution around the Bragg's positions is represented as a diffraction patterns. An example is given on Figure I.32. Its analysis provides information about the nature, the quantity (in case of mixtures) and micro-structure of the compound under study.

I.4.1.b Phase identification :

The diffraction pattern of a powder obtain by XRPD is a signature of the powder composition. Each phase that is present is responsible for a given diffraction pattern, with specific peaks position, width and intensity. The thinner the peaks of the diffractogram are, the more well crystallized the sample under study is, therefore the larger the crystallites are. Indeed, the X-ray source is a not monochromatic, but a doublet emitting at two close wavelengths. Because the diffraction effect varies with the wavelength, it results in two Bragg's direction slightly shifted that may be discernible in such cases. The phase identification is carried out by minimizing the difference between experimental data and referenced diffractograms provided in the PDF4+(2018) reference database.

An example of phase identification is given on the Figure I.32. In this sample, two crystalline phases have been identified : hematite and goethite. The intensity of the peaks is related to the amount of the corresponding phase in the studied sample. Here, the percentage of goethite appears to be much weaker compared to hematite : apart from hematite, only 7.17% of the total mass of the sample is caused by the goethite presence. [START_REF] Rietveld | A profile refinement method for nuclear and magnetic structures[END_REF] is a technique used to reproduce the experimental diffraction patterns of a powder whose crystal structure is known. In the case of hematite, the crystallized structure from [START_REF] Blake | Refinement of the hematite structure[END_REF] is chosen as a starting point. The fitting is done by optimizing the values of multiple structural parameters. The difference between the reproduced diffractogrmm and the experimental data is minimized using the root mean square methods :

I.4.1.c Rietveld refinements : Rietveld refinement

S = i w i |y cal i -y exp i | 2 (I.81)
with w i the weight of every phase i in the sample, y cal and y exp the theoretical and the experimental intensities respectively.

As it has been done by Pailhé et al. [START_REF] Pailhé | Correlation between structural features and vis-nir spectra of α-fe2o3 hematite and afe2o4 spinel oxides (a= mg, zn)[END_REF][START_REF] Pailhé | Impact of structural features on pigment properties of α-fe2o3 haematite[END_REF], we proceeded to Rietveld refinements optimizing over the following parameters :

• The unit cell's dimensions : Since hematite can be described by an hexagonal cell, the optimization is performed along lenghts a and c of the crystalline axes. The unit cell's volume is evaluated from these two dimensions.

• Atomic positions and inter-atomic lengths : The octahedra FeO 6 that compose the hematite crystal are not regular but undergo some distortion [START_REF] Pailhé | Correlation between structural features and vis-nir spectra of α-fe2o3 hematite and afe2o4 spinel oxides (a= mg, zn)[END_REF][START_REF] Pailhé | Impact of structural features on pigment properties of α-fe2o3 haematite[END_REF]. To evaluate such a phenomenon, the relative positions of the atom Fe and of the atoms O in an octahedron are optimized.

• Occupancies of Fe atoms : According to the neutrality of a crystalline lattice, in the case of pure hematite, two third of the octahedra sites should be occupied by Fe 3+ . This parameters allows to consider that another type of ion is located in one of these occupied sites, creating a color center I.3.2.b, and thereby influence the color of the sample.

• The average crystallite size : It is calculated from the width of the Bragg peaks and corresponds to the size of domains over which diffraction is coherent. If the powder consists of single-crystals, crystallite size and grain size (entities seen under the microscope) are identical. If the crystallization is not perfect, crystallites can be much smaller than grains observed by microscopy. In order to take into account the possible size variations along the crystalline axes, an anisotropy factor is also optimized.

• The preferred-orientation effect : It occurs when powder is consisting of grains with no random orientation (anisotropic shape) and lead to incorrect intensity measurements.

Another parameter which is very important in the case of not perfectly pure samples is the mass proportion of goethite. It permits to quantify the proportion of residual goethite that can be identified in a mixture with hematite.

I.4.1.d Experimental and structural study :

XRPD data were collected on the dried powders with a Bruker Endeavour D8 diffractometer operated at Cu Kα (λ=1.5404 Å). DIFFRAC.EVA's (Bruker) search/match module, which performs searchers on the PDF4+ (2018) reference database, has been used for phase identification. The measured samples were prepared by depositing drops of a highly concentrated hematite powder suspension on a silicon wafer with orientation (553) (see Figure I.33). This substrate only shows one diffraction peak at very large angles, which cannot be observed using this diffractometer.

Once the water is evaporated, it results in a quite homogeneous polycrystalline layer.

The Rietveld refinements of the X-ray powder diffraction patterns were performed with the FullProf Suite software [START_REF] Rodríguez-Carvajal | Recent advances in magnetic structure determination by neutron powder diffraction[END_REF].

I.4.2 Microscopy : SEM and optical microscopy

Microscopy techniques are used in order to evaluate the morphology of the powders, that is to say the size and shape of the grains they are made of. The samples are observed at the microscopic scale using optical microscopy, up to the nanometric scale using scanning electron microscopy.

I.4.2.a Scanning electron microscopy (SEM) :

The basic principle of scanning electron microscopy (SEM) is explained by Golstein [START_REF] Goldstein | Scanning electron microscopy and X-ray microanalysis[END_REF]. It is an imaging technique that gives information about the sample's surface topography and its chemical composition.

The sample is irradiated and its surface is scanned by a focused electron beam named the primary electrons. As it interacts with the atoms of the sample and undergoes multiple absorption and scattering events, secondary electrons are ejected from sample. Because these secondary electrons have a low energy (< 50 eV), their mean free path is very short. Therefore they escape from a few nanometers below the surface of the sample and are highly localized at the primary beam impact point. After the conversion of the electrons into photons and the amplification of the signal, a high resolution image is obtained.

Even though SEM can also be used to characterized the chemical composition of the sample under study, we limit its use as a high magnification microscope. In that extent, SEM is a characterization method that gives a fine knowledge of the morphology of the grains that are composing a powder at the nanometric scale.

However the field of view is restricted so only a small amount of grains can be observed in a single image, which can make the monodispersity characterization of the powder quite hazardous. Indeed, if only a few grains in a sample are very different in size and shape they may be absent of the field of view and therefore be missed. A study at a larger scale should allow to overcome this issue. Moreover, it this microscopy techniques only gives results as grayscale images. No information about the sample's color can be drawn from such observations. I.4.2.b Optical microscopy : Optical microscopy permits the magnification of images of a sample through an ocular and an objective. The principle will not be further detailed.

Such technique is complementary to SEM imaging. As the observations are made at a larger scale, it accounts for possible large agglomerates effects, and attests of the size distribution of the grains constituting the powder on a wider field of view : if the sample contains a small amount of very large crystals they might be missed with SEM images, but will be easily observable using optical microscopy. With a correct color calibration of the camera, it also takes into account the color of the samples under study.

I.4.2.c Experimental instruments :

The morphology of the grains composing the dry samples were studied by SEM using a Zeiss Ultra+ microscope, with incident electrons of energy 3 KV.

The samples are prepared following the same process as for the XRPD analysis, but a suspension with a weak concentration of hematite powder is used instead. It is dispersed using ultrasonic technique before its deposition of the wafer. Thus, it permits to obtain very thin films (few grains thick) that will ease the observation of the individual grains composing the powders.

For the optical microscopy characterization, we used a Nikon Eclipse LV100ND microscope. In order to be observed using this technique, a small amount of dry hematite powders is deposited on a glass microscope slide, and slightly squeezed to make the sample more flat, but not packed.

I.4.3 UV-visible and Near-Infrared spectroscopy

Finally, the samples are characterized at the macroscopic scale using diffuse reflectance spectroscopy. It is a non-contact optical technique that can be conducted in a wide spectral range and that provides valuable pieces of information about the colour of a sample. In this study it is use from the visible to the near infrared (NIR) spectral range (400 nm to 4800 nm).

I.4.3.a Principle :

A sample is illuminated by a light source in a direction θ i . The amount of light that is scattered in a direction θ o by reflection on the sample is then collected by a detector, and results in a reflectance spectrum. A standard illumination/observation geometry is

θ i = 0 • /θ o = 30 • is often used.
The perceived color of a material depends on the light source and the observer experience. To get rid of these variables, the raw measurements I r of the sample are calibrated by a previous measurement on a material of known reflectance I r e f and corrected by the actual reflectance factor of the calibration target R r e f : a Spectralon® in the visible spectrum and very near infrared, and an Infragold in the NIR spectrum. The resulting Reflectance factor R only depends on the material's intrinsic parameters.

R = I r I r e f R r e f (I.82)
The diffuse reflectance spectrum is the signature of the material color and composition. Each absorption band is connected to specific electronic exchanges occurring between the atoms composing the material I.3.2.b.

To limit the study in the visible spectral range would be sufficient to describe the color perceived by a human eye. However, the extension to the NIR domain provides more information about the chemical composition of the studied sample.

One of the main advantage of this technique is that it can be done on any kind of material : powder, bulk with different surface textures... Instruments have been developed to be directly used on the field, which for example allows the study of archaeological sites without removing any sample from the paintings.

I.4.3.b Experimental instrument :

The diffuse reflectance of the hematite powders was measured using the spectro-gonio radiometer SHADOWS developed by Potin et al. [START_REF] Potin | Shadows : a spectro-gonio radiometer for bidirectional reflectance studies of dark meteorites and terrestrial analogs : design, calibrations, and performances on challenging surfaces[END_REF] at Institut de Planétologie et d'Astrophysique de Grenoble (IPAG). Thanks to this instrument's design, only few milligrams of matter is needed to be accurately measured. The illumination spot is 5.2 mm in diameter. The measurements were performed on dried powder layer sample sufficiently thick so that the substrate does not contribute to the measured reflectance (>1 mm), under monochromatic light from 400 nm to 4800 nm, with a normal incidence and an observation angle fixed at 30 • .

To accurately investigate fine color changes between different samples, measurements were conducted in the visible range from 400 nm to 1000 nm with a step of 1 nm. The spectral range and resolution is chosen to optimize the signal-to-noise ratio and the acquisition time. About 2h10 is needed to acquire one spectrum. These spectra are then analysed in the CIE L*a*b* color space.

In the NIR spectral range, the measurements are done from 400 -2600 nm with a sampling of 10 nm and from 2600-4800 nm every 20 nm. After optimization of the acquisition parameters, the measurement of one spectra takes about 1h12.

I.4.4 BRDF measurements

The appearance of an object depends on the geometry under which it is illuminated and observed, as well as its color. As explain earlier in I.2, it is described by the BRDF. For each geometry, the reflected intensity measurement I is calibrated by the reference Spectralon in the geometry 0 • /(30

• , 0 • ). R 30 (θ i , θ o , ϕ) = I(θ i , θ o , ϕ) R r e f (0 • , 30 • , 0 • ) I r e f (0 • , 30 • , 0 • ) (I.83)
where R r e f is the real reference reflectance factor given by Bonnefoy [START_REF] Bonnefoy | Développement d'un spectrophoto-goniomètre pour l'étude de la réflectance bidirectionnelle de surfaces géophysiques : application au soufre et perspectives pour le satellite Io[END_REF], and I r e f is the reference reflected intensity measured in the same conditions as the sample measurement. The calibrated BRDF in any geometry is then given by : It is only investigated in the visible spectral range, from 400 nm to 1000 nm. For material such as hematite, the bands is the visible domain are large enough to choose a sampling every 20 nm. A set of 180 geometries is chosen :

R(θ i , θ o , ϕ) = R 30 (θ i , θ o , ϕ) R r e f (θ i , θ o , ϕ) R r e f (0 • , 30 • , 0 • ) I r e f (0 • , 30 • , 0 • ) I r e f (θ i , θ o , ϕ) (I.
• θ i = 0 • , 20 • , 40 • , 60 • • θ o = -70 • , -60 • , -50 • , -40 • , -30 • , -20 • , -10 • , 0 • , 10 • , 20 • , 30 • , 40 • , 50 • , 60 • , 70 • • ϕ = 0 • , 30 • , 45 • , 60 • , 90 •
After the optimization of the acquisition parameters, about 18h is necessary to measure the BRDF of one sample.

Chapter II

Absorptive thin films

As a first approximation, we can consider a paint of layer as an homogeneous colored thin layer on top of a substrate, either a canvas, paper and more particularly rock in the case of parietal paintings. Different colors are reached depending on the pigments that are at used, and the thinness of the layer may involve the contribution of the substrate seen by transparency to the overall appearance of the work of art. Over the centuries, the parietal paintings have been damaged in many ways including weathering or human activities. These two particular deterioration examples must have result in a removing a some of the coloring matter originally applied by the artist, and therefore lead to partially transparent paint layers that can be observed nowadays.

Thin films are also commonly found as multilayer superposition. An interesting example in cultural heritage is the case of applied brocades [START_REF] Florence | brocarts appliqués» dans la sculpture savoyarde. vers une caractérisation interdisciplinaire[END_REF], a polychrome sculpture technique used to reproduce the appearance of precious fabrics embroidered with gold and silver thread. Among others layers, it involves thin layers metallic sheets of tin and gold that can be coated with a paint layer. Their composition, their thinness and their wear may allow some transparency as well.

Absorptive thin layers, made out of material with a complex optical index, can become transparent if their thickness is small compared to the extinction coefficient (imaginary part of the optical index). This happens with thin metallic layers, but also with many pigments that are semiconductors : their extinction coefficient is close to zero for part of the visible spectrum.

The appearance of dielectric thin films have already been modelized, and is well known for involving interference effect ( [START_REF] Hirayama | Visualization of optical phenomena caused by multilayer films based on wave optics[END_REF][START_REF] Belcour | A practical extension to microfacet theory for the modeling of varying iridescence[END_REF]). Its extension to absorptive coatings requires to take into account not only interference but also attenuation of light as it travels through the object.

In this chapter I introduce a Reflectance and Transmittance model I have developped for the appearance simulation of a substrate coated with a thin to thick absorptive film, accounting for both attenuation and interference phenomena occurring at this scale. For simplicity, the film is supposed to be smooth and homogeneous. The model is implemented in the Mitsuba renderer to reproduce material appearance through photo-realistic images. It is also physically validated by comparing the results against measurements on suitable samples.

II.1 Problem description

As we have explained in the Section I.1.2, materials are usually separated in two distinguishable categories : dielectric materials that are transparent to light, and conductive materials like metals that are supposed to be perfectly opaque in the whole visible spectrum domain. This is linked to the non null extinction coefficient defined as the imaginary part of their optical constants. For thick objects this assumption is verified.

However, with thin enough metallic layers (for example less than 30 nm for gold), a significant amount of incoming light is transmitted through the layer. Practical examples include glass panels covered with layers of aluminium or copper alloys in modern buildings. The layer has to be extremely thin. For standard metals most of the light is blocked after about 100 nm. With these dimensions we need to take into account the wave nature of light and interference effects. It also applies for semi-conductors whose extinction coefficient is null or close to 0 for part of the visible spectrum. The incoming light is then partially transmitted, even for relatively large layers. It happens for various pigments such as hematite (Fe 2 O 3 ), Cadmium sulfide (CdS), or vermilion (HgS), used as thin layers in paintings.

I developed a spectral model for conductors and semi-conductors, taking into account both attenuation and interference phenomena through a thin layer [START_REF] Gerardin | Rendering transparent materials with a complex refractive index : semi-conductor and conductor thin layers[END_REF].

We consider an object with an optical index η 3 that can either be real or complex, covered with a thin conductor or semi-conductor layer with a complex optical index η 2 = n 2 + ik 2 , of constant thickness d . As a first approximation, the layer is supposed to be uniform and homogeneous. The object is placed in a transparent host medium with refractive index η 1 (see Figure II.2).

II.2 Model

Multiple physical phenomena are involved in the appearance of an absorptive thin-film. It accounts for reflection and transmission at the interfaces between the material and both the host medium and the substrate. The light is attenuated as it undergoes several bounces inside the layer, and produces interference effect as it goes out of the layer towards the observer.

II.2.1 Reflection and transmission

As for any smooth interface the incoming light is specularly reflected and transmitted in the directions given by the Snell's law. Reflection and transmission coefficients are then given by the Fresnel's formulas (Section I.2.1.a).

Here because we consider a layer object made of three materials (1 :host medium, 2 :thin-film material, 3 :substrate), interactions at multiple interfaces have to be taken into account. We write r 12 , t 12 , r 21 , t 21 , r 23 and t 23 the reflection and transmission coefficients at the 1-2, 2-1 and 2-3 interfaces respectively.

II.2.2 Absorption

In an homogeneous medium, light is absorbed according the Beer-Lambert's law (Section I.1.3). As it goes through an optical thickness d of the material, the light with wavelength λ is attenuated through an exponential falloff. The transmittance can be written as

T = e -αd (II.1)
with α the attenuation coefficient linked to the imaginary part k of the optical index through the relationship : α = 4πk λ .

II.2.3 Interferences

With the small dimensions of the layer's thickness, we need to use the wave nature of light. When light enters a thin layer coating on a larger object, multiple reflections inside the layer produce subtle interference effect due to the difference in optical path length between outgoing rays. The resulting observed light can be assessed from geometric considerations. Part of the light is transmitted towards the camera after each bounce on the substrate. For each outgoing direction, we get the sum of multiple contributions : light that was reflected on the top interface (R 1 ), plus light that was transmitted at the top interface, reflected on the bottom interface and transmitted again at the top interface (R 2 ), and so on with multiple bounces inside the layer (R 3 , R 4 ,...).

Let's denote δ the optical path difference between each of these contributions, corresponding to the difference of distance travelled by the light along two consecutive rays. This optical path difference results in a phase difference φ between the different rays, and thus into interference effects. It corresponds to the difference between the optical path length light travelled on the first ray R 1 , going from I 1 to H and the optical path length on the second ray R 2 , going from I 1 to I 2 :

δ = (I 1 J 1 I 2 ) -(I 1 H) (II.2) = 2η 2 d cos θ 2 -2η 2 d sin 2 θ 2 cos θ 2 (II.3) = 2η 2 d cos θ 2 (II.4)
For an absorptive material, this complex optical path difference δ results in a complex phase difference between the rays φ = 2π λ δ. We sum the contributions from all the reflected rays R i , with i -1 bounces inside the conductor layer ; R 1 is the light reflected on the top interface, R 2 is the light leaving after one bounce inside the layer and so on. For each ray R i , its amplitude is the amplitude of the incoming light multiplied by the attenuated reflection coefficient r i :

r 1 =
r n = (-1) n-1 1 -r 2 12 r n 23 r n-1 12 e i(n-1)φ (II.8)
The total contribution is the sum of contributions from all the rays :

r = ∞ n=0 r n = r 23 1 -r 2 12 ∞ n=0
-r 23 r 12 e iφ n (II.9)

After simplification, we get the reflection coefficient : This expression is valid for each component of the polarized light, s and p. To get the real transmittance T of unpolarized light, we average the contributions of the s and p components :

r = r 12 +
T = η 3 cos θ 3 η 1 cos θ 1 |t s | 2 + |t p | 2 2 (II.20)
Note that the absorption of the material is well modelized by the Beer Lambert's law involved in the transmittance term II.20 through the exponential factor e -φ I = e -4πk 2 λ d cosθ 2 . As the layer thickness increases, d increases and φ I increases with it. In the limit case, we get |r | 2 = ρ 2 12 and |t | 2 = 0, which corresponds to a uniform and opaque metallic material.

II.3 Results

We implemented our model in the Mitsuba renderer [START_REF] Jakob | [END_REF] to visualize through physically based rendering the appearance of objects covered by a thin absorptive film. observed color. With gold or copper, above 100 nm, the object is almost opaque. With Cadmium sulfide, the object remains partially transparent even at 1 mm. For pigments, such as hematite and Cadmium sulfide, the layer thickness has a visible impact on the material hue.

For certain material combinations, interference effects result in visible color changing depending on the surface orientation (see Figure II.5). The effects are more visible with semi-conductors ; with metals, the attenuation is too strong and light is absorbed before it has the chance to do several bounces inside the layer.

This model can be used to render any kind of two-layered material, involving either a metallic, semi-conductive or dielectric substrate coated with a metallic, a semi-conductive or a dielectric layer.

II.4 Model validation

Our theoretical model is physically-based and suitable to simulate both attenuation and interference effect in an absorptive layer. To demonstrate that it can actually reproduce the appearance of real materials, I have compared its results to multiple transmittance and reflectance measurements on appropriate samples.

II.4.1 Comparison with existing studies

The model is first compared to former thin films studies, carried out in both reflection and transmission. To do so, we used the results respectively published by Yoo et al. [START_REF] Yoo | Ultra-thin films with highly absorbent porous media fine-tunable for coloration and enhanced color purity[END_REF] and Hamd et al. [START_REF] Hamd | Mesoporous α-fe 2 o 3 thin films synthesized via the sol-gel process for light-driven water oxidation[END_REF].

II.4.1.a In reflection :

The samples used by Yoo et al. [START_REF] Yoo | Ultra-thin films with highly absorbent porous media fine-tunable for coloration and enhanced color purity[END_REF] are Germanium deposits with various thicknesses (10 nm, 15 nm, 20 nm, 25 nm and 100 nm) on a gold substrate. The thicknesses are well controlled and allow to observe many colors in reflection, from pink to gray including different blueish hues (see Figure II.6). Different porosities were obtain by changing the deposition angle from 0 • up to 70 • which leads to less uniform samples as the angle is increasing. Because our model has been developed in the hypothesis of a perfectly uniform thin film, we will only compare our results to the samples made with a 0 • depositing angle. We simulate the reflectance of similar virtual sample using our model. Their appearance under a white light point source is also reproduced with the Mitsuba renderer. Our results can be compared to the measurements and the photographs from the authors on Figure II.6. The overall trend of the simulated reflectances as well as the color variations with the thickness of the layers are in good agreement with the measurements. Even though the hue variations between the actual samples and the simulated ones are the same, some discrepancies are noticeable between the spectra. First, the spectral positions of the measured and theoretical absorption bands are quite different : while the main band observed for the 10 nm thick layer is situated around 525 nm according to Yoo et al., it is shifted towards 625 nm using our model. Other absorption bands around 500 and 550 nm are also predicted by our model. These may be the consequences of different material's optical constants used in the reference and in our predictions. It has an important impact on the observed color, which is illustrated by color difference ∆E values II.1 larger than the just noticeable difference.

Unfortunately, the spectrum of the 100 nm layer sample has not been measured by the authors. Only a comparison of its photograph along with our model prediction could be done, which again leads to a quite large and highly noticeable color difference. The amplitude of absorbance and the increasing of attenuation with the thickness of the layer is well predicted by our model. However, as for the reflection study an error on the reproduced color may be involved since the spectral positions and the intensities of the absorption bands are not in agreement.

II.4.1.c Conclusions :

The changing of colors are well taken into account by our model, meaning that physical phenomena at stake (attenuation and interference) are correctly identified and well modelled. Nevertheless, the difference between the simulated colors and the samples' colors remains noticeable. Multiple sources of errors have been identified : • The different spectral positions of the absorption bands are closely related to the materials' optical constants (see Section I. 1.4.d). A poor knowledge of these values has repercussions on the samples' renderings, leading to an increasing of the color difference with the real samples.

• The lighting conditions (geometry and illuminant spectrum) used for the photographs of the samples are not well known, and probably different from the one used in the renderings. This also impacts the perception of the reflected or transmitted color.

• Finally, because it is difficult to experimentally produce a perfectly smooth and uniform thin layer the samples are slightly porous. A control of the Ge-Au samples' porosity will change the effective values of n and k which can drastically change the color of the samples. Our model is an approximation that does not take into account such a phenomenon.

II.4.2 Experimental validation

In order to better control the unknown parameters from previous thin-films studies such as optical constants, layer's thickness and lighting conditions, I proceed to my own experimental validation of the model. My study is limited to the case of a gold film in transmission. A thin coat of gold is deposited through an evaporation process on a glass plate. Two deposits with different and well controlled thicknesses are made : -50 ± 1.5 nm thick deposit -100 ± 2.3 nm thick deposit Theoretical thicknesses have been checked by Atomic Force Microscopy, on another sample made using the same process but also covered with a 5 or 10nm layer of Titane. This treatment permits to fix the gold deposit on the glass plate. Otherwise the deposit would be removed as we proceed to the thickness measurement. Photographs of the sample seen in both reflection and transmission is given on

II.4.2.a Transmittance measurements :

The transmittance spectra are measured using a Perkin Elmer Lambda 900 spectrometer. Measurements of each gold thin film are performed between 360 and 830 nm, with a sampling step of 1 nm. A smooth transmittance peak is observed around 500 nm, whose intensity depends on the layer thickness : as expected, the thinner the layer is, the more intense is the transmitted light.

II.4.2.b Gold optical constants :

Our model requires two parameters that depends on the material of the thin layer : the real and imaginary parts of the optical constant. These optical constant values are available in databases such as [START_REF]Refractive index database[END_REF]. However the accuracy of their measurements remains an open issue that is most of the time ignored. As the case of gold has been widely studied, over 14 measured sets of values of these parameters are available in the literature. In this section we proceed to an overview of these values, their similarities and differences, in order to deduce which one is better suited to modelize the appearance of our thin gold layers samples.

• Theoretical optical constants :

Three references are offering gold optical constant values using three different theoretical models, briefly introduced in Section I.1.4.d. Both the Lorentz-Drude model and its improvement known as the Brendel-Bordmann model have been employed by Rakic et al [86], while the Density functional theory (DFT) has been used by Werner et al. [START_REF] Werner | Optical constants and inelastic electron-scattering data for 17 elemental metals[END_REF].

The results from these methods are represented on the Figure II.10. For each set of values, the transmittance of a 50 nm and a 100 nm thin film of gold on a glass substrate is simulated using our model and compared to the measurements on the corresponding sample. FIGURE II.10 -Optical constants of gold computed using different theoretical models, and simulation of the transmittance of a 50 nm and a 100 nm thin gild layer on top of a glass substrate using our model [START_REF] Gerardin | Rendering transparent materials with a complex refractive index : semi-conductor and conductor thin layers[END_REF].

According to our results, the optical constants theoretically determined are not suitable and reliable. Whereas the measurements on our samples agree that the transmittance through a thin film of gold is maximum around λ = 500 nm, the results from the Brendel-Bordemann and the DFT models predict two very different spectral positions. We should also notice that the sampling used by Werner [START_REF] Werner | Optical constants and inelastic electron-scattering data for 17 elemental metals[END_REF] is too coarse to be accurate enough resulting in a flattening of the transmittance peak. As for the Lorentz-Drude model, it correctly predicts the position of the peak, but it does overestimate its width and therefore underestimate its amplitude.

• Experimental optical constants :

Two main experimental methods are used to measure the optical constant of a material (Section I.1.4.d) : spectrometry coupled with Kramers-Krönig analysis, and ellipsometry. These can be performed on either a bulk crystal or a semi transparent thin film. The latter is of great interest here since it corresponds to our studied samples.

The referenced results I found are summarized in table II.2, and some of them are represented on Figures II.11 and II.12 along with the transmittance predictions of our model using these values. The others are given in the Appendix V.3.

