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Previously, during my PhD thesis and my post-doctorate, I have also worked on research topics in uid mechanics such as the ow of thin liquid lms [Figliuzzi et al., 2012a,b], capillary rise [Figliuzzi and Buie, 2013;Joung et al., 2014] or non-linear electrophoresis [Figliuzzi et al., 2014[Figliuzzi et al., , 2016a]].

Morphological models and stochastic geometry

The development of morphological models aimed at describing the microstructure of heterogeneous materials is a well-established line of research at the Center for Mathematical Morphology (CMM), which oers strong potential for exploitation in the industrial sector. Stochastic geometry is a relatively mature eld scientically, but the development of innovative heterogeneous materials constantly supports the reection on new geometric models of microstructures.

A signicant part of my research work has focused on the topic of the geometry of microstructures, through several collaborations with industrials in particular. Between

2013 and 2016, I worked on a research project carried out in direct partnership with the Japanese tire company Yokohama Rubber Co. on the modeling of the microstructure and of the visco-elastic response of a multi-scale material used to manufacture pneumatic materials. In 2015, I obtained with François Willot (CMM) a funding from the Air Force Oce of Scientic Research to develop microstructure models for a copper lm deposited by coldspray. This study combined experimental and simulation aspects.

The research project was conducted in collaboration with Vincent Bortolussi as part of his PhD thesis work, conducted at the center for materials engineering of Mines Paris under the supervision of Prof. Michel Jeandin. Coldspray is a deposition process that works by projecting particles at very high speed onto a substrate at room temperature.

In the case of the study, this process is used to deposit copper particles on the fuselage of an aircraft to provide protection in the event of a lightning strike. Our objective was to investigate the inuence of the geometry of the substrate microstructure on the macroscopic properties of electrical conduction. We describe the development of this model, which is particularly representative of the researches that I conducted on the topic of morphological models, in chapter 2 of the manuscript.

More recently, I have been working with L'Oréal R&I on the development of new morphological models to simulate the microstructure of colloidal media or the distribution of melanosomes in the epidermis. Most of this work could not be published due to condentiality reasons. On a scientic level, this work allowed me nevertheless to develop a method for optimizing the parameters of these models based on the MCMC [Figliuzzi et al., 2021] algorithm. I present this approach in chapter 3 of the manuscript.

Supervised segmentation

The experimental campaigns carried out in materials science often lead to the obtaining of substantial image databases, the exploitation of which must be automated as much as possible. Currently, the most ecient segmentation methods use supervised learning algorithms based on convolutional neural networks. In the case of images of microstructures, it is however rare to have enough manually segmented images to train these architectures. Techniques such as transfer learning are also insucient to address the lack of training data, as microstructure images are often very distinct from the natural images typically used to train most existing convolutional network architectures.

The development of segmentation methods using only a limited amount of data therefore remains an important axis of research, which conditions the applicability of the most ecient segmentation methods to actual industrial problems.

My main contribution to this axis of research is probably the PhD thesis of Kaïwen

Chang between 2016 and 2019, during which we developed an algorithm based on the Eikonal equation to generate a superpixel segmentation of a given image based on color and texture information [START_REF] Chang | Machine learning for image segmentation[END_REF][START_REF] Alais | Fast macula detection and application to retinal image quality assessment[END_REF]. The superpixel segmentation can be thought as a rst step toward the complete segmentation of the image and can be progressively merged until the obtaining of an actual segmentation of the image.

By relying on carefully handcrafted features directly computed on the spatial supports dened by the superpixels, it is possible to learn a similarity measure between adjacent superpixels with a limited number of training examples. The problem of segmenting the image can thus be reformulated as a clustering problem on the region adjacency graph associated to the superpixel segmentation. To perform the clustering, we relied on a generalization of the Eikonal equation to graph structures. I describe this research in chapter 5 of the manuscript.

In the case of relatively homogeneous images, I also sought to explore a solution to the absence of training data consisting in developing morphological models allowing the joint generation of a base of simulated images and of a ground truth in order to train a supervised segmentation algorithm. In 2020, as part of David Paulovics' internship, supervised in collaboration with F. Blanc (Nice Physics Institute), we successfully used this approach to automate the segmentation of images of suspended particles. This research is described in chapter 6.

Hyperspectral imaging

The last axis of research I am working on is hyperspectral imaging and in particular its use for the chemical characterization of surfaces. The principle of this imaging modality is to record the intensity of the light received by the sensor over a wide range of wavelengths, usually in the infrared domain. Due to the low spatial resolution of hyperspectral sensors, the observed spectra generally correspond to a combination, linear or not, of the spectra of the dierent elementary materials present in the observation area. The identication of these elementary spectra largely remains an open problem. In particular, the potential variability of the elementary spectra within a given observation scene, the case of highly mixed observations and non-linear mixing models remain very dicult to deal with at present.

As part of Tarek Zenati's thesis, conducted in partnership with the steel company ArcelorMittal, we developed an algorithm to detect and characterize the presence and thickness of oxide layers on steel surfaces. From an industrial point of view, it is indeed important to prevent the formation of such layers, which can signicantly impact the wettability properties of steel sheets. Our approach is based on a parsimonious regression approach originally introduced for the processing of satellite images [START_REF] Iordache | Sparse unmixing of hyperspectral data[END_REF], which makes it possible to bypass the issue of the strong non-linearity of the optical interactions leading to the formation of the hyperspectral measurements that we are processing [Zenati et al., 2022b]. This work is presented in the nal chapter of the manuscript.

Part I Morphological models and stochastic geometry

Understanding the impact of the microstructure on the overall physical and mechanical properties of a material has recently emerged as a prominent research area [START_REF] Capasso | [END_REF][START_REF] Jeulin | Morphological models of random structures[END_REF][START_REF] Ohser | 3D images of materials structures: processing and analysis[END_REF][START_REF] Redenbach | Microstructure models for cellular materials[END_REF][START_REF] Redenbach | Statistical analysis and stochastic modelling of bre composites[END_REF][START_REF] Torquato | Random heterogeneous materials: microstructure and macroscopic properties[END_REF]. One intriguing aspect of this research area involves generating random microstructures that accurately replicate the geometric features of a material [START_REF] Altendorf | Random-walk-based stochastic modeling of threedimensional ber systems[END_REF][START_REF] Jeulin | Percolation of multi-scale ber aggregates[END_REF][START_REF] Liebscher | Statistical analysis of the local strut thickness of open cell foams[END_REF][START_REF] Moreaud | Tem image analysis and modelling: application to boehmite nanoparticles[END_REF][START_REF] Peyrega | 3d morphological modelling of a random brous network[END_REF]. These simulated microstructures can subsequently serve as a foundation for exploring the physical and mechanical properties of heterogeneous materials through extensive numerical simulations. This approach is particularly valuable for understanding the inuence of microstructure on the macroscopic properties of the material and holds signicant potential for practical applications in the industrial sector.

I describe in this part of the manuscript selected research topics that I conducted on the simulation of materials microstructures and on morphological models more generally. In chapter 2, after recalling the basic theory of morphological models, I present a research study that aimed at developing microstructure models for a copper coating deposited using the coldspray process, conducted in collaboration with Vincent Bortolussi as part of his PhD thesis at the Center for Materials engineering of Mines Paris. In this study, our objective was to examine how the geometry of the substrate microstructure aects electrical conduction at the macroscopic level. I conclude chapter 2 by presenting a research work establishing a link between the Eikonal equation and a class of tessellation models, which allows to generate random tessellations with irregular boundaries.

I discuss then in chapter 3 a novel approach for the parameterization of morphological models, which aims to determine a set of parameters for the model that allows him to accurately reproduce an experimental microstructure.

Chapter 2

Multi-scale models of random microstructures Most materials used in contemporary life and industry are heterogeneous and exhibit a complex internal microstructure. The microstructure is therefore a key feature of the global material that largely determines most of its physical properties at the macroscopic level [START_REF] Torquato | Random heterogeneous materials: microstructure and macroscopic properties[END_REF]. It can be interesting to generate random microstructures that reproduce accurately the geometrical characteristics of the original material, as the simulated microstructures can in turn serve as a basis to investigate the physical or mechanical properties of heterogeneous materials through extensive numerical simulations. This approach is in particular interesting to better understand the inuence of the microstructure on the physical properties of the material at the macroscopic scale.

We begin in this chapter by presenting basic notions on morphological models in section 2.1, before detailing in section 2.2 research work that I have carried out on the microstructures of coatings deposited by coldspray process in collaboration with researchers from the Center for Material engineering at Mines Paris [Bortolussi et al., 2018[Bortolussi et al., , 2020]]. This research subject is particularly representative of several studies that I have conducted on the simulation of material microstructures [Figliuzzi et al., 2016a;Gasnier et al., 2015]. We present nally in section 2.3 a research work that establishes a link between the resolution of the Eikonal equation on a domain and the generation of Voronoï and Johnson-Mehl tessellations [Figliuzzi, 2019]. This works introduces an original way to construct tessellations of space with irregular (rough) boundaries.

Basics of morphological models

Mathematical morphology and stochastic geometry provide ecient tools for both analysis and simulation of heterogeneous microstructures. Mathematical morphology is a theory for the analysis and processing of geometrical structures based on set theory and topology [START_REF] Soille | Morphological image analysis: principles and applications[END_REF]. It nds most of its applications in the eld of image processing and random structures simulation. The basic idea behind mathematical morphology is to analyze a set A of some topological space E by probing it with a compact set K, classicaly referred to as structuring element.

The basic operators of mathematical morphology are dilation and erosion, which are dened in the following manner. Let A be a subset of R d . The dilated of the set A by the structuring element K is the set

A ⊕ K = {x ∈ R d |K x ∩ A ̸ = ∅}, (2.1)
where K x is the translated of the compact K at x ∈ E. Similarly, the eroded of the set A by the structuring element K is the set

A ⊖ K = {x ∈ R d |K x ⊂ A}.
(2.2)

Dilation and erosion are dual operators with respect to the complement, in the sense that dilating the set A by the structuring element K is equivalent to erode A c by K.

Dilation and erosion can be seen as the fundamental bricks of mathematical morphology, from which derive almost all other operators. For instance, by combining erosion and dilation, we can dene two new morphological operators. Let A, B be subsets of E. The closing A B and the opening A B of the set A by B are dened as follows:

A B = (A ⊕ B) ⊖ B, (2.3)
and

A B = (A ⊖ B) ⊕ B, (2.4)
where B = {x ∈ R d | -x ∈ B} is the symmetric set of B.

Random set theory

Let us assume in this section that we dispose of a set of experimental images of some materials microstructure. Once segmented, the images of the microstructure still carry a huge amount of information. As a consequence, we need to develop mathematical techniques to capture the main features of the complex geometry under scrutiny. A natural way to describe inclusions in a matrix is for instance to consider a set A, representing the included particles, and its complementary set A c , representing the matrix.

The study of such models falls into the scope of stochastic geometry, a mathematical discipline which aims at providing a systematic description of random spatial patterns.

The development of this discipline is intimately correlated to the one of mathematical morphology.

Let's go back to our previous example of inclusions in a matrix. As mentioned previously, to study the obtained sets, the idea behind mathematical morphology is to dene a structuring element K (e.g. a point, or a disk with some given diameter) and to use it to probe the subsets A and A c . The simplest relations that one can build to study A and A c are the following ones: at a given point of the porous media, either the structuring element K hits the set A or it is disjoint from it. At each point, the knowledge of wheter or not some structuring element K hits a set A is sucient to completely characterize A.

The deterministic approach exposed here proves however untractable in practice.

Obviously, processing each one of the microstructures constituting heterogeneous media is generally out of scope, since this would involve a considerable amount of data.

Nevertheless, considering the limited amount of information available on the material, this issue can be entirely alleviated by relying on a statistical approach [START_REF] Matheron | Random sets and integral geometry[END_REF]. From this perspective, instead of determining at each point if some structuring element intersects A, we try to estimate the probability that the considered structuring element intersects A. A benet of this approach is that we can determine statistical laws on experimental samples and generalize these laws to larger portions of the same material as long as the samples are statistically relevant.

In this framework, the set A representing the solid phase becomes completely characterized by the functional T (K) dened for all structuring element K by

T (K) = P (A ∩ K ̸ = ∅) = 1 -P (K ∩ A c ).
(2.5) T (K) is called the Choquet capacity of the random closed set A [START_REF] Choquet | Theory of capacities[END_REF]. Interestingly, the Choquet capacity is closely related to dilation and erosion operators. For all compact set K ⊂ R d , we have indeed

T (K) = P (K ∩ A ̸ = ∅) = P (x ∈ A ⊕ Ǩ) (2.6)
Through the Choquet capacity, mathematical morphology provides a solid mathematical framework to investigate the microstructure geometry. Since each compact set K brings its own information of the studied set A, the choice of structuring element allows one to conduct very specic statistical measurements on the random set A. For instance, if one chooses K to be a single point, the choquet capacity yieds

T ({x}) = P ({x} ∩ A ̸ = ∅) = P (x ∈ A), (2.7)
which is the spatial law of the set A. Similarly, if one chooses K to be the set {x, x + h}, the choquet capacity allows to calculate the covariance of the random closed set.

T ({x, x + h}) = P (x ∈ A, x + h ∈ A).

(2.8)

The covariance of the set A at a given point x and for a distance h is the probability that x and x + h both belong to A. The covariance C provides useful information about the spatial arrangement of the random set A. In particular, it accounts for the presence of several scales in the studied set or for periodicity.

Morphological models

As stated previously, one is often interested in developing stochastic models of the microstructure that reproduce accurately the geometrical characteristics of the original material.

Poisson point processes The basic ingredients of stochastic geometry models are random point processes. A random point process P is a collection of random points of the space R d . A particular role is played by Poisson point processes. Let θ > 0 be a positive real number. A Poisson point process on R d is a point process such that the number N (K) of points contained in any region K of R d is a Poisson random variable with intensity θ:

P (N (K) = n) = θ(K) n n! exp(-θ(K)).
(2.9)

A = xn∈P A ′ xn ,
(2.10)

where A ′ xn denotes the translated of the primary grain A ′ at point x n :

A ′ xn = {x n + y, y ∈ A ′ }.
(2.11)

Any shape can be used for the grain A ′ , including convex, non-convex or even non connected sets.

Interestingly, the Boolean model is tractable analytically. For instance, one can easily prove that the number N (K) of primary grains intersected by any compact region K follows a Poisson distribution of parameter θ μ( Ǎ′ ⊕ K):

P (N = n) = θ n μ( Ǎ′ ⊕ K) n n! exp(-θ μ( Ǎ′ ⊕ K)).
(2.12)

In this expression, μ( Ǎ′ ⊕K) denotes the average Lebesgue measure (the average surface in R 2 or the average volume in R 3 ) of a primary grain A ′ dilated by the compact set K. This result guarantees that the number of primary grains in any bounded window remains almost surely nite, and enables us to calculate the Choquet capacity of the Boolean model

T (K) = 1 -exp(-θ μ( Ǎ′ ⊕ K)).
(2.13) Knowing an analytical expression for the Choquet capacity, we can easily determine the covariance of the Boolean model that yields

C(h) = P (x ∈ A, x + h ∈ A) = 1 -exp(-θ μ( Ǎ ⊕ l h )), (2.14)
where l h denotes the structuring element constituted by both points 0 and h, h being some vector of R d .

Random tessellations A tessellation or mosaic is a division of the d-dimensional

Euclidean space R d into polyhedra. Such geometrical patterns can be observed in many natural situations, as shown in gure 2.1. Random tessellation models have therefore been extensively used in physics, materials science and chemistry.

Denition 1. A tessellation in R d is a countable system T of subsets satisfying the following conditions:

T ∈ F lf (R d ), meaning that T is a locally nite system of nonempty closed sets.

The sets K ∈ T are compact, convex and have interior points.

The sets of T cover the space,

∪ K∈T K = R d (2.15)
If K and K' are two sets of T then their interiors are disjoint.

We denote by T the set of all tessellations.

The faces of a cell C of the tessellation are the intersections of C with its supporting hyperplanes. A k-face is a face of dimension k. Among all possible k-faces, the 0-faces, or vertices, and the 1-faces, or edges, are of particular interest. The k -1 dimensional faces of a k-dimensional polytope will be referred to as its facets.

C i = y ∈ R 3 , ∀j ̸ = i, ∥x i -y∥ ≤ ∥x j -y∥ .
(2.16)

In practice, Voronoï tessellations are characterized by one single parameter, namely the intensity of the underlying point process. A general description of Poisson-Voronoï tessellations in R d can be found in Møller [START_REF] Møller | Random tessellations in R d[END_REF][START_REF] Møller | Random johnson-mehl tessellations[END_REF][START_REF] Møller | Lectures on random Voronoi tessellations[END_REF].

C i = y ∈ R 3 , ∀j ̸ = i, t i + ∥x i -y∥ v ≤ t j + ∥x j -y∥ v .
(2.17)

Note that when all times are set to zero, we recover the classical Poisson-Voronoï tessellation model. The Johnson-Mehl tessellation model was introduced by Johnson and Mehl to describe crystallization processes [START_REF] Avrami | Kinetics of phase change. I. general theory[END_REF][START_REF] Gilbert | Random subdivisions of space into crystals[END_REF][START_REF] Johnson | Reaction kinetics in processes of nucleation and growth[END_REF]. The paper [START_REF] Møller | Random johnson-mehl tessellations[END_REF] of Møller provides a unied exposition of Random Johnson-Mehl tessellations.

Microstructure modelling with morphological models

We present in this section a study dedicated to the development of a morphological model used for studying the electrical characteristics of a bi-phased coating produced through thermal spraying [START_REF] Amsellem | Two-dimensional (2d) and three-dimensional (3d) analyses of plasmasprayed alumina microstructures for nite-element simulation of young's modulus[END_REF][START_REF] Beauvais | Study of the porosity in plasma-sprayed alumina through an innovative three-dimensional simulation of the coating buildup[END_REF][START_REF] Delloro | A morphological approach to the modeling of the cold spray process[END_REF]. This coating is specically designed to shield recent commercial aircraft bodies from lightning strikes by providing a conductive surface that can eectively dissipate electrical charges. ). This study led to several publications related to the morphological model itself [Bortolussi et al., 2018] and to its exploitation to study the electric conductivity of coldspray coatings [Bortolussi et al., 2020].

Context

Modern commercial aircrafts are constructed using carbon ber-reinforced polymers, which provide excellent mechanical properties while being signicantly lighter than traditional aluminum alloys. However, the polymer matrix in these materials is generally highly electrically insulating, which poses security concerns in the event of lightning strikes. Advanced aerospace composites are typically fabricated using a matrix of PEEK (Poly-Ether-Ether-Ketone), a thermoplastic polymer that oers good mechanical and thermal properties while being an excellent insulator. To address the need for electrical charge dissipation in the event of lightning strikes, a layer of copper mesh is typically applied to the composite body. Copper is an easily machinable and corrosion-resistant electrical conductor with low electrical resistivity.

Experimental data

To simplify the manufacturing and assembly of copper meshing, a new coating method was developed by Vincent Bortolussi that relies on copper powder being thermally sprayed onto composite parts. Specically, an adherent and electrically conductive layer is achieved by using a powder mixture containing 80% volumetric of spherical copper powder (10-35 microns) and 20% of irregular PEEK particles (35-65 microns).

The mixture is applied using the cold-gas dynamic spraying or cold spray process.

In our work, we relied on microscopic images of the coating to study the resulting microstructure. Prior to observation, the cold spray coatings underwent crosssectioning and polishing. To do this, the coating samples were cut in two directions:

along the spraying path and orthogonal to it. Unfortunately, cutting and polishing caused debonding of the copper particles due to poor mechanical anchorage in the matrix. This resulted in dark holes at the surface. It should be noted that manual polishing can have a signicant impact on the debonding phenomenon. To counteract this, the samples were metallized with a layer of Gold-Palladium that was only 2 nm thick, using a Cressington sputter coater. This was a crucial step as the layer modied the color of the PEEK matrix, greatly enhancing the color gradient between phases. To observe the cross-sections, we used a Leica optical microscope at ×20 magnication in bright eld. The microscope had a resolution of 0.2428 microns per pixel. We selected this observation scale to obtain a representative fraction of copper while also highlighting PEEK interstices.

The microstructure of the cold spray coating can be observed in Fig. 2.3, where yellow copper particles are embedded in a grey PEEK matrix that contains dark footprints.

The matrix is made up of irregular PEEK particles that become highly deformed upon high speed impact, resulting in a dense structure with no visible pores at this scale. The copper particles deform only upon impact with each other, resulting in limited plas- The rst step of our study consisted in developing a segmentation algorithm of the experimental images in order to identify the copper particles within the microstructures as well as the interstices separating these particles. We refer the reader seeking additional details on the segmentation algorithm to the original article [Bortolussi et al., 2018]. Segmentation results are displayed in Fig. 2.5. Overall, we performed the segmentation of K = 13 large images of the microstructure, and we used these images to compute experimental covariance and granulometry curves that characterize the coating microstructure.

Morphological model

We present in this section the multi-scale morphological model [START_REF] Jeulin | Morphology and eective properties of multi-scale random sets: A review[END_REF] developed for describing the microstructure geometry. To construct the model, we make the assumption that the microstructure consists of copper spheres embedded in a PEEK matrix, separated by thin PEEK interstices. The covariance, copper fraction, granulometry, and interstice thickness are measured on 2D slices of the coating. Therefore, the parameters of the 3D model need to be inferred from this 2D information. The nal two-scale model is based on a two-step simulation process:

1. The rst step of the simulation process corresponds to the generation of a Boolean model of spheres with intensity θ. The radii of the spheres follow a Gamma distribution law with parameters λ and a. This rst scale of the model intends to represent the set of copper particles, which form aggregates due to interpenetration.

2. The second step involves interstice implantation based on a modied Johnson-Mehl tessellation. This two-scale model provides a representation of the 3D microstructure of the coating based on the 2D information obtained from the slices.

Aggregates simulation We assume that the copper particles aggregates can be represented by a Boolean model of spheres and that the distribution of the radii of the spheres is described by a Gamma law. The probability density function of the Gamma law is given by

p(r, λ, a) = r a-1 Γ(a)λ a exp - r λ , (2.18)
where Γ denotes the Gamma function. The average radius of the typical sphere is kλ.

Its variance is kλ 2 . The average surface of the typical grain is

S v = +∞ 0 4πr a+1 Γ(a)λ a exp - r λ dr = 4πλ 2 a(a + 1).
(2.19)

Similarly, its average volume is

V v = +∞ 0 4πr a+2 3Γ(a)λ a exp - r λ dr = 4π 3 πλ 3 a(a + 1)(a + 2).
(2.20)

To determine the 3D parameters of the model using 2D measurements, we use the stereological formulae

θ 3 V v = θ 2 Ā, θ 3 S v = 4 π θ 2 L, (2.21)
where Ā is the mean area of the sliced spheres and L their perimeter. We need to relate the 2D measurements to the parameters of the Boolean model. To that end, we rely on Miles' formulae [START_REF] Chiu | Stochastic geometry and its applications[END_REF][START_REF] Miles | The random division of space[END_REF][START_REF] Schneider | Stochastic and integral geometry[END_REF]]

A a = 1 -e (-θ 2 Ā) , L a = θ 2 L(1 -A a ), (2.22)
where A a is the mean surface fraction of copper on segmented images and L a is the mean perimeter of the copper phase on segmented images divided by the total surface. Using

Miles' formulae in conjunction with stereological formulae [START_REF] Chiu | Stochastic geometry and its applications[END_REF][START_REF] Schneider | Stochastic and integral geometry[END_REF], we nd, for the Boolean model

A a = 1 -exp(-θV v ), L a = θ π 4 S V exp(-θV v ).
(2.23)

Overall, there are three unknowns in the model, namely the intensity θ of the Boolean model and the parameters a and λ of the Gamma distribution. Hence, we can express all parameters as functions of a. (1 -A a ) ln(1 -A a ).

(2.25)

To determine the parameters of the stochastic model, we rely on a maximum likehood approach to nd the parameters that minimize the least-square distance between the covariance of the simulated microstructure and the covariance that is measured on the available experimental dataset. However L a is highly inuenced by interconnection between particles. As many particles remain in contact due to interstices segmentation, computing θ only from λ and a provides a more robust algorithm.

Interstices implantation

C i = y ∈ A, ∀j ̸ = i, t i + ∥x i -y∥ G v + ζ i ≤ t j + ∥x j -y∥ G v .
(2.26)

In this relation, ∥ • ∥ G denotes the geodesic distance with respect to the realization of the Boolean model. With this denition, we note that some points of the aggregates do not belong to any class of the Johnson-Mehl tessellation. We consider that these points form the interstices between the grains of the microstructure.

A signicant question remains, which is how to select the initial germs of the tessellation and the germination times. While selecting the germs, our aim is to preserve the geometrical shape of the grains of the microstructure. Hence, we seek to set the germs in the center of connected components to simulate a granulary microstructure.

To the end, we rely on the h-maxima [Serra, 1982] of the distance function to generate the germs. The h-maxima of the distance function form connected components. For each component, we select its barycentre to be the location of a germ. The threshold for the h-maxima is selected after an optimization procedure that aims at minimizing the distance between the granulometries. For each germ n, we denote by d n the value of the Euclidean distance function at the location of the germ. The germination time associated to germ n is dened to be

t n = (max m d m ) -d n .
(2.27) With this choice of germination time, the cells border of bigger spheres are kept close from the edges of the spheres and we are able to preserve the geometrical shape of the grains constituting the microstructure.

Maximum likehood for parameters estimation

The parameters estimation follows a two-step process. For estimating the parameters a and λ of the rst scale of the model, we rely on a Nelder-Mead procedure [START_REF] Nelder | A simplex method for function minimization[END_REF]. At each iteration, the parameters a and λ are xed and θ is computed using Eq. (2.24). Then, we generate a 1000×1000 pixels slice, extracted from a larger For the second scale of the model, we rely on a similar procedure, this time involving the granulometry measurements, to estimate the threshold value h of the h-maxima distance that is used to select the germ location and the parameter of the exponential law describing interstice thickness. Note that to allow a better t of the covariances we slightly increase A a to counterbalance copper surface fraction decreasing due to interstices implantation. The usual value is 57% instead of 55.3%.

Results and discussion

A 2D slice of the simulated microstructure obtained with the parameters estimated with the maximum likehood approach is represented in Fig. 2.6. The parameters are reported in Table 2.1, with A a the measured surface fraction of copper on segmented images, θ the intensity of the poisson point process, λ and a the parameters of the Gamma distribution of the spheres radii and h the h-maxima of the distance function used for selecting the germs of the tesselation. The covariance and the granulometry of the simulated and of the experimental microstructures are shown in Fig. 2.7. Covariances and granulometries taken from simulated and segmented images show very good agreement, assessing the validity of the model.

The model that we developed makes it possible to reproduce accurately the microstructure of coldspray coatings containing copper particles. The model was subsequently used to optimize the microstructure of the coating with the aim of increasing its electric conductivity. This ultimately led to the manufacturing of coldspray coating with high electric conductivity, as described in [Bortolussi et al., 2020].

Eikonal based tessellations

We present in this section a research work that establishes a link between the Johnson-Mehl and Voronoï tessellation models and the Eikonal equation on a domain. A direct incentive for this study was the development of ecient algorithms for generating the tessellations used to simulate the interstices of the coldspray microstructure considered in section 2.2. We base the writing of this section on the article [Figliuzzi, 2019]

∀x ∈ Ω, ∇t(x) = 1 u(x)
.

(2.28)

In Eq. (2.28), ∇ denotes the gradient operator, and t(x) and u(x) represent the rst arrival time of the wave at point x and the velocity of the wave at location x, respectively.

In order to compute the image of a

   ∇t(x) = 1 u(x) t(x i ) = τ i , ∀i = 1, .., N.
(2.29) If the velocity u(x) is constant, then we can use the following proposition (see [START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF]): Proposition 1. On the domain Ω, the solution t of problem (2.29) satises, for all

x ∈ Ω, t(x) = inf i=1,...,N (t i + ||x -x i || 2 ).
(2.30)

Proposition 1 states that solving the Eikonal equation is an ecient way to compute the Johnson-Mehl distance on a grid of points, as it precisely matches the distance used for constructing the Johnson-Mehl tessellation. Ecient algorithms are available for this task, including the fast marching algorithm described in the next section [START_REF] Dejnozková | A parallel algorithm for solving the eikonal equation[END_REF][START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF].

Fast marching algorithm

The fast marching (FM) algorithm compute the arrival times in the domain iteratively by following the propagation of the waves coming from the germs. Here, we present a slightly adapted version of the fast marching algorithm that keeps track of the labels of the initial germs during the propagation, in order to facilitate the generation of the Johnson-Mehl tessellation. We restrict ourselves to the 2D framework to keep notations simple. We initialize the algorithm as follows:

1. We initialize the map of arrival times by setting the arrival at each voxel x i equal to t i = ∞, except if the considered voxel contains a germ from P. In this case, the arrival time at x i is set to be the germination time τ j .

2. We initialize a map of labels that keeps tracks of the wave that has arrived at each location rst. We aect the label j to the voxels x i containing one of the germs (g j ) 1≤j≤N are associated the label j. We aect the label 0 to the other voxels.

3. Finally, we group all voxels containing one of the germ in a set referred to as the narrow band.

At each iteration, we extract the voxel (X, Y ) of the narrow band with smallest arrival time and we aect it to the frozen set. Next, we compute the arrival times of the wave at all neighboring voxels and we add these voxels to the narrow band. At each neighbor location (x, y) of (X, Y ), we compute the arrival time by solving Eikonal equation ||∇T ||(x, y)u(x, y) = 1.

(2.31)

The main diculty at this point is to propose a discretization for the gradient term.

Following the work [START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF], we discretize Eikonal equation in the following manner

(t x,y -min(t x,y+1 , t x,y-1 )) 2 + (t x,y -min(t x+1,y , t x-1,y )) 2 = 1 u(x, y) 2 (2.32)
Equation (2.32) has two distinct solutions. To respect the consistency of the scheme, the arrival time must be higher than the time t(X, Y ) at the selected point in the narrow band. Therefore, we select for the arrival time t(x, y) the largest solution of (2.32).

Once the arrival time t(x, y) has been computed, we can encounter two distinct situations:

When the neighbor point (x, y) is in the narrow band, it means that it has already been aected an arrival time t old (x, y). If t(x, y) < t old (x, y), then we aect the arrival time t(x, y) to (x, y) as well as the label of point (X, Y ). On the contrary, if t(x, y) > t old (x, y), the label and the arrival time at (x, y) remain unchanged. When the neighbor point (x, y) is not in the narrow band, we aect to it the arrival time t(x, y) as well as the label of (X, Y ), and we add it to the narrow band.

At each iteration of the algorithm, it is necessary to extract the element of the narrow band with the smallest arrival time. To sort the elements of the narrow band in an ecient manner, we use a binary heap to store them. We refer the reader interested by more details on the fast marching algorithm implementation to the original articles [START_REF] Malladi | Shape modeling with front propagation: A level set approach[END_REF][START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF][START_REF] Sethian | Fast marching methods[END_REF].

Applications

We describe in this section two potential applications of the proposed approach for generating a tessellation: the rst application describes a computationally ecient method for generating an image of a Voronoï or Johnson-Mehl tessellation from a set of germs.

The second application is related to the generation of tessellation with rough boundaries that it is made possible by the use of Eikonal equation.

Generation of images from vectorial simulations

For materials engineering applications, it is often necessary to simulate images of a generated Voronoï or Johnson-Mehl tessellation. In practice, the tessellation is entirely characterized by the location of its germs and by the corresponding implantation times in the case of the Johnson-Mehl tessellation. The image generation is achieved by discretizing the domain Ω and by evaluating the implicit functions associated with the tessellation's cells at each voxel location. However, this approach can be computationally expensive, particularly when high resolution is required. If we denote the number of voxels used to discretize Ω as P , the average number of Poisson points in P is proportional to P . As a result, the image generation algorithm's complexity is O(P 2 ), which can represent a signicant amount of computation when using this method for a vectorial simulation. By contrast, using the approach based on the Eikonal equation yields a complexity in O(N log N ), which is particularly appealing for large values of N .

Tessellation with rough boundaries

We have considered so far a constant velocity eld u in the Eikonal equation (2.29) for constructing the Johnson-Mehl tessellation. However, when using the Eikonal equation to build the mosaic, it is possible to use any velocity elds on the domain Ω, as long as u(x) > 0 for all x ∈ Ω. This observation opens the way to the development of tessellation models that cannot be directly obtained with classical approaches including the Voronoï and the Johnson-Mehl tessellations described previously. A potential application is for instance the generation of tessellations of space with rough boundaries between adjacent cells.

Let P be the realization of a marked point process with intensity θ on an open domain Ω, for which the marks are drawn according to the uniform distribution U([0, L]) on an interval [0, L] ⊂ R + . The points in P as well as their respective marks allow to compute a Johnson-Mehl tessellation on the domain Ω by solving problem (2.29). We construct a random velocity eld u on domain Ω by 1. computing a Voronoï tessellation of Ω based upon a Poisson point process P v with intensity θ v , and 2. setting a random velocity drawn according to some distribution D in each cell of the Voronoï tessellation.

By solving the problem

   ∇t(x) = 1 u(x) t(x i ) = τ i , ∀i = 1, .., N (2.33)
we obtain a tessellation of the domain Ω ressembling the Johnson-Mehl tessellation generated by solving (2.29), but with rough boundaries between adjacent cells. It is worth noting that the roughness observed at the boundaries is closely linked to the characteristic length of the uctuations in the velocity eld. This characteristic length is dependent on the intensity of the Voronoï tessellation, denoted by θ v , which is used to compute the velocity eld. A larger value of θ v results in a greater number of smaller Voronoï cells, leading to velocity variations over shorter distances. As a result, the generated Johnson-Mehl tessellation displays boundaries with a high concentration of roughness patterns with relatively small amplitudes. In contrast, Johnson-Mehl tessellations generated from velocity elds constructed with a smaller value of θ v exhibit boundaries with a lower density of roughness patterns with larger amplitudes. In the original article [Figliuzzi, 2019], we developed a multi-scale tortuosity descriptor, which is able to characterize the tortuosity of the boundary between two adjacent cells by iteratively computing the tortuosity of smoothened versions of the boundary. This descriptor allowed us to establish a direct correlation between the value of θ v and the tortuosity of the boundaries at distinct scales. We refer the reader interested by additional details to the original article [Figliuzzi, 2019].

To illustrate our algorithm, we present in gure 2.8 simulations of Voronoï tessellations with rough boundaries computed with dierent values of velocity parameter θ v .

In this gure, we generated four distinct Voronoï tessellations on a domain with size 50 × 50 discretized on a 500 × 500 regular grid. To generate the tessellation, we started from the same realization of a Poisson point process P with intensity θ = 5 × 10 -2 . We generated a random velocity eld for each one of the tessellations by rst constructing a Voronoï tessellation of the domain, and then selecting a random velocity drawn from the uniform distribution U on the interval [0.2, 1.8] for each cell of the tessellation. The characteristic length λ v of the velocity variation can be related to the intensity θ v of the point process used to construct the germs of the tessellation through the relationship

λ v ≃ 1 θ v .
We note that when λ v is small, it leads to the obtaining of small roughness patterns at the cell boundaries. Conversely, large values of λ v lead to larger sizes of roughness pattern. The proposed method therefore allows to control, at least qualitatively, the amplitude and the size of the roughness patterns observed at the boundaries.

Conclusion

I discussed in this chapter several morphological models used to describe the microstructure of cold spray coatings or to generate tessellations of space with rough boundaries.

These models share the characteristic of being described by a small number of parameters that are xed in order to match statistical features estimated from realizations of the model with features computed from experimental images of the microstructure.

In the next chapter of the manuscript, we present a general methodology for selecting these parameters based on the Monte Carlo Markov Chains (MCMC) algorithm.

Related works

Microstructure models for rubber materials Between 2013 and 2016, I worked on a research project carried out in direct partnership with the Japanese tire company Yokohama Rubber Co. on the modeling of the microstructure and of the visco-elastic response of a material used to manufacture pneumatic materials. This project was conducted in collaboration with Dominique Jeulin, Franccois Willot and Matthieu Faessel (Center for Mathematical Morphology, Mines Paris).