Crystal or thick film Thin film Ellipsometry Kramers-Krönig Analysis Johnson [START_REF] Johnson | Optical constants of the noble metals[END_REF] x x Palik [START_REF] Palik | Handbook of optical constants of solids[END_REF] x x Gao [START_REF] Gao | Comparison of different dispersion models for single layer optical thin film index determination[END_REF] x x McPeak [START_REF] Mcpeak | Plasmonic films can easily be better : rules and recipes[END_REF] x x Babar [START_REF] Babar | Optical constants of cu, ag, and au revisited[END_REF] x x Yakubovsky [START_REF] Yakubovsky | Optical constants and structural properties of thin gold films[END_REF] x x Yakubovsky [START_REF] Yakubovsky | Ultrathin and ultrasmooth gold films on monolayer mos2[END_REF] x x Rosenblatt [START_REF] Rosenblatt | Nonmodal plasmonics : controlling the forced optical response of nanostructures[END_REF] x x It appears that for the studies on thin films, the thinner the films are the further our simulations using these optical constant values are from the actual transmittance measurements. This is particularly bad for thicknesses smaller than 10 nm for which the experimental transmittance measurements are way underestimated by our model. This error should be explained by a worse control of the thin layer homogeneity and thickness at this scale. The results from Yakubovsky [START_REF] Yakubovsky | Ultrathin and ultrasmooth gold films on monolayer mos2[END_REF] lead to simulations completely different from what was expected, because of a MoS 2 substrate contribution. As a consequence, we considered their results as not representative of pure gold.

Even for thicker layers from 10 nm to 100 nm, the simulation tends to underestimate the real transmittance of the gold layers with both 50 nm and 100 nm thickness.

We can conclude that the measurements of the optical constants should be more reliable if done on an optically thick sample such as a bulk instead of thin films. This would ensure the homogeneity and the opacity of the sample so that no substrate contribution is included while proceeding to the measurements. Some results from studies on large crystals of gold are represented on the Figure II.12. Among all the references, Palik's results [START_REF] Palik | Handbook of optical constants of solids[END_REF] is the only study which shows a second transmission peak around λ = 650 nm, probably caused by some impurity in the measured sample. For this reason it is not considered to be a reliable and representative result for pure gold.

The other ones lead to satisfying transmittance prediction. FIGURE II.12 -Optical constants of gold measured on a bulk of gold, and simulation of the transmittance of a 50nm and a 100nm thin gold layer on top of a glass substrate using our model [START_REF] Gerardin | Rendering transparent materials with a complex refractive index : semi-conductor and conductor thin layers[END_REF].

Although most of the referenced values agree on the presence of one main absorption band in the visible spectrum around 500 nm, they show a different amount of bands (Palik [START_REF] Palik | Handbook of optical constants of solids[END_REF] and Yakubovsky [START_REF] Yakubovsky | Ultrathin and ultrasmooth gold films on monolayer mos2[END_REF]) with various intensities.

The one that better fits the measurements of our samples at both 50 nm and 100 nm thicknesses is the set of values given by Babar et al. [START_REF] Babar | Optical constants of cu, ag, and au revisited[END_REF]. However the peak is not well defined using these optical constants. This is due to the poor spectral sampling the authors used during their measurements.

In the following, we will use the values from Babar et al. to validate our semi transparent thinfilms model.

To conclude about this overview, we must notice that the multiplicity of measurements of gold optical constants leads to a multiplicity of different results. Some of them agree on the absorption bands spectral positions but the evaluation of its intensity remains inaccurate.

Yet, gold is a widely studied material for which numerous sets of optical constants values, predicted or measured, can be discussed and confronted against each other. Among all the referenced values we used, such a comparison is performed for the more recent measurements against the oldest ones from Johnson [START_REF] Johnson | Optical constants of the noble metals[END_REF] or Palik [START_REF] Palik | Handbook of optical constants of solids[END_REF]. Some of these values such as [START_REF] Yakubovsky | Ultrathin and ultrasmooth gold films on monolayer mos2[END_REF] and [START_REF] Rosenblatt | Nonmodal plasmonics : controlling the forced optical response of nanostructures[END_REF] are not even compared to any other reference. For less commonly used materials the reference values may be less numerous and not confronted at all, as we will show in the case of hematite later in this work III.3.

II.4.2.c Simulation using our model :

The comparison between the measured and the simulated transmittance spectra using our model is given on the Figure II.13. The spectra are very close, for the 50 nm thick layer as well as for the 100 nm thick one. Indeed the measured spectra for the thicker sample lies in the margin of error defined by the uncertainty on the sample real thickness we used in our predictions. It is not the case for the thinner gold layer, but the two spectra (measured and predicted) remain quite similar.

We performed a colorimetry analysis in the CIE L*a*b* color space, using a D65 illuminant. The color observed in transmission has a greenish hue which is well reproduced by our model. The color difference between the theoretical and the measured spectrum is even smaller than the just noticeable difference in the case of the 100 nm thick coating (E00 < 1). In agreement with our previous observations of the spectra, the color difference is slightly more important and therefore noticeable for the 50 nm gold layer. This can be seen on

II.4.2.d Conclusions :

This experimental study allows a more accurate validation of my model. Although the coating is not perfectly smooth, the better accuracy on the thickness and the transmittance measurements conditions leads to nice color reproductions. As the layer becomes thicker, the transmitted light becomes darker and its color so similar to our color predictions that the difference between them turns out to be unnoticeable. A larger gap slightly larger than the just noticeable color difference is obtained for thinner layers, which remains satisfactory. Nevertheless, the results are still highly depending on the optical constants at use. In the case of gold or other material whose optical constants have widely been studied, one may choose a set of values that better suits the experimental datas. But this can be an issue when it comes to less commonly used materials with poorly known optical constants.

II.5 Conclusions & Limitations

I have presented a new model for rendering materials with a complex optical constant. Example include thin metallic layers but also many colored pigments. Any kind of two layered material can be modelled using my model: metallic or dielectric layer on a metallic or dielectric substrate. It reproduces the spectral variations caused by multiple bounces inside a uniform layer: the color and aspect of the object change dramatically with the thickness.

The model has been validated along with existing thin films studies. Although numerous sources of errors remain such as the material optical constants and the layers' thickness and uniformity, the color changes are well reproduced by our model. To overcome some of these uncertainties, an even finer validation has been performed in the case of a thin coating of gold. My model is able to reproduce the appearance of such an object with a error smaller than the just noticeable color difference (E 00 < 1).

Taking into account interference in pigment layers causes a dramatic change in hue, which corresponds to experimental measurements.

Our model is still too simple compared to reality: pigment layers are not made of a continuous layer ; they are more likely to be an aggregation of small particles. This causes more color variations, due to scattering and the interference between the constituents. Similarly, the local structure and porosity of metallic layers also plays a role on their color appearance.

Chapter III

Color of Hematite

The approximation of an homogeneous absorptive layer on top of a substrate is too simplistic to be representative of an actual painting. Because it only considers absorption as being the main phenomenon responsible for the color of a material, it only relies on the chemical composition of the material the thin film is made of. Consequently, two paint coatings made out of a material with the same crystalline phase and the same thickness will have the exact same appearance. This does not agree with the observations made on parietal paintings.

Let's focus on only one material, hematite α-Fe 2 O 3 , the iron oxide identified in parietal paintings as responsible for reddish hues. It appears that various shades are observed, going from light red to purple, which is also corroborated by the hematite powders we have synthesized. The explanation for these color variation is given by the scattering phenomena, which has been completely neglected in our previous model. Because the painting layer is not an uniform and smooth bulk crystal but a quite dense collection of grains, light that is incoming of the material is deviated as it goes through the layer and encounters the scatterers. Depending on the size and shape of the grains, different appearance and color will be described.

In this chapter, I proceed to the full characterization of the natural and synthetic hematite crystalline powders in our possession, from the nanometric to the macroscopic scale. The color variations measured using spectroscopy are compared to the physico-chemical properties, including the structural parameters determined through X-ray powder diffraction and the morphology of the grains characterized by microscopy.

III.1 Hematite samples

This study is focused on hematite, the pigment that is responsible for red to purple-ish hues observed on parietal paintings. Although it has been identified as very pure on some paintings in Lascaux for example [START_REF] Chalmin | Les blasons de lascaux[END_REF], it is often mixed with other minerals, such as quite colorless clays or other coloring agent. In this study, I get rid of the coloring contribution of any impurity by only considering several pure hematite samples, covering a wide range of colors and different appearance from matte light red to shiny grey-ish hues.

Thirteen of them are coming from the syntheses routes described earlier in Section I.3.2.c and shown on the photographs on Figure I.30. The main issue of syntheses is that only small amounts, up to a few milligrams of powder are obtained.

I also used the four natural samples introduced in Section I.3.2.c, obtained by milling pure hematite bulks into powder.

All the studied samples are optically characterized, at multiple scales and using multiple methods in order to establish a relationship between their color and their physico-chemical properties.

III.2 Characterization of the samples

The samples are more accurately characterized using optical techniques to correlate their chemical and physical properties with their appearance.

In order to preserve the small quantities of synthesized powders, non-contact and non-invasive characterization methods are used to describe the different samples at multiple finer scales. First of all, the appearance of the samples is described at both macroscopic and microscopic scale by their observation with a naked eye and with an optical microscope. We also reached higher magnifications using the Scanning Electron Microscopy technique to visualize the morphology of the grains composing the samples.

In order to make sure that our samples are pure and to investigate the structural parameters of the powders, we used X-ray Powder Diffraction (XRPD).

Finally, we proceed to reflectance and BRDF measurements to fully and accurately characterize the color and more generally the appearance of hematite powders. Their appearance at both a microscopic and a macroscopic scale are also investigated using respectively microscopy techniques and Reflectance as well as BRDF measurements.

All the methods that we have used are described in more details in Section I.4.

III.2.1 Visual observation of the samples :

The color and the surface appearance of the samples are first described by two kind of visual observations : It is first done at the observer scale with a naked eye, and then at a higher magnification using optical microscopy. This is done for every sample, synthetic or natural.

Although those techniques mainly involve our perception therefore do not lead to accurate quantified results, such observations are mandatory to get a good idea of the wide range of color that can be reached by a single material, and to first intuit on the physical phenomena responsible for the material appearance.

III.2.1.a Observations with a naked eye :

Before any further and precise characterization, we can make the following observations on the samples' appearance perceived at the macroscopic scale with a naked eye.

First of all, we must observe that a wide range of colors is obtained throughout the natural samples and the syntheses.

Let's start with the description of the natural samples, summarized on the Figure III.2. Among the two bulks coming from the same seam, the smaller sample N Ls appears to be darker than the bigger one N Lb . The later also shows purple and red areas, and is more friable so easy to reduce to powder. When manually milled as a pigment, both of them appear to be dark purple-ish and glittery. As for the powder obtained from N Ls using the planetary grinder, it is much more uniform in color and shows a brownish hue that was not present in the sample obtained manually.

The black and specular plane sheets that are constituting the sample N I being very hard to manually mill, its powder is composed of grains that are still quite large therefore specularly reflective, and inhomogeneous in size.

A more reddish hue has been obtained with the sample N F , extracted from the red and rough areas that can be seen on the bulk.

Much more colorful observations are made on the synthetic samples, going from light red to purple. Most of the hematite powders show a uniform color. A more complex one is the commercial sample C that have two different color types of grains: although the main observed hue is purple-ish, it also contain traces of red. This can be seen on the photographs of Among the synthetized samples, the sample 1 3 draws attention because of its sparkling appearance. Such an effect but much more subtle can also be observed on the sample 6. Because the same glittery effect is much more visible in the manually milled natural samples, it suggests that it is caused by the presence of larger specular grains. A study at a higher magnification is needed to validate or refute this hypothesis (see Section III.2. 1.b).

Distinction between the samples can be done on their texture as well. While most of the powders are quite fluid, the samples P 1 and P 2 tend to agglomerate as larger packs. For the natural samples, the microscopic characterization is in agreement with the observations using the naked eye (Figure III.2). The grains are large and not uniform in size, especially for the samples N Ls and N I that were particularly hard to manually mill into powder. As expected, the sample milled using the planetary grinder has grains of size inferior to 50 µm but remains inhomogeneous. It appears that only the finest grains contribute to the slight red coloration of the powder, the biggest ones being specularly reflective crystals.

Among the synthetic samples, I identified similarities between some of the powders : samples 1, 1 2 , 2, 5 and 6 are all mainly composed of red small grains, and a few bigger ones of sizes ≈ 50 µm (see Figure III.4). The rough and matte aspect of these large grains does not affect the uniform appearance observed at a larger scale, which made us considering them as agglomerates of smaller grains. 1 3 also contains larger crystals that, contrary to the agglomerates in the former identified group, appear to be specular and glossy crystals (Figure III.3). It explains the sparkling effect of this sample.

The sample 3 is quite uniform at this scale. As a consequence, the smoothing of the surface can cause a re-orientation of some grains, responsible for the shiny areas that are visible on the photograph.

HT, P 1 and P 2 are also uniform, but are different from the former sample with their texture. These three powders have a tendency to agglomerate as large packs of powder. This can be observed at a small magnification (×5) where only well defined blocks or layers are visible. Higher magnifications allow us to discern the individual grains that are composing these ensembles. Their micrometric size will be confirmed by SEM.

The observation of both red and purple hues with a naked eye on the commercial sample C is confirmed by optical microscopy : distinct grains of different colors are composing this powder. The grains are much larger than for any other synthetic sample. It results in a free powder that does not show any tendency to agglomerate.

For almost every samples, this microscopy technique does not allow to characterize the morphology of the grains. Even if the samples HT, P 1 , P 2 seem to be composed of larger grain, their precise size and shape cannot be deduced from these observations. Such an investigation is done for synthetic samples at an even finer scale using SEM. Because the grains of the natural powders are not uniform in size and are too large to show interesting color variations, we did not investigate them any further.

III.2.1.c Scanning Electron Microscopy (SEM) observations :

The main principle of this technique is explained in Section I.4.2. It allows observations up to the nanometric scales, which is equivalent to the scale of the grains composing the synthetic hematite powders. For each sample, we can estimate the monodispersity, and the morphology of these grains.

I observed various shapes and sizes going from few nanometers to microns (Figure III.4). Except for the sample 3 which is composed of two bi-pyramid shaped crystal populations different in size (a smaller one 3 s and a larger one 3 l ), the monodispersity of all the samples seems to be well controlled. Only two of the powders have a particularly anisotropic shape of grains : AG which shows flat needle shape whose thickness is much smaller than its width and its length, and HT. The latter is of big interest because its hexagonal platelet shape is quite similar to the morphology of some natural hematite-based pigment found in Lascaux's painting [START_REF] Chalmin | Les blasons de lascaux[END_REF].

As we already saw using optical microscopy, the samples 1, 1 2 , 2, 5 and 6 are quite similar : they all show a diamond shape of slightly different but nanometric sizes. The different aging times between the samples 1 and 1 2 does not significantly modify the morphology of the resulting powders. The change of precursor on the other hand does. Indeed, 1 3 has a complete different morphology which consists in quite large sticks. The big crystals that have been observed with optical microscopy in this sample are not visible in the presented SEM images.

The finest powder that has been synthesized is the sample 4 which is composed of spherical particles with radius < 10 nm. P 1 and P 2 were expected to have a peanut-like morpholgy but show an ellipsoid shape instead.

According to [START_REF] Sugimoto | Preparation of monodisperse peanuttype α-fe2o3 particles from condensed ferric hydroxide gel[END_REF], some variations of the synthesis parameters would involve a narrower radius at the center of the grains. The observed grains do not only differ from one another from their shape and size, but also from surface appearance : some of them such as in P 1 , P 2 and 1 3 look quite rough. It may also be the case for 4 but the powder is too fine to confirm it.

Finally, the description of the sample C is quite complex because it can be done at two different scales : small spheres seem to have been annealed so that they melted and assembled to form micrometric grains. The average dimensions of the grains were deduced from SEM images and reduced to only one dimensional parameter per sample by computing the radius of an equivalent sphere r eq . In order to better take into account the shape anisotropy of certain samples (AG and HT), I computed this radius so that the ratio of the surface S over the volume V of the grain remains constant III.1 :

S spher e, eq

V spher e, eq = S g r ai n

V g r ai n so eq = 3 V g r ai n

S g r ai n (III.1) Specific precautions have been taken for the samples 3 et C. These two have two size populations. Two radius are then determined. Results are listed in Table III.1.

Grain size and average crystallite size we determined through XRPD are in good agreement for all samples : it shows that the powders consist of single-crystals. Larger discrepancies are found for samples AG, C, P 1 and P 2 for which average crystallite size are very smaller than grain size. In this case, grains are therefore agglomerates of small single-crystals, as the SEM images of peanuts samples P 1 and P 2 seem to show.

III.2.2 XRPD & Rietveld refinements

To ensure the purity of the samples, every powder is characterized by X-ray powder diffraction (XRPD). Hematite α-Fe 2 O 3 is identified in all of the samples.

Only one of the natural sample is very pure, N Ls . The three other ones contains another phase than hematite. The sample N Lb is a mixture of hematite and quartz which is a quite commonly found natural material in the coloring matters used for parietal art, while the second but minor phases in the sample N I and N F could not be identified. N F is particularly not pure since it is also composed of an oxide that does not appear to be natural. These impurities comforts us to only focus on the synthetic samples thereafter.

A minor phase of goethite FeO(OH) is identified in the sample 5, 6 and HT (see Figure III.5). This is confirmed by the Rietveld refinements which estimates that less than 10% of the mass of these samples is due to goethite. We assume that this is sufficiently small to be a negligible amount, and will consider in the following that all of the samples are composed of pure hematite. We can note that, according to the thin and even duplicated diffraction peaks, the samples 3 and HT are particularly well crystallized. On the contrary, the larger peaks obtained for the samples 4 and AG demonstrates that they are less well-crystallized. For AG one may notice that the peaks are asymmetric and of various width. This is linked to some anisotropy of the crystallites, the smallest mono-crystals of the sample that are diffracting the incoming X-rays. Several structural parameters, introduced in Section I.4.1.c are optimized during this analysis, such as the unit cell's dimension, the atomic positions, the occupancies of the iron cations, or the average crystallite size. Constraints are applied to the Rietveld's refinements so that the optimized crystallites are shaped in a plausible way. Among the parameters at stake it is the crystallite size parameter that seems to vary the most with the sample under study. Details are listed in the Table III.1. The complete refinements' results are given in Appendix.

III.2.3 UV-vis & Near-Infrared (NIR) spectroscopy

The color of the hematite samples is accurately measured by reflectance spectroscopy. Since the electronic exchanges between a material's constituents only occur at specific and well identified wavelengths, the absorption bands represented on a reflectance spectrum are the signature of the material's chemical composition. Then, spectroscopy also allows to corroborate the XRDP results about our samples' purity.

I measured the diffuse reflectance spectra of the hematite powders using the spectro-gonio radiometer SHADOWS developed by Potin & al.[84] at Institut de Planétologie et d'Astrophysique de Grenoble (see Section I.4.3).

We remind that two measurements are performed on each sample :

• A first measurement is done from the visible spectral range to the Near Infrared domain 400-4800 nm to roughly identify the absorption bands positions and width. As the reflectance spectra can be seen as a signature of the material composition, this study allows to verify the purity of our samples. A step of 10 nm is chosen between 400-2600 nm, and a step of 20 nm between 2600 -4600 nm. It requires about 1h12 to measure one spectrum.

• The second measurements is focused on the visible range 400 -1000 nm with a step of 1 nm to more accurately quantify the spectral differences between the samples. It permits to focus on their perceived color. Such spectrum is measured in 2h10.

III.2.3.a In the Near Infrared spectral range :

We already know that our samples are composed of pure hematite α-Fe 2 O 3 and may contain some traces of goethite FeO(OH) III.2.2 and water if we are dealing with hydro hematite. The identification of such slight impurities can be done by the study of the absorption bands occurring in the infrared spectral range I.3.2.c.

The measurements are done under monochromatic light from 400 -2600 nm with a sampling of 10 nm and from 2600-4600 nm every 20 nm. The spectral range and resolution is chosen to optimize the signal-to-noise ratio and the acquisition time. Results for the synthetic and the natural samples are respectively given on Figures III.6 and III.7.

First, let's note that in the case of the synthetic samples, the characteristic absorption bands of hematite are observed in all of our samples which validates the main chemical composition determined by X-ray powder diffraction.

Then we can investigate the differences between these spectra. Two groups are identified, one composed of the samples HT and C showing a flat reflectance from 1200 to 2700 nm, and a second one with all the other samples. We can assert that the later are hydro-hematite samples. Indeed, the absorption bands specific to the water's vibration modes are quite important which testifies of its presence in the samples (see the blue lines on Figure III.6). This is also confirmed by the greater saturation of the broad absorption band around 3000 nm. Such a difference between our samples can be explained by their synthesis conditions according to Schwertmann [START_REF] Schwertmann | Iron oxides in the laboratory : preparation and characterization[END_REF]. Except for the samples HT and C that are synthesized under hydrothermal conditions, all of our samples have been synthesized in aqueous solutions at quite low temperatures, facilitating the trapping of water in the crystal structure.

Finally, the impurity of samples 5, 6, and HT that has already been identified through the XRPD and Rietveld refinements analysis can be verified by this analysis of the 3150 nm absorption band which is more of a goethite feature. It is more noticeable for these three samples which attests of a small amount of a goethite crystalline phase.

Regarding the natural samples, the flat reflectance continuum as well as the non saturated bands around 3000 nm might be interpreted as the result of hematite formed under hydrothermal FIGURE III.6 -Reflectance spectra of the synthetic samples in the Near Infrared spectral range. The spectral positions schematized by the red lines represent the characteristic absorption bands of pure hematite, while the blue lines represent the absorption bands caused by water either adsorbed or trapped in the crystal structure. Most of our samples are actually hydro hematite, while HT and C appear to be purer. Note that the large and intense band between 4180 and 4340 nm is caused by corbon dioxide gas. Because this gas is strongly absorptive and fluctuating as we enter the experimental room, it is not well corrected. conditions. However this interpretation is questioned by the weakness of the detected reflectance signal. Indeed, it does not permit to nicely distinguish the characteristics of the other crystalline phases that were identified through XRPD in most of these samples. We can intuit a mineral phase of quartz in the sample N Lb thanks to OH bands at 2750 nm, which would match the previous results. But the quadruplets of bands around 2700 nm which suggests that N F contains kaolinite has not been identified by XRPD. Finally we can also note traces of organic material in the samples from Lesquerde, that are causing absorption bands between 3400 and 3500 nm.

III.2.3.b In the UV-visible spectral range :

In order to better study the color that is perceived by a human eye, a more precise measurement is made in the visible spectral range, from 400 to 1000 nm with a sampling every 1 nm.

As expected I.3.2.c, we observed three main absorption bands that are typical of hematite in the visible spectral range, on both synthetic and natural samples (see Figures III.9 and III.10). Two other bands that are described in the literature [START_REF] Sherman | Electronic spectra of fe3+ oxides and oxide hydroxides in the near ir to near uv[END_REF][START_REF] Marusak | Optical absorption spectrum of hematite, αfe2o3 near ir to uv[END_REF] at smaller respectively 430 nm and 310 nm are too saturated to be of particular interest. The latter could not be measured using the SHADOWS instrument, but explains the continuum of the reflectance spectra at small wavelengths.

Further study is carried out on the synthetic samples. Although every one of them involve the same coloration processes, their spectra differ from one another with their intensity, the width and the slope of the absorption bands, as well as the shift around their theoretical positions III.9. Those variations are very fine so they need to be determined with an accuracy of less than 1 nm. This is not possible from the raw measurements since the chosen sampling is of 1 nm.

To overcome this issue, the reflectance spectrum of hematite samples can be modelized using a sum of Gaussian functions : each absorption band is represented by one Gaussian. As I chose to carry out this study on a larger spectral domain than the one a human eye can perceive (from 380 nm to 780 nm), taking into account the absorption bands right before and right after the visible spectral range (respectively at 310 and 850 nm) allows a nice fitting of the measured spectrum over all the spectral domain of interest. Otherwise it would have led to discrepancies on its edges. Such a theoretical model reproduces the spectra with a RMS error of magnitude 10 -6 , and permits to predict the reflectance of every sample with any sampling even finer than 1 nm. The example of this modelization results for the sample 1 is given on the Figure III.8. Similar fits are obtained for all the measured samples and given in the Appendix. FIGURE III.8 -Modelization of the reflectance spectrum of sample 1. The spectrum is measured over the 400 to 1000 nm spectral domain and can be reproduced by the sum of five Gaussian functions : every one of the five absorption bands that have an impact in the visible spectral range is modelized by one Gaussian. Athough the bands at both 310 and 850 nm are not centered in the visible domain, they impinge on it because of their width. Without their accounting the modelization would lead to discrepancies on the edges of the considered spectral domain. This modelization allows us to reproduce the experimental spectrum with a RMS error of 10 -6 .

The conversion of reflectance spectra towards the CIE L*a*b* color space offers a simpler visualisation of the color differences between the samples (see Figure III.9). This is done using the formalism introduced in Section I. Such study involving Gaussian modelization of the spectra and their conversion towards colorimetric coordinates has not been performed on the natural samples. Since the grains composing these powders are quite big, the samples remain very dark and therefore highly absorptive in the visible spectral range. This results in a weak and noisy reflectance which cannot be accurately reproduced using the Gaussian model. Nevertheless, we can conclude that the sample N F definitely does not appear as pure hematite since its spectrum shows important differences from the other ones (larger absorption bands, and a steeper slope at 550 nm). FIGURE III.10 -Reflectance spectra of the natural samples in the visible spectral range. The absorption bands of hematite are well noticeable for all the samples, but less narrow in the case of sample N F . This may be caused by the impurity of this sample (presence of kaolinite) stated by XRPD analysis.

III.2.4 Analysis

The available samples include four natural samples and thirteen pure ones that have been synthesized using various routes.

We decide to not consider the natural samples for two reasons : first they are more complex as they show inhomogeneities in the morphology of their grains. Moreover, some of them are not pure hematite and contains complex or unidentified crystalline phases (see Appendix V.3), which could mislead our analysis.

Thereby, only the synthetic powders are studied in the following.

The different characterization methods described earlier allows to determine various parameters at multiple scales : the size and shape of the grains are deduced from SEM observations, and the average crystallite size are determined by XRPD and Rietveld's refinements analysis along with other micro-structural parameters such as atomic positions and occupancies. The mass proportion of goethite in the sample is also investigated in case of mixtures. Finally the perceived color of each sample measured by vis-NIR spectroscopy are studied using various their coordinates in the CIE L*a*b* color space.

The main results of theses characterizations are summarized in Table III These various parameters have been studied alongside with each other to find a correlation between them that would explain their different appearance. We especially looked for a similar trend found by Pailhé et al.[79] between the distorsion, characterized by the atomic positions deduced from Rietveld's refinements I.4.1.c, of the FeO 6 octaedra composing hematite crystals and the color variations. However, we couldn't conclude such a correlation with our samples (see Appendix V.3). This difference can be explained by two reasons :

• The sample's origins : Three of the samples used by Pailhé et al. [START_REF] Pailhé | Impact of structural features on pigment properties of α-fe2o3 haematite[END_REF] are synthetized using the same route but with different heating times and temperatures (samples named P 400 , P 600 and P 800 on the Figure III.9). Their color in a CIE L*a*b* diagram are therefore very close to each other. Because we followed various routes, our obtained samples may show much more appearance differences.

As we are looking for a more general trend that includes the thirteen synthetic hematite samples coming from numerous routes, we cannot conclude on a relationship between the octaedra distorsion and the colour.

I have investigated the variations of numerous parameters including the positions and ampltitude of the three main absorption bands, as well as the color coordinates. All of these tests are given in the Appendix V.3. Among the possible correlations we have studied, the more convincing results were obtained between grain size and the color parameters of the samples.

III.2.4.a Hue vs Grain size :

It is possible to distinguish different trends of the color parameters for two distinct groups of samples : the one with very small particles (< 80 nm in diameter) and the bigger ones, as shown in the Appendix V.3. But most of the time this trend is not well defined in the case of very fine powders. The most striking results are obtained for the study of the hue against the grain sizes (see Figure III.11). The hue, defined as the angular component of the colour in the polar representation, is first increasing for small particles, then the trend is reversed after reaching a critical diameter (≈ 80 nm). FIGURE III.11 -Hue against the equivalent sphere diameter deduced from SEM experiments. The hue is increasing for very small particles until a critical diameter (about 80 nm, dotted line)) where the trend is reversed.

As we will see later, there are several physical phenomena depending on the grain size that are active at this scale (Section IV.3). The reversal of the hue trend along with the grain size seems to be the result of a change in the phenomena at stake : change of scattering regime, absorption phenomenon that is becoming predominant over scattering or vice versa.

III.2.4.b Reflectance modelization using Mie or

Rayleigh scattering : I tried to simulate the reflectance spectra of an hematite powder with grains of diameter 25 nm using common scattering models. Since the material under study is quite absorbing, I carried this study using the single scattering approximation. I investigated three different phase functions : the Mie phase function from exact scattering theory, its approximation for non-absorbing particles small compared to the wavelength (Rayleigh phase function) and the empirical Henyey-Greenstein phase function I.2.3.b. For the latter, scattering is supposed to be isotropic (g = 0). The results can be compared on Figure III.12a to the measured reflectance of the sample 1 whose grains equivalent radius is equal to 12.5 nm. All the cited models lead to an underestimation of the reflectance spectra. As the grain size is much smaller than the wavelength, the Rayleigh theory provides a good approximation of the Mie theory. A more complete modeling would involve the addition of the multiple scattering contribution which requires a large amount of calculations. To avoid such computations, we used an estimation of multiple scattering proposed by Jensen et al. [START_REF] Jensen | A practical model for subsurface light transport[END_REF] (Figure III.12b). Nevertheless, as the absorption phenomenon is much stronger, the multiple scattering does not induce a significant improvement in the reflectance simulation.

III.2.4.c Limitations & Conclusion :

Classical scattering models are not sufficient to modelize the color of dried pigments. We can identify multiple issues that need to be improved if not solved.

One major assumption made for the Mie or Rayleigh scattering theory is that the particles are quite dispersed into a host medium. Since dried powders are made of densely packed particles, this assumption is broken in our samples. Moreover, the interface between the powder layer and the air is rough so it can induce some surface scattering effect. Other physical phenomena should be taken into account.

In that extent, the reflectance spectrum is not sufficient to fully study the scattering and the absorption occurring in pigments. A deeper study of the BRDF is needed (Section IV.2) to identify and modelize the physical phenomena responsible for their appearance.

Moreover, to faithfully describe the observed colour of the powders the appearance models involve the optical constants (refractive index n and extinction coefficient k) of the material under study. In the case of hematite, there are few available values in the litterature and they do not fully agree with each other (see Section I.1.4.d). As we would need a more precise measurements of this parameters, we try to overcome this issue in Section III.3.

III.3 Hematite optical constant

The optical constant of the material η = n + i k, as introduced in Section I.1.4.d is a couple of values that reflects the chemical composition of a material. n is the refractive index characterizing the reflectivity of the material while the extinction coefficient k describes its absorptivity. These two values are theoretically connected through the Kramers-Kronig relationships (Equation I.11), so when one is known the second can be computed.

For hematite α-Fe 2 O 3 , several values obtained from different measurement methods are available in the literature, but as we already discussed for gold II.4.2.b, these results are not compared to each other. In this section we propose to investigate the optical constant of hematite by comparing the different referenced values with measurements performed on our powder samples.

III.3.1 Optical constant from the litterature

I found 6 different optical constants values determined through measurement on hematite bulk samples in the literature, using the different methods introduced in Section I.1.4.d.

One of the oldest one is given by Bedidi et al. [START_REF] Bedidi | Light scattering by spherical particles with hematite and goethitelike optical properties : effect of water impregnation[END_REF] who used the method of Königsberger. The reflectance of the hematite sample R a and R h tabulated by Criddle et Stanley [START_REF] Criddle | Quantitative data file for ore minerals[END_REF] are measured for an hematite sample immersed into two different host media, air and oil, whose respective refractive indices η a and η h are well known [START_REF] Cervelle | Détermination de l'ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. application à la pyrrhotite hexagonale[END_REF].

The Kramers-Kronig analysis has also been used by both Querry and Longtin [START_REF] Querry | Optical constants[END_REF][START_REF] Longtin | A wind dependent desert aerosol model : Radiative properties[END_REF], resulting in an estimation of the optical constant in from the visible to the infrared spectral range.

Ellipsometry has been used by Couka et al. [START_REF] Couka | Optical response of a hematite coating : ellipsometry data versus fourier-based computations[END_REF] and Callet in the visible spectrum for two different natural hematite crystals : a pseudo-cubic therefore isotropic one and an anisotropic one made of hexagonal petals (see Figure Since the methods previously cited require an homogeneous and pure bulk crystal, it is possible to take into account the anisotropy of the sample. This has been done in some of the references [START_REF] Querry | Optical constants[END_REF][START_REF] Couka | Optical response of a hematite coating : ellipsometry data versus fourier-based computations[END_REF], leading to two differentiated set of values for the ordinary and the extraordinary axis of the crystal.

We also found another hematite optical constant values from the database [START_REF]Aerosol refractive index archive[END_REF] that have been measured by Triaud [108] but the experimental method is not specified here.

All the results from literature are represented on the Figure III.14.

We observe that the results from the different references cited beforehand are not consistent with each other : discrepancies are noticeable about the amplitude but also about the spectral positions of the main absorption bands which must imply color variations.

First, we define the measurements from Bedidi [START_REF] Bedidi | Light scattering by spherical particles with hematite and goethitelike optical properties : effect of water impregnation[END_REF] and Longtin [START_REF] Longtin | A wind dependent desert aerosol model : Radiative properties[END_REF] as not usable because the former is not covering the complete visible spectral range and the later is not resolved enough. All the other measurements have two main absorption bands in the visible domain whose amplitude and spectral position vary with the sample and the method in use. These uncertainties could be explained by impurities in the crystals measured instead of errors from the measurement method. Indeed, it is interesting to note that for two samples with the same assumed chemical composition and measured by Callet following the same protocol, different results are obtained as shown on Figure III.14. FIGURE III.14 -Comparison of hematite optical index found in the literature. Except for Longtin [START_REF] Longtin | A wind dependent desert aerosol model : Radiative properties[END_REF] which is not enough spectrally resolved, and Bedidi [START_REF] Bedidi | Light scattering by spherical particles with hematite and goethitelike optical properties : effect of water impregnation[END_REF] that is not covering the whole spectral domain of interest, the majority of the presented results have two main absorption bands in the visible domain. However, the spectral position and the amplitude of those bands are not well defined.

It is not possible from this state of the art to identify which referenced optical constant is better suited to be representative of our hematite sample. Nonetheless, we note that all the optical constants previously cited have been deduced from measurement on hematite bulks, while our samples are powders. For these reasons, I proceed to my own measurements.

III.3.2 Optical constant from our measurements

Here I propose a new estimation of the complex optical constant of hematite pigment, that can be used for any very absorptive material. Even if it is not leading to very accurate results, it might gives some clue about which referenced optical constants are better suited to be representative of our hematite samples. It is a study inspired from the work of Marra et al. [START_REF] Marra | Fine hematite particles of martian interest : absorption spectra and optical constants[END_REF], and based on pellets containing different concentration of hematite powder. The authors compared their results with the ones from Querry's [START_REF] Querry | Optical constants[END_REF] and explained the discrepancies by the fact that absorption bands that are not involved in a bulk crystal may be caused by surface modes which occur in small particles (typically r < 10 nm).

The absorption coefficient k is first investigated through spectrometry. The refractive index would then be computed using the Kramers-Kronig's relationships.

In order to only consider the absorption and neglect the scattering of light caused by the particles, the pigment is dispersed into a KBr matrix, a material which is perfectly transparent and whose refractive index is well known in the visible spectral range. Numerous pellets of KBr containing different Fe 2 O 3 concentrations are produced following the protocol III.3.2.

Since the powder used for making the pellets cannot be recovered aftewards, only samples 1 2 and C are in sufficient quantity to be considered. Because the grains of the sample C are micrometric they are not easily and homogeneously dispersed into the KBr medium, and their absorption power is too strong so the pellet appears very opaque even for small concentrations of powder. Finally, I chose the sample 1 2 to carry out this study which, thanks to its fine grains, permits to obtain absorptive but still transparent pellets while reducing the possible scattering phenomena if used at low concentrations.

With such a small amount of hematite mixed with the KBr, it is important to note that the real concentrations of the pellets may be very different from the theoretical ones.

III.3.3 Measurements and analysis

I first measured the transmittance of each pellet in the infrared spectral domain to check if scattering phenomena are well avoid, and then in the visible domain to estimate the absorption coefficient in this spectral range. Part of the light that is incoming onto the pellet is reflected at its interfaces and therefore results in a loss on the transmittance spectrum. This reflection is evaluated using Fresnel's coefficients for a thin plate of KBr illuminated under normal incidence, and corrected. We used the optical constant of KBr from Palik [START_REF] Palik | Potassium bromide (kbr)[END_REF] for this correction. A fully corrected transmittance spectrum would reach an horizontal asymptote at 1 for a non scattering pellet, meaning that all the incoming light is transmitted except for some wavelengths where absorption is at stake. According to our tests, it appears that only pellets with very low concentrations of hematite powder can actually be considered as purely absorptive. Indeed, purely absorptive pellet would be characterized, apart from the absorption bands of the material it is made of, by a flat and maximum continuum of the transmittance factor. For example, in the case of the pellet n • 1, no absorp-tion band is detected over the 4000 to 8000 cm -1 domain which implies that there is no absorption phenomena in this spectral range. However the continuum is not maximum and therefore cannot be corrected by the accounting of fresnel's reflection on the pellet interfaces. This gap is due to scattering phenomena at stake in the sample. The slope observed in the visible spectral range is also representative of some scattering phenomenon.

We can make a similar observation, but with much lesser amplitude, for the pellet n • 3. The other ones with weaker concentrations, n • 4, 5 and 7, show a rather constant and maximum offset. As a consequence we consider that only absorption phenomena is responsible for the colored appearance of those pellets.

The spectra can then be converted into absorbance using the Beer Lambert's law A =log(T). The signature of hematite is given by the absorption bands detected between 400 and 700 cm -1 .

We define the Abundance factors as the amplitude factors that need to be applied on each sample's spectrum so that the amplitude of the main absorption band at 560 cm -1 is the same whatever pellet is studied. We chose the pelet 3 as a reference pellet, since it shows the more important absorption and almost no scattering. It abundance factor is fixed to 1. The abundance factors of the other pellets are determined comparatively to this reference.

Abundance factor Values

III.3.3.b Study in the visible spectral domain :

Transmittance spectra of the pellets of interest are measured in the visible spectral range (190 to 1100 nm) using a JASCO V-730ST UV/VIS spectrophotometer. The spectrum obtained for the pellet 3 is completely saturated at wavelengths smaller than 400 nm. This corroborates the analysis from the study in IR spectral range that its concentration in hematite is too high to be purely absorbing. The offset caused by the reflections of light on the pellet interfaces is corrected, and abundance factors are applied to the corresponding spectra. If the pellets were perfectly uniform and absorptive, their absorbance spectra should overlapp. Among them, the results from the pellet 4 does not fit any other measurement. However there is a well superimposition between the samples 5 and 7 or between the samples 3 and 6 (see Figure III.17).

Because of the previous observations, we can therefore suppose that only samples 5 and 7 are purely absorptive. Their absorbance can then be described by the Beer-Lambert's law.

III.3.3.c Absorption coefficient :

Let's consider one of the purely absorptive pellets, either 5 or 7. The incoming light is absorbed while going through an effective thickness x of hematite. As the optical index is supposed to be well known in the infrared, it is possible to deduce this thickness x from the Beer-Lambert's law (Section I.1.3) applied at the wavelength of the main absorption band (560 cm -1 or 17857 nm). Two of the referenced optical constants give the values in the infrared domain : Querry [START_REF] Querry | Optical constants[END_REF] The effective thickness of hematite that is crossed by the light as it goes through the pellet is deduced at the wavelength 17857 nm using the known values of the absorption coefficient κ given at this wavelength by the Querry [START_REF] Querry | Optical constants[END_REF] and Triaud [108]. The new value of the absorption coefficient in the visible range in then derived from Beer-Lambert law (Equation I.2 using this thickness x and the transmittance spectrum T :

κ = - λ 4πx lnT (III.2)
Depending on which referenced values of k in the infrared is used, we found slightly different thicknesses : x = 0.86 and x = 0.91 for respectively Querry's and Triaud's optical constants. Such a small difference is explained by the similarity of these published values in the infrared spectral range.

Thus, two spectra of the coefficient k are deduced in the visible spectrum, that are shown and compared to the one from Querry and Triaud on the The positions of the two main absorption bands obtained from our transmittance measurements are in agreement with the results from Triaud, and are more shifted into the small wavelengths than for any other results from the previously cited references (Section III.3.1) including the one from Querry. The difference of amplitude may come from a wrong estimation of the effective thickness x. Indeed, a way much smaller one (x ≈ 0.027) would lead to comparable amplitudes. In such a case, the slope of k from 600 nm to 1000 nm is also comparable with Triaud's results.

The other parameter of the optical constant, the refractive index n can be computed using the Kramers-Kronig relationship. Since our results do not appear to be pretty accurate, such computation has not been performed.

III.3.3.d Limitations & Conclusion :

We have proposed a new estimation of the optical constants of a powder material. Our protocol is based on the transmittance in both infrared and visible spectral range of absorptive but non scattering pellets containing low concentrations of hematite powders. If the results allow us to draw some conclusion about the spectral position of the absorption bands, they remains inaccurate and not reliable regarding the amplitude of the expected constants.

Several sources of errors have been identified throughout the process. First, the scale used to control the hematite concentration of the pellets is not reliable enough for such small amount of powder. To enhance the accuracy of the weighing a very small amount of hematite is obtained by dispersing the pure material twice before the pellet is formed : A first step consist in mixing 1 mg of pure hematite into 300 mg of KBr. Then few milligramms of this mixture is blend with 300 mg of KBr again. However this process also increases the inhomogeneity of the pellets. To verify if hematite is uniformely dispersed into the KBr matrix two of the pellets (4 and 5) have been made with the same proportions and out of the same mixture 300 mg KBr + 1 mg Fe 2 O 3 . The appearance of these two pellets is different for a naked eye (Figure III.15), and so are their transmittance spectra. It does prove that either the weighing is not reliable, either the mixing is not homogeneous, or both.

We should also note that the presence of fine particles in the pellet is still responsible for some scattering in the visible spectral range. A correction using the Mie or the Rayleigh theory instead of using an offset correction would improve the results since it accounts for a wevelength dependency.

Finally the conservation conditions of the pellets can be of an issue, since they highly depends on the humidity of the surrounding environment. Indeed, after 24 hours in the open air, they tends to opacify and to become more diffusing. To avoid this problem we took care to proceed to the measurements on the pellets as soon as their preparation is done. Between 3 to 4 hours are needed to perform both the preparation and the measurements of the pellets, which is already enough for the pellets to turned more opaque. Some residual scattering observed on the transmittance spectra may be implied by such a phenomenon.

Based on the transmittance measurements on pellets with low concentration of hematite, it appears to us that the optical constant values given by Triaud [108] are more suitable to describe our hematite powders.

On the other hand, the absorption bands observed on the reflectance spectra of optically thick powder layer (Section III.2.3, Figure III.9) seems to more closely fit with the one measured by Querry [START_REF] Querry | Optical constants[END_REF].

The measurement of hematite optical constant remains an open problem since it appears to not only depend on the chemical composition of this material but also on its morphology. Although several measurements have been performed on bulk samples they do not agree with each other. As for measurements on a powder, it requires the manufacturing of low concentration pellets which imply many sources of errors.

As we could not conclude on which referenced values were the more accurate ones, we decided to use the ones from Querry [START_REF] Querry | Optical constants[END_REF] in the following. We believe that the close spectral position of its absorption bands to the one we observed through reflectance measurements on optically thick hematite layers makes it more suitable to describe the case of dry powders.

Chapter IV

Hematite Reflectance Function

Our main objective is to be able to predict the appearance of paint layers containing pure hematite pigments as a function of the morphology of their grains.

In the previous chapter we studied the color of hematite for a single incidence and a single observation direction in the incident plane. This does not permit to describe other appearance characteristics such as the shininess or diffuse, the smooth or rough look of the material. To do so it is necessary to proceed to the study of the material reflectance in various geometries : light can reach the material under any incoming direction and be perceived by the observer in any outgoing direction. This is known as the Bidirectionnal Reflectance Distribution function (BRDF).

The appearance of dry pigments is the results of multiple physical phenomena including absorption and scattering. The description of their appearance suggests the full understanding of the interactions of light with the material. Because we do not consider any binder, the dry grains composing the powders are too close to each other to be modelized using classical scattering theories such as Mie scattering.

In this chapter, I am aiming to identify the numerous scattering phenomena at stake in the BRDF of an optically thick layer hematite powders, and to modelize them using analytical and physically based models. I will first explain the BRDF measurements I performed on the synthetic samples, and then detail three different models we used to reproduce such functions. Two of them are models from the literature, introduced in Chapter I (Spectralon's model I.2.3.e and RTLSR model I.2.3.d), that I adapt to the case of dry powders. I also propose a new physically based model which accounts for the coherent scattering phenomena introduced in Section I.2.3.f, which is able to accurately reproduce the BRDF of optically thick layer of dry powders with various grains' morphologies.

IV.1 Light scattering in an heterogeneous optically thick hematite layer

As we have shown in the previous chapter, powders composed of only one chemical entity can result in a several shades of one color. In the case of hematite, although the powders mostly show red hues they can reach colors going from light red to purple. To account for these appearance variations, and besides the chemical composition of the pigment, the BRDF modelization should involve parameters relative to the morphology of the grains composing the powder.

I simplify this problem according to the Figure IV.1 by making the following assumptions : First let's consider that the paint layer is composed of only one pure material. Since we are interested in parietal painting we will limit the study to pure hematite samples. A more realistic approach would be to consider ochers, that is to say a mixture of ferruginous rocks such as hematite or goethite along with clays, sands, and other non-coloring minerals.

Moreover, the powders are composed of spherical grains that are uniform in size. Though, the case of more complex morphology involving two different dimensions for the description of the FIGURE IV.1 -Representation of an optically thick hematite layer. The powder is made out of pure hematite spherical grains that are uniform in size. The layer is sufficiently thick and absorbing so that the substrate on top of which the pigment is spread out does not contribute to the observed appearance of the layer. grains size will also be briefly discussed. This assumption permits us to approximate the surface of the layer as a collection of microfacets with a uniform normal distribution.

Finally powder layer is supposed to be optically thick. This allows us to neglect the contribution of the substrate on which the powder is spread out, as well as possible interference effects caused by multiple reflections between the two interfaces of the painted layer.

For this study, I used the synthetic powders of hematite α-Fe 2 O 3 presented in Section III.1 with various sizes and shapes of grains resulting in different shades of red.

First, the BRDF of our hematite samples is measured. It is done in two steps : the main part of the BRDF is measured in collaboration with Bernard Schmitt (IPAG) using the gonio-spectrophotometer SHADOWS (Section I.4.4) primarily used to perform measurements for astrophysics studies. We have designed a specific instrument in collaboration with Corinne Félix at Institu Néel to proceed to a fine sampling in and around the exact back-scattering direction.

The appearance of dry hematite powders is then simulated using two adapted and optimized BRDF models : the Spectralon's model (Section I.2.3.e) and the RTLSR model (Section I.2.3.d). Because the former is empirical and the later not precise enough to modelize fine details variations among the BRDF, we have developed another model accounting for coherent back-scattering (Section I.2.3.f) in the layer of powder that we applied to our samples.

IV.2 BRDF measurements of the hematite samples IV.2.1 Gonio-spectrophotometry :

The major part of the BRDF has been measured using the SHADOWS instrument developed by Potin et al. [START_REF] Potin | Shadows : a spectro-gonio radiometer for bidirectional reflectance studies of dark meteorites and terrestrial analogs : design, calibrations, and performances on challenging surfaces[END_REF], as explained in Section I. When it comes to a geometry where incident and observation directions are identical, because of the design of such an instrument, the sensor is obstructed by the light source. These geometries are then impossible to measure (see Figure IV.2). A specific device needs to be designed to evaluate the amount of light scattered is these directions.

IV.2.2 Back-scattering measurements :

I have noticed that when it comes to dry powders, an important amount of the incoming light is scattered in the backward direction. It has been observed and denoted as the opposition effect in astrophysics [START_REF] Beck | Photometry of meteorites[END_REF], and is also known as the Back scattering. Its measurement is quite challenging since it requires the light source and the sensor to be at the exact same position which, in the case of a goniometer, would lead to the former obscuring the latter or vice versa. The devices that have been designed to overcome this constraint by [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF] or [START_REF] Belcour | Bidirectional reflectance distribution function measurements and analysis of retroreflective materials[END_REF] require vertical samples. We adapted the concept of these devices to measure in the visible spectral range the back-scattered light from dried powders, materials that require to be horizontally hold (see Figure IV.3).

We use an halogen lamp as a light source to normally illuminate the sample after being reflected by a beam splitter at 45 • . The size of the illumination spot is minimized by focusing the light with a 50 mm focal lens. An additional diaphragm is added to clean the focused beam. The direction of the incident light θ i is adjusted by positioning the beam splitter at an angle of 45

• + θ i /2.
As the light interacts with the powder, it is scattered in every directions and preferably back into the incident direction. It is then partially transmitted through the beam splitter and collected by a Thorlabs F950FC-A collimator. The latter is fixed on a vertical rotating plate so that we can scan different observation directions around the exact back-scattering one in the incident plane. We use a fiber spectrometer OceanOptics USB650 along with the SpectraSuite software in order to detect and visualize the measured signal.

To ensure the accuracy of the device, we proceed to an optical alignment using an HeNe laser beam as a light source upon a plane mirror. The laser beam is controlled to be parallel to the experiment table by being reflected on two consecutive mirrors whose orientations' at 45 • is finely adjusted.

The ensemble lens+diaphragm is positioned so that the beam is focused on the sample. The orientation of the beam splitter is then adjusted by auto-collimation : the beam that is specularly reflected by the mirror sample should travel exactly the same path as the incident beam. When their spots on the second 45 • mirror are superposed, the incident beam reaches the beam splitter with an angle of exactly 45 • , and so the sample with a perfectly normal incidence. At this point, the laser can be replaced by the white source.

The tilts of the collimator are set to maximize the detected flux in the specular direction. The mirror sample can then be replaced by the sample on which we want to proceed to the measurement.

When we want to study another incident direction, some steps of the alignment process should be renewed. First, the beam splitter should be turned with the desired angle. As the light spot is slightly shifted with the increase of the incidence, the position of the collimator should also be shifted. The position and the tilts of the collimator are adjusted to maximize the signal resulting from the reflection of light on the sample mirror in the specular direction. The bench is ready to be used on the wanted sample.

For the calibration of the measurements, a Spectralon® from Labsphere is used as a white reference. This instrument allows us to consider the incidences from 0 • up to 40 • in the incident plane (ϕ o = 0 • ). At higher incidences, two issues will hamper the measurements : because of the compactness of our device, the collected light is partially transmitted through the beam splitter and partially directly scattered by the sample without passing through the glass plate. This may involve errors in the magnitude of the detected flux. The second issue appears at larger incident directions : the rotation of the collimator is blocked by the ensemble lens+diaphragm. This could be improved by choosing another focal lens, but then at grazing angles the collimator will be hidden by the mount of the lens, becoming unreachable for the light scattered in the desired direction.

As it has been done for the BRDF measurements on the SHADOWS intrument, we will study the incidences 0 • , 20 • and 40 • . Observation angles are scanned in the range [θ i -10 • , θ i +10 • ] with a step of 2 • . The integration time is chosen to maximize the detected signal and a averaging over 5 spectra is performed to minimize the noise. A compromise between these two parameters is done to optimize the acquisition time. Less than 20 seconds in necessary to measure one spectrum.

We performed the measurements on sufficiently thick layers of dry powders to avoid any contribution of the substrate. For the measurements on the SHADOWS instrument, the powders are placed in an aluminium sample holder. As for the back scattering measurements, the powders are deposited on a silicon wafer so that they cover an area larger than the illumination spot. The surface of the samples is not flatten in order to keep a more natural and rough surface appearance.

The spectrum of the noise I n that may be added by undesired light from the surrounding environment is measured and retrieved to any further experiment. As it is mostly due to indirect lighting, we can make the assumption that it is the same in any direction. Only one measurement is done. Using a Spectralon, the calibrated spectrum of the sample is then deduced as follows :

R(θ i , θ o , ϕ o = 0 • ) = I(θ i , θ o , ϕ o = 0 • ) -I n I r e f (θ i , θ o , ϕ o = 0 • ) -I n R r e f (θ i , θ o , ϕ o = 0 • ) (IV.1)
with I and I r e f respectively the raw measurements on the sample and on the spectralon, and R r e f the real reflectance of the spectralon given by the manufacturer. Since our device requires a small quantity of powder, all of the samples have been measured : the natural and the thirteen synthetic ones. For the specific case of the sample 1 3 which shows some sparkling effect, measurements have also been performed around the specular direction.

IV.2.3 Complete measurements :

The BRDF and back-scattering measurements are complementary to one another. As the goniospectrophotometer measurement requires a certain volume of powder, the complete BRDF's measurement can only be performed on the samples 1, 1 2 , 3, 5, 6, P 2 , and C. Nevertheless, our backscattering device does not have such a strong limitation : the measurements can be performed using very small quantities of powder so that each of our synthetic samples can be studied.

The back scattering measurements at θ i ± 10 • should match the BRDF measurement at the same geometry. However this is not always the case : for some samples, the measured points from the back-scattering are shifted in amplitude. An example is given on the Figure IV.4 for the sample 5.

One explanation could be that the two measurements have been done independently. Indeed each sample has been prepared twice : once for the BRDF measurement, and once for the backscattering one. The surface topography of the sample is obviously not exactly the same and may induce slightly different surface scattering.

The same error was obtained by Levesque et al. [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF] as the authors also used two different instruments for the measurement of different part of the BRDF. As the difference between the two measured points at the θ i + 10 • is not very important (about 5%), we have performed the same correction they proposed. The back-scattering measurements are rescaled in order to fit with the previous and more accurate BRDF measurements. The BRDF's variations of the samples are quite smooth : the spectra we measured in various geometries all show the same characteristics with slight intensity changes from one geometry to a close another one (see Figure IV.5). This is coherent with the samples' uniform appearance observed with a naked eye. Measured points can then be linearly interpolated in order to complete our values in the geometries we didn't cover with our sampling (see Figure IV.6). This permits us to proceed to further studies with more data points without increasing the measurement time.

The BRDF measurement results of all the synthetic samples are given in the Appendix V.3.

IV.2.4 Analysis :

Once the BRDF measurements are completed, we can identify two peaks (see Figure IV.7). The main one is found in the backward direction, and becomes more and more important along with the increasing incidence. The second and weaker one accounts for the light specularly reflected. It becomes unnoticeable for powders with the largest grain size (>100 nm).

Because of the sampling I chose, there are few measurement geometries around the backward direction. The interpolation is then misled, resulting in an unusual spreading of the main identified peak along the azimuth directions while we were expecting a more symmetrical one. This is even more visible for higher incidences.

As illustrated on the Figure IV.8, a different conclusion can be drawn for the sample 1 3 . The specular contribution is much more important in that sample than in any other one. The backscattering contribution is as important as in the other samples, but quite small compared to the specular. It illustrates the sparkling appearance that we observed with a naked eye. This sample will therefore not be used for the BRDF modelization in the Section IV.3.