Rubber are classically constituted by a polymer matrix lled by black carbon particles. The physical properties of rubber are therefore strongly inuenced by the volume fraction and spatial distribution of llers [START_REF] Moumen | Eect of overlapping inclusions on eective elastic properties of composites[END_REF][START_REF] Ferté | Étude et analyse de couches minces par techniques multi-spectroscopiques pour une application sur une ligne de galvanisation[END_REF][START_REF] Jean | A multiscale microstructure model of carbon black distribution in rubber[END_REF]. In particular, when nano-sized llers form extended percolating networks, they have a signicant impact on the macroscopic mechanical properties of rubbers. To predict the mechanical properties of rubber materials, we combined the use of morphological models that describe the multi-scale microstructure of rubber with ecient Fourier-based numerical algorithms that allow to compute the eective visco-elastic properties of the material based on an accurate description of its microstructure. The morphological models were developed from experimental transmission microscope images of the microstructure and take into account the non-uniform distribution of llers observed within these images.

In this study, we proposed various types of multi-scale models to describe the ob- The morphological model was exploited by François Willot along with a novel Fourier-based algorithm to estimate the viscoelastic properties of linear heterogeneous media under harmonic conditions, which demonstrated that the proposed numerical method is eective in calculating the viscoelastic response of microstructures containing rubbers and llers. This work was published in 2016 in Technische Mechanik [Figliuzzi et al., 2016a] and presented at the European Conference on Constitutive Models for Rubbers X in 2017 [Koishi et al., 2017].

Microstructure models for pressed explosive materials In [Gasnier et al., 2015],

we developed a numerical approach to homogenization specically designed for pressed explosives based on an energetic molecular crystal referred to as TATB (1,3,5-triamino-2,4,6-trinitrobenzene). Our approach combined virtual microstructure modeling and Chapter 3

Morphological models parameterization

We presented in the previous chapter dierent morphological models allowing to reproduce the microstructure of materials from experimental imaging data. Typically, these morphological models rely on a set of parameters Θ that must be carefully selected to align with experimental observations. One common approach to determine these parameters is by minimizing an objective function, often representing the dierence between measurements obtained from simulations and actual experimental data.

Numerous methods have been proposed in literature for this purpose.

A classic technique involves minimizing the objective function using a gradient descent algorithm. Gradient descent algorithms are iterative optimization algorithms of the rst order that aim to locate local minima of a dierentiable function by progressively advancing along steps in the direction of the negative gradient in the parameter space. These algorithms are highly eective for minimizing strongly convex functions, but applying them to the problem at hand presents several challenges. Firstly, for morphological models, computing the gradient of the objective function is often analytically intractable and necessitates the use of nite dierence schemes to evaluate the gradient.

This process requires computing at least D + 1 evaluations of the objective function, where D represents the dimension of the parameter space. Moreover, the inherent statistical variability of morphological models often results in noisy gradient estimates.

Secondly, the objective function is often non-convex, implying that a gradient descent algorithm will converge only to a local minimum of the objective function.

Alternatives to the gradient descent algorithm exist in the literature. Notably, in [START_REF] Wang | Modelling mesoporous alumina microstructure with 3d random models of platelets[END_REF][START_REF] Wang | Numerical simulation of hindered diusion in γ-alumina catalyst supports[END_REF][START_REF] Wang | Modelling of the microstructure of mesoporous alumina constrained by morphological simulation of nitrogen porosimetry[END_REF] and [Figliuzzi et al., 2016a], the authors suggest employing the Nelder-Mead algorithm for the optimization, an heuristic approach used to nd the minimum or maximum of an objective function in a multidimensional space.

It is a direct search method that relies solely on iterative evaluations of the objective function, making it suitable for cases where analytical computation of the gradient is not feasible.

In this chapter of the manuscript, I present researches that I conducted on the relatively overlooked question of morphological models parameterization. This study was partly inspired by issues encountered in the context of a research project with L'Oréal R&I, and was published in [Figliuzzi et al., 2021]. In this study, we introduce an approach based on a Bayesian formulation of the problem. This allows us to come up with a posterior distribution model for the parameters of the microstructure model conditioned on the experimental observations. To generate samples from the posterior distribution, we employ a Monte Carlo Markov Chains (MCMC) algorithm. This approach oers several advantages compared to other methods. Specically, the MCMC algorithm produces samples that accurately represent the actual posterior distribution, enabling the detection of parameter correlations within the model and quantication of their respective inuences.

3.1 Mathematical model

Problem statement

Let us assume that we dispose of a set of previously segmented experimental images of the microstructure of a material, referred to as the observations O. We can then characterize the geometry of the microstructure by computing a number of statistical features including the covariance, the granulometry or the granulometry of the complementary image. The covariance of a random set A was dened in chapter 2. It is the function

C A dened on R d × R d by C A (x, x + h) = P (x ∈ A, x + h ∈ A), (3.1)
where h is some vector of R d . For a stationary random set, the covariance is a function of the distance h only:

C A (x, x + h) = C A (h). (3.2)
The granulometry is an other statistical characteristics of the microstructure, constructed as follows. Let K be a convex set. We consider the family of structuring elements (K λ , λ > 0), where K λ = λK. The operator dened for all closed set A ⊂ R d by

Φ λ (A) = (A ⊖ Ǩλ ) ⊕ K λ , (3.3)
where ⊖ is the morphological erosion and ⊕ the morphological dilation, is a granulometry. The granulometry by openings of a random set A provides a characterization of the size distribution of the elements of A. The random set A can be characterized by recording, for distinct values of λ, the volume fraction of the residual set A\Φ λ (A).

Granulometry measurements can be computed on the random set A, as well as on its complementary set A c .

In the remainder of this chapter, we will denote by m O and m Θ the statistical measurements conducted on the observations and the statistical measurements conducted on simulated microstructure, respectively. The parameters Θ of the model are typically selected to minimize the dierence between m O and m Θ . The determination of the parameters of morphological model can therefore be formulated as the following constrained optimization problem:

Θ := arg min Θ ∥m O -m Θ ∥ 2 2 subject to Θ ∈ D.
(3.4)

In (3.4), D denotes the set of admissible parameters for the model.

Bayesian formulation

Starting from measurements obtained from a morphological model and from experimental observations of the studied microstructure, we use a Bayesian approach to determine the parameters of the model. The likelihood that the observations were obtained with the set of parameters Θ can be dened as follows:

p(O|Θ) = 1 2π|Λ| exp((m O -m Θ ) T Λ -1 (m O -m Θ )), (3.5)
In Eq. (3.5), we usually consider a diagonal covariance matrix Λ and we assign a weight to the parameters corresponding to their respective importance. The arbitrary choice of a Gaussian distribution is motivated by its simplicity, but other distributions could potentially be considered for the likelihood.

We resort to physical considerations to dene a prior distribution on the parameters Θ. Usually, we are able to dene a range of variation for all parameters of the model based on their physical validity:

Θ min ⪯ Θ ⪯ Θ max , (3.6)
where ⪯ is a component-wise inequality. We simply consider a uniform distribution between the lower and the upper bound for each parameter as prior distribution:

p(Θ) := U Θ min ,Θmax .

(3.7)

Thanks to Bayes formula, we obtain an expression for the posterior distribution of the parameters Θ knowing the observations:

p(Θ|O)p(O) = p(O|Θ)p(Θ).

(3.8)

Since the (unknown) probability law for the observation is independent from Θ, the posterior distribution is proportional to the product of the likelihood and of the prior distribution:

p(Θ|O) ∝ p(O|Θ)p(Θ).

(3.9)

The problem of the model parameterization becomes the determination of a set of parameters maximizing the posterior probability p(Θ|O):

Θ = argmax Θ p(Θ|O).
(3.10)

Sampling from the posterior distribution

A question now arises, which is how to solve the optimization problem from Eq. (3.10) in practice? An obvious solution could be to use a gradient ascent algorithm to nd local maxima in the parameters space. In this section, we propose a distinct approach consisting in using a Monte Carlo Markov Chains (MCMC) algorithm to generate samples from the posterior distribution [START_REF] Andrieu | An introduction to mcmc for machine learning[END_REF][START_REF] Robert | Monte Carlo statistical methods[END_REF]. This approach oers several benets compared to gradient ascent. Computing the gradient of the measurements with respect to the parameters is often challenging due to two primary factors. Firstly, there is usually no readily available analytical formula to calculate the gradient. Secondly, the function m Θ is a stochastic function dependent on the parameter set Θ, making it dicult to estimate the gradient using a nite dierence scheme. Moreover, the MCMC approach possesses an additional advantage as it enables the generation of samples that accurately represent the posterior distribution.

This capability allows for the capture of noteworthy characteristics of the microstructure, including parameter correlations and characteristic ranges of parameter variation.

The algorithm that we proposed works as follows:

Initialization: At initialization, we rst generate a set of parameters from the prior distribution:

Θ 1 ∼ U Θ min ,Θmax .
(3.11)

n-th iteration: At each iteration, we repeat the following steps:

1. Parameters sampling we generate a set of parameters Θ from the current state Θ n according to a proposal distribution q:

Θ ∼ q( Θ|Θ n ) (3.12)
2. Simulation We compute a simulation of the morphological model with this set of parameters, and perform measurements on the computed microstructure.

The measurements m Θ are used to compute the posterior distribution, up to the constant factor p(O):

p( Θ|O) ∝ p(O| Θ)p( Θ).

(3.13) 3. Accept/reject step We compute the so-called Hastings ratio r, dened by:

r = min 1, p( Θ|O)q(Θ n | Θ) p(Θ n |O)q( Θ|Θ n ) (3.14)
Noteworthy, the Hastings ratio r is independent of the quantity p(O). The new set of parameters Θ is accepted or rejected with probability r:

Θ n+1 = Θ if u < r Θ n otherwise (3.15)
where u is a random variable sampled from the uniform distribution U(0, 1).

3.2

Numerical experiments

Experiments

To illustrate our methodology, we consider two examples of microstructures generated with morphological models. Our objective is to recover the parameterization of these models from statistical measurements conducted on realizations of the models. To that end, for each example, we generate a set of N = 10 random microstructures in a domain of size 30 by 30 in R 2 . Realizations of both morphological models are displayed in Fig. 3.1. To obtain a set of measurements m O for each morphological model, we averaged the covariance, the granulometry and the granulometry of the complementary as computed on 10 model realizations.

The upper and lower bounds for the parameters used to dene the prior distribution are indicated in table 3.1 for the Boolean model and in table 3.2 for the Cox-Boolean model.

We consider the same diagonal covariance matrix Λ in the likelihood function (3.5)

for both morphological models. To put more emphasis on the correlations at the smallest scales, we employ the following expression for the diagonal coecients of Λ:

Λ n,n = λ 1 + 1 n -1 , (3.16)
where λ = 0.1. Next, we initialize the Metropolis-Hastings algorithm by sampling a rst set of parameters Θ 1 := (θ 1 , R 1 ) from the prior distribution. At each step of the algorithm, the proposal distribution for each parameter is the truncated normal distribution:

q(θ|θ n ) = N tr (θ n , σ θ , θ min , θ max ), (3.17)
where θ min (resp. θ max ) is the minimal (resp. maximal) possible value of the parameter as established for dening the prior distribution, and the standard deviation σ is set to:

σ θ = θ max -θ min ∆ .
(3.18)

We x ∆ = 20 to get an acceptation rate of 0.3 in the algorithm.

We stop the algorithm after 500 iterations, and we discard the rst 100 parameters sampled by the algorithm, as they remain highly dependent on the initial conditions.

Finally, we estimate the set of optimal parameters for the microstructure by considering the maximal a posteriori (MAP) value found for the sampled parameters:

Θ := arg max n=100,...,500 p(Θ n |O).

(3.19)

Results and discussion

The optimal set of parameters Θ found by the Metropolis-Hastings algorithm are displayed in Tab. parameter has experienced only minor variations throughout the iterations of the algorithm, it implies that when this parameter deviates from its optimal value, there is a substantial deviation from the targeted measurements. In other words, even small changes in the parameter can have a signicant impact on the agreement between the predicted measurements and the desired outcome. Overall, we can note that there is a good agreement between the parameters identied by the algorithm and the ones corresponding to the experimental microstructure. A comparison between the measurements as conducted on the target microstructures simulated with both morphological models and as obtained with the optimal set of parameters Θ is displayed in Fig. 3.2 and 3.3.

θ e [L -1 ] R e [L] θ [L -1 ] R Target 0.
Again, we note a fairly good agreement between the measurements. Fig. 3.4 displays a scatterplot of the parameters sampled by the Metropolis-Hastings algorithm for both morphological models. Interestingly, we note that the parameters θ and R sampled by the algorithm are strongly correlated in the case of the Boolean model of disks. We performed a linear regression between the parameters θ and 1/R 2 , to nd a coecient of determination equal to 0.81. For a Boolean model of disks, the volume fraction v is known to be given by v = 1 -exp(πR 2 θ).

(3.20)

This leads to the linear relationship

1 R 2 = - π log(1 -v)
θ.

(3.21)

The covariance measurements are highly sensitive to the volume fraction. Hence, by construction, the Metropolis-Hastings will usually select a set of parameters that allows to obtain a volume fraction similar to the one measured on the experimental samples.

As evidenced in Fig. 3.4 , the parameters are indeed distributed around a straight line.

For the Cox-Boolean model, we note that the parameters θ and R sampled by the algorithm remain correlated, but the correlation is less obvious than for the Boolean model. A linear regression performed between the parameters θ and 1/R 2 yields a coecient of determination equal to 0.51. Due to the presence of exclusion zones in the microstructure, Eq. (3.21) is not valid anymore, which explains this reduced correlation.

Experiments

In this section, we demonstrate the application of our proposed optimization approach to investigate an experimental microstructure. The microstructure under examination consists of pigments embedded in a matrix, forming a colloid lm. These pigments possess an elongated ellipsoidal shape similar to that of a needle. To capture the microstructure for analysis, a dataset of experimental images was acquired using transmission electron microscopy (TEM) imaging. Figure 3.6 provides an illustrative example of one such TEM image. The analysis of the experimental TEM images reveals a noticeable tendency of the pigments to aggregate within the microstructure, resulting in the formation of clusters. These clusters exhibit a characteristic size that signicantly exceeds the individual pigment size.

To describe the microstructure, we use a two-scale model as follows:

The The description of the microstructure model requires eight parameters. These parameters include the intensity θ i of the Poisson point process for the aggregates, their radius R i , the lengths of the semi-axes L and l of the pigments, along with their corresponding standard variations σ L and σ l , the intensity θ of the point process used for the pigments and the hardcore distance h.

To determine the model parameters, we employ the Bayesian approach explained in the preceding section. We initiate the Metropolis-Hastings algorithm by setting λ to 1 during the simulation, and we terminate the algorithm after 200 iterations. Similar to before, we estimate the optimal parameter set for the microstructure by selecting the sample that exhibits the highest posterior probability among all the sampled parameters. Throughout the MCMC run, we generate TEM images of the microstructure by rst creating a 3D sample of the model, then extracting a thick slice from the 3D volume that matches the thickness of the experimental slices used for obtaining the TEM images, and nally projecting the thick slice onto a single plane. An illustrative example of the microstructure generated using this approach is presented in Figure 3.6. simulated with the morphological model using the optimal set of parameters identied by the Bayesian approach [Figliuzzi et al., 2021]. Covariance and granulometry measurements are subsequently performed on both the experimental and simulated TEM images.

θ i (µm -3 ) R i (µm) θ (µm -3 ) L (µm) l (µm) h (
The optimal set of parameters Θ determined by the Metropolis-Hastings algorithm for the morphological model is shown in Table 3.3. A comparison is presented in Fig- ure 3.5 between the measurements conducted on the experimental microstructure and those obtained using the optimal set of parameters. We observe that the optimal parameter set enables us to achieve a good agreement between the experimental measurements and the measurements performed on simulations of the morphological models.

Conclusion

In this chapter, we presented a Bayesian approach for determining the parameters of morphological models of microstructure using measurements obtained from experimental microstructure images. Through various examples, we demonstrated that this approach eectively identies the optimal parameters for dierent morphological models and detects potential correlations among the model parameters. Our proposed methodology oers several advantages compared to other parameter determination approaches.

Specically, it eliminates the need for computing the gradient of the model with respect to the parameters and provides a sequence of parameter samples, enabling the quantication of the parameters sensitivity and the identication of potential correlations between them. given image based on color and texture information [START_REF] Chang | Machine learning for image segmentation[END_REF][START_REF] Alais | Fast macula detection and application to retinal image quality assessment[END_REF].

Any superpixel segmentation can be represented by a graph whose nodes represent the superpixels of the image and whose edges represent the adjacency relations between superpixels. Using dierent features extracted from each superpixel, we showed that it was possible to learn a distance characterizing the dissimilarity between neighboring superpixels with a limited number of images. We nally relied on a generalization of the Eikonal equation to graph structures to merge the superpixels of the image into an actual segmentation. I describe these researches in chapters 4 and 5. In chapter 6, I present a study conducted in collaboration with the Institut de Physique de Nice, which explores the second line of research mentioned previously, and aims to segment images in a supervised way from a dataset of images synthesized with a morphological model.

Chapter 4

Eikonal-based superpixels Superpixel algorithms are a class of techniques that seek to divide an image into smaller regions consisting of similar pixels, providing more meaningful and analyzable representations than raw pixels. One major advantage of superpixels is that they signicantly reduce the amount of calculations required for further processing, as their number is much lower than that of the original pixels. Moreover, due to their homogeneous nature, superpixels form subregions of the image that are particularly relevant for feature computation. As a result, superpixel algorithms are frequently used as a pre-processing step in various applications such as object classication, image segmentation, or depth estimation, as demonstrated by studies like [START_REF] Zitnick | Stereo for image-based rendering using image oversegmentation[END_REF] and [START_REF] Fulkerson | Class segmentation and object localization with superpixel neighborhoods[END_REF].

A signicant amount of literature in computer vision and image analysis is dedicated to the topic of superpixels [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF][START_REF] Li | Superpixel segmentation using Linear Spectral Clustering[END_REF][START_REF] Stutz | Superpixels: An evaluation of the state-of-the-art[END_REF].