IV.2.5 Limitations & Conclusion :

I have characterized the appearance of our samples in the visible spectral range through BRDF measurements and complementary measurements around the back-scattering direction using an optical bench of our own design. All the results from our BRDF measurements are given in the Appendix V.3.

Apart from the back-scattering, 180 geometries with various incident, observation and azimuth angles have been chosen to sample the BRDF over the whole reflection hemisphere. By using the SHADOWS instrument, although the measurements on one sample is quite long, they are also quite accurate and fully automatic. ). The measurements (large dots, in grey for the SHADOWS' measurements and in orange for the rescaled back scattering measurements) have been linearly interpolated (small dots). Two peaks can be identified, a major one in the back-scattering direction and a weaker one in the specular direction.

As we were not expecting the light to be mainly scattered in the backward direction, we decided to chose more geometries in the forward half-space. In the case of fine powders it would have been better to chose reachable geometries around the exact back-scattering direction instead.

The latter BRDF measurements are completed with back-scattering measurements, over 33 geometries around exact back-scattering directions in the incident plane. We have designed a specific instrument for this, and optimized it making some compromises. Because our sample are quite dark, our back-scattering measurement has been designed in a compact way to maximize the detected flux. It allows a fast acquisition of the reflectance spectrum of any kind of horizontal and flat material, including powders. The accuracy of such an instrument is convenient since it closely matches the measurements at same geometries performed using the SHADOWS instrument. A better precision could be reach by increasing the dimensions between the instrument's components, but then a longer integration time would be needed to detect a sufficient amount of signal. Some improvements can be added to our device in order to reach a better accuracy and to ease its use. First, the scanning of the observation directions is manual with a precision of ±1 • . An automating would lead to both a faster scan and a more precise setting of the desired direction.

When the incident direction is changed, the position of the collimator needs to be adjusted. During the process its position can be shifted out of the incident plane, which may cause some errors on the amplitude of the measured spectra.

Finally, our device is not able to reach grazing angles for incident and observation. For large incidences, the illumination spot is spread out so it would require a larger amount of powder and a larger collimator diameter to correctly collect the scattered light. Because of the compact design we chose, the collimator is blocked by the ensemble lens + diaphragm for incident angles larger than 40 • .

From the measurements, we can conclude that the BRDF of dry hematite powders is composed of an overall diffuse contribution, and two directional scattering phenomena including mainly back-scattering and some specular scattering. The BRDF of the characterized samples vary in intensity and width of the specularly and back scattered lobes. We are aiming to modelize these variations while taking into account the physico-chemical properties of the studied powders determined in Section III.2.

IV.3 BRDF modelization

Based on the previous BRDF measurements of pure hematite powder samples, we have identified some scattering phenomena responsible for their appearance. For all the samples the incoming light is mainly scattered backward (Back scattering), and for some of them a small amount is also scattered forward (Specular Scattering).

I assume that all powder samples involve the same scattering phenomena in different propor-tions. Their apperance dissimilarities are related to the morphology of the grains they are composed of. In order to be able to predict the appearance of a given material powder, each phenomena has to be analytically modelled. For a better understanding of the scattering phenomena at stake, a physically based model would be preferred to an empirical approach. In this manner the model will explicitly take into account the chemical composition of the material through its optical constants, as well as the size and shape of the powder's grains.

The proposed BRDF modelization is relevant for any kind of powder, and is validated against the measurements in the particular case of pure synthetic hematite powders with grain size smaller than 10 µm described in Section I.3.2.c. I first proceed to the BRDF modelization using two existing models in the literature : an empirical model specifically developped for the Spectralon material [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF], and the RTLSR model initially expressed for the appearance of canopies in teledetection [START_REF] Lucht | An algorithm for the retrieval of albedo from space using semiempirical brdf models[END_REF]. Both of these models consider the BRDF to be the sum of two independent scattering contributions, one occurring in the volume of the material and the second one at its surface. We made some adjustments to these models so they can be applied for the dry powders BRDF's simulations.

As the results obtained from these models are not satisfying enough, I developed a new physicalbased model better suited for the reproduction of the hematite powders's BRDF we measured in the previous section IV.2.

IV.3.1 Spectralon's model for dry powders

This model is described in Section I.

BRDF = D(θ

i , θ o ) + S S (θ i , θ o ) + B(θ i , θ o ) + S(θ i , θ o ) (IV.2)
This model has specifically been developed for the Spectralon material, the most lambertian and white material that exists (see Figure I.15). The fine knowledge of its BRDF is required to be used as a reference for the calibration of any reflectance measurements. This material appears to be quite different from our powdered samples : it shows a smoother surface, and a higher albedo than our pigments.

Moreover, the given mathematical expressions are only expressed for a modelization in the incident plane (ϕ o = 0). Out of this plane, it is necessary to take into account the azimuth angle ϕ o as described on the Figure IV.9. Some adjustments of the Spectralon's model are needed to correctly reproduce the measured BRDF.

IV.3.1.a Model adaptation :

The main part of the BRDF is described by the Deep scattering contribution whose analytical expression in the incident plane has been introduced in the chapter one using the Equation I.48. This modelization proposed by [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF] is inspired from an empirical one given by [START_REF] Liang | Retrieval of surface brdf from multiangle remotely sensed data[END_REF]. The latter involves the azimuth angle through a cosine factor. We adjust the formula from Equation I.48 in the same way :

D(θ i , θ o , ϕ o ) = d 0 + d 1 (θ 2 i + θ 2 i ) + d 2 (θ 4 i + θ 4 o ) + d 3 θ i θ o cosϕ o + d 4 θ 2 i θ 2 o (IV.3)
The surface scattering phenomena, specular and back scattering are respectively modelized by a Gaussian and a Lorentzian function using the previously introduced Equations I.55 and I.54 where Ω i and Ω o are defined on the Figure IV.9, and Ω i is the symmetric vector of Ω i with respect to the surface normal. In a more explicit way, these angles can be projected according to the Figure IV.9 so that they can be related to the incident/observation geometry by the equations :

cos Θ = cos θ i cos θ o + sin θ i sin θ o cos ϕ o cos Θ = cos θ i cos θ o -sin θ i sin θ o cos ϕ o (IV.5) (IV.6)
We then extend both the Back and Specular scattering models for any azimuth geometry as follows :

B(θ i , θ o , ϕ o ) = b 0 b 2 1 Θ 2 + b 2 1 (IV.7)
and

S(θ i , θ o , ϕ o ) = a f (θ i )exp - 1 2 Θ s 1 2 (IV.8)
with Θ, Θ defined by the Equations IV.4 and a f given by the Equation I.56.

Two expressions have been proposed to simulate the contribution called subsurface scattering, depending on the desired accuracy. By definition, this contribution only plays a role at grazing angles. As it does not appear to be of such an importance in our measurements we chose to use the simplest but less accurate model (Equation I.49), which involves only two variable parameters. As for the surface contributions, it is adapted to account for the azimuth angle :

SS(θ i , θ o ) = ss 0 exp(ss 1 |Θ |) (IV.9)
with Θ defined by the Equation IV.4.

Our model is then the sum of the four contributions IV.2 whose expressions are given by the previous Equations IV. 3, IV.7, IV.8, and IV.9.

The twelve parameters involved in the model and summarized in the Table IV Those parameters are empirical, do not have any known physical meaning and a priori do not depend on the incident and observation directions θ i and θ o . They are optimized by minimizing the difference between the measured points y exp and the theoretical result from the model y t h using least squares method. Mathematically, it is equivalent to minimize the error E RMS such as :

E RMS = N i =0 |y exp, i -y t h, i | 2 (IV.10)
with i specifying the considered discretized incidence/emergence geometry. Only one average error value per sample is computed, by uniformly taking into account the contributions of all the geometries used in the optimization.

It is highly possible that the parameters depend on other acquisition parameters such as the wavelength. We performed the optimization over all the measured geometries for each wavelength.

For better results, I performed the optimization process twice. Because of their differences in chemical composition and appearance, the magnitude of the parameters that would describe hematite powders should be quite different from the ones describing the Spectralon material. To correctly optimize the model to fit our measurements, we first need an estimate of the order of magnitude of all the variable parameters. To do so, a first optimization decomposed into four consecutive steps is performed. In the first step, only deep scattering is optimized. Its modelization is then retrieved from the measurement points, leading to the residual y exp, /D . The Back scattering is optimized in the second step to fit these residual points.

y exp, /D, i = y exp, i -D(d 0 , d 1 , d 2 , d 3 , d 4 ) (IV.11)
In the same way, specular and subsurface scattering are respectively optimized on the residuals y exp, /D/B and y exp, /D/B/S . Note that the different contributions are optimized in order a specific order : first come the contributions that appear to be the more significant (deep and back scattering), and then the contributions that bring finer and less noticeable details (specular and subsurface scattering). The complete simulation of the BRDF is finally obtained by summing the four contributions modelized using the optimized parameters.

Although the resulting values of the variable parameters are quite accurate in magnitude, they are not optimal. Indeed, because this process is done in multiple steps, the errors occurring at each steps propagates towards the final one. The sum of the four independently optimized contributions tends to overestimate the real BRDF of the studied sample.

That is why I also perfom the full optimization of the model accounting for the four contributions at once. It uses parameters with the previously determined magnitude as an input and returns final values that better match the experiments.

Results obtained for the samples whose BRDF has been completely measured are given in the next section.

IV.3.1.b Results :

We are aiming to reproduce the measured BRDF of our samples using the adapted Spectralon's model described earlier IV.3.1.a. Although only the results obtained for the sample 5 are given in this Section, all the synthetic samples whose BRDF has been measured by gonio-spectrophotometry have been studied. The optimization results of the Spectralon's model over all the samples are given in the Appendix V.3.

According to the reflectance spectra of our samples, in the visible spectral range the maximum reflectance is reached around 700 nm. Since there is less absorption phenomena, we chose to perform the optimization of the model at this wavelength.

A former optimization in four steps is performed to deduce the magnitude of every parameters. For simplification purposes, it is done in the incident plane (ϕ o = 0 • ) over one incident direction and for outgoing angles from -70 • to 70 • . In order to easily distinguish the specular from the back-scattering contribution, we chose θ i = 40 • . Indeed for small incidences these two contribution may overlap or even be superimposed which could lead to errors in the optimized parameters. Results at every steps of the process are represented on Every contribution properly matches the measured points. The specular contribution being very weak, its is not easy to accurately fit as a single contribution. However the fit obtained for the subsurface scattering using the less accurate model proposed by [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF] is good enough. The second and more complex model the authors proposed to better modelize such contribution at grazing angles will therefore not be considered in the rest of our study.

The sum of the four optimized contribution is equivalent to the complete BRDF model, and closely fits the original measurements as it is shown on Figure IV.11.

The magnitude of the variable parameters obtained throughout the first optimization process are then used as a starting point for a finer optimization of the full model at once, in the same geometries. By comparing the results of the two optimization process on Figrue IV.11, we can see some improvements of details of the BRDF, especially for the specular contribution which is much more well simulated after the second refinement. As the optimized parameters values suggest, it also slightly modified the subsurface scattering amplitude. The difference between the measured and the simulated BRDF is less than 1% of the Reflectance (see Figure IV.12) which is consistent with the prediction of [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF]. This is confirmed by an RMS value smaller than 7.10 -4 after the optimization. The optimization is done in only one step accounting for the four contributions using the previously determined parameters magnitude as an input.

The result better fits the experiment, especially for the specular contribution.

FIGURE IV.12 -Error between the BRDF measurements and simulations using the spectralon's model on sample hematite sample 5. As described in the reference [START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF], the error is less than 1%. An even finer result is obtained from the full model optimization once the magnitude of the variable parameters are known.

For the simulation of the whole BRDF, the model is optimized over all the measured geometries using the results of the first 4-steps optimization process's results as an input. Because it was not possible with our device to measure the back scattering contribution for an incidence of 60 • , these geometries are set aside for now. The optimization is then made for incidences 0 • , 20 , 90 • and still for outgoing angles between -70 • and 70 • . Most of the refined parameters are similar whatever optimization is used, either the four steps or the one step one (Table IV increasing the subsurface amplitude ss 0 and the width of the specular scattering lobe through the parameters s 1 and s 2 .

Results of the BRDF simulations are presented on the Figure IV.13. As expected the specular contribution is almost unnoticeable compared to back scattering one. The measured BRDF is a bit underestimated by the model especially for large incidences. However the RMS value is smaller than 4.10 -2 which is still very satisfying, and is mainly caused by measurements errors for an incidence θ i = 0 • and an azimuth ϕ o = 135 • . Indeed, the detector of the SHADOWS instrument may be partially obstructed by the light source at these geometries. The weighing of the geometries' contribution to the error may be more reliable.

IV.3.1.c Limitations & Conclusion :

The Spectralon's model offers a nice and interesting decomposition of the physical phenomena at stake, that can be extended to any dense particulate material such as dry powders. The investigation of volume scattering (deep and subsurface scattering) and surface scattering (specular and back scattering) is made independently, leading to analytical and quite simple expressions for each of the four identified contributions.

The 12 parameters give the model its ability to fit the measurements of any of the sample we are studying with a pretty good accuracy. The RMS values between the complete BRDF measurements and optimized simulations are of magnitude 10 -2 which is satisfying.

According to the results, this model is able to predict with a good accuracy the appearance of powders in geometries that have not been sampled during the BRDF measurement. All the results of such modelization on the synthetic samples are given in the Appendix V.3.

On the other hand, this model is entirely empirical and involves parameters devoid of physical meaning. The chemical composition, described by the wavelength dependency of the optical constants of the material, as well as the morphology of the grains composing the powder under study are not explicitly required for the model optimization.

Hence, the appearance modelization can only be retrieved by fitting the BRDF measurement : it is not possible to use the Spectralon's model to predict the appearance of a dry powder whose BRDF hasn't be measured beforehand. I would prefer to seek for a physically based model that involves less but physically meaningful parameters, and still allows some flexibility to modelize a wide range of different samples with good precision.

IV.3.2 RTLSR model for dry powders :

The RTLSR (Ross thick Li Sparse Reflectance) model, first developed for the BRDF modelization of canopies in teledetection is fully described in Section I.2.3.d. It is said to also be able to simulate the BRDF of bare soils [START_REF] Roujean | A bidirectional reflectance model of the earth's surface for the correction of remote sensing data[END_REF], so we applied in for the reproduction of our hematite samples' BRDF measurements.

This model separates the complete BRDF into two independent contributions : volume scattering caused by multiple interactions of light with the particles inside the medium and surface scattering that depends on the geometry of the interface between the incident material (air) and the particulate medium.

As far as we know some adjustments have been made for the case of snow [START_REF] Jiao | Development of a snow kernel to better model the anisotropic reflectance of pure snow in a kernel-driven brdf model framework[END_REF], but not for the specific case of soils made out of dry powder such as sand or dirt. I propose some modifications for this purpose.

IV.3.2.a Model adaptation :

Both volumetric and geometric contributions need to be adapted for the case of powders. Different expressions of this model are given depending on the density of the considered medium [START_REF] Wanner | On the derivation of kernels for kernel-driven models of bidirectional reflectance[END_REF]. For dry powders, it is assumed that the density is high enough to only consider the dense formulation of the RTLSR model.

The part of the BRDF due to volume scattering R vol is deduced from the radiative transfer theory and results in the analytical expression given by the Equation I.34. It involves parameters intrinsic to the physico-chemical composition of the sample under study : a scattering phase function p(Θ), a diffuse reflectance s depending on the optical constants of the material, and the factor LAI (Leaf Area Index) accounting for the density and the morphology of the scattering elements. Since we are studying grains composing hematite powders observed at a short distance, instead of tree leaves observed at a large distance, some parameters involved in the model initial formulation must be adapted.

First of all, we need to account for the morphology differences between leaves and particles. Leaves are two-dimensional elements whose size, shape and density are described by the Leaf Area Index LAI. In this case I consider spherical particles which leads to the introduction of the equivalent Particle area index PAI such as :

PAI = N zσ (IV.12)
with N the density of particle per unit volume, z the thickness of the powder layer, and σ the scattering cross section of one spherical particle. One may note that the spherical particle hypothesis actually better satisfies the assumption of isotropic normal distribution of the micro-facets description of the interface between air and the scattering medium than is the case of 2 dimensional leaves.

To account for the change in chemical composition of the scatterers, the lambertian reflectance s is replaced by the single scattering albedo ρ of a spherical particle.

Finally, as we already established that both chemical composition and morphology of the considered scatterers changed from the original canopy's model, new scattering properties should be taken into account. This is equivalent to modify the phase function at use. If the phase function for a plant stand is mainly used to describe a canopy, we will consider other phase functions better suited for particles such as Mie, Rayleigh or Henyey-Greenstein phase functions. Different optimization of the model will be done in order to identify which phase function would be a better choice to describe light scattering by dry powders.

In the case of hematite and any other highly absorptive or sufficiently thick materials, the PAI increases so that the exponential term accounting for the substrate contribution tends to zero. It can therefore be neglected which simplifies the model even more as follows :

R vol = ρ 2 p(Θ) µ i + µ o (IV.13)
By assuming that scatterers are spherical particles, the geometric contribution R g eo for dense medium given by the Equation I.46 is also modified.

To do so, I simplified the geometry of the medium. As the RTLSR model was developed for teledection that is to say for objects seen from a long distance, the geometric contribution of the BRDF has been developed using the interface between air and trees. The trees are considered as collection of scatterers, among which only the outside ones contributes to surface scattering as is it shown on Figure IV.14a.

For dry powders seen from a much smaller distance, this contribution occurs at the interface between air and the homogeneous particles. However, for a sufficiently thick and dense layer the bottom particles are completely occulted by the top ones. As for trees, only the upper one-particle thick layer of the powder actually contributes to the geometric part of the BRDF. The scatterers that are under are approximated as an homogeneous layer that is completely occulting any other soil substrate (see Figure IV.14c). The grains that are composing the powder are supposed to be spherical with radius r . (c) For a sufficiently thick layer, the shadows that are observed are only cause by the upper one-particle layer of the medium. The powder is then approximated as one oneparticle thick layer of spherical particles on an homogeneous substrate that is completely occulting any other soil substrate.

The new substrate is then made out of the same material as the scattering particles, so we have the equality between the reflectance of the illuminated particle and the reflectance of the illuminated soil C = G.

Because of the equality b = h = r in the case of spheres, the incident and observation shadows can respectively be written A

i (θ i ) = πr 2 sec θ i and A o (θ o , ϕ o ) = πr 2 sec θ o .
These simplifications allows us to rewrite the proportion of the substrate K G that is both illuminated and viewed as follows :

K G = e -D(A i (θ i )+A o (θ o ,ϕ o )-O(θ i ,θ o ,ϕ o )) (IV.14)
where O is the overlapp between the illuminated and observed projected shadows.

O(θ i , θ o ) = 1 π (t -sin t cos t )(sec θ i + sec θ o ) (IV.15)
The other parameters involved in the expression IV.14, originally given by the Equations I.39 to I.41 are now slightly simplified :

cos t = D 2 + (tan θ i tan θ o sin ϕ o ) 2 sec θ i + sec θ o (IV.16) D = tan 2 θ i + tan 2 θ o -2 tan θ i tan θ o cos ϕ o (IV.17)
The proportion of illuminated material K C is also modified using these new hypotheses. For a dense layer, we already know that :

K C = F(1 -K G ) (IV.18)
with K G given by Equation IV.14. The factor F is now expressed from the Equation I.44 using the b = r equality :

F = 1/2(1 + cos Θ) sec θ o sec θ i + sec θ o -O(θ i , θ o , ϕ o ) (IV.19)
Now, for a dense medium we can assume that K G is pretty small which allows us to expressed the complete geometric contribution using the Equations IV.14 and IV.18 :

R g eo = CF = C 2 (1 + cos Θ) sec θ o sec θ o + sec θ i -O(θ i , θ o ) (IV.20)
The complete BRDF I adapted from the RTLSR model to be applied on dry powder is the sum of the previous volume and surface contributions respectively given by Equations IV.13 and IV.20 , weighted by an empirical proportion parameter α such as in the Equation I.2.3.d.

The values of only two variable parameters are refined in order to minimize the difference between the measurements and the simulations. The parameter α can be seen as a proportion parameter describing the amount of volume scattering compared to surface scattering. The second optimized parameter is the reflectance C, described as the reflectance of the illuminated material. If we had information about the geometry it corresponds to, it could be fixed using the measurements presented in Section III.2.3 to reduce the complexity of the model. By definition these parameters satisfy the relationships :

0 < α < 1 0 < C < 1 (IV.21a) (IV.21b)
A third parameter g , accounting for the anisotropy of the scattering by one particle may be added depending on the scattering phase function at use.

IV.3.2.b Results :

The adapted RTLSR model is optimized for each hematite sample whose BRDF has previously been measured. As for the Specralon's model simulations, only the case of sample 5 is presented in this section. The results for the other samples are given in the Appendix V.3. A first optimization is done in order to find out what scattering phase function is better suited to modelize dry powders. Four phase functions are tested : the one for plant stand originally in the RTLSR model, the Mie, Rayleigh and Henyey-Greenstein phase functions. This is done for only one incident direction θ i = 40 • at the wavelength 700 nm. The comparison of the results is presented on the Figure IV.15.

For such small grains Rayleigh phase function in a very good approximation of the Mie phase function. Indeed the results obtained with these two are very similar. They are also quite similar to results using the originally phase function developed for plant stands. With either of these three scattering phase functions, the amplitude of the peak in the backward direction is in good agreement with our measurements, but we notice large errors in the simulated slope of the BRDF especially for grazing angles. The computed RMS errors between measurements and optimized simulations are of magnitude 10 -1 .

It appears that the Heynyey-Greenstein allows the model to better fit the measurement. Although the back-scattering peak is underestimated, the slope is better taken into account and more likely able to predict the BRDF at larger observation angles. This is preferable since, as it has been discussed earlier, the back scattering measurements are less accurate than the measurements in other geometries. It is confirmed by the RMS error whose magnitude falls to 10 -3 . This is mainly due to the additional anisotropy parameter g ; contrary to the other functions tested this one has not two but three parameters, which reasonably increase the complexity of the model but also its flexibility leading to sensibly better results.

From these former optimizations, we can conclude that the Henyey-Greenstein scattering phase function is suitable to modelize BRDF of dry powders. I will only use this one in the following simulations. The respective impact of both volume and geometric scattering on the complete appearance of the studied material is shown on the Figure IV.16. The volume contribution reproduces an overall and almost constant scattered amplitude while the surface contribution accounts for more variations. In particular, it simulates the peak that is expected in the backward direction. Another physical phenomena that I was expecting to be modelized by the geometric scattering is the specular reflection. We should note that it is not taken into account by this model.

The optimization of the full model over all the geometries used for the BRDF measurement has been performed. Only the geometries with incidence equal to 60 • for which the back-scattering peak could not be measured are not included. The resulting refined parameters α and C for the sample 5 are given in Table IV We notice that the values of the refined parameters are very different from the one obtained when only one incident direction was taken into account. This influences the simulations as shown on the Figure IV.16. When the optimization is performed on all the geometries, the slope of the BRDF introduced by the geometric contribution is similar to the measurement, but the overall intensity mainly modelled by the volume contribution is either overestimated or underestimated depending on the considered incident direction. Only the simulations for the incidence 20 • are satisfying.

This might be explained by a dependency of some parameters on the geometry. By definition of the anisotropy parameter g of the scattering phase function, this parameter does not depend on the geometry. Nevertheless, it is possible that either α or C depends on the incidence direction. Indeed, α is empirical so it can depend on the incident direction without any contraindication. As for C it is only defined as a reflectance that we assumed to be fixed. But unless the material is perfectly isotropic its reflectance varies with the geometry.

The complete simulations of the BRDF using the new RTLSR model are presented on the Figure IV.17. The possible dependency on the geometry of some parameters results in important errors for the incidences 0 • and 40 • . For the mean incidence 20 • the simulations are satisfying. Other errors occur at grazing angles.

IV.3.2.c Limitations & conclusion :

The adapted RTLSR model is of great interest since it is established from physical and geometrical considerations. It does involve parameters describing the physical and chemical properties of the material under study. Among them we can cite the radius of the grains composing the powder, and the optical constants of the material the powder is made of.

It is quite simple and only involve one empirical parameter α that quantifies the amount of volume scattering compared to surface scattering. Two other parameters (C and g ) depending on the material can also be refined, allowing a better flexibility and adjustments for the simulation of samples with fine differences in appearance.

Nevertheless the RTLSR model also has some drawbacks. It results in a good approximation of the BRDF for a given incident direction, but becomes quite useless when numerous incidences are taken into account. Indeed it does not fit the measurements very well, and leads to large and unsatisfying RMS errors of magnitude 10 -1 . This is due to some of the variable parameters that may depends on the geometry.

The small number of variable parameters can also be seen as a disadvantage. The adjustment of the model to differentiate the appearance of two powder samples of different but very close morphological parameters is then very limited. For these reasons, we could not establish how the variable parameters of the model vary with the morphology of the sample. As a consequence, it is not possible to accurately predict the appearance of a powder using this model. It is impossible to do for samples whose BRDF has not been measured beforehand, and only feasible in the measured geometries for samples with known BRDF.

Both volume and surface contributions need to be improved in order to better modelize the appearance of dry powder. Although the geometric contribution account for the back scattering phenomena, it tends to under estimate its amplitude. Even if the measured specular contribution is quite negligible for most of the samples, we should note that is not even taken into account using this model. Yet, this phenomenon is noticeable in some samples with very fine grains, such as the sample 5. It would require the addition of another term in R g eo contribution to be accounted for.

The volume scattering which controls the mean amplitude of the BRDF is either underestimated or overestimated depending on the considered geometry. A finer investigation of this term by optimizing the model for each geometry would be needed to deduce an empirical law between the parameter α and the geometry.

IV.3.3 BRDF modelization : My model

None of the two previous models is good enough to simulate the appearance of dry powders. The spectralon model is empirical but efficient to reproduce the measured BRDF while the RTLSR is partially physically-based but doesn't lead to satisfying results.

To overcome these issues I have designed my own BRDF model. It is physically based and account for chemical and morphological parameters of the studied powders. It is also able to accurately reproduce the BRDF of dry powders.

IV.3.3.a My model :

As for the Spectralon and the RTLSR model, my model is divided into two contributions, respectively accounting for volume and surface scattering.

BRDF = A v BRDF v + A s BRDF s (IV.22)
These two contributions are weighted with amplitude parameters denoted as A v and A s .

The main difference with the two previously introduced models is that, based on the coherent back scattering theory (Section I.2.3.f), we consider back scattering as part of the volume contribution. The surface contribution is then reduced to specular scattering.

Volume scattering : coherent back scattering : Volume scattering includes single scattering and multiple scattering that can be deduced from the radiative transfer theory. If only these two phenomena are accounted for, it tends to underestimate the back scattering.

I identify the important peak in the backward direction as the result of constructive interference described by the enhanced back scattering theory (Section I.2.3.f).

Let's remind that the resulting BRDF using this theory is the sum of one single scattering term and two multiple scattering terms as described by the Equation I.57 :

BRDF v = BRDF L 1 + BRDF L 2+ + BRDF C (IV.23)
The analytical expression of each term has been well investigated [START_REF] Tsang | Scattering of electromagnetic waves : advanced topics[END_REF] so it can be used for either isotropic and anisotropic scattering. As far as we know, the complete BRDF model is only established for isotropic scattering. This greatly simplifies the multiple scattering contribution by considering the scattering phase function as constant over the whole scattering hemisphere. Anisotropy can then eventually be accounted for by choosing another varying phase function in the single scattering contribution.

The extension of the model for taking into account anisotropic multiple scattering has only been done for a normal incidence.

In the following we expose the few calculations that have been made in order to consider anisotropic scattering in both single and multiple scattering terms for any θ i /(θ o , ϕ o ) geometry. The final model only involves two variable parameters that rely on the physical and chemical properties of the material that is under study.

• Single scattering BRDF L 1 :

This contribution accounts for light that is only scattered once before being observed and is similar the one introduce in I.2.3.f. Indeed, the anisotropy is taking care of by the scattering phase function involved in the analytical expression of BRDF L 1 in Equation I.66.

BRDF L 1 = 4πr 2 L 1 Aµ i µ o (IV.24)
• Multiple ladder scattering BRDF bs,L 2+ :

Because of some averaging hypotheses, the possible anisotropy of the scattering by particles is neglected in the multiple scattering terms.

Let's remind that the scattering by one particle can be described by a transition operator T( r ) according to the Equation I.58, which is related to the scattering phase function p(Θ). With the assumption that the phase function is not isotropic, and therefore not constant, it is involved at every scattering event that light encounters as it travels the particulate medium. We denote P i and P o respectively the first and the last particles that are responsible for light scattering.

At the second order scattering, the light is first incoming on P i , scattered with an angle Θ i towards the particle P o , and finally scattered towards the observer with an angle Θ o as illustrated by the Figure IV.18.

For the n-th order scattering, light undergoes n scattering events. As for the second order, it undergoes a first one described by the angle Θ i while encountering the particle P i and last one with the particle P o and the angle Θ o . But it also interacts with n -2 other particles in between, resulting in n -2 scattering events respectively described by the scattering angles Θ k = θ i k -θ ok , difference between the incoming direction θ i k and outgoing direction θ ok on the k-th particle, with k = 2 ; ∞ as it is shown on the Figure IV.18.

By averaging over all the possible paths that light can follow inside the medium while encountering n scatterers, we can write the n-th order ladder term as follows :

L n = (4π) n-2 (ρκ) n r 2 |p(Θ n )| 2 |G m11 (r n , r n-1 )| 2 ...|G m11 (r 2 , r 1 )| 2 |p(Θ 1 )| 2 exp κ z 1 µ o + z n µ i dr 1 ...dr n
(IV.25) with r the distance from the sample to the observation point.

The total ladder term accounting for multiple scattering can be written in a recursive way, following the same reasoning as in Section I.2.3.f. It gives : Note that contrary to the ladder term defined by the Equation I.70, our new model involves a varying scattering phase function whose expression is chosen to better modelize the material under study. If one wants to describe isotropic scattering a constant phase function such as p(Θ) = p should be used which leads back to the expression originally introduced in Section I.2.3.f. The resulting BRDF of the ladder contribution is then given by :

L 2+ = exp(κ( z i µ o + z o µ i )A(r i ,
BRDF L 2+ = 4πr 2 L 2+ Aµ i µ o (IV.28)
where A is the area of the sample that is studied.

• Multiple cyclical scattering BRDF C :

As for the ladder term, we are aiming to take into account the anisotropic scattering of particles in the cyclical term. We will use the same assumptions as illustrated by the Figure IV.18. We consider a ray of light that is scattered by n particles as it takes a path inside the medium. By accounting for the correlation between this path and its reversal and averaging over all the possible paths the n-th order cyclical term can be written : The first scatterer that light encounters as it travels inside the medium is denoted as P i and involves a scattering event described by the angle Θ i . The last scattering event before light goes out of the medium is caused by the scatterer P o with an angle Θ o . In between these two scatterers, light is scattered with an angle Θ k at each interaction with a particle k.

C n = (4π) n-2 (ρκ) n r 2 |p(Θ n )| 2 |G m11 (r n , r n-1 )| 2 ...|G m11 (r 2 , r 1 )| 2 |p(Θ 1 )| 2 × cos k(µ i -µ o )(z 1 -z n ) exp κ 2 1 µ o + 1 µ i (z 1 + z n ) dr 1 ...dr n (IV.29)
The complete cyclical contribution is obtained by summing over the n-th orders and can be simplified replacing G m11 by its definition (Equations I.25) and using the new expression of A(r i , r o ) given by Equation IV.27 as :

C = cos(2k(µ i -µ o )(z o -z i ))exp κ 2 z i µ o + z o µ i A(r i , r o )dr i dr o (IV.30)
Once again, the scattering properties of each particle that plays a role in the n-th order scattering is explicitly involved through the varying phase function. The use of a constant phase function p(Θ) = p brings back to the isotropic case already discussed in Section I.2.3.f. Finally, the cyclical term contributes to the complete appearance of the dry powder through the partial BRDF :

BRDF C = 4πr 2 C Aµ i µ o (IV.31)
In my model, the complete volume scattering of a dry powder accounting for the anisotropic scattering properties of particles is the sum of the previously determined single scattering, ladder and cyclical contributions respectively given by Equations IV.24, IV.28, and IV.31. These three contributions have been deduced under physical considerations, and therefore involve parameters describing physical properties of the material such as scattering through the phase function p, as well as its morphology and chemical composition respectively through the grain radius r eq and the optical constants that are implicitly involved in the albedo ρ.

Nevertheless such equations cannot be analytically solved. Under statistical approximations known as Diffusion approximation described in section I.2.3.f, the different terms of the complete BRDF are reduced to the analytical formulas given by the Equations I.75, but with a new definition of the parameters it involves.

With this approximation, the scattering properties of the particles are described by the anisotropy parameter g that depends on the scattering phase function with the relationship :

g = 2π π 0 p(Θ) cos Θ sin Θd Θ (IV.32)
This parameter corresponds to the variable parameter g used in the Henyey-Greenstein phase function. It takes its values in ] -1, 1[, and is equal to zero in the case of isotropic scattering.

Based on the work of [START_REF] Tsang | Scattering of electromagnetic waves : advanced topics[END_REF], but extending their theory for any incident angle, we first define the parameters u, v and α so that the model is valid for any geometry including various incident directions :

                 u = k 0 κ (µ i -µ o ) v = 1 2 1 µ i + 1 µ o α = k 0 κ f (IV.33a) (IV.33b) (IV.33c)
where f 2 = f 2 x + f 2 y allows the BRDF modelization in any azimuthal plane, according the its former definition in Chapter I, Figure I.8.

Then we also introduce the parameters accounting for the scattering inside the medium :

         c = 3(1 -ρ)κ t r + α 2 κ t r = 1 -g ρ Q = 1 + µ i c -(1 -µ i c)exp(-2cτ o ) /4c (IV.34a) (IV.34b) (IV.34c)
where τ o , defined in Section I.2.3.f, depends on the optical constants of the powder material. This enhanced back scattering model adapted for either isotropic and anisotropic scattering relies on three physical contributions with simple analytical expressions. An example of its application is given on the Figure IV.19 to better understand the influence of each of these contribution. The shape of the BRDF is roughly reproduced by the single scattering, while finer details are modelized by the multiple scattering contributions. The incoherent term leads to an overall and almost constant diffuse reflectance, and the coherent term justifies the sharp peak that is observed in the backward direction.

There are two parameters whose value is not depending on the geometry but only on the material properties : FIGURE IV.19 -Contributions of the single scattering, ladder and cyclical terms to the complete BRDF. The rough shape of the BRDF is reproduced by the single scattering contribution and is mainly controlled by the fitting of the anisotropy parameter g . The Henyey-Greenstein phase function is at use here. Finer details are simulated using multiple scattering : while the incoherent term accounts for an overall diffuse contribution, the coherent one models the sharp peak in the backward direction whose width is controlled by the variable parameter κ.

• First there is the anisotropy parameter g accounting for the scattering phenomenon induced by the grains. It mostly influences the rough shape of the BRDF through the single scattering contribution. As an assumption we can expect that this parameter is related to the morphology of the grains and more probably their shapes.

• Secondly, there is κ which is known as the extinction rate and related to the absorption inside the medium. It mainly controls the width of the back scattering peak. It might depend on the morphology of the powder at stake or its chemical composition, but we don't have any further information about its value.

Surface scattering : specular scattering : Since the back scattering is now identified as part of the volume scattering, surface scattering is reduced to the specular contribution. The light that is incoming on a rough surface such as the interface between air and a dry powder is scattered in a lobe around the perfectly specular direction (Section I.2.2).

Two options have been considered to include such a phenomenon in our model : an empirical and physically based one.

The former is to model the specular peak by a gaussian function. This is an empirical but simple and good enough method to reproduce the specular contribution of dry powders. By assuming a uniform distribution of the surface normals, it requires only one parameter which is the gaussian half width σ. With this modelization the surface scattering BRDF is written as :

BRDF s = 1 σ 2π exp( -Θ 2 2σ 2 ) (IV.35)
The second option relies on the physical approach from Harvey-Shack [62], introduced in Section I.2.2, that accounts for diffraction effects caused by the surface asperities. Using this theory, the incoming light on a rough surface is then scattered around the specular direction according to the following BRDF :

BRDF s = 16π 2 λ 4 cos θ i cos θ o PSD( f x , f y ) (IV.36)
where the power spread function (PSD) modelizes a rough surface topography.

In the case of hematite dry powders, the measured peak in the specular direction is so small if not unnoticeable that it can be approximated by this Gaussian model without loosing a significant amount of precision. However for other samples that may show an important specular contribution, such as the sample 1 3 for example, we believe that the Harvey-Shack model is more suitable to accurately simulate the surface BRDF.

Our complete model is the weighted sum of volume and surface contributions according to Equation IV.22 respectively given by the adapted enhanced back scattering model and a specular model. In order to deal with only physically based models, we will use the Harvey-Shack model in the following. For optimization purposes its number of parameters is reduced to one instead of two by including the standard deviation σ into its associated amplitude parameter A s . By summarizing all the previous equations, our complete model can be written as follows :

BRDF =A v [BRDF L 1 (g ) + BRDF L 2+ (g ) + BRDF C (g , κ)] + A s BRDF s (l c ) (IV.37) BRDF =A v ρp(Θ) µ o + µ i 1 -exp -κd 1 µ o + 1 µ i + 3ρ 2 1 -µ 2 i c 2 2Q 1 -µ o c - µ 2 i µ i + µ o (IV.38) + 3ρ 2 [c + v(1 -exp(-2cτ o ))] 2µ i µ o cv[(c + v) 2 + u 2 ] + A s 4π 2 λ 4 (µ i + µ o ) 2 πl 2 c exp - π 2 l 2 c λ 4 f 2 (IV.39)
with parameters introduced in Equations IV. [START_REF] Fleming | Visual perception of thick transparent materials[END_REF] and IV.34.

The model then depends on a total of 5 empirical and physical parameters summarized in Table IV

IV.3.3.b Results

: I applied the model to all the hematite samples of previously measured BRDF through the refinement of its 5 variable parameters. The optimization is performed at the wavelength 700 nm for all the geometries of incidence smaller than 60 • . Only the case of some of the samples are actually fully described in this section. The complete results are given in the Appendix V.3. Our model is particularly efficient in the case of samples whose morphology is easily characterized, that is to say for the samples with well crystallized mono-crystals. It includes the samples 1, 1 2 , 5, 6 and also P 2 in some extent that will be discussed later.

Let's begin with the samples whose appearance is well and easily reproduced (Figure IV.20a). After the optimization of its parameters, our model fits the measured BRDF quite well in the case of samples 1 and 1 2 . Both the specular and the back scattering contributions are detected and modelized, even though later is a bit underestimated. It is less obvious for the sample 5 whose study is detailed below, as well as for the sample 6. Indeed, a small change in the magnitude of the parameters to refine implies a drastic change in the optimized result. As shown on the Figure IV.21 the back scattering simulated peak is either too sharp and overestimated or too large and underestimated, leading to unsatisfactory RMS errors of magnitude 10 -1 .

To overcome this issue without making my model more complex, I removed one constraint which is the fixed value of the grain size, 25 nm in the case of sample 5. Until then, the used grain radius are the one retrieved from the grains dimensions observed in MEB images and reduced to an equivalent sphere. Because of the real shape of the grains, these radius have been defined with a margin of error (see Table III.1).

Let's define the effective radius as the mean size that is seen by the incoming light and therefore responsible for scattering. A new optimization of the model is performed including the effective radius as a parameter to refine. This allows more flexibility to the model without a significant increase of its complexity. A total of 6 parameters per sample are now optimized.

The optimized effective radius is slightly out of the determined margin of error but remains plausible. The results are much more pleasing : the width and amplitude of the back scattering peak are well modelized, and even a small specular contribution is now spotted. The simulation of the BRDF of sample 5 in the incident plane for mutliple incident direction is shown on the Figure IV.22. We also note, on Figure IV.20b that accounting for an effective radius leads to even better simulations of the back scattering peak in the case of previously well fitted samples (1 and 1 2 ).

Since the refined parameters permit my model to quite precisely fit with the measurements in any geometry, it appears that none of them depends on the incoming, outgoing and azimuth directions. The model is then able to predict the appearance of the studied powder in any other geometry that hasn't been considered during its optimization. The results are quite convincing even for more grazing angles such as incidence θ i = 60 The difference between the measurements and the simulation does not exceed 8% of the reflectance factor. We obtain errors around the back scattering peak because of the measurements' bad sampling and interpolation at these geometries. Errors also become larger with grazing angles. This is comparable to the results obtained using the Spectralon's model. Similar efficiency of the model is obtained for the BRDF simulation of the samples 6, 1, an 1 2 when taking into account the effective radius.

The samples 3, and C are more difficult to modelize because of their complex morphology. Each of these samples can be described using two sizes of grains.

A first option is to apply the model while taking into account only one effective size population. Unfortunately this does not lead to nice and convincing results, as shown on Figures IV.24a and IV.25a. The fitting is particularly bad at grazing angles. It confirms that my model needs to be adapted to account for both sizes.

In the particular case of sample 3 two distinct size populations have been identified (Section III.2). This allows me to consider two completely independent contributions from each of the population. According to the SEM images, it is composed with a proportion of about 90% of small grains (r s = 47.1 nm) and about 10% of larger ones (r l = 83 nm).

Because of their large amount, the small grains are close to each other which facilitates the multiple scattering phenomena to occur. Their contribution to the sample's BRDF is then described by our full model from Equation IV.39, including single, multiple and surface scattering.

On the contrary, large grains are much disperse into the powder. The distance between them is too long and strewn with smaller grains. So the probability of multiple scattering to happen between them and only them is negligible. However large grains are still responsible a single scattering contribution BRDF L 1 (g l ) of amplitude A v,l .

A total of five variable parameters are needed to simulate the volume scattering in such a sample : the amplitude parameters of each population contribution, the anisotropy parameters involve in single scattering, and the width κ of the back scattering peak.

Both populations are involved in surface scattering. However, according to the measurements the specular contribution is unnoticeable. So it can be neglected in order to keep a low complexity of the model.

Finally we have the following extension to our original model :

BRDF = A v,s BRDF L 1 (g s ) + BRDF L 2+ (g s ) + BRDF C (g s , κ) + A v,l BRDF L 1 (g l ) (IV.40)
involving 5 to 7 parameters to refine if the effective size of the populations is also optimized.

The same adapted model can be used for the case of the sample C, but not for the same reasons. This sample does not show two distinct grains populations, but very large grains (r l > 1 µm) which are agglomerates of smaller ones (r s = 49 nm). As for the sample 3, the small grains are numerous and close enough to produce multiple and coherent scattering. But the large grains cannot be described as disperse into the powder. Indeed they are in contact with each other as observed by optical microscopy (Section III.2). Nevertheless, because their size is much larger than the wavelength, they do not induce multiple scattering and especially interference phenomena. The model reproduces the BRDF of both of these samples with a RMS error of magnitude 10 -2 . Its prediction ability appears to reach its limitations as it tends to underestimate the reflectance at larger incidences such as 60 • (Figures IV.24d and IV.25d).

The last sample I studied is the sample P 2 . It could also be described at two scales since the large peanut grains seem to be agglomerates of much smaller ones. However our model leads to nice results without necessarily taking into account these two descriptions. It seems that it is sufficient to apply the model for one effective size of same magnitude as the peanut grains' size.

Although the parameters involved in this model have been defined with a physical meaning, their values remains unknown before they get to be optimized to fit a BRDF measurement. Thus it is not possible to predict the appearance of any powder with known grains' morphology and chemical composition whose BRDF hasn't been measured beforehand. To do so, the possible dependency of each parameters with the grain size and the wavelegnth are necessary.

• Spectral dependency of the parameters : I first attempt to establish a relationship between the refined parameters and the wavelength at which the BRDF is simulated.

To do so, the model is optimized for different wavelength in the visible spectrum using the effective radius previously deduced from the first optimizations at 700 nm.

Nice fitting reflectance is obtained for wavelengths between 650 nm and 850 nm. But for shorter wavelengths the model is not able to accurately reproduce the measurements as it shown on the Figure IV.26.

Let's remind that a fixed physical parameter that is involved in the model already depends on the wavelength : the optical constants. As discussed in Section I.1.4.d, they've been measured several times in the case of hematite but none of the results agree with each other. One particular difference is the spectral positions of the main absorption bands. The results may vary depending on the values chosen for these parameters.

Two sets of values have been tested : the one from Querry [START_REF] Querry | Optical constants[END_REF] and the one from Triaud [108]. Even if the later leads to even nicer results at large wavelengths, some issue remains under 650 nm. It appears that in the spectral domain 400 -600 nm the measured intensity is quite low and tends to decrease with the wavelength. This issue can be solved doing a similar study with a brighter sample whose reflectance is quite high for the whole visible spectrum. If the measurements can then be accurately fitted at these wavelengths, the spectral dependency of the refinement parameters could be drawn. Such a study has not been conducted yet, so we could not reach any empirical correlation between the model's parameters and the wavelength.

• Morphological parameters and refined parameters correlation :

The refined parameters are also expected to depend on the morphological and chemical properties of the samples. In order to be able to predict the appearance of other powders, we are looking for a correlation between them. Since the specular is negligible, we restricted our study to trends for the variables involved in the volume scattering contribution have been studied (see Figure IV.27). Only the grains' sizes of each sample that are responsible for multiple scattering are considered. Two groups of samples can be distinguished, especially for the study of the parameters g and κ : the samples with smaller grains and the one with large grain sizes. These two parameters seem to increase with the effective radius. This can be expected for g which describes the scattering anisotropy since it also increases with the scatterer size in the case of Mie theory. However the variation of this parameter should not be linear. Indeed it is defined in the range -1 < g < 1 and therefore may reach an asymptote for even larger grains. κ describes the extinction in the medium and characterizes the width of the back scattering peak. As it goes through larger grains, the light that propagates in the medium is more absorbed, resulting in a larger κ, and therefore the intensity of scattered light A v decreases.

Unfortunately, the lack of experimental points at intermediate sizes prevents us to conclude on a precise mathematical correlation. Indeed, among the synthesis routes we used only the one resulting in the sample AG permits to reach grain sizes in this range (r = 38 nm). But the amount of synthesized powder is too small to perform the BRDF measurements. It would be possible to obtain suitable samples using other routes. For example, assuming that it leads to mono-crystals, the hydrothermal synthesis followed by Pailhé et al. [START_REF] Pailhé | Impact of structural features on pigment properties of α-fe2o3 haematite[END_REF] results in grains with radius r ≈ 36 nm. Another solution would be to increase the temperature at which the fine powders (samples 1 to 6) are synthesized. In fact, for the same protocol Pailhé et al. [START_REF] Pailhé | Impact of structural features on pigment properties of α-fe2o3 haematite[END_REF] obtained powders whose grain size increases with the temperature. A similar result might be obtained by annealing the previously synthesized samples.

As a consequence it is not possible to predict the appearance of a powder whose BRDF is com-pletely unknown based on its morphology and chemical composition.

IV.3.3.c Limitations & Conclusion :

Our model is able to reproduce the measured BRDF of hematite powder samples with an accuracy comparable to the one of the Spectralon's model, but with a much lower complexity. Only 6 parameters are refined against 12 for the Spectralon's model. Moreover, these parameters are related to actual physical properties of the samples such as the morphology of the grains and the surface roughness. They do not appear to depend on the incidence/observation geometry, so the model can be used to predict the reflectance of the studied sample in any geometry that was not take into account during its optimization. It provides a physical explanation and modelization for the important and sharp back scattering peak that is observed on the BRDF of such materials.

This model can be applied in its original form Equation IV.22 for the simulation of monodisperse powders' appearance, but also for more complex morphologies through its extension given by the Equation IV.40. All of our samples have been modelized with RMS errors of magnitude 10 -2 .

Nevertheless, our model also has some limitations. First of all it is not completely physically based. We introduced two empirical (A v and A s ) parameters to characterize the proportions of the volume and the surface scattering, as it has been done in the RTLSR model. To allow more flexibility, these two have been chosen to be independent from each other.

Then, even though the refined parameters are linked to morphological properties of the samples through their physical meaning, it was not possible to establish any clear correlation. Only some assumptions can be done, like the increasing trends of the anisotropy parameter and the parameter κ with the grain size. This is mainly due to the lack of known BRDF of samples with intermediate or much larger sizes such as HT, AG or 1 3 .

For this reason it is not possible yet to predict the appearance of another powder whose BRDF has not been measured beforehand.

The accuracy of the specular modelization could not be certified because of the very weak contribution measured in this direction. More sparkling samples like the sample 1 3 should be studied to validate such a representation. Further study could be done on eyeshadow for example.

Other extensions should be brought to this model in order to account for more complex materials present in parietal paintings. First we will have to deal with the case of mixtures instead of pure hematite. The assumption of an optically thick layer is also too restrictive. Indeed, the painted layers are much more thin and even semi-transparent. This involves a contribution of the substrate (rock), but also possible interference phenomena caused by multiple reflections between the two interfaces of the powder layer.

Finally, if no binder is nowadays detected on the paintings it is highly possible that one was used to ease the application of the pigment on the rock. This results in a different appearance than if the dry powder was directly applied.

Although it can be used to simulate the appearance of any pure crystalline powder, we did not take care of the case of mixtures which is more representative of the coloring matters used on the paintings. One of the most common materials identified on the paintings are ochres, a powder coming from the alteration of ferruginous rock through weathering. Ochres are then a mixture of different minerals and clays, mostly uncolored or white, and a coloring agent (chromophore). It results in both red hues when hematite α-Fe 2 O 3 is at stake or yellow hues when goethite FeO(OH) is involved.

Moreover, a paint of layer may be very thin. This is especially true for parietal paintings, that may have undergo some alterations over the years. For example, the paint peel off causes the removing of coloring matter and results in thinner paint coatings that allow the substrate to be seen through and thus to contribute to the overall appearance.

A more realistic modelization of a painting layer would be to consider a powder composed of multiple minerals deposited as a thin film on a rock substrate. This is schematized on the Figure V.1.

In this chapter, I introduce some preliminary results and studies that should be completed to modelize the appearance of a more complex dry powder paint layer. First we investigate the chemical composition of some ochre powders used as coloring matter on parietal paintings, and study its impact of the ochre color. We also show that depending on the thickness of the layer, the substrate on top of which the paint is applied may have an important contribution to the appearance we perceive. Finally we discuss the importance of the way the painting has been made, by showing that its appearance is impacted when a binder is used.

V.1 Mixtures : the case of ochres

Among other ferruginous rocks, ochres are commonly identified as coloring matters used on parietal paintings. They are mixtures mainly composed of clays and colorless minerals that gets its coloration from an iron oxyde and/or hydroxide : hematite α-Fe 2 O 3 in the case of red ochres, and goethite FeO(OH) for the yellow ones. Because of this more complex chemical composition, the appearance and especially the color of red ochres is different from the one of the pure chromophore powder it involves.

As an illustration, we studied five natural ochres coming from Burgundy, France and from Roussillon, France, respectively commercialized by Kremer and Okhra. We characterized these powders using the same characterization methods previously used for pure hematite.

I first made observations using both the naked eye and optical microscopy. I did not proceed to SEM observations and therefore do not have more precise pieces of information about the size and morphology of the grains the ochres are made of. Nevertheless, we investigate their chemical composition and more specifically the mass proportion of each identified crystalline phase through XRPD. I finally proceed to a fine measurement of the color and the appearance of these ochres using spectroscopy in the visible a near-infrared spectral range.

V.1.1 Observations

From a first observation with a naked eye, the five ochres can be separated in two groups depending on the perceived color : two of them (G K11573 and G O2400200 ) show yellow hues which must be caused by the presence of goethite, while the three other ones (H K11575 , H K48600 and H O3200800 ) show red hues which must involve hematite.

Although at the appearance of these dry powders observed at the macroscopic scale is quite uniform, some inhomogeneities are visible using optical microscopy, that suppose the presence of mutliple minerals. The results of these observations are given on the Figures V.2 and V.3.

These impurities are well identified on the samples H K11575 and H O3200800 that mainly appear to be red but show some white and bright yellow grains. It is more difficult to notice in the case of FIGURE V.2 -Observation of the yellow ochres. On the left, a photograph of the sample. On the right, its observation using optical microscopy. The observed color is not uniform which may suppose the presence of multiple phases.

the yellow ochres because the color difference between the white impurities and the yellow grains responsible for the color of the sample is less noticeable. We do not spot the same inhomogeneities in the sample H K48600 which appears to be much darker than the other ones. Only few grey-ish grains are observed in this sample.

We can conclude that the presence of other minerals than pure hematite or goethite has a strong impact on the color of the ochres. According to additive color synthesis, the presence of white mineral powders results in a clearer hue.

We can also give some insights about the complexity of the morphology of these ochres samples. The powders are far from being monodisperse, as this can be seen on the sample G K11573 which shows various size of grains. Many shapes are also obtained. The more particular one is observed for the sample H K48600 which seems to be composed of spherical grain agglomerates with different sizes.

We did not perform SEM observations of the ochres samples. A further study using this characterization technique would bring more information about the nature and the morphology of the grains composing the powders. Knowing the size and shape of the chromophore grains, it will be possible to compare the appearance of the ochre it is involved in with the appearance of the pure chromophore crystalline powder.

V.1.2 X-ray Powder Diffraction

The XRPD of these samples has already been performed before the beginning of our project, in order to accurately identify the crystalline phase at stake in the ochres samples as well as their respective mass proportion, we proceed to XRPD. We did not proceed to a new measurement, but we have exploited the results from this characterization. Details about these results are given on the following table V.1. FIGURE V.3 -Observation of the red ochres. On the left, a photograph of the sample. On the right, its observation using optical microscopy. The observed color is not uniform which may suppose the presence of multiple phases. The sample H K48600 also shows a particular morphology with large and perfectly spherical agglomerates of smaller grains.

In most of the samples both quartz SiO 2 and Kaolinite Al 2 Si 2 O 5 (OH) 4 are identified. The chromophore, either goethite or hematite, only represent a small percentage of the powder (between 8 to 25 % of the mass). The sample H K48600 appears to be quite pure hematite, with only a minor phase of maghemite, another iron oxide with a brownish hue, that is probably responsible for the inhomogeneities observed using optical microscopy. Although the purity of the sample agrees with the manufacturer data, the nature of the impurities is not consistent. According to the manufacturer, this small mass percentage is mainly caused by the presence of quartz. Either way, its amount is sufficiently small to be consider as negligible.