Graph based methods rely on a graph representation of the image. These methods include the normalized cut algorithm (NC) of Shi and Malik [START_REF] Malik | Contour and texture analysis for image segmentation[END_REF][START_REF] Shi | Normalized cuts and image segmentation[END_REF] or the ecient graph-based segmentation algorithm (GS) of Felzenswalb and Huttenlocher [START_REF] Felzenszwalb | Ecient graph-based image segmentation[END_REF]. By contrast, clustering based methods proceed by iteratively rening clusters of pixels until some convergence criterion is met. These methods notably include mean shift [START_REF] Comaniciu | Mean shift: A robust approach toward feature space analysis[END_REF],

watershed [START_REF] Beucher | Use of watersheds in contour detection[END_REF][START_REF] Beucher | The morphological approach to segmentation: the watershed transformation[END_REF]Figliuzzi et al., 2017;[START_REF] Meyer | Morphological segmentation[END_REF][START_REF] Vincent | Watersheds in digital spaces: an ecient algorithm based on immersion simulations[END_REF], turbopixel [START_REF] Levinshtein | Turbopixels: Fast superpixels using geometric ows[END_REF] or waterpixel [Cettour-Janet et al., 2019;Machairas et al., 2015] algorithms, respectively.

This chapter presents in a synthetic manner the work conducted during the PhD thesis of Kaïwen Chang on the development of a novel superpixel algorithm [START_REF] Chang | Machine learning for image segmentation[END_REF]. Our objective was to use superpixels as a rst segmentation step to perform the complete segmentation of an image by region merging techniques. We noticed during this work that the quality of the resulting segmentation was strongly related to the quality of the initial superpixel segmentation. Hence, we dedicated part of our eorts to the development of a clustering-based algorithm for generating a superpixel partition, which will be referred to as fast-marching based superpixels (FMS) algorithm in the remainder of this manuscript. In this algorithm, following an idea originally introduced for the Eikonal-based region growing for ecient clustering algorithm (ERGC) of Buyssens et al. [Buyssens et al., 2014a,b,c], we used the Eikonal equation to compute the superpixel partition of the image. This work resulted in two scientic publications [START_REF] Chang | Machine learning for image segmentation[END_REF][START_REF] Alais | Fast macula detection and application to retinal image quality assessment[END_REF], which we summarize in the present chapter.

Algorithm

The FMS algorithm operates by likening the growth of regions on an image to waves propagating through a non-uniform medium, where the growth rate depends on the local color and texture. The wave propagation is governed by the stationary Eikonal equation.

Hence, the task of constructing the superpixel partition essentially involves solving the Eikonal equation on the image domain using a velocity eld that is determined by the local color and texture. A similar approach was introduced by Buyssens et al.

in 2014 [Buyssens et al., 2014a,b,c], but there are notable dierences between the two methods, such as the expression for the local velocity as a function of the image content, the use of texture features, and the region update mechanism during propagation.

The Eikonal equation on a bounded domain

We recall in this paragraph basic notions on the stationary Eikonal equation, already discussed in section 2.3 of this manuscript. Let Ω denote some bounded domain in R 2 .

In what follows, we consider a wave front emerging from a set of points on the domain and propagating at a velocity u := u(x) specied at every location x ∈ Ω. Let us denote by x → T (x) the function associating with each point x ∈ Ω the rst arrival time of the propagation front. It can be shown [START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF] that T is solution of the so-called stationary Eikonal equation

∥∇T (x)∥ = 1 u(x)
, ∀x ∈ Ω.

(4.1)

The stationary Eikonal equation (4.1) must be complemented by boundary conditions specied on ∂Ω. One usually considers a function g dened on the boundary ∂Ω so that T (x) = g(x) for all x ∈ ∂Ω. Generally, the function g is taken to be identically 0.

Hence, the stationary Eikonal equation becomes

   ||∇T (x)|| = 1 u(x) , ∀x ∈ Ω T (x) = 0, ∀x ∈ ∂Ω. (4.2)
For all x in Ω, the solution T (x) of Eq. (4.2) can be interpreted as the minimal time required to travel from x to the domain boundary ∂Ω. In other words, the Eikonal equation allows us to compute the shortest distance between any point x of the domain Ω and the boundary ∂Ω. Ecient numerical methods can be found in the literature to solve the Eikonal equation on a domain. Among these methods, the fast marching algorithm, originally introduced by Sethian et al., ranks among the most popular. It works by iteratively following the wavefront propagation and computing the rst arrival time step by step [START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF].

Region growing

We describe in this section the practical implementation of the superpixel generation algorithm. Let us rst introduce some notations. A pixel in image I is denoted by p and its coordinates in the image by (x, y). We denote by C(p) the color at pixel p in the CIELAB color space. Similarly, we denote by T(p) a vector of features characterizing the local texture at pixel p. The proposed algorithm is equivalent to solving the stationary Eikonal equation on the image domain with the fast marching algorithm for a velocity eld depending upon the local content of the image.

Let K denote the requested number of superpixels. We initialize the algorithm by selecting K seeds on a regular grid. To avoid placing a seed on a boundary, following [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF], we place the seed at the local minimum of the gradient in a 3 × 3 neighborhood of the nodes of the grid.

For each seed s i , we dene a velocity eld u i (p), which depends on the color and on the texture of both the seed and the location p := (x, y) in the image. The labels of the seeds are then gradually propagated from the labeled pixels to the unlabeled pixels according to the local velocities (u i (p)) i=1,..,K . The propagation is described by the Eikonal equation

   ||∇T (p)|| = 1 u(p) , ∀p ∈ I T (p) = 0, ∀p ∈ ∂I. (4.3)
In this expression, u(p) is the local velocity at pixel p, ∂I corresponds to the subset of the image I constituted by the seeds {s 1 , s 2 , ..., s K } and T (p) is the minimal traveling time from ∂I to p. Once seeds have been selected, we solve Eq. (4.3) using the fast marching algorithm, whose cost is known to be in O(N log N ) [START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF][START_REF] Sethian | Fast marching methods[END_REF], N

being the number of pixels in the image.

Local velocity model

We can use the algorithm described in the previous paragraph with any non-negative velocity model u(p). One of the main diculties related to superpixel segmentation is that certain regions of the image are extremely textured. In these regions, the colorimetric distance between two pixels presumes in no way that these two pixels belong to the same cluster or not. To account for texture, we considered a velocity model incorporating both local color and texture information. This velocity model is based on a distance D between the image pixels and the seeds dened by

D(p, s i ) = w 0 ∥C p -C i ∥ 2 + w 1 d(T p , T i ), (4.4)
where T p is a vector of features characterizing the local texture at pixel p, T i is the corresponding vector of features for the i-th cluster center s i , d(T p , T i ) a texture distance between the pixel p and the i-th cluster center, and w 0 and w 1 are positive coecients weighting color and texture contributions, respectively.

Several approaches can be implemented to obtain the features vector T p . In our study, we constructed a texton map based on some texture classier and we associated with each pixel in the image an histogram of textons computed in a local window of xed size. We could then dene a texture distance between two pixels of the image by considering the χ 2 distance between the textons histograms associated with both pixels.

The local velocity u i (p) associated with the propagation of region i is computed by relying on the exponential kernel

u i (p) = exp(-w 0 ∥C p -C i ∥ 2 -w 1 d(T p , T i )).
(4.5)

In the FMS algorithm, the velocity is at its maximum and equals 1 when both the pixel and the cluster center have identical color and texture characteristics. The weighting parameters w 0 and w 1 that are used to compute the color and texture distances need to be chosen carefully. Increasing these parameters leads to better adherence to the image boundaries because the propagation velocity becomes more sensitive to variations in color and texture. However, using high values of w 0 and w 1 (i.e. > 8) can cause the local velocity to become very low. This, in turn, can result in the creation of many small, isolated superpixels that correspond to low-contrast regions of the image, which can degrade the compactness of the nal superpixels partition. Therefore, the choice of parameters involves a trade-o between boundary adherence performance and topological considerations.

Renement

Images often contain regions of interest that vary greatly in scale, making it challenging to select seeds for superpixel segmentation using a uniform grid. To achieve better boundary adherence in the superpixel segmentation, we employed a renement strategy that utilizes the map of arrival times.

The map T (p) obtained at the end of the superpixel generation algorithm contains information about the arrival time at each pixel p, and can be used to rene the superpixel partition by adding additional seeds. When the arrival time at a pixel p within a region R i associated with seed s i is high, it indicates that the region boundary propagating from s i has passed through pixels that are highly dissimilar to s i . Therefore, the arrival times in each region R i can be used as a criterion to determine if that region should be split further. A similar renement strategy is employed in [START_REF] Buyssens | Eikonal-based region growing for ecient clustering[END_REF], where the distance map is used to generate new superpixels for renement at the end of the over-segmentation process.

The renement is conducted as follows. Let us denote by B the set of pixels belonging to the superpixel boundaries and by δ(B) the dilated set of B by a disk of radius 2 pixels. Then, the maximal arrival time in region R i is dened to be

t i = max p∈R i ∩δ(B) c T (p), (4.6)
where δ(B) c is the complementary set of δ(B). To further rene the superpixel segmentation, we select the k regions with highest t i and we add seeds at the corresponding locations before re-propagating, k being a parameter xed by the user. We exclude the pixels that belong to the region δ(B) to avoid implanting a seed directly on a boundary. This procedure is repeated iteratively until the desired number of superpixels is obtained and signicantly improves the boundary adherence of the resulting superpixel partition.

It must however be noted that the renement strategy usually reduces the compactness of the obtained superpixel segmentation by increasing the density of superpixels in selected regions of the image.

Comparison with other algorithms

It is interesting to point out the main dierences between the FMS and the ERGC algorithms [Buyssens et al., 2014a,b,c]. ERGC is a clustering-based algorithm that computes superpixel partitions by relying upon the Eikonal equation. The most signicant dierence between both algorithms is related to the velocity models that are employed in both cases. For ERGC, the local velocity eld is simply given by:

u i (p) = 1 ||C p -C i ∥ 2 . (4.7)
In particular, the velocity is very high when the pixel and the seed have similar color characteristics. Another important dierence is that ERGC only considers local color information to compute the local velocity, while FMS is also able to incorporate texture information. FMS is also very close to an algorithm referred to as Iterative Spanning Forest (ISF) [START_REF] Vargas-Muñoz | An iterative spanning forest framework for superpixel segmentation[END_REF], which constructs superpixels by iteratively merging pixels with the aim of maximizing intra-clusters color and spatial distance proximity. The maximization is conducted through a greedy procedure. Even if ISF cannot be directly interpreted in terms of waves propagating according to the stationary Eikonal equation, the greedy procedure for merging superpixels is similar to the fast marching algorithm used to compute the superpixel partition in FMS and ERGC.

Overall, the main dierences between the proposed approach and both ISF and ERGC are the incorporation of texture information and the choice of the velocity model to compute the clustering, as well as the renement procedure and the choice of initial seeds, that are not implemented for ISF and ERGC, respectively.

Experiments and discussion

We evaluate in this section the performance of the FMS algorithm and compare it with several state of the art superpixel algorithms including SLIC [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF],

ERGC [Buyssens et al., 2014a,b,c] and ISF [START_REF] Vargas-Muñoz | An iterative spanning forest framework for superpixel segmentation[END_REF]. These algorithms share the characteristics of generating superpixels through clustering procedures, either by relying on a K-means algorithm, like SLIC, or on iterative agglomeration procedures, like ERGC, ISF and FMS. To perform the evaluation, we use the Berkeley Segmentation Dataset 500 (BSDS500) [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF][START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF]. This dataset contains 500 images and provides several ground truth manual segmentations for each image. We point out that we also presented in [START_REF] Chang | Machine learning for image segmentation[END_REF]Chang and Figliuzzi, 2020] an evaluation of the performance of FMS on a database of images recombining texture patches extracted from images representing stripes, constructed by Giraud et al. with the intent to evaluate a superpixel algorithm adapted to highly textured images [START_REF] Giraud | Texture-aware superpixel segmentation[END_REF].

Experiments on the BSD500 dataset

To evaluate the performance of the algorithms, we considered the following metrics:

boundary recall, undersegmentation error, compactness and contour density. These metrics are classic in the eld of image segmentation and we refer the reader seeking more details to the articles [START_REF] Chang | Machine learning for image segmentation[END_REF][START_REF] Alais | Fast macula detection and application to retinal image quality assessment[END_REF][START_REF] Schick | Measuring and evaluating the compactness of superpixels[END_REF]. These metrics are represented for a number K of superpixels ranging from 100 to 600.

We also display in Fig. 4.1 superpixel partitions obtained with these algorithms on an image of the BSDS500. In our results, we present several versions of the FMS algorithm: two versions using only color information, and a third one incorporating texture information through an additional texton channel computed with a bank of Gabor lters [START_REF] Jain | Unsupervised texture segmentation using gabor lters[END_REF]. We selected the parameter w 0 = 3 for the color term w 0 , we also displayed the metrics as obtained with the set of parameters w 0 = 1 and w 1 = 0. Finally, we computed the superpixel partition using the parameters w 0 = 3 and w 1 = 3 to evaluate the texture inuence. Additional experiments aiming at characterizing the inuence of the renement and of the seeds initialization are also presented in the original article [START_REF] Chang | Machine learning for image segmentation[END_REF][START_REF] Alais | Fast macula detection and application to retinal image quality assessment[END_REF] related to the present work.

In terms of recall, we can note that the FMS algorithm yields better results than SLIC and ERGC algorithms, and performs slightly better than ISF when w 0 = 3.

However, the boundary recall should be considered along with compactness, and we can note that the compactness of the FMS algorithm is signicantly lower than the one of ISF when w 0 = 3. Interestingly, when specifying w 0 = 1 and therefore favoring the construction of compact superpixels, we are able to obtain a similar boundary recall than the one reached by ISF, but with higher compactness. The gain in terms of superpixel compactness is also clearly visible in Fig. 4.1. The compact FMS algorithm also exhibits a smaller density when compared to all other algorithms except SLIC. The performances of SLIC appear to be signicantly below the ones of the other algorithms. Regarding the compactness, we can note that the compactness decreases between K = 100 and K = 200 for the FMS algorithm. This is caused by the fact that no renement steps are applied when computing the partition with K = 100 superpixels. Due to the renement procedure, the size of the superpixels constructed with the FMS algorithm is locally adapted depending on the content of the image. Therefore, the renement is performed at the expense of the superpixels uniformity: the renement yields superpixels that are heterogeneous in scale and size. We can nally note that in terms of undersegmentation, all considered superpixel algorithms are relatively similar.

Conclusion

During Kaïwen Chang's PhD thesis research [START_REF] Chang | Machine learning for image segmentation[END_REF], we developed a fast-marching based algorithm for generating superpixel partitions of images, building upon an idea initially used in the ERGC algorithm by Buyssens et al. [Buyssens et al., 2014a,b,c].

The signicant contribution of this study was the introduction of a new expression for the velocity term, which allowed for texture information to be incorporated into the computation of the superpixel partition. In addition, dierent strategies were also proposed to rene the segmentation.

We evaluted the FMS algorithm on the Berkeley Segmentation Database 500 and we found that it constructs superpixel partitions with slightly higher recall and similar undersegmentation error compared to similar superpixel algorithms such as ERGC, ISF, and SLIC. Moreover, we demonstrated that the inclusion of texture information improved the compactness of the partition without compromising boundary recall.

One potential extension of this work could involve incorporating gradient information into the local velocity model to better account for contours and image discontinuities. It would also be interesting to attempt to automatically estimate optimal parameters, such as w 0 and w 1 , based on the processed image, a strategy that has been successfully employed in several other superpixel algorithms [Achanta and Susstrunk, 2017;[START_REF] Giraud | Texture-aware superpixel segmentation[END_REF]. Additionally, developing a multiscale version of the algorithm, where superpixels are computed and rened at dierent scales using a pyramid representation of the image, could be useful. This multiscale approach could generate a hierarchy of superpixel partitions and a corresponding saliency map, which could be utilized as a criterion for superpixel merging.

Related works

In addition to this work on the generation of superpixels, we also worked during Kaiwen Chang's PhD thesis on a segmentation algorithm based on the watershed algorithm [START_REF] Beucher | The morphological approach to segmentation: the watershed transformation[END_REF][START_REF] Vincent | Watersheds in digital spaces: an ecient algorithm based on immersion simulations[END_REF]. In general, applying a watershed algorithm directly on the image gradient results in an over-segmented image.

To overcome this problem, markers are classically employed to indicate the approximate locations of the objects to be segmented [START_REF] Meyer | Morphological segmentation[END_REF]]. The primary challenge in marker-controlled segmentation is therefore to determine appropriate markers locations. Our algorithm relies on a multi-scale representation [START_REF] Burt | The laplacian pyramid as a compact image code[END_REF] of the image to select relevant markers at dierent scales, which enables the obtaining of a multi-scale segmentation of the image. Our approach uses the discrete decimated wavelet transform to obtain successive approximations of the image [START_REF] Mallat | A wavelet tour of signal processing[END_REF]. For each approximation, the minima of the gradient image are propagated back to the original image space and selected as markers for the watershed transform. Interestingly, this technique establishes a hierarchical structure for the detected contours, since the most important contours in the image are usually persistent through the subsequent Chapter 5

Eikonal-based region merging

In chapter 4, we presented a superpixel algorithm using the Eikonal equation to compute the superpixel partition of an image. We go one step further in this chapter and present a generalization of this algorithm to the framework of graphs, which allows to merge image regions into an actual segmentation of the image. The chapter is organized as follows: in section 5.1, we introduce a generalization of the Eikonal equation to the setting of graphs. In section 5.2, we explain how the Eikonal equation can be leveraged to perform region merging in the image. We present nally in section 5.3 the results obtained with our algorithm on the Berkeley Segmentation Dataset and compare these results to the ones obtained with a classical algorithm for performing graph clustering:

the normalized cut [START_REF] Shi | Normalized cuts and image segmentation[END_REF]. The research works presented in this chapter was conducted during the PhD thesis of Kaïwen Chang between 2016 an 2019 [START_REF] Chang | Machine learning for image segmentation[END_REF].