To conclude on this study, the complex chemical composition has a strong impact on the appearance of the powder. We observed that pure hematite powder is darker than red ochres involving Quartz and Kaolinite. Indeed, when milled into powder, these other minerals are mainly white or colorless. When it is involved in a mixture with a colored pigment, it tends to brighter the color perceived at the macroscopic scale (samples H K11575 and H O3200800 ). turated band at 3150 nm. Other absorption bands than the one noticed in the spectrum of the pure hematite sample are observed. The group of bands around 1410 and 2200 nm characterizes the presence of hydroxyl groups OH, and the narrow band at 2730 nm which is also observed in the spectra of pure quartz and kaolinite [START_REF] Chalmin | How to distinguish red coloring matter used in prehistoric time ? the contribution of visible near-infrared diffuse reflectance spectroscopy[END_REF], which corroborates the XRPD analysis. These bands are not involved in the sample H 48600 which once again testifies of its purity. Finally, we should note that the bands around 4200 -4300 nm are caused by the presence of CO 2 in the experiment room. The amount of this gas easily changes during the experiment depending, for example, on how many people are in the room while the measurement is performed, so the intensity of such bands is not well corrected. These bands are not representative of the powders under study.

V.1.4 Back scattering measurement

To fully describe the appearance of the ochres, we would need the measurement of their BRDF. I did not perform the measurement of the complete BRDF, but I used the instrument we designed to at least measure their back scattering contribution, in the incident plane for the different incidences θ i = 0 • , 20 • and 40 • . Then I compare these measurements with the result obtained for the synthetic pure hematite powder 1. Only the results in incidence 20 • are given on the Figure V.6. The other results are in the Appendix V.3.

Although the complete BRDF is not known, we can draw some conclusion from the only measurements around the back scattering direction. We observe an important peak in the backward direction, similar in width and magnitude to the ones observed on synthetic pure hematite samples. The main difference are observed in amplitude : the ochres samples tend to be clearer than the pure hematite sample 1. The results we obtained for a purer sample such as H K48600 are very close to the one from the synthetic sample. We therefore have good reasons to believe that the appearance description we made on synthetic samples can be extended to natural samples.

As expected, the change of chemical composition does not involve a change of the physical phenomena identified as responsible for the powder appearance. Therefore, my appearance model developed for optically thick hematite crystalline powders should be able, by carefully choosing the parameters values, to reproduce the BRDF of other more complex powders such as ochres.

V.1.5 Conclusion

The mineral composition of the powder has a strong impact on its appearance. Different colors are obtained depending on which mineral is used. In the case of ochres, either red or yellow is obtained if respectively hematite or goethite is involved. The addition of other crystalline phases to a pigment also influences the hue of this pigment's color. For example, compare to a pure hematite powder, a red ochre also involving white minerals such as quartz and kaolinite shows a brighter color.

We can also conclude that it is important to take into account the proportion of each mineral involved to accurately describe the color of an ochre. Two ochres made out of the same minerals but with different proportions such as the samples H K11575 and H O3200800 show slightly different colors, probably because of some differences in the size of the hematite or quartz grains.

V.2 Thin layers

Another parameter that we did not take into account in our modelization of a powder appearance is the thickness of the powder layer. We assumed that the layer is optically thick, that is to say that the substrate on which the pigment is spread out does not contribute to the overall appearance.

However, when it comes to paintings, the layer of crystalline powder may be quite thin and semi-transparent. We highlight this phenomenon with observations and reflectance measurements on either thick and thin paint layers observed on the Trou de la Féclaz archaeological site.

V.2.1 Measurements on a archaeological site : Trou de la Féclaz

We performed reflectance using a SVC HR-1024 spectrometer measurements on actual paintings from the 'Trou de la Féclaz' archaeological site located in Saint-Jean-d'Arvey (73230, France). This shelter is one of the 316 documented archaeological sites in southern France and western Alps that shows schematics paintings. This kind of representations are assigned to the Neolithic era [START_REF] Defrasne | Saint-jean-d'arvey-le trou de la féclaz. relevé d'art rupestre[END_REF]. On theses paintings, several shades are observed from orange to dark red.

To proceed to the measurements, we used a 35 W/220 V halogen lamp as an light source, and a 1 meter long fiber optic to detect the reflected signal. We were confronted to several issues while performing measurements in situ, because of both the inaccuracy of the measurement set up and the environmental conditions.

The current lighting and observation parts of the instrument are not easy controllable which prevents us to precisely define the measurement geometry or to accurately spot the point of the painting that is actually measured. To overcome these issues, we need to improve the design. It would include a more uniform and concentrated lighting, a finer control of the incident angle, as well as a better definition and a minimizing of the observation spot.

Because we are working in a natural environment, on a non planar surface, the lighting conditions from one measurement point to another one can be very different. To minimize the photometric errors between the measurements, the acquisitions performed at each point of the wall are calibrated by a measurement of the spectralon material at the same point.

Here we focus on one measurement that justifies the importance of the paint layer thickness in the perceived appearance.

V.2.2 Analysis

The measurements on the painting exhibit the same absorption bands than those that characterize the substrate. The main differences are noticed in the visible spectral range. For a very thin layer of paint, the reflectance is pretty close to the one of the substrate, and therefore implies its important contribution. The thicker the layer is, the more different the spectra are. At some point, for a thick enough layer the substrate is completely covered and unreacheable for the incoming light. In that case the absorption of the powder film is predominant over scattering from the substrate, which is visible on the reflectance spectra by the increase of the absorption bands specific to the pigment material such as the one seen around 850 nm on the Figure V.8, which characterized the presence of hematite in the coloring matter.

V.2.3 Conclusion

Because they have been altered over the years, the parietal paintings we can observe on archaeological sites such as Trou de la Féclaz undergo peel off, erasing or removing which reveals the rock substrate. With reflectance measurements we showed that in the regions where the remaining paint layer is thin, the substrate actually plays an important role in the color we perceive.

It is therefore necessary to take into account the substrate contribution to the appearance of these paint layers. Further study would allow to extend our appearance model in the case of thin layers.

V.3 Binder

The last appearance variations I would like to discuss is not connected to the physico-chemical properties of the material but to the way it is applied on the substrate. Although the paintings that are visible nowadays only show the presence of dry powders, it is possible that for their creation, the pigments were dispersed into a binder to ease their application on the wall.

Pommerol at al. [START_REF] Pommerol | Photometric properties of mars soils analogs[END_REF] showed that the appearance of a powder that has been wet and dry is different from the appearance of the dry powder that has never been wet. They explained these effects by a re-organization of the grains composing the powder as it is wet that remains unchanged after it dries. We are aiming to observe such appearance differences and to quantify them so as to determine if it is noticeable enough be of interest in our model.

To do so, I prepared two different samples involving the same red ochre H 3200800 applied on white paper. The first paint layer is made out of the original dry powder. For the second one, the powder is first dispersed in a small amount of water in order to obtain a paste that is then applied On the left the dry ochre is applied on the paper. On the right, the powder is applied as wet on the paper. We wait for this sample to be completely dried before proceeding to the measurements. (b) Reflectance measurement of red ochre layers spread on white paper, in the geometry 0 • /30 • . The powder is either dry or mixed into water before it is spread with a finger. We perform the measurement once the wet layer has completely dried. The reflectance of the dry powder layer is very similar to the one measured on a thick layer, showing the good covering power of the coloring matter. The wet and dried sample shows a darker look responsible that results in a weaker reflectance. on the paper. I waited for the sample to completely dry before doing any further measurements. The photographs of the samples after complete drying is given on the Figure V.9.

Reflectance measurement are then performed in the visible spectrum on each of these samples. The thickness of the layer is not well controlled here. We already showed in Section V.2 that the appearance may differ depending on the layer thickness. In order to avoid any misinterpretation of the appearance change that would not be caused by the use of a binder but by a thickness difference, we compared the measurements with the reflectance spectra measured on an optically thick layer of the studied powder.

From our perceptual point of view, we can observe that the wet-and-dry paint layer tends to be darker with a smoother and therefore a more specularly reflective surface than the one made from dry powder. This is in agreement with the observations of Pommerol et al. [START_REF] Pommerol | Photometric properties of mars soils analogs[END_REF], and validated by reflectance measurements on Figure V.9.

The reflectance of the dry powder applied on paper is very similar to the one of the optically thick layer. This indicates that the layer we put on the paper is thick enough so that the substrate is not involved in its reflectance measurement. The spectral position of the absorption bands are the same whatever sample is studied, with or without a binder. Only the amplitude changes : they are more intense for the sample for which we used water as a binder. This darkening is then not due to a larger thickness but because of the difference in the painting process process.

Conclusion and Future work

Hematite α-Fe 2 O 3 crystalline powders identified in parietal paintings show a very different appearance than an homogeneous bulk crystal, and can be found as responsible for different shades of red. During this thesis, our main objectives were to connect the appearance of such dry powders with their chemical composition and the morphological and microstructural parameters of their grains, and to modelize it with analytical models that can be used in computer graphics to reproduce photo-realistic images of those materials. This is done by the investigation of the optical phenomena occurring as incoming light encounters the material's surface that we identified as absorption and scattering.

We first considered a paint layer as an homogeneous absorptive but non scattering thin film deposited on top of a rocky substrate. Such a modelization involves the attenuation of light as it travels in the layer, which is a wavelength dependent phenomena responsible for the coloration of the material. It is mainly described by the complex optical constant n, intrinsic parameter describing the propagation of light in the material. In the case of a transparent or thin enough layer, two other contributions should affect the observed color : the contribution of the substrate, and an interference effect caused by the multiple reflections of light between the interfaces of the film. Because there is no scattering, the light is supposed to follow a linear propagation inside the material. Under these geometric optic considerations, we have deduced an analytical model allowing the description of any two layered material. The validation has been done in reflection by comparing photographs of layered material with different thin film thicknesses with photo-realistic renderings obtained using our model. Although the color difference between the real samples and the simulated ones are noticeable for a standard observer, the color change with the thickness is well reproduced. The validation is even more convincing in transmission : the difference between the experiments and the modelization is not discernable any more.

But this modelization only considers the chemical composition of the material through its optical constant. Therefore it does not explain the different shades a single material such as hematite is able to reach. A paint layer is not an homogeneous bulk but a dense collection of small grains, which implies that light is deviated, and therefore scattered, as it travels in the medium. Depending on the size and the morphology of the grains, scattering may differs from one hematite powder to another one. Several hematite crystalline powders that have been synthesized in order to mimic the color range observed on actual paintings are characterized using different non destructive techniques : XRPD and microscopy (optical and electronic). They allow the determination of the size, morphology and microstructure of the grains composing the powders. The color is described through reflectance spectra and their conversion in the CIE L*a*b* color space. All of these characterizations made it possible to investigate the change of color with the grains' size. It appears that beyond a critical grain diameter of about ≈ 80 nm the trend that the color follows is reversed.

From the BRDF measurements, we highlight that in the particular case of dry powders, the light is mainly back scattered. Two existing models are first adapted to reproduce the appearance of our samples : the spectralon's model which is very efficient but fully empirical, and the semi-empirical but inaccurate RTLSR model. We propose a new BRDF model accounting for three different scattering interactions : single scattering, incoherent multiple scattering and coherent multiple scattering resulting in the sharp peak in the backward direction. It is a semi-empirical model based on the propagation of light inside a granular medium that involves the size of the grains and only 5 refinement parameters with physical meaning. Unfortunately the available samples we used did not permit to conclude on an explicit correlation between the model's parameters and the structural and size ones. Yet, the model is not able to predict the appearance of a powder whose BRDF has not been measured beforehand. But it does reproduce the measured appearance with a good accuracy, with RMS errors of magnitude smaller than 5% of the reflectance.

Our model still faces some limitations that might be solved by a further study. First of all, we would like our model to be able to predict the appearance of any hematite powder based on its chemical composition and the size of its grains. This requires the estimation of a correlation relationship between the refinement parameters of the model and the grains' size and shape, and therefore to extend our study to other pure hematite samples with grain's radius ranging between 25 and 45 nm. The validity of the model's wavelength dependency is also a main issue that could not be solved during this work. Indeed, it is mainly described by the optical constant at stake whose value appears to be quite doubtful according to the discrepancies between the results in the litterature. Such uncertainties are often ignored in material science and computer graphics whereas it can really mislead the conclusion. Thus, we believe that only more reliable values would allow to conclude on the spectral accuracy of our model. However we have shown that the measurement of such parameters can be quite painstaking and therefore remains a major issue nowadays. This works offers many future work perspectives in order to better reproduce the appearance of paint layers, regarding both their chemical composition and their physical modelization. Indeed we limited our study to the single crystalline phase α-Fe 2 O 3 responsible for the reddish hues observed on parietal paintings. But yellow caused by the presence of another mineral which is the iron hydroxide FeO(OH) known as goethite. The methodology we developped in the chapters III and IV could be applied to pure goethite powders in order to describe their color variations from light yellow to slightly orange hues. The combination of the study on these two pigments will allow to reproduce a wider range of color by considering the case mixtures whose color is the result of the additive color synthesis.

However pure hematite or goethite crystalline powders are quite rare. An even more representative approach would be to investigate the color of ochres, mixtures that are much more commonly found on parietal paintings, involving one chromophore and other crystalline phases. Preliminary results have been investigated in the case of natural ochres. Although their appearance is quite uniform at the macroscopic scale, inhomogeneities observed through optical microscopy highlight the presence of the other constituents identified through XRPD, with possible different morphologies and therefore different contributions to the resulting color : because these minerals show a whitish color once milled into powders, the color of an ochre should therefore be lighter than the color of the pure iron oxides powder it contains. As the same sharp back scattering peak identified on pure hematite samples is measured on ochres, we believe that our model should be well suited to reproduce the appearance of any powder with a more complex chemical composition. In order to correctly modelize the contribution of each crystalline phase in the mixture, it would require a fine knowledge of their proportions. As for the validation of this extended model, the appearance simulations should be compared to complete BRDF measurements of the ochres samples under study.

The strong assumption of an optically thick powder layer we used to establish our model should also be questioned. Indeed the paint layers and especially the more erased ones are very thin (< 5 µm) and therefore implies a contribution of the substrate reflectance to the overall appearance. This is well visible on in situ measurements, by the comparison of the reflectance measured on a paint layer and on the bare rock substrate right next to it. An extension to the model of enhanced back scattering by isotropic scatterers proposed by Van der Mark [START_REF] Van Der | Light scattering in strongly scattering media : multiple scattering and weak localization[END_REF] may be a good starting point, that would need to be adapted for anisotropic scatterers. Nevertheless some issues might be encountered as for the validation of such a model. Thin films of dry powder with constant and controlled thicknesses will be necessary for the BRDF measurements and the precise modelization of light absorption but are difficult to manufactured.

Moreover, some attributes appearance of the paint layers may depend on the process that was used to apply the coloring matter onto the substrate. It is possible that the pigments were dispersed into a binder such as water to obtain a paste that is easier to spread on a rock wall. Nowadays this binder has completely disappeared, which enabled us to approximate the painting as a collection of randomly organized grains. Yet, one may observe that two layers made out of the same powder, a first one obtained after the drying of humidified powder and the second one directly using the dry powder, look quite different. Depending on their shape, the grains may have a preferential orientation they tend to adopt (on one face for platelet grains for example) leading to a more compact layer and a smoother surface. As a consequence a wet then dried powder is more specularly reflective than the originally dry powder. This effect should impact the specular contribution of our model, but might also influence back scattering. The tool that was used to spread the powder may also have an influence on the organization of the grains, and therefore on the appearance of the paint layer : a powder that is deposited using a finger would result in a thicker and denser paint layer than if it is projected on the substrate by blowing.

As a future work, I would like to implement our dry powder BRDF model in a physically based renderer so that photo-realistic renderings of the powders can be made. This would first allow the validation of the model by comparing the rendered images with photographs of the real samples. But it would also permit, by chosing the values of the variable parameters to either reproduce the appearance of other powder materials, or to create new ones with any chemical composition and morphology of grains.

Finally, I would like to investigate the deteriorations undergone by parietal paintings because of their interaction with the environment. By identifying the materials that are nowadays on parietal paintings, the previously mentioned extensions to our appearance model would allow to accurately reproduce its current appearance. However, because they have been painted a long time ago, they may have undergone multiple alterations either physical such as peeling off and erasure, or chemical such as encrustation or the appearance of organic constituents (lichens or animal contribution). With the study of such phenomena and the modelization of their impacts on the BRDF of a paint layer, we would be able to predict the appearance of the paintings at the time they were painted, and over the years.

(c) Values from Werner et al. [START_REF] Werner | Optical constants and inelastic electron-scattering data for 17 elemental metals[END_REF]. FIGURE 10 -Optical constants of gold computed using different theoretical models, and simulation of the transmittance of a 50 nm and a 100 nm thin gild layer on top of a glass substrate using our model [START_REF] Gerardin | Rendering transparent materials with a complex refractive index : semi-conductor and conductor thin layers[END_REF].

Experimental optical constants

The experimental values of the optical constant of gold are either measured on bulk crystals or on thin films. Both ellipsometry and Kramers-Kronig analysis have been studied in the literature, according to the Table 2.

Crystal or thick film Thin film Ellipsometry Kramers-Krönig Analysis Johnson [START_REF] Johnson | Optical constants of the noble metals[END_REF] x x Palik [START_REF] Palik | Handbook of optical constants of solids[END_REF] x x Gao [START_REF] Gao | Comparison of different dispersion models for single layer optical thin film index determination[END_REF] x x McPeak [START_REF] Mcpeak | Plasmonic films can easily be better : rules and recipes[END_REF] x x Babar [START_REF] Babar | Optical constants of cu, ag, and au revisited[END_REF] x x Yakubovsky [START_REF] Yakubovsky | Optical constants and structural properties of thin gold films[END_REF] x x Yakubovsky [START_REF] Yakubovsky | Ultrathin and ultrasmooth gold films on monolayer mos2[END_REF] x x Rosenblatt [START_REF] Rosenblatt | Nonmodal plasmonics : controlling the forced optical response of nanostructures[END_REF] x x The results already given in II.4.2.b are here completed with the optical constant values measured on very thin layers. For each set of values, the comparison between the measured transmittance of gold thin films and their simulation using our mode [START_REF] Gerardin | Rendering transparent materials with a complex refractive index : semi-conductor and conductor thin layers[END_REF] are also given in the following Figures 11 and12 The results obtained of crystal bulks are the same as the one introduced in Section II.4.2.b, represented in Figure 12.

-Optical constants of gold measured on a bulk of gold, and simulation of the transmittance of a 50 nm and a 100 nm thin gold layer on top of a glass substrate using our model [START_REF] Gerardin | Rendering transparent materials with a complex refractive index : semi-conductor and conductor thin layers[END_REF].

Analysis

As far as we know, the previously cited references have not been compared to each other : While the results from Yakubovsky et al. [START_REF] Yakubovsky | Ultrathin and ultrasmooth gold films on monolayer mos2[END_REF] and Rosenblatt et al. [START_REF] Rosenblatt | Nonmodal plasmonics : controlling the forced optical response of nanostructures[END_REF] are not confronted to any other existing results, the others are only compared to pretty old measurements from Johnson et al. [START_REF] Johnson | Optical constants of the noble metals[END_REF] or Palik [START_REF] Palik | Handbook of optical constants of solids[END_REF]. Yet, the latter is not reliable since it is the only one that shows a second peak at 650 nm, probably caused by some impurities in the measured bulk.

We have proceeded to such a confrontation in Section II.4.2.b, and observed many differences between all of these values. It leads us to the following conclusions : Among the theoretical models, only the Lorentz-Drude model is able to correctly predict the spectral position of the absorption bands. The experimental models are more reliable, but tends to underestimate the real transmittance. This underestimation is even more important as the layer from which the optical constant is measured is thin. For extremely thin layers used by Yakubovsky [START_REF] Yakubovsky | Ultrathin and ultrasmooth gold films on monolayer mos2[END_REF], not only gold but also the MoS 2 substrate is involved in the proposed results.

Most of the time, the optical constants of a material that are used in material science come from quite old measurements, and are supposed to be well defined. The more recent sets of values, supposedly obtained with more precise and efficient measurement devices, are sometimes confronted and validated against these older ones ( [START_REF] Johnson | Optical constants of the noble metals[END_REF] and [START_REF] Palik | Handbook of optical constants of solids[END_REF]), but not against each others. Yet, a multiplicity of different results are observed, with no clue on which one is better suited to faithfully characterize the material under study.

The reliable measurement of the optical constants of one material is still a major issue nowadays, especially for materials that have not been extensively studied.

X-ray powder diffraction (XRPD)

The principle of this method is described in Section I.4.1. It has been used for every synthesized samples, as well as the natural ones. The resulting diffractograms, phase identifications and Rietveld analyses are given below.

Synthesized samples

Except for the samples 5, 6 and HT for which a small amount of goethite is identified, the synthesized samples are pure hematite. According to the narrowness of the Bragg's peaks, HT and 3 are particularly well crystallized.

Natural samples

Among the two samples from the Pigmentothèque [START_REF] Salomon | Construction d'une Pigmentothèque : un outil pour comprendre l'approvisionnement en matériaux colorants durant la Préhistoire[END_REF] coming from the same seam, the denser one N Ls is only made out of pure hematite while the larger one N Lb also shows some quartz.

The sample N I is quite pure as well, even if it contains a minor unidentified phase.

The last natural sample N F is mainly composed of hematite and another phase identified as Cobalt Germanium Titanium oxide.

Rietveld refinements

The measured XRPD diagrams are fitted by optimizing over structural parameters listed in Table 3 Because the band D is partially defined in the infrared spectrum, we did not study the variations of its width.

FeO 6 octaedra distorsion

Pailhé et al. [START_REF] Pailhé | Impact of structural features on pigment properties of α-fe2o3 haematite[END_REF] observed a correlation between the distortion of the octahedra composing the hematite crystals, described by the x O and z Fe atomic positions, and the crystallite size. It also impacts the gap∆E between of the absorption bands CT1 and CT2, influencing the color of the powders. A similar study on our synthetic hematite powders is in agreement with the results of Pailhé et al. [START_REF] Pailhé | Impact of structural features on pigment properties of α-fe2o3 haematite[END_REF] regarding the crystallite size. However we did not find such a correlation with the relative positions of the bands CT1 and CT2.

• Coordinate on the green/red axis a* :

205

Hematite BRDF : Spectralon's model

The adapted Spectralon's model IV.3.1 is used to modelize the BRDF of the measured synthetic samples. This model involves 12 empirical parameters that are optimized in order to minimize the root mean square difference between the measurements and the simulations.

The optimization is performed at the wavelength λ = 700 nm by taking into account all the measurement geometries of incidence θ i = 0 • , 20 • and 40 • . For an incidence of 60

Sample 3 2

This powder is not monodisperse. We estimate the average radius of the large grains composing this sample at 83 nm, and the average radius of the smaller grains at 47 nm. Two simulations are performed by taking into account either the large or the small grain size.

• For grains with radius 47 nm :

Parameters Optimization on all geometries α 0.95210561 C 0.33879183 g 0.50000225 E RMS 0.7095070630244724 • For grains with radius 81 nm :

Parameters Optimization on all geometries α 0.95963218 C 0.33613119 g 0.50988416 E RMS 0.7095044399981142

Sample P 2

This sample shows large grains (r = 687 nm) that seem to be agglomerates of much smaller ones (r ≈ 10 nm). Two simulations are performed by taking into account either the grain size deduced from SEM observations or the average crystallite given by the XRPD analysis. • For grains with radius 687 nm :

Parameters Optimization on all geometries α 0.48213159 C 0.53921339 g 0.49601901 E RMS 0.4922019944519871

Sample C

This sample is not mono-disperse. It is composed of large grains (r ≈ 2222 nm) that are agglomerates of much smaller spherical grains (r ≈ 100 nm).

• For grains with radius 49 nm :

Parameters Optimization on all geometries α 0.772167757 C 0.281667485 g 0.0 E RMS 0.2965853749059329 • For grains with radius 2222 nm :

Parameters Optimization on all geometries α 0.70298246 C 0.30938841 g 0.05 E RMS 0.2965853750382204

Sample 1

This sample is mono-disperse. It is composed of grains that are mono-crystals with a radius we estimate at 12.5 nm. We proceed to two simulations using either the 5 or the 6-parameters model (ie using a fixed or an optimized radius).

Parameters

Optimization with fixed radius Optimization with variable radius r (nm) 12.5

Sample 1 2

This sample is mono-disperse. It is composed of grains that are mono-crystals with a radius we estimate at 12 nm. We proceed to two simulations using either the 5 or the 6-parameters model (ie using a fixed or an optimized radius).

Parameters

Optimization

Sample 5

This sample is mono-disperse. It is composed of grains that are mono-crystals with a radius we estimate at 25 nm. We proceed to two simulations using either the 5 or the 6-parameters model (ie using a fixed or an optimized radius).

Parameters

Optimization

FIGURE I. 2 -

 2 FIGURE I.2 -Surface scattering in transmission. On the right : the pot is made of a perfectly transparent material which allows to clearly see the nuts inside. On the left : the pot is not perfectly transparent. Because of light scattering the image of the nut seen through the pot is blurry.

FIGURE I. 3 -

 3 FIGURE I.3 -Optical index for several materials as a function of the wavelength in the visible spectral range. (a) The optical index of a dielectric material is real. (b,c) The optical index of a conductor material is complex. For a semi-conductor the imaginary part is null over part of the spectrum.

•

  The chroma, which describes the color as bright or dull. C * ab = a * 2 + b * 2 (I.6) • The saturation : in theory, this parameter is defined in the CIE L*u*v* color space as the chroma relative to the lightness : s * uv = C * uv L * . It can be defined by analogy in the L*a*b* color space : s * ab = C * ab L * .

FIGURE I. 4 -

 4 FIGURE I.4 -Color gamuts of the RGB and a CMYK color representation. All the reachable colors are contained in the corresponding polygon. The colors outside of the polygons can be perceived by a human eye but cannot be reproduced using these color representations. The CMYK gamut is obtained for Fogra39 characterization data set[3].

FIGURE I. 5 -

 5 FIGURE I.5 -Diagramm L * a * b * for L * =40. The axis a * represents the colors from green to red, and the axis b * represents the color from blue to yellow. The Chroma C * is defined as the position of a color from the centre, and the hue is the polar angle from axis a * .

  FIGURE I.7 -Light/matter interactions at the interface between two homogeneous materials. (a)(b) For specular materials light is reflected or transmitted towards a unique direction given by the Snell Descartes' laws. (c)(d) For diffusing materials light is reflected or transmitted in every directions of the corresponding hemisphere.

  θ o . According to its representation on the Figure I.8, f x = sin θ i + sin θ o cos ϕ and f y = sin θ o sin ϕ.

FIGURE I. 8 -

 8 FIGURE I.8 -Definition of the vector f.

FIGURE I. 10 -

 10 FIGURE I.10 -Definition of the scattering angle Θ according to the incident geometry θ i and the observation geometry (θ o , ϕ).

FIGURE I. 11 -

 11 FIGURE I.11 -Comparison of the different phase function for spherical particles of radius r = 25 nm, at λ 0 = 700 nm. Only the Mie scattering explicitly takes into account the scatterer size. At this scale Rayleigh theory is a good approximation. This is not the case for larger sizes (r ∼ λ 0 ).

  [START_REF] Florence | brocarts appliqués» dans la sculpture savoyarde. vers une caractérisation interdisciplinaire[END_REF] with p the scattering phase function, s the lambertian reflectance of a leaf, and ρ 0 the lambertian reflectance of the soil underneath the canopy. The LAI term stands for the Leaf Area Index defined as a function of the leaves density per unit volume N , the thickness z of the canopy and the area of a single leaf σ : LAI = N zσ.

FIGURE I. 12 -

 12 FIGURE I.12 -Scattering in a canopy : light is either scattered by a leaf (solid line) or reflected by the soil (dotted line).

  3.2), here we use the ellipsoids representation proposed by Strahler et Jupp [101]. The ellipsoid is centered at a height h above the ground, with b the half length of the major axis, and r the half length of the smaller axis (see Figure I.13).

FIGURE I. 13 -

 13 FIGURE I.13 -Canopy modelization. Trees of the canopy are represented as ellipsoids with dimensions b and r , centered on a stick at the height h above the ground.

FIGURE I. 15 -

 15 FIGURE I.15 -Photograph of a Spectralon material from Labsphere

  FIGURE I.16 -Volume scattering phenomena (a) at small angles, (b) at large angles.

FIGURE I. 17 -

 17 FIGURE I.17 -Surface scattering phenomena (a) in the specular direction, (b) in the backward direction.

FIGURE I. 18 -

 18 FIGURE I.18 -Example of a BRDF generated using the spectralon's model [68], for the incidence t het a i = 30 • , in the incident plane v ar phi = 0 • . The 12 parameters model is used with the parameters d 0 = 0.961,d 1 = -1.143e -5 ,d 2 = -3.3e -9 ,d 3 = 1.8e -5 ,d 4 = 8e -9 ,ss 0 = 1.4e -6 ,ss 1 = 0.095,s 0 = 0.011,s 1 = 9,s 2 = 3.3,b 0 = 0.034,b 1 = 7, optimized by Levesque et al.[START_REF] Lévesque | Measurement and modeling of the spectralon spectropolarimetric bidirectional reflectance distribution function[END_REF] to modelize the BRDF of the Spectralon material. An offset of 0.7 has been added to the single contributions of subsurface, specular and back scattering in order to better vizualize their individual contribution to the complete reflectance.

FIGURE I. 19 -

 19 FIGURE I.[START_REF] Cervelle | Détermination de l'ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. application à la pyrrhotite hexagonale[END_REF] -Point scatterer approximation. The scatterers are identical spheres of radius r . We consider that the scattering event occurs at the center point r n rather than at the surface of the scatterer.

  FIGURE I.20 -Multiple scattering involving two scatterers

FIGURE I. 23 -

 23 FIGURE I.23 -At a natural state, organic pigments come from animals and plants. (a) For example, the red pigment carmine is obtained from the cochineal. (b) The indigo pigment is named after the plant it comes from.

  FIGURE I.25 -Crystal structure of hematite. (a)Hematite belongs to the R 3c symmetry group and crystallized as described by Blake & al. [11]. (b) Details of an distorted octaedron composing the unit-cell. The chromophore ion Fe 3+ is surrounded by six O 2 -ions.

  FIGURE I.26 -Color in hematite. (a) Reflectance spectra of a sample synthetized using the method 1 (see Section I.3.2.c). Three main absorption bands are responsible for the red hue of hematite powder. (b) According to the description from Pailhé et al. [79], the bands CT1 and CT2 are caused by charge transfers between O 2 -and Fe 3+ ions while the band D is the consequence of intra-atomic electronic exchanges between the degenerated energy levels of the d-orbitals of Fe 3+ cations.

FIGURE I. 30 -

 30 FIGURE I.30 -Synthetic hematite samples. Every samples have been synthesized at Institut Néel, France.Only the sample C comes from a commercial manufacturer. Its production process is not well known. The photographs are calibrated using an X-rite 24 patches Classic Colorchecker.

FIGURE I. 31 -

 31 FIGURE I.31 -Bragg's law. The incoming waves are coherently scattered by the atoms of the crystal's lattice in discrete directions θ B . The optical path length difference between two interfering rays is represented in red.

  FIGURE I.32 -Rietveld refinements of a powder synthesized using the method {6} : This sample is mainly composed of hematite, and also shows a small amount of goethite.

  Multiple objectives are available to reach magnifications x5, x10, x20 and x50. The microscope has a built-in camera of 2558x1920 pixel resolution. With this microscope it is possible to assemble different images taken at different focal lengths to create a composite image. Although it loses the depth of view, it permits to observe the details of a surface that is not perfectly flat, which is particularly useful in the case of dry powders. A demonstration is shown in Figure I.34.

FIGURE I. 34 -

 34 FIGURE I.34 -Hematite powder synthetized following the method 1, seen under optical microscopy with a magnification x5. Four images at different focal lengths are combined to create a sharper image.

I. 4

 4 .4.a Principle : BRDF measurements consists in a set of reflected intensity measurements in various illumination/observation geometries covering the whole reflection hemisphere. For each measured spectrum the light is incoming with a direction Ω i = θ i on the sample, and is collected in a direction Ω o = (θ o , ϕ) where ϕ is the azimuth (see Figure I.35).

  FIGURE I.35 -BRDF measurements :The light is coming from a direction θ i and is collected in any direction (θ o ,ϕ). The angle between the incident and the observation direction is known as the scattering angle.

FIGURE II. 1 -

 1 FIGURE II.1 -Schematization of a paint layer of a parietal painting from the Hall of Bulls, Lascaux, France. The red painted areas of the painting are modelized as an homogeneous thin semi-transparent films of hematite on top of a substrate.

  FIGURE II.2 -Multiple bounces inside a uniform thin layer of thickness d . The interfaces are supposed to be perfectly smooth.

FIGURE II. 3 -

 3 FIGURE II.3 -Objects made with transparent glass and covered with a thin gold (Au, top) or copper (Cu, bottom) layer, with thickness increasing from left to right.

Figure

  Figure II.3 and II.[START_REF]Refractive index database[END_REF] show an object made of glass, covered with a layer of varying thickness. The change in color and appearance of the object is related to the layer thickness. The observed color is the sum of the color that is reflected at the surface of the object and the color that is transmitted through the object. The thicker the object is, the less the transmitted light contributes to the

FIGURE II. 5 -

 5 FIGURE II.5 -Interference effects with thin layers. Left : air-Cadmium sulfide-air, thickness : 250 µm. Middle : air-Hematite-glass, thickness 500 µm. Right : air-Copper Oxide (CuO)-Aluminium, thickness 500 µm.

  FIGURE II.6 -Validation of our model in reflection, for gold plates covered with thin Germanium films with various thicknesses (10 nm, 15 nm, 20 nm, 25 nm et 100 nm) [121]. (a,b) Comparison of measured and modeled reflectances. (c) Comparison of samples' photographs and renderings using Mitsuba.

II. 4 . 1 .

 41 FIGURE II.7 -Validation of our model in transmission, for glass plates covered with thin hematite films with various thicknesses[START_REF] Hamd | Mesoporous α-fe 2 o 3 thin films synthesized via the sol-gel process for light-driven water oxidation[END_REF]. The thickness of one layer is supposed to be equal to 40 nm.

  FIGURE II.8 -Thin gold film on glass, seen in reflection and transmission. On the left, the deposit is 50nm thick. On the right it is 100nm thick. The colors of the photographs have been calibrated using a X-rite 24 patches Classic Color-checker.

  Figure II.8. As expected the deposits are quite transparent with a gold-ish hue in reflection while the transmitted light has a green to dark green one (see Figure II.9).

  FIGURE II.9 -Thin gold film on glass, seen in reflection and in transmission. The observed color is obtained for the 50 nm thick coat.

  FIGURE II.11 -Optical constants of gold measured on gold thin films, and simulation of the transmittance of a 50nm and a 100nm thin gold layer on top of a glass substrate using our model[START_REF] Gerardin | Rendering transparent materials with a complex refractive index : semi-conductor and conductor thin layers[END_REF].

  (a) Values from Johnson et al. [57]. (b) Values from Palik [81]. (c) Values from McPeak et al. [72]. (d) Values from Babar et al. [6].

  Figure II.3.

FIGURE II. 13 -

 13 FIGURE II.13 -Comparison of the measured and simulated transmittance spectra. (a) for a 50 nm thin layer, (b) for a 100 nm thin layer.

  Figure III.1, and will be confirmed by the optical microscope observations on Figure III.4.

FIGURE III. 1 -

 1 FIGURE III.1 -Observation of the commercial sample C. Contrary to the other samples, this one does not show a uniform coloration. Althought it mainly appears to be dark purple, it also contains red grains that may influence the overall perceived color.

  FIGURE III.2 -Natural samples as bulks, powders observed using optical microscopy. The photographs are calibrated using an X-rite 24 patches Classic Colorchecker.

  FIGURE III.3 -Samples 6 and 1 3 seen under optical microscope with a magnification ×50. The sparkling appearance of the sample 1 3 is explained by the presence of large sapecular crystals. The sample 6 also shows larger grains, but their rough and diffuse aspect make us consider them as agglomerates of smaller grains instead of crystals. The same conclusion is drawn for the samples 1, 1 2 , 2, and 5.

FIGURE III. 4 -

 4 FIGURE III.4 -Visualization and observations of the synthetic samples. From left to right are repectively shown a calibrated photograph of the sample, its observation using optical microscopy, and using SEM.

  FIGURE III.5 -Phase identification of the synthesized samples. (a) Most of the samples are only made of pure hematite α-Fe 2 O 3 (1, 1 2 , 1 3 , 3, 4, AG, P 1 , P 2 , C). Only the diffractogram from the sample 3 is shown here. The duplication of each representative peak testifies of the well crystallisation of the sample. (b) The samples 5, 6 and HT contains a small amount of goethite FeO(OH). Only the diffractogramm of sample 5 is given here. (c) The asymmetry of the peaks measured for the sample AG is link to some anisotropy of the crystallites. The other diffractograms are given in the appendix.

FIGURE III. 7 -

 7 FIGURE III.7 -Reflectance spectra of the natural samples in the Near Infrared spectral range.

  FIGURE III.9 -Reflectance spectra of the synthetic samples in the visible spectral range (left), calibrated by a Spectralon®, and position of the corresponding colours in the CIE L*a*b* diagram for L*=30 (right, black dots). The samples from Pailhé et al.[79] are plotted in white.

  FIGURE III.12 -Reflectance simulations of the sample 1 using different scattering regime described by the Mie, Rayleigh and Henyey-Greenstein phase functions.

  .13).

  FIGURE III.13 -Natural hematite crystal from Musée de Minéralogie de l'école des MINES Paristech whose optical constants have been measured by ellipsometry. (a) Anisotropic sample used by Callet and by Couka & al. [26]. (b) Isotropic sample used by Callet.

III. 3 . 3 .

 33 a Study in the IR spectral domain : Our pellets are first studied in the infrared spectral range. The transmittance spectra are measured using a Bruker Vertex 70V FTIR spectrometer from 400 to 8000 cm -1 , which is equivalent in terms of wavelength of the spectral range from 1250 nm to 25µm. The results are presented on the Figure III.16.

  FIGURE III.16 -Transmittance spectra of the KBr pellets in the IR spectral domain.

  and Triaud [108]. Their results are indeed quite similar in the infrared specral range and very different in the visible spectral range (Figure III.18).

  FIGURE III.17 -(a) The measurements have been corrected by taking into account the reflections at the interfaces air/KBr or the pellets. (b) The abundance factor evaluated in the IR spectrum are applied in the visible spectral range. For purely absorptive pellets, the absorption spectra should superimposed.

  Figure III.19.

FIGURE III. 19 -

 19 FIGURE III.19 -Comparison between the absorption coefficient predicted from our method (red) computed for different effective thicknesses x with the results from Querry [85] and Triaud [108] (orange)

4 . 4 .

 44 The measurement of one sample's BRDF is made over 180 geometries : the scattered light is observed between -70 • and 70 • , for four incidences (0 • , 20 • , 40 • , 60 • ) and the azimuthal angles 0 • , 30 • , 60 • , 90 • , and 135 • (see Figure IV.2). The spectral range has been limited to 360 nm -860 nm with a step of 20 nm. Every point is measured with an integration time of 300 ms and averaged over 10 spectra. Considering the delays for mechanical movements, it takes around 18 hours to measure one BRDF.

FIGURE IV. 2 -

 2 FIGURE IV.2 -BRDF measurements with the SHADOWS instrument. The incident direction θ i , the observation direction θ o and the azimuth ϕ o are chosen to scan the whole scattering hemisphere. 180 geometries have been chosen to correctly sample the BRDF. This instrument is not able to measure back-scattering, represented by the turquoise dot for the incidence θ i = 0 • , in green for the incidence θ i = 20 • , in yellow for the incidence 40 • and in red for θ i = 60 • .

FIGURE IV. 3 -

 3 FIGURE IV.3 -Back-scattering measurement device. The beam of a white source is focused on the sample's surface at an incidence θ i using a 50 mm focal lens and a beam splitter. The reflected light is collected through a collimator in different obsevation directions θ o .

  FIGURE IV.4 -Correction of the shift between the back-scattering and the BRDF measurements at θ i = 20 • for the sample 5. The rescaling is done to superpose the points measured at θ i + 10 • .

FIGURE IV. 5 -

 5 FIGURE IV.5 -Spectra measured on the sample 1 2 for θ i = 0 • at different observation geometries θ o in the incident plane ϕ o = 0 • . The reflectance is smoothly varying with the geometry changes

FIGURE IV. 6 -

 6 FIGURE IV.6 -Complete BRDF of the sample 1 2 at the wavelength 700 nm for an incidence θ i = 40 • . The measured geometries are spotted with black dots, and the intermediate values are interpolated. The light is mostly reflected in the backward direction, and a very weak specular scattering peak may be observed as well. The spreading of the back scattering peak with the azimuth angle is due to the interpolation between the measured points.

FIGURE IV. 7 -

 7 FIGURE IV.7 -Complete BRDF measurements of sample 1 2 , for λ = 700 nm, with incidence θ i = 40 • , in the incident plane (ϕ o = 0 • ). The measurements (large dots, in grey for the SHADOWS' measurements and in orange for the rescaled back scattering measurements) have been linearly interpolated (small dots). Two peaks can be identified, a major one in the back-scattering direction and a weaker one in the specular direction.

FIGURE IV. 8 -

 8 FIGURE IV.8 -Specular and back scattering contributions of the samples 1 2 and 1 3 , for θ i = 20 • , ϕ = 0 • and λ = 700 nm. Contrary to any other sample, the powder 1 3 has a much more important specular contribution, which describes the sparkling effect of this particular sample. The back scattering on the other hand is of comparable intensity.

  2.3.e. Its decomposition into four physical components is of particular interest in the case of powders since it coincides with our observations on measured BRDF of such materials. The overall BRDF corresponds to volumetric scattering phenomena (Deep and Subsurface scattering illustrated on the Figure I.16). The specular and back scattering peaks identified on the measurements are supposed to be the result of light interaction with the surface of the material (Specular and Back scattering illustrated on the Figure I.17).

  FIGURE IV.9 -The light is incoming in the direction Ω i with an angle θ i , and outgoing in the direction Ω o described by the outgoing angle θ o and the azimuth angle ϕ o . In the incident plane ϕ o = 0 • . We note Ω i the direction with an angle -θ i , symmetric vector to Ω i .

FigureFIGURE IV. 10 -

 10 FIGURE IV.10 -Four steps optimization using the Spectralon model on the hematite sample 5. It is performed for an incidence of θ i = 40 • in the incident plane ϕ o = 0 • , at the wavelength 700 nm. (a) The optimization is first done only considering the deep scattering contribution. The resulting simulation of this contribution is retrieved from the measurement. The optimization is then performed for (b) back-scattering, (c) specular scattering and (d) subsurface scattering on the respective measurement residuals.

  FIGURE IV.11 -Comparison of the optimization results. (a) The optimization is done in four steps to estimate the magnitude of the variable parameters. Because of error's propagation, the sum of the four optimized contributions tends to overestimate the experiment BRDF. (b)The optimization is done in only one step accounting for the four contributions using the previously determined parameters magnitude as an input. The result better fits the experiment, especially for the specular contribution.

FIGURE IV. 13 -

 13 FIGURE IV.13 -Simulation of the BRDF of sample 5 using the Spectralon's model. (a)(b)(c) Incidence 0 • , (d)(e)(f) Incidence 20 • , (g)(h)(i) Incidence 40 • . The measured points of the BRDF are represented as black dots on the measurements figure. The complete BRDF is computed using cubic interpolation. The inadequate sampling around the backward direction results in a spreading of the back scattering peak with the azimuth angle.

FIGURE IV. 14 -

 14 FIGURE IV.14 -Modelization of a canopy and of a powder on a soil.(a) Trees are modelized as ellipsoids with dimensions r and b on a stick of height h above the soil. (b)The grains that are composing the powder are supposed to be spherical with radius r . (c) For a sufficiently thick layer, the shadows that are observed are only cause by the upper one-particle layer of the medium. The powder is then approximated as one oneparticle thick layer of spherical particles on an homogeneous substrate that is completely occulting any other soil substrate.

3 -

 3 Parameters involved in the RTLSR model to refine.

FIGURE IV. 15 -

 15 FIGURE IV.15 -Comparison of the RTLSR model optimization using different scattering phase function.

  FIGURE IV.16 -Comparison of the model optimizations on the hematite sample 5. (a) The optimization is done for the incidence θ i = 40 • in the incident plane. (b) The optimization is done over all the available geometries. Only the simulations in the incident plane are represented on the figure.

FIGURE IV. 17 -

 17 FIGURE IV.17 -Simulation of the BRDF of sample 5 using the adapted RTLSR model. (a)(b)(c) Incidence 0 • , (d)(e)(f ) incidence 20 • , (g)(h)(i) incidence 40 • . The measured points of the BRDF are represented as black dots on the measurements figures. The complete BRDF is computer using cubic interpolation. The inadequate sampling around the backward direction results in a spreading of the back scattering peak with the azimuth angle.

  FIGURE IV.18 -Mutliple scattering with two and n particles. The first scatterer that light encounters as it travels inside the medium is denoted as P i and involves a scattering event described by the angle Θ i . The last scattering event before light goes out of the medium is caused by the scatterer P o with an angle Θ o . In between these two scatterers, light is scattered with an angle Θ k at each interaction with a particle k.

  FIGURE IV.20 -Simulation of the BRDF of sample 1 in the incidence plane (ϕ o = 0) for incidences θ i = 0 • , θ i = 20 • , θ i = 40 • using our model. On the left (a)(c) our model is optimized with a fixed radius. On the right (b)(d) it is optimized also on the radius. The results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

  FIGURE IV.21 -Simulation of the BRDF of sample 5 in the incidence plane (ϕ o = 0 • ) for incidences θ i = 0 • , 20 • and 40 • using the refined parameters obtained with a fixed radius r = 25 nm. The back scattering peak is either overestimated or underestimated depending on the intial magnitude of the refinement parameters. The Results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

  • . Example of the complete BRDF simulations over the whole half-space are given on the Figure IV.23 and compared to the interpolated BRDF measurements.

FIGURE IV. 22 -

 22 FIGURE IV.22 -Simulation of the BRDF of sample 5 in the incidence plane (ϕ o = 0 • ) for incidences θ i = 0 • , θ i = 20 • , θ i = 40 • using the refined parameters obtained with a variable radius. The radius originally fixed at r = 25 nm is optimized to the value r = 14 nm. Results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

FIGURE IV. 23 -

 23 FIGURE IV.23 -Simulation of the BRDF of sample 5 using our model. (a)(b)(c) Incidence 0 • , (d)(e)(f ) incidence 20 • , (g)(h)(i) incidence 40 • . The measured points of the BRDF are represented as black dots on the measurements figures. The complete BRDF is computed using cubic interpolation. The inadequate sampling around the backward direction results in a spreading of the back scattering peak with the azimuth angle. The simulations are made choosing a finer sampling of the scattering hemisphere and completed using cubic interpolation.

FIGURE IV. 24 -

 24 FIGURE IV.24 -Simulations of the BRDF of sample 3 using (a) our original model (b) our adapted model. The large amount of small grains is responsible for the enhanced back-scattering phenomenon. The large grains are too far away from each other so that multiple scattering can occur. Only single scattering is induced by such grains. The optimization is done over all the measured geometries, only simulations in the incident plane (ϕ o = 0 • ) for incoming directions θ i = 0 • , 20 • and 40 • are represented here. Surface scattering is neglected in both simulations. Results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

FIGURE IV. 25 -

 25 FIGURE IV.25 -Simulation of the BRDF of sample C using (a) our model (b) our adapted model. The large amount of small grains is responsible for the enhanced back-scattering phenomenon. The large grains are too big for interference effects to occur. Only single scattering is induced by such grains. The optimization is done over all the measured geometries, only simulations in the incident plane (ϕ o = 0 • ) for incoming directions θ i = 0 • , 20 • and 40 • are represented here. Surface scattering is neglected in both simulations. Results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

  FIGURE IV.27 -Variations of the volume scattering's refined parameters against the effective grain size. Because there is a lack of points for intermediate grain sizes, it is not possible to conclude on any precise correlation.

FIGURE V. 5 -

 5 FIGURE V.5 -Ochres reflectance spectra in the infrared spectral range, compared to the reflectance of pure hematite powder from the synthetic sample 1.

FIGURE V. 6 -

 6 FIGURE V.6 -Back scattering measurements of the ochre samples for an incidence θ i = 20 • in the incident plane ϕ = 0 • . The measurements are compared to the results obtained on the pure synthetic hematite powder 1.

  Three measurements are made at the points represented on the Figure V.7 : a first one is performed on the thickest part of the paint dot, a second one on the thinnest part in the middle of the dot, and a last on on the bare rock substrate next to the paint dot. The measurements are made between 400 and 2500 nm, and the resulting spectra are given on the Figure V.8

FIGURE V. 7 -

 7 FIGURE V.7 -Measurement points of the Trou de la Féclaz archaeological site (Saint Jean d'Arvey, 73230 France). Three measurement are made on a thick paint layer (dark red), on a thin layer (light red) and on the rock substrate.

FIGURE V. 8 -

 8 FIGURE V.8 -Reflectance spectra from the measurements points on Figure V.7.

  FIGURE V.9 -Comparison between a dry red ochre powder and a wet then dried red ochre powder spread on paper. (a) Photograph of the two samples. On the left the dry ochre is applied on the paper. On the right, the powder is applied as wet on the paper. We wait for this sample to be completely dried before proceeding to the measurements. (b) Reflectance measurement of red ochre layers spread on white paper, in the geometry 0 • /30 • . The powder is either dry or mixed into water before it is spread with a finger. We perform the measurement once the wet layer has completely dried. The reflectance of the dry powder layer is very similar to the one measured on a thick layer, showing the good covering power of the coloring matter. The wet and dried sample shows a darker look responsible that results in a weaker reflectance.

  .

  (c) Values from Yakubovsky et al.[START_REF] Yakubovsky | Ultrathin and ultrasmooth gold films on monolayer mos2[END_REF] measured on gold thin films with thicknesses (from top to bottom) 4 nm, 6 nm and 9 nm. The films are deposited on a MoS 2 substrate, probably responsible for the numerous absorption bands that are more and more noticeable as the thickness decreases.(d) Values from Rosenblatt et al.[START_REF] Rosenblatt | Nonmodal plasmonics : controlling the forced optical response of nanostructures[END_REF] measured on gold thin films with thicknesses (from top to bottom) 11 nm, 21 nm and 44 nm.

FIGURE 11 -

 11 FIGURE 11 -Optical constants of gold measured on gold thin films, and simulation of the transmittance of a 50 nm and a 100 nm thin gold layer on top of a glass substrate using our model[START_REF] Gerardin | Rendering transparent materials with a complex refractive index : semi-conductor and conductor thin layers[END_REF].

  (a) Values from Johnson et al. [57]. (b) Values from Palik [81]. (c) Values from McPeak et al. [72]. (d) Values from Babar et al. [6].

FIGURE 13 -Sample 1 FIGURE 14 -

 13114 FIGURE 13 -Sample 1

FIGURE 15

 15 FIGURE 15 -Sample

FIGURE 18 -Sample 5 FIGURE 19 -Sample 6 FIGURE 20

 18519620 FIGURE 18 -Sample 5

FIGURE 24 -

 24 FIGURE 24 -Sample N Ls

FIGURE 25 -FIGURE 27 -

 2527 FIGURE 25 -Sample N Lb

(a) Sample 1 ,FIGURE 30 -

 130 FIGURE 30 -Mesurements for θ i = 40 •

FIGURE 57 -

 57 FIGURE 57 -Position of the CT2 band against the grains morphology

FIGURE 58 -FIGURE 59 -

 5859 FIGURE 58 -Illustration of the correlation between the particle diameter atomic positions relative to the octahedral site distortion

FIGURE 61 -•

 61 FIGURE 61 -Variation of the coordinate a * against the grains morphology. The gray dotted line represents a critical grain size of 80 nm in diameter.

FIGURE 63 -

 63 FIGURE 63 -Variation of Chroma C * in the CIE L * a * b * color space against the grains morphology. The gray dotted line represents a critical grain size of 80 nm in diameter.

FIGURE 64 -

 64 FIGURE 64 -Variation of Hue h * in the CIE L * a * b * color space against the grains morphology. The gray dotted line represents a critical grain size of 80 nm in diameter.

FIGURE 65 -

 65 FIGURE 65 -Variation of the saturation s * in the CIE L * u * v * color space against the grains morphology. The gray dotted line represents a critical grain size of 80 nm in diameter.

FIGURE 66 -

 66 FIGURE 66 -Simulated BRDF of the sample 1 in the incident plane (ϕ = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 67 -

 67 FIGURE 67 -Comparison between the BRDF measurements and simulations of the sample 1 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 68 -

 68 FIGURE 68 -Simulated BRDF of the sample 1 2 in the incident plane (ϕ = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 69 -

 69 FIGURE 69 -Comparison between the BRDF measurements and simulations of the sample 1 2 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 70 -

 70 FIGURE 70 -Simulated BRDF of the sample 3 in the incident plane (ϕ = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 71 -

 71 FIGURE 71 -Comparison between the BRDF measurements and simulations of the sample 3 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 72 -

 72 FIGURE 72 -Simulated BRDF of the sample 5 in the incident plane (ϕ = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 73 -

 73 FIGURE 73 -Comparison between the BRDF measurements and simulations of the sample 5 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 74 -

 74 FIGURE 74 -Simulated BRDF of the sample 6 in the incident plane (ϕ = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 76 -

 76 FIGURE 76 -Simulated BRDF of the sample P 2 in the incident plane (ϕ = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 77 -TABLE 23 -

 7723 FIGURE 77 -Comparison between the BRDF measurements and simulations of the sample P 2 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 78 -

 78 FIGURE 78 -Simulated BRDF of the sample C in the incident plane (ϕ = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 79 -

 79 FIGURE 79 -Comparison between the BRDF measurements and simulations of the sample C for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 80 -

 80 FIGURE 80 -Simulated BRDF of the sample 1 in the incident plane (ϕ o = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 81 -

 81 FIGURE 81 -Comparison between the BRDF measurements and simulations of the sample 1 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 82 -

 82 FIGURE 82 -Simulated BRDF of the sample 1 2 in the incident plane (ϕ o = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 83 -

 83 FIGURE 83 -Comparison between the BRDF measurements and simulations of the sample 1 2 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 84 -

 84 FIGURE 84 -Simulated BRDF of the sample 3 in the incident plane (ϕ o = 0 • ). The verage radius is fixed at 47 nm. The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 85 -

 85 FIGURE 85 -Comparison between the BRDF measurements and simulations of the sample 3 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference. The average radius is fixed at 47 nm.

FIGURE 86 -

 86 FIGURE 86 -Simulated BRDF of the sample 3 in the incident plane (ϕ o = 0 • ). The average radius is fixed at 81 nm. The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 87 -

 87 FIGURE 87 -Comparison between the BRDF measurements and simulations of the sample 3 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference. The average radius is fixed at 81 nm.