Eikonal equation on an undirected graph

We describe in this section how to extend the continuous Eikonal equation to the setting of graphs. A graph is a structure employed to depict a collection of objects, wherein certain objects are connected. The objects are denoted by vertices or nodes, while the connections between pairs of vertices are represented by edges.

Eikonal equation

In what follows, we consider an undirected, path-connected graph G := (V, E). We assume that each edge (i, j) in E connecting vertices V i and V j carries a weight w ij that describes the similarity between nodes V i and V j , and that t : V → R is some function dened on the set V of all vertices of the graph. To keep notations simple, we note t u the value of the function t at vertice u.

Denition 2. Let v be some vertex of the graph. We denote by N v the set of all neighbor vertices of v i.e. the set of all vertices that are connected to v. For all vertices u in N v , the morphological derivative of t at v with respect to u is:

Dt(u, v) := w uv (t u -t v ) + , (5.1)
where the quantity (t u -t v ) + is dened by (t u -t v ) + = max(0, t u -t v ).

CHAPTER 5. EIKONAL-BASED REGION MERGING

Denition 2 allows us to dene the gradient of t in the following manner:

Denition 3. The gradient of t at vertex v is the vector ∇t(v) := (Dt(u, v)) u∈Nv .

(5.2)

Based on denitions 2 and 3, we can propose a formulation of the Eikonal equation adapted to graph structures by using an analogy with the continuous setting. In the continuous setting, the Eikonal equation relates the L p norm of the local gradient to the local velocity u in the open domain Ω:

u(x)∥∇t(x)∥ p = 1, ∀x ∈ Ω.
(5.3)

For a graph structure, the equivalent formulation is therefore

u(v)∥∇t(v)∥ p = 1, ∀v ∈ V, (5.4)
where u(v) denotes a local velocity associated to node v.

Here, we focus on the case where p = ∞. In this case, the Eikonal equation is given at each vertex v of G by u(v) max u∈Nv w uv (t u -t v ) + = 1.

(5.5)

The choices of p = ∞ and of the morphological derivative in Eq. ( 2) are not accidental and expression (5.5) arises naturally if we consider a wave propagating on the entire graph G and arriving at a given vertice v ∈ G. For all neighbor vertices u ∈ N v , we have necessarily

t v ≤ t u + 1 w uv , (5.6)
and there exists a particular neighbor û of v such that t u = t û + 1 w ûv .

(5.7)

If we put these requirements together, we obtain the single equation ∥∇t(v)∥ := max u∈Nv w uv (t v -t u ) + .

(5.8)

It is nally straightforward to dene boundary conditions for the Eikonal equation, by simply selecting a subset ∂G = (v i ) 1≤i≤k of k vertices in V and specifying that ∀i = 1, ..., k, t v i = 0.

Eikonal equation and shortest path distance between two vertices

Denition 4 (Shortest path distance between two vertices). Let us denote by P(u, v)

the set of all paths in G that connect vertices u and v. Since G is path-connected, P(u, v) contains at least one element. A path p in G(V, E) is a collection of edges in E. The distance D(p) associated to path p is simply the sum of the weights of the edges constitutive of p. The shortest path distance between two vertices u and v in G is:

D(u, v) = min p∈P(u,v)

D(p)

(5.9)

Building upon this denition, we can dene the distance function between any vertex v of the graph and the subset ∂G = (v i ) 1≤i≤k to be with boundary conditions t(v i ) = 0, ∀i = 1, ..., k.

d(v, ∂G) = inf i=1,...,k D(v, v i ).

Fast marching algorithm on graphs

We present in this section the generalization of the fast marching algorithm already encountered in chapters 2 and 4 to undirected graph structures. To that end, let us consider an undirected, path-connected graph G := (V, E) such that each edge (i, j) in E carries a weight w ij . The fast marching algorithm seeks to determine the solution of the Eikonal equation. However, instead of iteratively solving this equation for each vertex of G until convergence, it works by following the front propagation within the graph to compute the arrival times. During the procedure, the vertices of G are divided into three distinct subsets:

The frozen set groups all vertices already reached by the propagation front.

Vertices adjacent to the frozen points but not reached by the front yet are grouped in the narrow band.

The remaining vertices constitute a subset referred to as the far away set.

Initialization 1. We aect the arrival time t = 0 to all vertices in ∂G and we add them to the narrow band.

2. We label all other vertices as far away and we aect them the arrival time ∞.

The frozen set is initially empty.

3. In order to keep track of the shortest paths between each vertex in G and the boundary ∂G, we aect the label i to each vertex in {v i , i = 1, ..., k}. All other vertices are labelled 0.

Iteration At each iteration, we extract the vertex v of the narrow band with the smallest arrival time and we label it as frozen. Next, we compute the arrival times for each neighbor u of v not belonging to the frozen set, by solving equation (5.5) and by considering that the arrival times at the neighbor nodes w of u are t w if w ∈ C i and ∞ otherwise, where C i is the subset of G containing the points reached by the front that emerged from the i-th vertex in ∂G. We assume obviously that v ∈ C i . Once the arrival time t of a neighbor point u has been computed, two situations can be encountered:

When u is in the narrow band, it has already been aected an arrival time and it is aected to one of the subsets C j , j = 1, ..., k. If the new arrival time is smaller than the current one, the arrival time is updated and the vertex u is aected to the subset C i .

When the neighbor vertex u is in the far away set, we add it to the narrow band with the computed arrival time and to the subset C i .

Stopping condition

The fast marching algorithm stops when the narrow band is empty.

Application to superpixels merging

We describe in this section an algorithm that allows to perform the clustering of a similarity graph based on the resolution of the Eikonal equation in order to perform the segmentation of color images. A similar idea was proposed in 2014 in [START_REF] Buyssens | Eikonal-based vertices growing and iterative seeding for ecient graph-based segmentation[END_REF].

Let I be a color image. We denote by I(p, q) the color of the pixel located at position (p, q) in the image. In what follows, we assume that I is represented as the union of N disjoint superpixels (S i ) i=1,...,N :

I = ∪ 1≤i≤N S i , (5.11) with S i ∩ S j = ∅ if i ̸ = j.
Region adjacency graph We can associate to the superpixel partition of I a graph referred to as its region adjacency graph G. This region adjacency graph is a representation of the image I as an undirected graph, whose vertices (V i ) i=1,...,N are associated with the superpixels (S i ) i=1,...,N . Two vertices V i and V j are linked by an edge of the graph if and only if the corresponding superpixels S i and S j share a common boundary in the image. In the following, we will adopt the notation G := (V, E) when referring to the region adjacency graph, where V is the set of all vertices (superpixels) in G and E the set of all edges.

We can specify a weight for each edge in E by dening a function w : E → [0, 1] which associates to the edge e ij joining vertices v i and v j a quantity w ij ∈ [0, 1], interpreted as a dissimilarity measure between superpixels S i and S j . Several approaches have been considered in the literature to compute the dissimilarity weights w ij , including the color distance between S i and S j or the strength of the gradient at the boundary. Here, we will assume that, for each pair (S i , S j ) of adjacent superpixels, we were able to estimate the probability p ij that S i and S j belong to the same segment of the image.

It is clear that the quantity

w ij = exp(-p ij ),
(5.12) then denes a dissimilarity measure between vertices V i and V j of G.

Graph clustering According to proposition 2, it is possible to partition a graph G into K ≥ 1 subgraphs by relying on the Eikonal equation. To that end, we start by selecting K vertices (v k ) 1≤k≤K of G. Then, solving the Eikonal equation on G with boundary conditions set to be t(v k ) = 0, ∀k = 1, ..., K allows to compute the distance w(v, ∂G) of the shortest path linking each vertex v ∈ V to the closest vertex in the subset ∂G) := (v k ) 1≤k≤K . Since the graph G is path-connected, for k = 1, ..., K, the subsets

C k = {v ∈ V, w(v, v k ) = w(v, ∂G)}
(5.13) constitute a partition of G into K connected subgraphs. We can use this approach to compute a partition of the region adjacency graph associated to a superpixel segmentation and therefore coarsen the segmentation.

Algorithm The merging algorithm that we proposed works by iteratively solving the Eikonal equation on the region adjacency graph of the superpixel segmentation S of I and adapting the boundary conditions. The algorithm start with an initial segmentation S 0 containing N superpixels. Typical values for N are in the order of 500 to 800 superpixels. Our objective is to signicantly reduce the number of segments in the image to a value around 50 -100. The superpixel merging is conducted as follows:

1. To initialize the algorithm, K vertices (v k ) 1≤k≤K are chosen randomly in the region adjacency graph.

2. The Eikonal equation is solved for the region adjacency graph with boundary conditions t(v k ) = 0, ∀k = 1, ..., K. After this step, the graph is clustered into K separated subgraphs (G k ) k=1,...,K . For k = 1, ..., K, we denote by V k and E k the set of the vertices and of the edges of G i , respectively.

3. For each subgraph (G k ) k=1,...,K , we search for the edge e k in E k with maximal weight w k . We denote by n 0,k and n 1,k the vertices in V k linked by e k . Then, we select the subgraph G j whose maximal internal weight is the highest and we add the nodes n 0,j and n 1,j to the boundary conditions. This step allows to rene the previously obtained segmentation.

4. We solve the Eikonal equation for the region adjacency graph with the updated boundary conditions t(v k ) = 0, ∀k = 1, ..., K + 2, where v k+1 = n 0,j and v k+2 = n 1,j .

5. We iterate between steps 3 and 4 until some stopping criterion is met.

Two distinct stopping criteria can be used in the algorithm. A rst stopping criterion consists in stopping the algorithm iterations when a specied number of segments are obtained, the advantage of this approach being that it enables to control the number of segments obtained in the nal segmentation. However, this approach can potentially yield a segmentation with segments still containing highly dissimilar superpixels. A second stopping criterion consists in specifying a probability threshold t and in iterating between steps 3 and 4 until no subgraph contains weights higher than this threshold.

Results and discussion

We present in this section results obtained on the Berkeley Segmentation Dataset (BSDS500) with the proposed merging approach. Starting from a superpixel partition with roughly 500 superpixels, our objective is to reduce the number of superpixels to around 60-80 by merging them. The oversegmentation that results from this process contains a reasonable number of segments and can serve as a solid foundation for implementing a classication algorithm depending on higher-level features to accomplish the segmentation.

Experiments

Our proposed merging method is agnostic to the choice of superpixel generation algorithm. In our experiments, we started from an image partition containing K = 500 superpixels computed with the fast marching based algorithm described in Chapter 4, and we used this initial partition to construct a region adjacency graph (RAG) on the image. The merging procedure was completed in two steps. The rst step allows to reduce the number of superpixels from N = 500 to N = 100 -200. After the rst step, the weights of the RAG are updated and a second merging procedure is conducted in order to obtain an oversegmentation with N = 50 -100 regions.

Dissimilarity measure

The edges of the RAG must carry a dissimilarity measure constructed through Eq. ( 5.12) from the probability that the adjacent regions associated with the edge belong to the same segment. To compute the dissimilarity measure, we learned the similarity measure between regions by using a regression algorithm, which takes as input features extracted from the pair of adjacent superpixels and returns a score that can be interpreted as the probability of merging each pair of adjacent regions. These features incorporate in particular various color and texture distances between the adjacent superpixels and information on the gradient strength at the boundary separating the superpixels. We refer the reader interested by additional details on the selected features to the manuscript [START_REF] Chang | Machine learning for image segmentation[END_REF].

To train the classier, we use Berkeley Segmentation Dataset 500 (BSDS500) [START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF] in the following manner. For each training image in the BSDS500, we dispose of K distinct manual segmentation. Let T k := (T k i ) i=1,...K k be the k-th segmentation mask in the ground truth, which contains K k segments. We can associate each superpixel S := (S i ) i=1,...N to the region T k = (T k i ) i=1,...K k that it intersects the most:

A k i = arg max j=1,...,Kn A(S i ∩ T k j ),
(5.14)

where A(S i ∩ T k j ) is the area of the intersection and A k i denotes the index of the region associated with S i in the k-th manual segmentation. For a given edge (S i , S j ) of the RAG, we dene the target similarity measure ŵij to be:

ŵij = exp - 1 K K k=1 1 {A k i =A k j } .

Results and discussion

To evaluate the performance of our algorithm, we compared it to a classical approach for performing graph clustering, the normalized cut algorithm (Ncut) [START_REF] Shi | Normalized cuts and image segmentation[END_REF]. In addition to classical metrics including boundary recall and precision, we also report the results in terms of the segmentation covering metrics. Segmentation covering provides a measures of the average matching between a segmentation and a given ground truth. It is dened by

SC(S, S g ) = s i ∈S |s i | |P| max s j ∈Sg |s i ∩ s j | |s i ∪ s j | .
(5.15)

In this expression, S and S g are the segmentation and the ground truth segmentation, respectively. P is the set of pixels, so that |P| corresponds to the total number of pixels in the image. The segmentation covering metric computes, for each segment s i ∈ S, the area of the largest intersection over union with the corresponding ground truth segment s j . The calculation is weighted by the number of pixels within segment s i and normalized by the total number of pixels in the image. The maximum possible value 1 can be achieved when S is identical with S g .

The results obtained with our merging approach are provided in Tab. 5.1. We display one example of obtained segmentation in Fig. 5.1. Overall, we can note that the results obtained with the Eikonal equation on graphs are better than the ones obtained with the normalized cut. In particular, the Eikonal based approach brings signicant improvement over the normalized cut algorithm with respect to the boundary precision and to the segmentation covering metrics. In our experiment, we opted for the second stopping criterion for the Eikonal algorithm, which consists in specifying a probability threshold t and in iterating between the last two steps of the algorithm until no cluster contains weights higher than t. A drawback of this approach is that it makes it dicult to control the number of segments obtained at the end of the procedure. It is interesting to discuss in greater detail the stopping criterion used for Eikonal algorithm, in particular the choice of the threshold t. This threshold can indeed be xed in an adaptative manner depending on the set of weights as observed on the adjacency graph. In our experiments, we followed the following procedure to select the threshold value:

1. Sort the edges increasingly according to their weight.

2. Arbitrarily select a proportion of edges considered to be actual contours, and set the corresponding weight as threshold.

Another interesting thing to notice is that in spite of the renement step, the obtained segmentation depends on the initial choice of germs. Due to the initial random selection of germs, certain areas of the image may become articially over-segmented. To address this issue and minimize the impact of the initial seed selection, a straightforward post-processing operation can be performed at the conclusion of the algorithm. This operation involves extracting the highest merging probability observed at the boundaries between adjacent regions, thereby establishing a dissimilarity measure between these regions. Subsequently, the pairs of regions are sorted in ascending order based on their dissimilarity values. As the algorithm progresses, pairs of adjacent regions are evaluated, and if their dissimilarity falls below a specied threshold for the renement step, they are merged together. It is important to note that each region can only be merged once during this process to prevent the creation of adjacency graph regions with edges exceeding the threshold weight. By implementing this post-processing step, the number of clustered regions can be eectively reduced, while the overall quality of the segmentation is only minimally aected. This post-processing was applied in our ex-periment, which explains why the number of segments obtained with the Eikonal based algorithm and the normalized cut are slightly dierent in Tab. 5.1.

Conclusion

In this chapter, we presented a novel algorithm that performs region merging on the superpixel segmentation of an image. This algorithm is based upon a region adjacency graph representation of the superpixel partition. Each edge in the graph corresponds to a pair of adjacent superpixels and carries a weight accounting for the dissimilarity between these superpixels. The algorithm uses a generalization of the Eikonal equation to the framework of graphs to perform the graph clustering, an idea originally introduced by [START_REF] Buyssens | Eikonal-based vertices growing and iterative seeding for ecient graph-based segmentation[END_REF]. During Kaiwen Chang's PhD thesis, our main contribution was to propose to learn the weights of the graph based on annotated data.

We evaluated the performance of our Eikonal-based approach by comparing it to a classical graph clustering algorithm, namely the normalized cut [START_REF] Shi | Normalized cuts and image segmentation[END_REF]. Our results demonstrated improvements over the classical algorithm, particularly in terms of segmentation covering and boundary precision metrics.

The initial goal of the algorithm was to process images from experiments in materials science. For these images, it is dicult to obtain annotated training data, and therefore it is crucial to develop segmentation algorithms requiring a low number of data to be trained. So far, our merging algorithm has only been applied to perform the segmentation of natural images taken from the Berkeley Segmentation Dataset. Hence, an obvious short term perspective of the work presented in this chapter would be to evaluate it on images obtained during physics experiments. We expect these images to be more homogeneous in terms of content than natural images, so that they might require a lower amount of training example to be segmented using a supervised approach.

Chapter 6

Supervised segmentation from synthesized data

As pointed out in the introduction of this part of the manuscript dedicated to image segmentation, state-of-the-art image segmentation techniques currently rely on supervised learning algorithms including convolutional neural networks. However, segmenting images obtained during physics experiments with these algorithms requires to train them on manually segmented images that are often not available. In addition, the images that we want to study are signicantly dierent from natural images, therefore making The measurement of the concentration eld relies on the detection and segmentation of the particles present in the image [START_REF] D'ambrosio | Viscous resuspension of non-brownian particles: determination of the concentration proles and particle normal stresses[END_REF][START_REF] Snook | Dynamics of shear-induced migration of spherical particles in oscillatory pipe ow[END_REF].

Currently, the algorithms that are utilized for this specic task are relatively classical image processing algorithms, that often rely on mathematical morphology tools [START_REF] Blanc | Microstructure in sheared nonbrownian concentrated suspensions[END_REF][START_REF] D'ambrosio | Viscous resuspension of non-brownian particles: determination of the concentration proles and particle normal stresses[END_REF][START_REF] Dijksman | Refractive index matched scanning and detection of soft particles[END_REF][START_REF] Dougherty | Mathematical morphology in image processing[END_REF]; Kimme 6.2 Segmentation algorithm

Network architecture

To perform the image segmentation, we use the Context Aggregation Network (CAN)

introduced in [START_REF] Yu | Multi-scale context aggregation by dilated convolutions[END_REF]. This network is entirely composed of convolutional layers, making it adaptable to any size of input image. Its main particularity is that it gradually aggregates contextual information without losing resolution through the use of dilated convolutions whose eld of view increases exponentially over the successive network layers. This exponential growth yields global information aggregation with a very compact structure [START_REF] Chen | Fast image processing with fully-convolutional networks[END_REF][START_REF] Yu | Multi-scale context aggregation by dilated convolutions[END_REF].