FIGURE 88 -

 88 FIGURE 88 -Simulated BRDF of the sample 5 in the incident plane (ϕ o = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 89 -

 89 FIGURE 89 -Comparison between the BRDF measurements and simulations of the sample 5 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 90 -

 90 FIGURE 90 -Simulated BRDF of sample 6 in the incident plane (ϕ o = 0 • ). The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 91 -

 91 FIGURE 91 -Comparison between the BRDF measurements and simulations of sample 6 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 92 -

 92 FIGURE 92 -Simulated BRDF of the sample P 2 in the incident plane (ϕ o = 0 • ). The average radius is fixed at 18.3 nm. The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 94 -

 94 FIGURE 94 -Simulated BRDF of the sample P 2 in the incident plane (ϕ o = 0 • ). The average radius is fixed at 687 nm. The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 93 -

 93 FIGURE 93 -Comparison between the BRDF measurements and simulations of the sample P 2 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference. The average radius is fixed at 18.3 nm.

FIGURE 95 -

 95 FIGURE 95 -Comparison between the BRDF measurements and simulations of the sample P 2 for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference. The average radius is fixed at 687 nm.

FIGURE 96 -

 96 FIGURE 96 -Simulated BRDF of the sample C in the incident plane (ϕ o = 0 • ). The average radius is fixed at 49 nm. The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 98 -

 98 FIGURE 98 -Simulated BRDF of the sample C in the incident plane (ϕ o = 0 • ). The average radius is fixed at 2222 nm. The result for incidence θ i = 60 • is predicted by the model using the parameters optimized for incidence θ i = 0 • , 20 • and 40 • .

FIGURE 97 -

 97 FIGURE 97 -Comparison between the BRDF measurements and simulations of the sample C for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference. The average radius is fixed at 49 nm.

FIGURE 99 -

 99 FIGURE 99 -Comparison between the BRDF measurements and simulations of the sample C for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference. The average radius is fixed at 2222 nm.

FIGURE 100 -

 100 FIGURE 100 -Simulation of the BRDF of the sample 1 in the incidence plane (ϕ o = 0 • ) for incidences θ i = 0 • , θ i = 20 • , θ i = 40 • using our model. On the left (a)(c) our model is optimized with a fixed radius. On the right (b)(d) it is optimized on the radius. The results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

FIGURE 101 -

 101 FIGURE 101 -Comparison between the BRDF measurements and simulations on the sample 1 using our model with an optimized radius r = 12.5 nm, for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

FIGURE 103 -

 103 FIGURE 103 -Simulation of the BRDF of the sample 1 2 in the incidence plane (ϕ o = 0 • ) for incidences θ i = 0 • , θ i = 20 • , θ i = 40 • using our model. On the left (a)(c) our model is optimized with a fixed radius. On the right (b)(d) it is optimized on the radius. The results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

  FIGURE 108 -Simulation of the BRDF of the sample 3 in the incidence plane (ϕ o = 0 • ) for incidences θ i = 0 • , θ i = 20 • , θ i = 40 • using our model adapted for two scatterer sizes. Two radii are optimized. The results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

FIGURE 110 -

 110 FIGURE 110 -Simulation of the BRDF of the sample 5 in the incidence plane (ϕ o = 0 • ) for incidences θ i = 0 • , θ i = 20 • , θ i = 40 • using the refined parameters obtained with a variable radius. The radius originally fixed at r = 25 nm is optimized to the value r = 14.1513098 nm. Results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

FIGURE 118 -

 118 FIGURE 118 -Simulation of the BRDF of the sample P 2 in the incidence plane (ϕ o = 0 • ) for incidences θ i = 0 • , θ i = 20 • , θ i = 40 • using our model. Two radii are optimized. The results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

FIGURE 122 -

 122 FIGURE 122 -Simulation of the BRDF of the sample C in the incidence plane (ϕ o = 0 • ) for incidences θ i = 0 • , θ i = 20 • , θ i = 40 • using our model adapted for two scatterer sizes. Two radii are optimized. The results are also given in polar representation, along with the prediction of the model for an incidence of 60 • .

r o )dr i dr o

						(I.70)
	with
	A(r i , r o ) =	ρpκ 4π	2 exp(κ|r i -r o |) |r i -r o | 2	+	4π ρpκ	exp(κ|r

i -r n |) |r i -r n | 2 A(r n , r o )dr n

(I.71)

  • C. It actually reproduces a quite common and natural process of hematite formation. The obtained sample is named HT. 40ml NaOH (8M) + 40ml FeCl 3 • 6 H 2 O (2M) 2h,180 • C, autoclave Fe 2 O 3 {9}

  This computation is valid for each component of polarized light, s and p, taken individually and produces a complex number. To get the real reflectance R and transmittance T, we need to sum the square modulus of the polarized coefficients (see Equation I.17) : R = (|r s | 2 + |r p | 2 )/2.To get the real coefficient R, we need the modulus and phase of r 12 and r 23 , as well as the real and imaginary part of the phase difference φ : This expression is numerically unstable for large values of the imaginary part of the phase difference φ I , due to the exponential term. We divide numerator and denominator by e φ I to get a stable version : -2φ I + 2ρ 12 ρ 23 e -φ I cos(ϕ 23 -ϕ 12 + φ R ) 1 + ρ 2 12 ρ 2 23 e -2φ I + 2ρ 12 ρ 23 e -φ I cos(ϕ 23 + ϕ 12 + φ R ) (II.16) This formula is valid for both polarizations, r s and r p , substituting the respective coefficients, r 12s and r 23s or r 12p and r 23p as needed. We then get the real reflectance R with : R = (|r s | 2 + |r p | 2 )/2. Similarly, to evaluate the transmittance T we write the modulus and phase of the transmission coefficients t 12 and t 23 :

	|r | 2 =	ρ 2 12 + ρ 2 23 e t 12 = τ 12 e iψ 12	(II.17)
			t 23 = τ 23 e iψ 23	(II.18)
	and get the complex transmission coefficient across the layer :
	|t | 2 =	1 + ρ 2 12 ρ 2 23 e -2φ	τ 2 12 τ 2 23 e -φ I
			r 23 e iφ 1 + r 12 r 23 e iφ	(II.10)
	Similarly for transmission, we have :
		t =	2 t 12 t 23 e i φ 1 + r 12 r 23 e iφ	(II.11)
		r 12 = ρ 12 e iϕ 12	(II.12)
		r 23 = ρ 23 e iϕ 23	(II.13)

φ = φ R + iφ I (II.14) With these, we get |r | 2 = r r : |r | 2 = ρ 2 12 e φ I + ρ 2 23 e -φ I + 2ρ 12 ρ 23 cos(ϕ 23 -ϕ 12 + φ R ) e φ I + ρ 2 12 ρ 2 23 e -φ I + 2ρ 12 ρ 23 cos(ϕ 23 + ϕ 12 + φ R ) (II.15) I + 2ρ 12 ρ 23 e -φ I cos(ϕ 12 + ϕ 23 + φ R ) (II.19)

  .1.

		Grains'	Average	Average	Goethite
		shape	grain size	crystallite	mass
			(nm)	size (nm) ±	proportion
				anisotropy	(%mass)
	1 3	Stick	244 ± 66.2	206 ± 43.2	--	[36.2, 26.1, 16.7]
	HT	Platelet	320 ± 22.8	399 ± 242	0.91 ± 0.09	[29.3, 24.8, 10.6]
	P 2	Ellipsoid	1374 ± 128	18.3 ± 0.40	--	[32.7, 21.8, 7.65]
	P 1	Ellipsoid	1914 ± 698	16.6 ± 2.30	--	[32.1, 13.7, 3.10]
	C	Sphere or	98.0 ± 25.0	526 ± 53.3	--	[32.4, 30.3, 17.8]
		stick	4444 ± 2108

Color [L * , a * , b * ] 4 Sphere 16.0 ± 2.88 10.9 ± 0.40 --[27.1, 28.2, 22.3] 2 Diamond 20.0 ± 3.28 20.1 ± 3.00 --[27.2, 26.2, 19.0] 1 2 Diamond 24.0 ± 3.22 25.4 ± 4.90 --[25.0, 23.2, 15.9] 1 Diamond 25.0 ± 5.90 24.5 ± 4.

80 --[27.5, 27.6, 21.6] 6 Diamond 44.0 ± 7.40 61.2 ± 10.2 7.17 ± 0.18 [29.6, 23.0, 19.7] 5 Diamond 50.0 ± 8.20 46.3 ± 8.80 8.98 ± 0.16 [31.4, 26.6, 23.3] AG Flat needle 76.2 ± 23.2 7.30 ± 3.60 --[31.4, 29.6, 31.1] 3 Bi-pyramid 94.2 ± 10.5 166 ± 67.0 197 ± 50.8 --[32.4, 30.3, 17.8]

TABLE III .

 III

1 -Features of the hematite crystals (or agglomerates of crystals) and colours of the powders. The samples have been organised by increasing grain size. Grain shapes and sizes are determined through SEM images. Average crystallite sizes and mass proportions of goethite are optimised through Rietveld refinements. Colours are calculated from the reflectance spectra in the CIE L*a*b* colour space

  .1 have been refined in order to modelize the BRDF of the 7 synthetic samples we fully characterized (samples 1, 1 2 , 3, 5, 6, P 2 and C).

	Contribution	Number of parameters Parameters' names
	Deep scattering	5	d 0 , d 1 , d 2 , d 3 , d 4
	Subsurface scattering	2	ss 0 , ss 1
	Back scattering	2	b 0 , b 1
	Specular scattering	3	s 0 , s 1 , s 2

TABLE IV .

 IV 1 -Parameters involved in the Spectralon's model to refine.

TABLE IV .

 IV 2 -Optimized parameters obtained for the Spectralon's model applied on the sample 5. Both sets of parameters resulting from the four steps optimization process and from the full model optimization are given here.

  .4.

	Parameters Optimization on 1 geometry Optimization on all geometries
	α	0.6485296	0.266077
	C	0.91399483	0.29440486
	g	0.65400398	0.79620892

TABLE IV .

 IV 4 -Optimized parameters obtained for the RTLSR model applied on the hematite sample 5.

  .5 to accurately simulate the BRDF of any of our hematite powder.

	Contribution	Parameter' description	Parameters' names
	Back scattering	Intensity of back scattering	A v
		Scattering anisotropy	g
		Extinction rate	κ
	Specular	Intensity of specular scattering	A s
	scattering
		Correlation length	l c

TABLE IV .

 IV 5 -Parameters involved in my model to refine.

TABLE IV .

 IV 6 -Optimized parameters obtained for our model applied on the hematite sample 5. When using a fixed radius deduced from MEB images the model tends to either overestimates or underestimates the back scattering contribution. This is solved by introducing the effective radius optimized along with the other five parameters.

TABLE 2 -

 2 Measurements of the optical constants of gold.

  .

	a (Å) c (Å) zFe xO Occ(Fe) (%) Preferential direction Goethite (%m)			0.89134 -5.033746 13.756426 0.35471 0.31129 0.91491	0.96142 -5.033013 13.754228 0.35483 0.31372 0.91533	0.93361 -5.035601 13.780084 0.35495 0.32216 0.93855	0.94328 -5.035864 13.763653 0.35442 0.30768 0.93441	0.87468 -5.020934 13.716597 0.35514 0.30671 1.000(fixed)	0.73267 -5.022277 13.730134 0.35385 0.31833 0.9111	0.77925 8.98(0.16) 5.019995 13.720441 0.35453 0.31369 0.90219	0.87792 7.17( 0.18) 5.020706 13.716195 0.35473 0.31065 0.93018	--5.019788 13.716421 0.35335 0.33055 0.81195	0.70051 0.91( 0.09) 5.022289 13.720375 0.35510 0.31068 0.9375	0.90825 -5.022263 13.740641 0.35420 0.31383 0.9213	0.93619 -5.022190 13.741387 0.35455 0.30813 0.92439	--5.036474 13.753274 0.35544 0.31100 0.96663	5.038000 13.771999 0.35530 0.30590 1.000
	Mean crystallite	diameter ± anisotropy	(nm)	24.332±4.783	25.371±4.928	205.637±43.167	20.065±3.008	197.118±50.797	10.855±0.408	46.325±8.765	61.168±10.182	7.327±3.639	399.427±242.430	16.625±2.343	18.302±0.364	526.281±53.302
	χ 2			1.82	2.07	3.78	1.20	2.12	1.85	3.15	2.36	1.89	3.96	4.21	2.14	2.43
				1	1 2	1 3	2	3	4	5	6	AG	HT	P 1	P 2	C	Référence

TABLE 3 -

 3 Rietveld refinements of synthetized hematite powders 167

  • , the BRDF is predicted by using the refined parameters.

	Sample 1
	Parameters	Optimization
	Deep scattering
	d 0	0.180316549
	d 1	-1.00172059 × 10 -7
	d 2	-5.25439117 × 10 -10
	d 3	-0.400545846 × 10 -7
	d 4	0.126732726 × 10 -10
	Back scattering
	b 0	0.125461664
	b 1	9.04140975
	Subsurface scattering
	ss 0	6.14460340 × 10 2
	ss 1	0.0282933573
	Specular scattering
	s 0	7.35644650 × 10 -14
	s 1	11.4606515
	s 2	4.91812611
	E RMS	0.05155065297503135

TABLE 17 -

 17 Optimized parameters obtained for the full Spectralon's model applied on the sample 1.

TABLE 18 -

 18 Optimized parameters obtained for the full Spectralon's model applied on the sample 1 2 .

TABLE 19 -

 19 Optimized parameters obtained for the full Spectralon's model applied on the sample 3.

TABLE 20 -

 20 Optimized parameters obtained for the full Spectralon's model applied on the sample 5.

TABLE 21 -

 21 Optimized parameters obtained for the full Spectralon's model applied on the sample 6.

TABLE 22 -

 22 Optimized parameters obtained for the full Spectralon's model applied on the sample P 2 .

  This sample is mono-disperse. It is composed of grains that are monocrystals with a radius we estimate at 12 nm.

	Sample 1 2
	Parameters Optimization on all geometries
	α	0.64602458
	C	0.2171363
	g	0.8
	E RMS	0.1694157516439828

TABLE 26 -

 26 Optimized parameters obtained for the RTLSR model applied on the hematite sample 1 2 .

TABLE 27 -

 27 Optimized parameters obtained for the RTLSR model applied on the hematite sample 3. The average radius is fixed at 47 nm.

TABLE 28 -

 28 Optimized parameters obtained for the RTLSR model applied on the hematite sample 5. The average radius is fixed at 81 nm.

  This sample is mono-disperse. It is composed of grains that are monocrystals with a radius we estimate at 25 nm.

	Sample 5 2
	Parameters Optimization on all geometries
	α	0.733923
	C	0.29440486
	g	0.79620892
	E RMS	0.2923067405614302

TABLE 29 -

 29 Optimized parameters obtained for the RTLSR model applied on the hematite sample 5.

  This sample is mono-disperse. It is composed of grains that are monocrystals with a radius we estimate at 22 nm.

	Sample 6
	Parameters Optimization on all geometries
	α	0.91267744
	C	0.19359928
	g	0.8
	E RMS	0.2475899381456139

TABLE 30 -

 30 Optimized parameters obtained for the RTLSR model applied on the hematite sample 6.

• For grains with radius 18.3 nm :

	Parameters Optimization on all geometries
	α	0.40418953
	C	0.64140556
	g	0.67498495
	E RMS	0.4931504041723403

TABLE 31 -

 31 Optimized parameters obtained for the RTLSR model applied on the hematite sample P 2 . The average radius is fixed at 18.3 nm.

TABLE 32 -

 32 Optimized parameters obtained for the RTLSR model applied on the hematite sample P 2 . The average raiuds is fixed at 687 nm.

TABLE 33 -

 33 Optimized parameters obtained for the RTLSR model applied on the hematite sample C. The average radius is fixed at 49 nm.

TABLE 34 -

 34 Optimized parameters obtained for the RTLSR model applied on the hematite sample C. The average radius is fixed at 2222 nm.

TABLE 37 -

 37 Optimized parameters obtained from our model applied on the hematite sample 1, with either a fixed or an optimized radius.

TABLE 38 -

 38 Optimized parameters obtained for our model applied on the hematite sample 1 2 , with either a fixed or an optimized radius.

TABLE 40 -

 40 Optimized parameters obtained for our model adapted for two scatterer sizes applied on the hematite sample 3.

TABLE 41 -

 41 Optimized parameters obtained for our model applied on the hematite sample 5, with either a fixed or an optimized radius.

TABLE 44 -

 44 Optimized parameters obtained for our model adapted for two scatterer sizes applied on the hematite sample P 2 .

TABLE 46 -

 46 Optimized parameters obtained for our model adapted for two scatterer sizes applied on the hematite sample C.

(a) Values from Rakic et al.[86] computed using the Brendel-Bormann model.

• The number of samples : From the five samples that were studied in the reference, only four are used to draw the trend. The hydrothermal sample P HT is omitted and would actually not fit in. In our work, no less than thirteen samples are at stake.

(a) Measurement (b) Simulation (c) Error (d) Measurement (e) Simulation (f) Error (g) Measurement (h) Simulation (i) Error FIGURE 75 -Comparison between the BRDF of the sample 6 measurements and simulations for incidences θ i = 0 • , 20 • and 40 • . The error is defined as their difference.

Remerciements

BRDF measurements

Complete BRDF

The measurement of the BRDF requires a certain amount of powder. Only the synthetic samples 1, 1 2 , 3, 5, 6, P 2 and C are in sufficient quantity to be measured.

The measurements are performed from 0.400 nm to 1000 nm, in the following geometries :

• , 45 • , 60 • , 90 • Results obtained for incidences θ i = 0 • , θ i = 20 • and θ i = 40 • are respectively given on the Figures 28, 29 and 30 169 (a) Sample 1,

FIGURE 29 -Mesurements for θ i = 20 • Amplitude of the bands • Band CT1 (around 550 nm) : FIGURE 53 -Position of the CT1 band against the grains morphology

Protocol Preparation of a pellet with concentration x% :

• 300mg of KBr dehydrated powder is milled and mixed using a planetary grinder for 30 minutes at 30Hz.

• The grind is mixed with a small amount of pure hematite powder (3x<1mg) depending on desired concentration.

• The ensemble KBr+Fe 2 O 3 is milled using a planetary grinder for 30 minutes at 30Hz.

• The grind is poured into a metal mould previously heated at 60 • C by a heating tape. In order to ease the demoulding of the pellet, a paper ring is placed in the mould before the grinding matter.

• The mould is placed under a press at 10bars for 30 minutes. The vacuum is created all along the press process.

• The pellet is demoulded and saved in a dry environment.

Numerous pellets are produce using this protocol, whose photographs is given on Figure III. [START_REF] Brendel | An infrared dielectric function model for amorphous solids[END_REF]. The details regarding their concentration in hematite are listed below. We chose concentration that are high enough so the absorption bands are easily observable, but small enough to avoid any scattering phenomena between the grains dispersed in the pellet.

Pellet n°0 : 0.3% (1mg) of C powder Pellet n°1 : 0.26% (0.79mg) of 1 2 powder Pellet n°2 : 0.13% (0.38mg) of 1 2 powder Pellet n°3 : 0.037% (0.11mg) of 1 2 powder Pellet n°4 & 5 : 0.0011% (0.0033mg) of 1 2 powder. Such a small amount of powder cannot be weighed accurately with the available scale. A first grind is made using 1mg of hematite powder. 1mg of it is then used along with 300mg of KBr to create the final pellet. Pellet n°6 : 0.013% (0.04mg) of 1 2 powder. Such a small amount of powder cannot be weighed accurately. A first grind is made using 1mg of hematite powder. 12mg of it is then used along with 300mg of KBr to create the final pellet. Pellet n°7 : 0.0037% (0.011mg) of 1 2 powder. Such a small amount of powder cannot be weighed accurately. A first grind is made using 1mg of hematite powder. 3.4mg of it is then used along with 300mg of KBr to create the final pellet.

.15 -KBr pellets with different hematite α-Fe 2 O 3 concentrations. From left to right, the pellets are theoretically composed of 0.0011%, 0.0011%, 0.0037% and 0.013% of pure hematite powder from synthesis 1 2 .

Chapter V Preliminary results on ochres

The modelization of an optically thick pure hematite powder layer I proposed in Chapter IV is obviously too simplifed enough to describe an actual paint layer as the one observed on parietal paintings. FIGURE V.1 -Representation of ochre layer observed in parietal paintings from the Hall of Bulls, Lascaux, France. The powder responsible for the colors on the painting is mainly made out of colorless or white minerals and clays, and is colored by a small amount of a chromophore, either hematite α-Fe 2 O 3 for the red hues or goethite FeO(OH) for the yellow ones. The layer may be quite thin so that the substrate on which it is spread out may contribute to the overall perceived appearance.

V.1.3 Reflectance measurement

I measured the reflectance spectrum of each ochre sample using the SHADOWS instrument in the geometry 0 • /30 • . The measurements are performed in the visible and in the Near infrared spectral range (from 400 to 4600 nm). The results are compared with the reflectance of the pure hematite synthetic sample 1 measured in the same conditions. In the visible spectral range (Figure V.4) it is quite easy to distinguish goethite from hematite, because it shows less saturated absorption bands at small wavelengths. As expected the spectrum of the red samples shows the same absorption bands as pure hematite, while the yellow samples show absorption bands that are characteristics to goethite. Some differences are also observed between the red ochres and a pure sample of hematite. The samples containing only a small mass proportion of hematite are mainly composed of white powders (quartz and kaolinite) and necessarily appear to be clearer which explain the amplitude differences of the absorption bands. The impurity also seems to affect the width of the bands. However, we can note that the sample H 3200800 which contains more hematite but about the same amount of white minerals than H 11575 is clearer than the later. The size of the grains may be responsible for this effect.

In the Near-infrared spectral range (Figure V.5), goethite is mainly identified by the more sa-

Appendices

Optical constants of gold

The optical constant of a material is very important to understand its appearance and other physical properties. It is a complex parameters whose real part n refers to the refractive index of the material and whose imaginary part k, also called extinction coefficient describes its absorption power.

Its measurement is difficult and still an issue nowadays. Various methods have been developed in that extent. In the following we give a non-exhaustive list of optical constant values referenced in the case of gold in the visible spectrum, that we used in Section II.4.2.b. We will distinguish between theoretical values computed from models and experimental values measured using different techniques.

Theoretical optical constants

Three theoretical models can be used to predict the optical constant of a material. The Lorentz-Drude and the Brendel-Bordmann models have been used by Rakic et al. [86] in the case of gold, while Werner et al. [START_REF] Werner | Optical constants and inelastic electron-scattering data for 17 elemental metals[END_REF] were interested in the DFT model.

The results from these methods, already given in Section II.4.2.b, are represented on the Figure 10. For each set of values, the transmittance of a 50 nm and a 100 nm thin film of gold on a glass substrate is simulated using our model [START_REF] Gerardin | Rendering transparent materials with a complex refractive index : semi-conductor and conductor thin layers[END_REF] and compared to measurements.

Back scattering measurements

We do not possess large enough amount of some samples to fully measure their BRDF. Only the back scattering contribution has been measured using the device developed at Institut Néel.

The back scattering of all the synthetic samples is measured. The ochres from Okhra et Kremer we used in this work are also measured.

The measurements are performed from 400 nm to 1000 nm, in the following geometries :

For the incidence θ i = 40 • , our device could not reach observation angles smaller than -42 • . The back scattering are less well defined in these geometries.

Synthetic hematite powders

Color of Hematite

Gaussian fits

The reflectance of every synthetic sample measured in the incidence/observation geometry (0 • /30 • ) is fitted using a sum of five Gaussians. Each absorption band, theoretically observed around 310 nm, 430 nm, 550 nm (band CT1), 650 nm (band CT2) and 850 nm (band D), is modelized using a Gaussian function such as :

with A its amplitude, λ 0 its mean value and σ the half width. The Root Mean Square (RMS) error between the measurement and the fit is also given.

Sample 1

Error RMS : E RMS = 4.9 × 10 -6

Absorption bands and morphology

We study the variation of the absorption bands' characteristics (amplitude, mean value and width) along with the morphology parameters of the synthetic samples (grain size from SEM images, and average crystallites size deduced from Rietveld's refinements).

We only take into account the bands that have an impact on the color of the samples, that is to say in the visible spectral range.

Position of the bands

In order to obtain comparable results with the study from Pailhé [START_REF] Pailhé | Correlation between structural features and vis-nir spectra of α-fe2o3 hematite and afe2o4 spinel oxides (a= mg, zn)[END_REF][START_REF] Pailhé | Impact of structural features on pigment properties of α-fe2o3 haematite[END_REF] about the spectral positions of the absorption bands around 550 nm (CT1), 650 nm (CT2) and 850 nm (D), they are defined as the extremum of the derivated reflectance function.

• Band CT1 (around 550 nm) :

Color parameters vs morphology

The reflectance spectra are converted into the CIE L*a*b* color space introduced in section I.1.4. The variation of the color parameters such as the coordinates L*, a*, b*, the chroma C*, the hue h* and the saturation is study against the morphology parameters of the samples (grain size from SEM images and average crystallite size deduced from Rietveld's refinements).

Most of the time, the samples can be separated into two groups that don't show the same variations : a first one composed of grains smaller than 80 nm in diameter, and a second one with grains larger than this value. The study against the average crystallite size is less convincing because some of the samples are composed of large grains that are not mono-crystals (P 1 , P 2 , C). The most striking result is obtained for the correlation between the hue and the grain size.

• Luminance L* :

Hematite BRDF : RTLSR model

The adapted RTLSR model IV.3.1 is used to modelize the BRDF of the measured synthetic samples. This model involves 2 physical and 1 empirical parameters that are optimized in order to minimize the root mean square difference between the measurements and the simulations.

Contribution

Parameter The optimization is performed at the wavelength λ = 700 nm by taking into account all the measurement geometries of incidence θ i = 0 • , 20 • and 40 • . For an incidence of 60 • , the BRDF is predicted by using the refined parameters.

Sample 1

This sample is mono-disperse. It is composed of grains that are monocrystals with a radius we estimate at 12.5 nm.

Parameters Optimization on all geometries α 0.46722825 C 0.36757953 g 0.8 E RMS 0.20740749635244643 also be considered as a refinement parameter. We will refer to the 6-parameters model in that case.

In the case of a polydisperse powder, or grains that are agglomerates, I would prefer the 7 parameters model which neglects the specular scattering but accounts for two different size populations. The parameters are then summarized as follows : The optimization is performed at the wavelength λ = 700 nm by taking into account all the measurement geometries of incidence θ i = 0 • , 20 • and 40 • . For an incidence of 60 • , the BRDF is predicted by using the refined parameters.

Sample 3

This powder is not monodisperse. We estimate the average radius of the large grains composing this sample at 83 nm, and the average radius of the smaller grains at 47 nm. Two simulations are performed by either using our 6-parameters model which optimizes on a single average radius or by using the 7-parameters model adapted for the case of multiple size populations.

Optimizing on a single radius :

Parameters Optimization with a variable radius r (nm) [START_REF] Nassau | The fifteen causes of color : the physics and chemistry of color[END_REF]

Sample 6

This sample is mono-disperse. It is composed of grains that are mono-crystals with a radius we estimate at 22 nm. We proceed to two simulations using either the 5 or the 6-parameters model (ie using a fixed or an optimized radius).

Parameters

Optimization with fixed radius

Sample P 2

This sample shows large grains (r = 687 nm) that seem to be agglomerates of much smaller ones (r ≈ 10 nm). Two simulations are performed by either using the 6-parameters model which optimizes on the large grain radius, or by using the 7-parameters to describe the two size populations.

When we optimize on the two radii, we observed that only the smaller grains are actually involved in the fitted spectrum.

Optimizing on a single radius :

Parameters Optimization with a va-

Sample C

This sample is not mono-disperse. It is composed of large grains (r ≈ 2222 nm) that are agglomerates of much smaller spherical grains (r ≈ 100 nm). Two simulations are performed by either using the 6-paremeters model which optimizes on a single average radius, or by optimizing on two different sizes.

Optimizing on a single radius : 265