The CAN architecture is composed of a set of basis layers {L (s) } 1≤s≤ℓ . We modify the output of the original network so that it is composed of an image with two channels corresponding to a segmentation mask M for the granular suspension particles and of an image C used to locate the centers of the particles, respectively.

The detailed architecture of the network is presented in Tab. 6.1. Each block L (s) for s ∈ [[2, ℓ -2]] is made of a 3 × 3 dilated convolution with kernel K (s) and dilation parameter r (s) = 2 s-1 , followed by an adaptive batch normalization layer Ψ (s) [START_REF] Chen | Fast image processing with fully-convolutional networks[END_REF] [START_REF] Paulovics | A supervised algorithm entirely trained on a synthetic dataset to segment granular suspension images[END_REF].

3 × 3 3 × 3 3 × 3 3 × 3 3 × 3 3 × 3 1 × 1 Conv. dilation r (
The penultimate layer of the network is a classic convolution layer with a lter with size 3 × 3. The nal layer is a 1 × 1 convolution used to perform dimension reduction. The neural network produces a segmentation mask M and an image C with bi-dimensional Gaussian functions placed at locations corresponding to the centers of the detected particles. We obtain a labeled image of the detected particles by applying a watershed algorithm [START_REF] Vincent | Watersheds in digital spaces: an ecient algorithm based on immersion simulations[END_REF] to the segmentation mask M , previously thresholded at the value 1/2, with the local maxima of C selected as markers.

Generation of synthetic training images

As mentioned previously, the use of convolutional neural networks can be challenging due to the substantial amount of annotated data required for training. Manual annotation is especially arduous when dealing with large quantities of particles in each image, which can number in the thousands.

To address this issue, we utilize synthetic images created using a morphological model to train the neural network. This approach enables us to acquire training images with corresponding ground truth information without the need for manual annotation of a subset of experimental images. However, generating synthetic images that closely resemble the experimental images is crucial to ensure that the trained neural network architecture has good generalization properties.

Our approach consists in generating gray level images encoded on 8 bits through the use of random morphological models. The image generation proceeds in several subsequent steps:

Step 1. We start by specifying the dimension w × h of the synthetic image and we build a mask specifying the location of the wall of the ow cell at the image borders. We assign distinct gray levels to the mask and to the interior to obtain an intensity image denoted Ī.

Step 2. The experimental images exhibit quasi-periodic stripes patterns. To simulate these patterns, we perturb the intensity at each pixel location [x, y] in the image according to the relationship:

Î1 [x, y] = Ī + 2 i=1 A i cos(2πf i ϕ(x, y)) . (6.1)
In this equation, the amplitudes A 1 and A 2 and the frequencies f 1 and f 2 are specied randomly for each generated image from uniform distributions on specied intervals. The quantities ϕ(x, y) dened at each location are independent random variables drawn from a normal distribution with mean x and standard deviation σ and are used instead of the coordinate x in order to add randomness to the geometry of the patterns.

Step 3. We use a Boolean model of disks to simulate a mask for the particles.

We recall (see Chap 2.1) that the Boolean model is a grain model obtained by implanting independent random primary grains G ′ on the germs {x k } of a Poisson point process with intensity θ. The resulting set G is

G = x k ∈P G ′ x k , (6.2)
where G ′

x k denotes the translated of the primary grain G ′ at point x k . In general, the grains of a Boolean model can overlap. To avoid this, we add the grains of the Boolean model sequentially. When a grain intersects a grain which is already present, we simply remove it from the simulation. The primary grains used to construct the model are random disks whose radii are drawn according to a normal distribution with mean R and standard deviation σ R specied for each image.

In practice, we draw R and σ R from pre-dened uniform distribution for each generated image. A gray level is nally selected independently for each particle according to an uniform law on a specied interval. The gray level background is set equal to 255. This results in the obtaining of a particle image P . The synthetic image is updated by taking the minimum value between the background image Î1 and the particles image P : Î2 [x, y] = min{ Î1 [x, y], P [x, y]} .

(6.3)

The particle image P is used to generate a binary mask image M indicating the presence of the particles in the generated image. In addition, we create an image Ĉ recording the centers (x i , y i ) 1≤i≤N of the N implanted suspension particles by setting:

Ĉ[x, y] = N i=1 1 2πs 2 exp - (x -x i ) 2 + (y -y i ) 2 2s 2 , (6.4)
where s is the size of the selected Gaussian kernel. In this image, each particle is identied by a normalized bi-dimensional Gaussian function. M and Ĉ constitute the ground truth images associated with the synthetic image.

Step 4. To complete the image generation, we add blur to the synthetic image by convolving it with a Gaussian kernel G with standard deviation set equal to 3 pixels, as well as white noise. The synthetic image is therefore described by:

Î[x, y] = max{0, ( Î2 * G)[x, y] + W [x, y]} , (6.5)
where the quantities {W [x, y]} 1≤x≤w, 1≤y≤h are independent centered Gaussian random variables with specied standard deviation.

All the parameters involved in the description of the model can be adapted depending on the set of experimental images under scrutiny. We display in Fig. 6.2 a synthetic image of the suspension constructed with the aforementioned procedure. We remark that synthetic images are visually very close to the suspension images obtained in the experiments.

Training of the neural network

To train the neural network architecture, we generated a training set and a validation set containing respectively 2240 and 360 synthetic images along with their corresponding ground truth images. We used the Euclidean distance between the output of the network and the ground truth images as loss function to train the algorithm, therefore formulating the segmentation as a regression problem. We relied on data augmentation techniques [START_REF] Shorten | A survey on image data augmentation for deep learning[END_REF] to improve the robustness of the network :

the network was fed with random crops of the training images with randomly distorted gray level histogram. To train the neural network, we used the Adam optimizer with a learning rate initially set to 0.1 and a batch size of 8, and we divided the learning rate by a factor of 2 every 50 epochs. We xed the maximal number of epochs to 400, and retained the weights of the epoch that led to the minimal error on the validation set.

CAN K-means

Image Particles Recall Prec.

D[px]

6.3

Results and discussion

Evaluation dataset

In order to quantitatively evaluate the results and investigate the generalization capability of the algorithm to real experimental images, we performed manual annotation on 5 experimental images. Each image was annotated by labeling all suspension particles with a disk, providing the center and radius of each particle. While 5 images may seem relatively small in number, these images are large in size and contain a signicant number of particles, as shown in Tab. 6.2. Consequently, the detection results were tested against a substantial number of particles, ensuring the statistical validity of the ndings. Moreover, we deliberately included a low-quality image in the experimental dataset, featuring a prominent illumination gradient and noticeable blurriness. This specic image is depicted in Fig. 6.3. On average, each image in the dataset contained 1037 particles. Manual annotation of an image typically required one to two hours, illustrating the signicant time investment involved. This highlights the practical challenge of manually annotating an entire set of images for training a convolutional network architecture, which is often unfeasible in many applications.

Detection metrics

In order to evaluate the segmentation results, we developed an approach that allows to establish a one-to-one correspondence between the particles detected by the algorithm (the detections) and the particles present in the ground truth. We refer the reader interested by more details on this approach to the original preprint [START_REF] Paulovics | A supervised algorithm entirely trained on a synthetic dataset to segment granular suspension images[END_REF].

Once the correspondence established between the particles and the ground truth, we determined the number fp of false detections by counting the number of detections not associated with any particle. Similarly, we determined the number fn of undetected particles in the ground truth by counting the number of particles in the ground truth left unassociated. The number tp of correct detections is to the number of correspondences established between the particles of the ground truth and the detections. The ability of the algorithm to properly detect the suspension particles is described in terms of precision and recall, dened by Recall = tp tp + fn , Precision = tp tp + fp .

(6.6) For all correct detections, we computed dierent metrics that characterize the quality of the segmentation including the distance D = ∥c p -ĉq ∥ 2 between the center of the detection and the actual center of the particle as annotated in the ground truth or the intersection over union (IoU) of the particle and the detection, dened by IoU(P p , D q ) = P p ∩ D q P p ∪ D q .

(6.7)

Results and discussion

We compare the results of the convolutional network architecture trained on synthetic data to results obtained with traditional algorithms, including Otsu thresholding [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF], adaptive thresholding [START_REF] Gonzalez | Digital Image Processing[END_REF] and K-means segmentation [START_REF] Bishop | Pattern recognition and machine learning[END_REF]. The results for the CAN neural network, the K-mean segmentation and the Otsu and adaptive thresholding algorithms for the segmentation metrics (precision P , recall R and average IoU between the particle and the detection mask) are reported in Fig. 6.5 and Tab. 6.2, where we also report the average distance D (in pixels) between the centers of the particles and the centroid of their corresponding detection masks.

In general, it can be observed that the convolutional neural network (CNN) performs signicantly better than traditional methods in terms of segmentation quality for all evaluation metrics. However, the algorithm shows slightly lower performance for image #3 in the test dataset, which can be attributed to its lower quality compared to the other images, making segmentation more challenging. Interestingly, all proposed approaches exhibit relatively similar precision results, with values systematically exceeding 0.95. It is mostly for the recall metric that the CNN signicantly outperforms other methods.

Furthermore, there is a substantial improvement in particle center localization, with an average below one pixel for test images using the CNN architecture. Additionally, the intersection over union (IoU) metric shows signicantly better results than conventional algorithms. Notably, adaptive thresholding outperforms K-means or Otsu thresholding, emphasizing the importance of local threshold adaptation rather than relying on global image information.

To illustrate the performance of the compared algorithms, segmentation examples are presented in Figures 6.3 and 6.4. Blue represents correctly detected particles from the ground truth, green indicates false negatives, and yellow highlights false positives superimposed on the ground truth image. It is worth noting that false positives and negatives often occur for particles with ambiguous or inaccurate manual segmentation. In practice, we can therefore consider that the CNN architecture can provide segmentation of similar quality to manual segmentation.

Conclusion and perspectives

In the study described in this chapter, we introduced a novel approach for segmenting experimental images of a suspension by training a convolutional neural network on synthetic images generated with a morphological model. We demonstrated in particular that the CNN exhibits good generalization properties and outperforms traditional segmentation algorithms when applied to real images of the suspension. From a broader perspective, ecient image processing techniques are essential for making the most of images collected during physical experiments. This study highlights the value of using morphological models to generate reliable training samples in situations where annotated images are unavailable, therefore enabling the use of current state-of-the-art supervised approaches in image segmentation.

Related work

In addition to this work on the segmentation of images from rheology experiments, I also worked as part of Robin Alais PhD thesis on the development of a convolutional neural network architecture that evaluates the quality of retinal images by assessing the visibility of the macular region. The algorithm deems an image to be of acceptable quality if the macular region is completely visible and within the eld of view. In addition, the method can pinpoint the location of the fovea with a maximal error of 0.34 mm for acceptable images. The algorithm is based on a lightweight fully-convolutional network, which is several thousand times smaller than state-of-the-art networks classically used for this specic task, and achieves near-human performance for assessing macula visibility and fovea localization. The main advantage of the method is that it can easily be integrated into portable retinographs, reduce the number of ungradable images and therefore save both patient and physician time. It is a signicant step towards automating the screening process for retinal pathologies, including diabetic retinopathy, which constitutes a major global healthcare concern. This work was published in the journal Biomedical Signal Processing and Control [Alais et al., 2020].

Part III Hyperspectral imaging

Hyperspectral imaging is an image modality that captures and processes images at many dierent wavelengths, allowing for the analysis of objects or scenes in great detail. A hyperspectral sensor collects information in the form of a set of spatial images where each image corresponds to a narrow band of wavelengths of the electromagnetic spectrum.

Its spatial resolution is a key characteristic of the sensor: when pixels are too large, the measured spectra is usually a combination of multiple elementary materials present in the observation scene. In this situation, sophisticated post-processing methods have to be applied to identify these materials. The development of these methods has led to the emergence of an active eld of research referred to as hyperspectral unmixing (HU). HU algorithms aim at identifying the spectral signatures or endmembers of the elementary materials and at quantifying their relative contributions or abundances to the hypersensor measurements.

A classical approach in HU is to assume that the measured spectra are linear combi- modeling endmembers as sets, or modeling endmembers as statistical distributions [START_REF] Zare | Endmember variability in hyperspectral analysis: Addressing spectral variability during spectral unmixing[END_REF].

Highly mixed observations. When the mixing is linear, the hyperspectral observations are embedded in a low-dimensional simplex. Geometrical methods try to identify this simplex by assuming the presence of observations either at the vertices or at the boundary of the simplex. These assumptions are however not satised when the observations are highly mixed. Statistical approaches have notably been proposed in the literature to deal specically with this issue [START_REF] Bioucas-Dias | Hyperspectral unmixing overview: Geometrical, statistical, and sparse regressionbased approaches[END_REF].

Nonlinear mixing models. In situations where the incoming electromagnetic wave interacts with more than one material present at the scene, the mixing model is usually nonlinear. The development of HU algorithms adapted to nonlinear mixing remains an active topic of research [START_REF] Heylen | A review of nonlinear hyperspectral unmixing methods[END_REF].

Hyperspectral imaging is traditionally associated with remote sensing applications, but is increasingly used in novel applications including surface characterization of chemical imaging. The optical interactions leading to the hyperspectral observations in these Chapter 7

Oxide detection using hyperspectral imaging

During their production, a number of alloying elements including manganese (Mn), aluminum (Al) or silicon (Si) are usually incorporated into steel sheets. These additives can subsequently cause the formation of selective oxides, which can signicantly impact the surface properties of the produced sheets. The wettability of the surface can for instance be altered by the presence of oxides, which is a signicant concern when additional coatings have to be deposited on the steel to protect it from oxidation.

Preventing the formation of selective oxides at the steel surface is therefore essential, which explains why the development of image processing techniques aimed at detecting their formation has been an active topic of research over the past few years in the steel industry [START_REF] Ferté | Étude et analyse de couches minces par techniques multi-spectroscopiques pour une application sur une ligne de galvanisation[END_REF]Ham et al., 2016;[START_REF] Ham | In-situ spectral emissivity measurement of alloy steels during annealing in controlled atmosphere[END_REF][START_REF] Masci | Steel defect classication with max-pooling convolutional neural networks[END_REF][START_REF] Masci | Multi-scale pyramidal pooling network for generic steel defect classication[END_REF]Zenati et al., 2022a,b,c].

During the course of the PhD thesis of Tarek Zenati, we investigated the possibility of using hyperspectral observations to perform the detection of selective oxides at the surface of steel sheets. One of the main issue is that due to the limited spatial resolution of hyperspectral sensors, the hyperspectral observations measured at each surface location are often highly mixed and potentially incorporate the contribution of several oxide species. The inability to distinguish between these oxides is a barrier to the development of surface analysis techniques based on hyperspectral observations. Our primary contribution during the PhD was to propose a novel approach for detecting the formation of oxide mixtures on steel surfaces and for estimating the thickness of the corresponding oxidized layers. Our approach involves the prior identication of all potential mixtures of oxides that can form on the surface and the computation of a comprehensive dictionary of all associated spectra. This allowed us to bypass the issue of the nonlinear mixing model by reformulating the problem of oxide determination as a sparse regression problem.

The outline of this chapter is as follows. In section 7.1, we present the optical model used to describe the formation of the hyperspectral measurements conducted on the steel. We introduce next the sparse regression based approach in section 7.2. Finally, we summarize in section 7.3 some experimental results obtained during the PhD study.

Optical model

We describe in this section the optical model used to describe the formation of the hyperspectral observations. This model relies on the assumption that the substrate is covered by homogeneous layers constituted of a mixture of several oxides.

Permittivity models A major issue related to the modeling is that it requires a precise knowledge of the permittivity of the oxides that can form at the surface for an extended range of wavelengths. During their formation, the oxide layers are intrinsically heterogeneous and contain several chemical species. As a consequence, the permittivity of the resulting heterogeneous material depends on the permittivities of the pure oxides, on their crystallinity and on the microstructure geometry all together. A potential solution to account for the presence of heterogeneous oxides could be to compute an eective permittivity from the permittivities of the elementary oxides from the mixture by using a homogenization model. This approach is however dicult to follow in practice, for two main reasons: 1. it is dicult to fully account for the complexity of the microstructure geometry with a simple homogenization model and 2. computing the exact permittivity of the pure oxides is in itself a dicult task.

To overcome these diculties, we used experiments conducted on oxidized samples in the PhD study [START_REF] Ham | Propriétés physico-chimiques des oxydes de surface et analyse des données de l'imagerie hyperspectrale[END_REF] to determine the permittivities of elementary mixtures of oxides that can form at the surface. To account for the potentially large range of chemical compositions of the oxide layers that form at the surface, we assumed that these layers contain several elementary mixtures. To compute the eective permittivity of the resulting heterogeneous oxide, we used the self-consistent model [START_REF] Tolstoy | Handbook of infrared spectroscopy of ultrathin lms[END_REF], for which the eective permittivity ϵ SC is dened implicitly through the equation

n i=1 p i ϵ i -ϵ SC ϵ i + 2ϵ SC = 0, with n i=1 p i = 1, (7.1)
where n is the number of elementary oxides, and p i and ϵ i the volume fraction and the permittivity of the i-th elementary oxide, respectively.

Reection on multiple layers The model describing the formation of the hyperspectral observations relies on a simplied representation of the surface geometry, described as a stack of homogeneous oxide layers parallel to the surface. We denote by xy the plane parallel to the surface and by z the orthogonal direction. According to the superposition principle, we can decompose the electric eld E onto a eld E f that propagates in the forward direction and a eld E b that propagates in the backward direction [START_REF] Tolstoy | Handbook of infrared spectroscopy of ultrathin lms[END_REF]]:

E = E f e ikzz + E b e -ikzz .
(7.2)

In Eq. (7.2), k z denotes the component of the wave vector along direction z. Let us denote by z k the location of the interface separating layers k and k + 1. At each layer interface, according to Young-Fresnel equations, we have

E f (z - k ) E b (z - k ) = 1 t k,k+1 1 r k,k+1 r k,k+1 1 T k,k+1 E f (z + k ) E b (z + k ) , (7.3)
where t k,k+1 and r k,k+1 are the Fresnel coecients for transmission and reection, respectively. These coecients can be directly linked to the complex permittivity ϵ of the oxides constituting the layers and to the incidence angle ϕ k at the interface. Let us denote by ξ k the generalized complex index of refraction, dened by

ξ k = n k cos ϕ k , (7.4)
where n k is the complex optical index of the oxide in the k-th layer. Then, when the electromagnetic wave is s-polarized, the Fresnel coecients are given by r (s)

k,k+1 = ξ k+1 -ξ k ξ k+1 + ξ k , t (s)
k,k+1 = 2ξ k ξ k+1 + ξ k .

(7.5)

Similarly, when the electromagnetic wave is p-polarized, the Fresnel coecients are

r (p) k,k+1 = ϵ k ξ k+1 -ϵ k+1 ξ k ϵ k ξ k+1 + ϵ k+1 ξ k , t (p) k,k+1 = 2n k n k+1 ξ k n 2 k ξ k+1 + n 2 k+1 ξ k . (7.6)
The propagation of the electric elds within the k-th layer is in turn described by

E f (z + k ) E b (z + k ) = e ikzh k 0 0 e -ikzh k T k (h k ) E f (z - k+1 ) E b (z - k+1 ) , (7.7)
where h k = z k+1 -z k is the thickness of the k-th layer. A global equation describing the eld propagation for the multi-layered geometry can be obtained by computing the product between all transmission matrices. For a geometry with K layers, we have, since there is no electric eld propagating backward in the steel,

E i (0 + ) E r (0 + ) = K k=1 T k-1,k T k (h k )T k,k+1 T (h 1 ,...,h K ) E t (z K+1 ) 0 , (7.8)
where E i , E r and E t correspond to the incident, reected and transmitted elds, respectively. In Eq. (7.8), the indices k = 0 and k = K + 1 are associated to the air and to the steel substrate. In the presence of a single layer of oxide on a steel substrate, the system is described by:

E i (0 + ) E r (0 + ) = T 0,1 T 1 (h)T 1,2 T (h) E t (h -) 0 , (7.9)
where h is the layer thickness. The global transmission and reection coecients t and r for the entire system can easily be computed from the coecients of T (h) as a function of the layer thickness:

r = [T (h)] 21 [T (h)] 11 , t = 1 [T (h)] 11 .
(7.10)

It is nally straightforward to compute the reectance, dened as

R(h) = |r(h)| 2 .
(7.11)

Sparse unmixing approach

We describe in this section our methodology to invert the optical model described in section 7.1. The problem can be summarized as follows: given a hyperspectral observation, is it possible to identify the chemical composition and the thickness of the oxide layers that led to this observation? In what follows, we make the assumption that the measured reectance spectra result from a linear combination of a small number of elementary spectra ⃗ S i corresponding to a layer of thickness h i of some mixture i of k oxides with respective volume fractions ⃗ v i := (v

(1) i , . . . , v (k)
i ):

⃗ S(λ) = n i=1 α i ⃗ S i (h i , ⃗ v i ; λ).
(7.12)

In expression (7.12), λ corresponds to the wavelength and the coecients (α i ) i=1,...,n are the abundances of the dierent mixtures of oxides and can be considered as effective surface fractions for the oxidized layers. This model can be interpreted as an approximation of the optical interactions occurring in the geometrical setting described in Fig. 7.1, where several mixtures of oxide with distinct thicknesses are present in the observation region.

Figure 7.1: Multiple layers of oxides on a steel substrat [Zenati et al., 2023].

A possible approach for characterizing the presence of an oxidized layer is to calculate, for each mixture of oxide potentially present at the surface, the reectance spectra corresponding to dierent reference thicknesses and dierent volume fractions for the elementary oxides present in the layer. The pre-calculated spectra can then be stored in a dictionary D, which takes the form of a matrix of size L × N . The columns of D stack the calculated spectra, L designating here the number of wavelengths. The problem of determining the elementary reectance spectra is then reformulated as the search for a linear combination involving a small number of elementary spectra from the dictionary D that allows to reconstruct the observed spectrum ⃗ S with good accuracy.

Let ⃗ S be an experimental reectance observation. Our aim is to nd a sparse vector ⃗ α such that ⃗ S = D⃗ α in order to represent ⃗ S as a linear combination of a small number of pre-calculated oxide spectra taken from the dictionary. From a mathematical perspective, this can be formulated as the following optimization problem:

⃗ α = arg min ⃗ α∈R N
∥ ⃗ S -D⃗ α∥ 2 + γ∥⃗ α∥ 0 .

(7.13)

In formulation (7.13), the quantity ⃗ α corresponds to the coecients of a vector selecting the elements of the dictionary D used for the reconstruction of the observed spectrum ⃗ S. ∥⃗ α∥ 0 refers to the number of non-zero components of the vector ⃗ α:

∥⃗ α∥ 0 = #{i ∈ 1, N , α i ̸ = 0}.

(7.14) Formulation (7.13) can be seen as the search for a compromise between a vector of coecients ⃗ α that properly reconstructs the reectance ⃗ S through the minimization of the term ∥ ⃗ S -D⃗ α∥ 2 , while remaining parsimonious through the minimization of the term ∥⃗ α∥ 0 . The positive parameter γ establishes a trade-o between the parsimony of ⃗ α and the accuracy ∥ ⃗ S -D⃗ α∥ 2 of the reconstruction: a high value of γ leads to parsimonious solutions to the detriment of the reconstruction accuracy. In practice, problem (7.13) is dicult to solve due to its non-convexity. To obtain an exact solution, it is necessary to consider all possible combinations of parameters, which leads to a combinatorial complexity. Dierent algorithms are nevertheless available in the literature to obtain approximate solutions to the problem, including the orthogonal matching pursuit (OMP) [START_REF] Bourguignon | Exact sparse approximation problems via mixed-integer programming: Formulations and computational performance[END_REF][START_REF] Mallat | Matching pursuits with time-frequency dictionaries[END_REF]. We discuss in the next sections two alternative reformulations of problem (7.13) that present the advantage of being computationally tractable.

Path LASSO approach

A rst alternative to formulation (7.13) is to penalize the problem by the ℓ 1 norm instead of the ℓ 0 norm, leading to the new optimization problem:

⃗ α = arg min ⃗ α∈R N ∥ ⃗ S -D⃗ α∥ 2 + γ∥⃗ α∥ 1 , (7.15)
where the norm l 1 of ⃗ α is given by ∥⃗ α∥ 1 = N i=1 |α i |. The main advantage of this second formulation, known under the name of Least Absolute Shrinkage and Selection Operator (LASSO [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]) is that (7.15) is a convex optimization problem.

Ecient algorithms are therefore available to solve it [START_REF] Boyd | Convex optimization[END_REF][START_REF] Bubeck | Convex optimization: Algorithms and complexity[END_REF][START_REF] Efron | Least angle regression[END_REF].

The Lasso formulation is only relevant when applied to normalized observations. All spectra are therefore subtracted their mean and divided by their standard deviation:

⃗ S(λ) ← ⃗ S(λ) - S σ S , (7.16)
where

S = 1 L L i=1 ⃗ S(λ i ), σ 2 S = 1 L L i=1 ( ⃗ S(λ i ) -S) 2 .
(7.17)

In practical situations, xing the value of the parameter γ that controls the trade-o between the parsimony of the solution and the reconstruction accuracy is not straightforward. The most obvious solution is to x the value of γ a priori. An alternative is to start from a high value of γ and to solve problem (7.15) for decreasing values of the parameter γ, an approach referred to as path Lasso in the literature. Starting with a high value of γ yields a highly parsimonious ⃗ α(γ) solution. By progressively reducing the value of γ, we obtain solutions that are less sparse but that lead to higher reconstruction accuracy. We retain the solution satisfying

N i=1 ⃗ α(γ) i ≃ 1. (7.18)
The fact that the coordinate i of the solution ⃗ α(γ) is non-zero indicates that it was necessary to select the oxide corresponding to the column i of the dictionary D to reconstruct the observed spectrum. As the reconstruction problem is penalized by the ℓ 1 norm, only a small number of coecients are non-zero, and only a small number of oxides are involved in the reconstruction of the spectrum. The criterion (7.18) is relatively easy to interpret: when it is satised, the quantity D ⃗ α(γ) can be seen as a weighted average of dierent elements of the dictionary.

Group LASSO approach

A second alternative to formulation (7.13) is to rely on the group Lasso algorithm. The group Lasso algorithm, introduced in [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF], allows to perform a sparse selection of groups of variables. In the problem that we consider, the hyperspectral observation in the dictionary D can naturally be divided in G distinct groups (G 1 , . . . , G G )

by selecting all spectra obtained with a given mixture of oxides, regardless of the thickness of the oxide layer and of the volume fraction of the elementary oxides constituting the mixture. The group Lasso formulation of the oxide detection problem amounts to solve the following optimization problem:

⃗ α = arg min ⃗ α∈R G ∥ ⃗ S - G g=1 D⃗ α g ∥ 2 + γ G g=1 √ p g ∥⃗ α g ∥ 2 .
(7.19)

In expression (7.19), the quantity ⃗ α g corresponds to the restriction of the vector ⃗ α to the entries of the dictionary D that belong to the g-th (7.20) where the quantities (p g ) g∈ 1,G denote the cardinal of each group. We can note that when each group contains only one variable, the formulation (7.19) reduces to the Lasso (7.15). The main interest of the group Lasso formulation is that it allows to impose the sparsity on the selection of the oxide mixtures that can be present at the surface, regardless of their thicknesses and of the relative contribution of the elementary oxides to the mixture. The group Lasso algorithm therefore tries to recover the hyperspectral observation with a limited number of oxides mixtures, but does not impose sparsity within the spectra corresponding to similar oxide mixtures. It is possible to impose a sparse selection of the spectra within a given group by considering the sparse group Lasso formulation, which consists in adding a ℓ 1 penalty to the vector ⃗ α. The sparse group Lasso formulation reads

group ∀g ∈ 1, G , ⃗ α g [i] = ⃗ α[i] if i ∈ G g 0 otherwise,
⃗ α = arg min ⃗ α∈R G ∥ ⃗ S - G g=1 D⃗ α g ∥ 2 + γ 1 G g=1 √ p g ∥⃗ α g ∥ 2 + γ 2 ∥⃗ α∥ 1 (7.21)
and can be used to perform the spectra identication. Its main dierence with the path Lasso algorithm is that it imposes sparsity on both the selection of the oxides mixtures that can intervene in the observation recovery through the group penalty and on the selection of spectra within a given group through the ℓ 1 penalty.

Algorithm

Before applying the proposed methodology, a prerequisite is to pre-compute, for each mixture of oxides potentially appearing at the substrate surface, the reectance spectra corresponding to dierent reference thicknesses and dierent chemical compositions, before stacking all pre-computed spectra in a dictionary D. The estimation and characterization of the oxides present on the observed reectance spectrum are then carried out in two steps:

1. The rst step of the algorithm consists in identifying a small number of layer of oxides in the dictionary that allows to reconstruct the spectral observation.

This can be done either with the path Lasso formulation or the group Lasso formulation. The purpose of this step is to identify a subset of spectra present in the observation.

2. In the second step of the algorithm, we use the subset S of spectra from D identied by the path or group Lasso algorithm to perform the recovery of the hyperspectral observation ⃗ S. This is done by considering the least-square minimization problem

⃗ α(γ) = arg min ⃗ α ∥ ⃗ S -D S ⃗ α∥ 2 2 s.t ⃗ α ⪰ 0, Card(S) i=1 ⃗ α i = 1. (7.22)
As a general rule, nothing prevents the path or group Lasso algorithms from selecting columns of D corresponding to the same mixture of oxides, but for dierent thicknesses and chemical compositions. We therefore estimate the thickness and the chemical composition of each mixture of oxides eectively selected by calculating their average, weighted by the value of the coecients of ⃗ α(γ). More precisely, for a given oxide O, let us denote by ⃗ α(γ, O) the vector such that:

∀i ∈ 1, N , α(γ, O) i = α(γ) i if the i-th column refers to oxide O 0 otherwise.

(7.23)

Let h ∈ R N be the vector containing, for all n ∈ 1, N the thicknesses of the precalculated spectra in the columns of D. For a given mixture of oxides O, we estimate the thickness of the oxide layer to be: .24) 7.3 Experiments

h(O) = h T ⃗ α(γ, O) ∥α(γ, O)∥ 1 = 1 ∥ ⃗ α(γ, O)∥ 1 N i=1 h i ⃗ α(γ, O) i . ( 7

Numerical experiments

We present and discuss in this section numerical experiments conducted to evaluate the proposed algorithm. In these experiments, we use the optical properties of K = 5 distinct oxides, including aluminum oxide (Al 2 O 3 ), silicon dioxide (SiO 2 ), manganese oxide (MnO) and mixtures of manganese and aluminum oxides (Mn-Al-O, Mn*-Al-O) with dierent stochiometries. The permittivities of the MnO, Mn-Al-O and Mn*-Al-O oxides were obtained using the approach described in section 7.1, while the permittivities of Al 2 O 3 and SiO 2 were directly obtained from the literature [Brun et al., 2013; Meneses et al., 2014]. Based on the optical properties of the oxides, we construct a dictionary D for 8 reference thicknesses ranging from 20 to 300 nm. For each pair of oxide and thickness, we compute the corresponding reectance for 186 wavelengths ranging uniformly from 6.5 to 12.6 microns. We incorporate the reectance of the substrate in the dictionary. The shape of the dictionary D is therefore 41 × 186.

In our experiment, we randomly select k oxides, with k selected randomly between 1 and 3, and k random thicknesses (h 1 , . . . , h k ) between 10 nm and 350 nm according to a uniform law. Surface fractions (α 1 , . . . , α k ) are drawn for the selected oxides according to a uniform Dirichlet distribution with dimension k. Based upon model (7.12), we compute then the optical response of the corresponding geometry. This procedure allows us to simulate the obtaining of experimental reectance spectra. Next, using the dictionary D, we rely on the algorithm described in section 7.2 to estimate the proportion of each one of the considered oxides at the surface and their respective thicknesses. The estimation is then compared to the actual geometric setting. The results of the experiments are summarized in table 7.1. We obtain a precision of 0.91 and a recall of 0.92 when multiple oxides are present at the surface with the path Lasso algorithm, and a precision of 0.92 and a recall of 0.89 with the group Lasso algorithm.

We display in Fig. 7.3 the results of the estimation on two generated samples. The rst sample corresponds to a mixture of Al 2 O 3 (5.7%, estimation: 5.8%) and SiO 2 (94.3%, estimation: 94.2%) with respective thicknesses 234 nm (estimation: 200 nm) and 215 nm (estimation: 214 nm). The second sample corresponds to a mixture of MnO (31.3%) and Mn*-Al-O (68.7%) with respective thicknesses 179 nm and 216 nm. However, our estimation yields a mixture of substrate (10.3%) and Mn*-Al-O (89.7%) with thickness 182 nm. In this case, the proposed approach fails to properly estimate the correct proportion of oxides and even to identify one of the actual oxides present at the surface. However, as can be observed in gure 7.3, the spectrum is still reconstructed with high accuracy. This example illustrates one intrinsic diculty related to our approach: the diculty arises when the elements of the dictionary are highly correlated, meaning that they share similar features. In such cases, multiple combinations of spectra from the dictionary can result in the same reconstruction, which can lead to inaccuracies in estimating the correct volume fraction of oxides in the observed reectance spectrum. The ℓ 1 penalization may not be sucient to overcome this difculty and ensure the accuracy of the reconstruction. This highlights the importance of carefully selecting a suitable dictionary and regularization technique when applying this approach. mixture of SiO 2 and Mn-Al-O; right: mixture of MnO and Mn-Al-O*) [Zenati et al., 2023].

Experiments Samples preparation and measurements

We summarize in this section experiments conducted on steel samples to validate the algorithm. During these experiments, we subjected steel sheets samples to two distinct thermal cycles, referred to as HP1180 and DP980, to control the formation of distinct species of oxides at their surface, while varying the value of the dew point to obtain dierent structures of oxides at the surface.

X-ray photoelectron spectrometry (XPS) and Glow-discharge optical emission spectroscopy (GD-OES) analysis were conducted to characterize the surface composition.

The XPS analysis enabled us to obtain quantitative information on the atomic species directly located at the surface of the samples. The GD-OES analysis, conducted by and SiO 2 oxides referred to as Mn-Si-O mixture in the following, while the thermal cycle DP980 favors the growth of manganese oxides (MnO) at the surface. The high resolution spectra from XPS analysis conrm the presence of these oxides. Scanning electron microscopy (SEM) images of cross-sections of the samples were also registered during the experiments to provide a qualitative view of geometry of the oxide layers.

Results and discussion

The patterns of oxides in the experimental samples are complex and cannot be accurately described by the simple geometric model presented in section 7.1. This model assumes that the oxides form layers with uniform thicknesses on the surface, which only partially approximates the actual geometry. Furthermore, comparing the estimates obtained from our approach to the results of X-ray photoelectron spectroscopy (XPS) measurements is challenging because XPS only characterizes the presence of oxides over a few nanometers, making it dicult to quantitatively compare the results.

Instead, our goal is to qualitatively verify that the oxides detected on the samples are also present in XPS and glow discharge optical emission spectroscopy (GD-OES) analyses and that the geometry matches the observations made using scanning electron microscopy (SEM) at dierent depths and at the surface. For instance, SEM images can reveal the presence of oxide covering layers, which should also be detected by our estimation. It is also possible to qualitatively compare the estimations obtained on different samples. Samples 19 and 22 are made of the same steel grade but have undergone dierent thermal cycles, resulting in a signicant amount of silica oxides detected by XPS, while other samples should contain more manganese oxides.

Our proposed approach requires the construction of a dictionary that stores reference reectance spectra pre-computed with the optical model. To construct the dictionary, we assume that the oxides that can form at the steel surfaces can be of three distinct natures: 1. a mixture of substrate, SiO 2 and Mn-Si-O, 2. a mixture of substrate, Mn-AlO and Al 2 O 3 , and 3. a mixture of substrate and MnO. We assume that these mixtures can be observed with dierent thicknesses and volume fractions for each one of the individual oxide components and we use the self-consistent approximation to compute the eective permittivity of the material. We compute the dictionary of spectra used for the inversion procedure by calculating the optical response of a layer of each considered mixture of oxides for a given range of 8 thicknesses and dierent combinations of volume fractions. This leads to a dictionary containing K = 246 spectra. We present and discuss below the estimation results obtained for one experimental sample for each thermal cycle. We refer the reader to the original article [Zenati et al., 2023] for a complete overview of our results.

HP1180 thermal cycle We present here the estimation results obtained for one of the samples obtained through the thermal cycle HP1180 (sample 19). The estimation conducted from the hyperspectral measurements indicates the presence of a layer constituted of a mixture of steel (v f = 0.27), Mn-Si-O (v f = 0.69) and SiO 2 (v f = 0.04).

The thickness of the layer is estimated to be 139 nm, and its relative abundance is α = 0.89, which indicates that the oxide layer almost covers the entire sample. Traces of Al 2 O 3 are also detected at the surface, with a relative abundance of 0.11. DP980 thermal cycle We present here the estimation results obtained for one of the samples obtained through the thermal cycle DP980 (sample 5). The estimation from the hyperspectral measurements indicates that the main layer present at the surface is a mixture between steel and MnO, with respective volume fractions 0.37 and 0.63.

A mixture of steel, Mn-Si-O and SiO 2 is also detected by the algorithm, as well as a layer of aluminum oxides. The presence of these oxides is conrmed by the XPS analysis. In comparison with samples 19 and 22, the additives used for sample 5 contain more manganese, which is well detected by the algorithm. However, the observations conducted on the SEM images seem to indicate that the proportion of steel at the surface is under-estimated by our analysis. We attribute this under-estimation to the fact that there is a high correlation between the characteristic spectra observed for MnO layers and for the substrate, which leads the estimation algorithm to signicantly over-estimate the presence of MnO oxides to the detriment of steel. Overall, we note that there is a good agreement between the estimations obtained with the proposed approach and the observations that are conducted on the experimental samples. We note in particular that the algorithm succeeds in detecting the presence of a covering oxide layer for sample 19, which is the only one to be completely covered with oxides on the surface. The estimated thickness is also consistent with the observations made on the sample. We can also note that the estimates made on sample 5, and more generally for all samples associated with the DP980 thermal cycle show a relative preponderance of manganese oxides compared to silicon oxides. This is again in agreement with the experimental observations conducted on the samples.

Finally, it should be noted that the estimated reectance curves are close to the curves measured experimentally for all the studied samples. The algorithm has nevertheless important limitations due to the strong correlations that can occur between some of the reectance spectra present in the dictionary. The strong correlation that exists at the wavelengths considered between the reectance of the manganese oxides and that of the steel substrate explains for instance the over-estimation of the proportion of manganese oxides MnO for the samples with the DP980 thermal cycle.

Conclusions and perspectives

In this chapter, we presented a novel methodology that utilizes hyperspectral measurements to detect selective oxide surface layers on steel sheets. Our approach involves pre-computing optical reectance associated with heterogeneous oxide layers of distinct thickness and chemical composition to create a reference spectrum dictionary. This enables us to formulate the highly nonlinear unmixing problem as a sparse regression problem, which can be solved using various approaches, including the path and group Lasso algorithms. The group Lasso algorithm is particularly suitable for the dictionary's intrinsic structure, which contains groups of heterogeneous oxides with the same elementary components but dierent thicknesses and stochiometries. Our numerical experiments demonstrate that our approach can accurately estimate the presence of selective oxides with high recall and precision. However, we also identied a major limitation of our method: when spectra in the dictionary are strongly correlated, dierent oxide combinations can potentially produce similar reectance spectra. To address this issue, incorporating spatial information to enforce consistency between estimates at adjacent locations could be helpful. Total variation based algorithms have been proposed for this purpose in the literature [START_REF] Chouzenoux | Algorithme primal-dual de points intérieurs pour l'estimation pénalisée des cartes d'abondances en imagerie hyperspectrale[END_REF][START_REF] Iordache | Total variation spatial regularization for sparse hyperspectral unmixing[END_REF][START_REF] Moussaoui | Primal dual interior point optimization for penalized least squares estimation of abundance maps in hyperspectral imaging[END_REF], which we will explore in future research. Finally, we applied our estimation method to experimental samples previously characterized using XPS analysis and scanning electron microscopy. The results were qualitatively consistent with the XPS and SEM observations, providing further evidence for the eectiveness of our approach.

Related work on hyperspectral imaging

In addition to Tarek Zenati's PhD thesis work, I had the opportunity to contribute to the eld of hyperspectral unmixing within the European project M3S. This project aimed to develop a multimodal microscopy platform to improve the diagnosis and prognosis of chronic lymphocytic leukemia. As part of the platform, Raman spectroscopy imaging was incorporated.

In this study, our objective was to design a Bayesian algorithm for extracting endmembers and estimating their abundances in situations where the observations were highly mixed but prior information on the abundances was available. Bayesian approaches provide a robust mathematical framework for incorporating prior knowledge into problem descriptions. We extended the Bayesian algorithm initially described in [START_REF] Dobigeon | Joint bayesian endmember extraction and linear unmixing for hyperspectral imagery[END_REF] to handle scenarios where informative priors were readily accessible for the endmember abundances.

Specically, we introduced bound parameters that allowed us to incorporate qualitative prior information on the abundances, which could be estimated using a simulated annealing algorithm. The development of this method was primarily motivated by the challenge of unmixing Raman spectra obtained from B-lymphocyte cells. Due to the limited spatial resolution of the acquisition, the Raman spectra often contained contributions from the nucleus, cytoplasm, and cell membrane. However, it was known that the contributions of the membrane and cytoplasm to the measured spectra were limited, with abundances typically ranging around 0.2 or 0.3 at most. Therefore, their respective spectra were never observed as distinct entities within the available hyperspectral observations. This research study resulted in a conference publication [Figliuzzi et al., 2016b]. on image processing and its applications, in particular in the eld of materials science.

To conclude the manuscript, I would like to provide some perspectives on the dierent research topics addressed in this work. properties of the skin. We also intend to adapt the MCMC approach presented in chapter 3 in order to determine, for given optical properties, which biological parameters of the skin allow for the obtaining of these properties.

Morphological models

Image segmentation My main research focus in the eld of image segmentation is to advance the development of supervised approaches that utilize synthesized datasets for segmenting sets of similar images obtained during physics experiments. These images are often available in large quantities, but annotating them is a time-consuming task. Consequently, using supervised algorithms for image segmentation is generally not feasible. As discussed previously, one potential solution to address this issue is to jointly generate synthesized images that closely resemble the experimental ones and ground truth annotations. This approach is challenging in general, but it could be viable for the specic types of images that we aim to synthesize, such as suspension or microstructure images. These images are indeed characterized by relatively simple geometries. There is a substantial body of literature accurately describing material microstructures using morphological models. Hence, leveraging morphological models could enable the generation of synthesized training datasets, allowing us to train supervised segmentation algorithms for various practical problems, as demonstrated in the research work presented in the manuscript. It remains however to be determined to what extend the approach presented in Chap. 6 can be utilized for other types of image datasets.

In the short term, I am exploring this approach in the eld of medical imaging as part of Pablo Jeken Rico's thesis, which I co-supervise in collaboration with Elie Hachem (Centre de mise en forme des matériaux, Mines Paris, PSL University). The primary objective of this research project is to develop a segmentation model for threedimensional MRI images of aneurysms located at the base of the brain. The limited availability of annotated images poses a signicant challenge for developing the model.

To overcome this challenge, we plan to investigate the generation of synthetic MRI images using morphological models, similar to the approach presented in Chap. 6 for segmenting images from rheology experiments. By articially increasing the size of the available training data through synthetic image generation, we hope to be able to subsequently apply advanced supervised segmentation techniques to these images.

Looking ahead, in the long term, I aim to explore the combined use of morphological models and generative models, such as generative adversarial networks (GANs) [START_REF] Goodfellow | Generative adversarial nets[END_REF], to facilitate the synthesis of training images. For example, it may be of interest to employ a GAN on top of a morphological model to generate realistic noise and blur in the synthesized images.

Hyperspectral imaging The last axis of research described in this manuscript revolves around hyperspectral imaging. As emphasized previously, hyperspectral imaging has traditionally been linked to remote sensing applications, but its usage is expanding into new domains, such as surface characterization and chemical imaging. The optical phenomena that lead to the hyperspectral observations in these applications dier signicantly from those encountered in traditional remote sensing, necessitating the development of specialized signal processing methods. Interestingly, these developments intersect current research topics that remain largely open and generate signicant interest in the hyperspectral unmixing literature: highly mixed observations, nonlinear mixing models and endmembers variability. To address these issues, I would like to explore two distinct approaches.

1. Optimal transport based hyperspectral unmixing A classical approach to hyperspectral unmixing is to formulate the problem within the framework of non-negative matrix factorization (NMF). In that case, the problem boils down to determining a factorization of the hyperspectral observation matrix into two non-negative matrices corresponding to the endmembers and to their respective abundances, respectively. The main issue with NMF is that the factorization is not unique, which triggers the need for some kind of regularization. A common approach to perform the regularization is to exploit the geometry of the observations, which are known to lie in a low dimensional simplex in the observations space. Here, I would like to explore an alternative NMF-based formulation of blind hyperspectral unmixing, where the regularization term is the optimal transport distance between some discrete probability distribution directly related to the abundances and an arbitrary probability distribution [START_REF] Genevay | Stochastic optimization for large-scale optimal transport[END_REF][START_REF] Genevay | Learning generative models with sinkhorn divergences[END_REF][START_REF] Peyré | Computational optimal transport: With applications to data science[END_REF]. By enforcing the algorithm to identify endmembers whose associated abundances follow some pre-specied distribution, this kind of regularization could be useful to deal with highly mixed observations.

2. Variational Autoencoders Let us assume that we dispose of a set (⃗ x i ) 1≤i≤n of n hyperspectral observations in R d resulting from the mixing of k distinct endmembers characterized by some degree of variability. The observations (⃗ x i ) 1≤i≤n can be interpreted as random realizations of a conditional probability distribution p(⃗ x i |S i , ⃗ a i ) describing the mixing model, where S i = (⃗ s i 1 , . . . , ⃗ s i k ) ∈ R d×k and ⃗ a i = (a 1 i , . . . , a k i ) T ∈ R k are the endmembers matrix and the abundances vector that led to the i-th hyperspectral observation, respectively. The quantities S i and ⃗ a i are latent variables of the model and have to be estimated for each observation ⃗ x i . To account for endmembers variability, we can dene a prior distribution on the endmembers matrices (S i ) 1≤i≤n by considering for instance that for i = 1, . . . , n, the endmembers ⃗ s i k ∈ R d are realizations of a multivariate Gaussian distribution with mean ⃗ µ k and specied covariance matrix. The task of hyperspectral unmixing can be summarized as the estimation of 1) the parameters (⃗ µ 1 , . . . ⃗ µ k ) ∈ R d×k characterizing the prior endmembers distribution and 2) of the latent variables (⃗ s i 1 , . . . ⃗ s i k ) ∈ R d×k and ⃗ a i ∈ R k associated with the observation (⃗ x i ), for all i = 1, . . . , n.

My objective is to explore the use of a variational autoencoder (VAE) architectures to perform these estimations [START_REF] Kingma | Auto-encoding variational bayes[END_REF][START_REF] Kingma | An introduction to variational autoencoders[END_REF].

VAEs are constituted of two independent parameterized models: an encoder or recognition model, and a decoder or generative model. Here, we will use the generative model leading to the observations as decoder. This model is parameterized by the quantities (⃗ µ 1 , . . . , ⃗ µ k ). For the encoder g, we will consider a probability distribution parameterized by the output of a neural network f with a set of parameters ϕ, which will be trained to approximate the conditional distribution of the latent variables (S i ) 1≤i≤n and (⃗ a i ) 1≤i≤n knowing the observations. The interest of this approach is 1) that it incorporates the mixing model directly into the decoder, which enables to consider nonlinear models and 2) that it models the endmembers by statistical distributions, which allows to account for endmembers variability [START_REF] Zare | Endmember variability in hyperspectral analysis: Addressing spectral variability during spectral unmixing[END_REF].

  characterized by complex geometries at the microscopic scale. Quantifying the inuence of the microstructure on the macroscopic physical and mechanical properties of a material has therefore become a major research topic in materials science. Furthermore, the exploitation of experimental images of microstructures and the increasing use of new imaging modalities like hyper-spectral imaging open interesting research perspectives in the eld of image processing. I present in this Habilitation manuscript the research work that I have been conducting since 2014 at the Center for Mathematical Morphology at Mines Paris, which revolves around these questions and in particular around the following three axes: 1. The development of stochastic geometry models to characterize and simulate the geometries of microstructures, 2. The development of segmentation algorithms dedicated to material microstructures, 3. The use of hyper-spectral imaging for characterizing the surface of materials.

Figure 2

 2 Figure 2.1: Steel polycristal microstructures are naturally organized in the form of a tessellation.

Figure 2

 2 Figure 2.2: Voronoï (left) and Johnson-Mehl (right) tessellations in R 3 generated with the software VtkSim [Faessel, 2016]. The centers of the tessellation cells are represented in grey. The rst implanted germs are represented with larger radii for the Johnson-Mehl tessellation.

Figure 2

 2 Figure 2.3: Optical microscope cross-section of the coating microstructure with debonded particles in black (2560 × 1920 pixels representing a surface area of 620 × 476 µm 2 ) [Bortolussi et al., 2018].

Figure 2

 2 Figure 2.4: Magnied optical microscope cross-section of the microstructure of the coating (2560 × 1920 pixels representing a surface area of 248 × 186 µm 2 ) [Bortolussi et al., 2018].

Figure 2

 2 Figure2.6: 500×500 pixels slice of a microstructure simulated with the optimized parameters. This slice is taken from a larger 3D volume of size 500×500×500 pixels. A pixel corresponds to a resolution of 0.2428 microns[Bortolussi et al., 2018].

Figure 2 . 7 :

 27 Figure 2.7: Simulated and experimental microstructures comparisons between covariances (a) and granulometries (b)[Bortolussi et al., 2018].

Figure 2

 2 Figure 2.8: Rough Voronoï tessellations (left) and corresponding local velocity eld (right). The tessellations were generated with the same set of germs P on a 50 × 50 domain, discretized on a 500 × 500 regular grid. The local velocity elds are realizations of Voronoï tessellations with respective intensities θ = 0, θ = 0.1, θ = 1 and θ = 10 [Figliuzzi, 2019].

  served microstructure. Specically, aggregates of black carbon particles were positioned outside of an exclusion polymer, simulated through two distinct approaches. The rst approach utilized a Boolean model of spheres, while the second model employed a mosaic model constructed from a Johnson-Mehl tessellation. The aggregates were then characterized as the intersection between a Boolean model of spheres and the complementary region of the exclusion polymer. We simulated the individual carbon black particles within the aggregates by using a Cox-Boolean model of spheres. The multi-scale model was parameterized based on the experimental covariance and on the cumulative granulometry measured on the experimental images.

Fourier-based computations

  to compute eective thermoelastic properties for the material. In particular, we used a Johnson-Mehl tessellation model with anisotropic grain growth to simulate the microstructure of the material. Extensive Fourier computations were conducted on microstructures generated with the model to determine eective thermoelastic properties for the heterogeneous material, which demonstrated excellent agreement with experimental data, particularly in relation to the overall thermal expansion coecient, and adhere to established bounds for polycrystalline anisotropic thermoelasticity.2.6Related publications B. Figliuzzi. Eikonal-based models of random tessellations. Image Analysis and Stereology, 38(1), 15-23, 2019.

Figure 3

 3 Figure 3.1: Sample of the microstructure generated with the Boolean model of disks (left) and with the Cox-Boolean model (right)[Figliuzzi et al., 2021].

  Figure 3.2: Comparison between the experimental measurements and the measurements obtained with simulations conducted with the MAP parameters identied by the MCMC algorithm for the Boolean model of disks [Figliuzzi et al., 2021].

  Figure 3.3: Comparison between the experimental measurements and the measurements obtained with simulations conducted with the MAP parameters identied by the MCMC algorithm for the Cox-Boolean model of disks [Figliuzzi et al., 2021].

Figure 3

 3 Figure 3.4: Scatterplot of the parameters 1/R 2 and θ sampled by the Metropolis-Hastings algorithm for the Boolean model of disks (left) and for the Cox-Boolean model (right). The color intensity of each point corresponds to the frequency of the corresponding parameters within the samples [Figliuzzi et al., 2021].

  rst scale of the simulation describes the pigments aggregates through a Boolean model of inclusion spheres characterized by two parameters, namely the intensity θ i of the Poisson point P process used to construct the Boolean model and the radii R i of the spheres. The second scale of the model simulates the pigments within the aggregates using a Boolean model of ellipsoids whose largest semi-axis follows a normal law with mean L and standard deviation σ L , and whose smallest semi-axes follow a normal law with mean l and standard deviation σ l . A hardcore distance h is introduced in the underlying point process. Ellipsoids are implanted at locations sampled from a Poisson point process with intensity θ restricted to the set constituted by the spheres constitutive of the rst scale of the model. Ellipsoids belonging to the same inclusion sphere share the same orientation, sampled uniformly from Euler's angles.

  Figure 3.5: Comparison between the covariance and the granulometries as obtained onthe experimental TEM images (dashed lines) and on the simulated TEM images for the "needles" microstructure[Figliuzzi et al., 2021].

Figure 3

 3 Figure 3.6: Needles microstructure: experimental TEM image (left) and binary image

  , A. Montaux-Lambert, F. Willot, G. Naudin, P. Dupuis, B. Querleux, E. Huguet. A Bayesian Approach to Morphological Models Characterization. Image Analysis and Stereology, 40(3), 171-180, 2021. Part II Image segmentation In materials science, experimental campaigns can generate large databases of images that require automatic processing for being eciently exploited. Supervised learning algorithms based on convolutional neural networks constitute the state-of-the-art approaches for performing image segmentation. However, training these architectures depends upon the availability of images segmented manually, which are often not available in sucient quantity. This challenge is further compounded by the fact that images from materials science experiments are markedly dierent from the natural images that are typically used for training convolutional neural networks. Consequently, transfer learning techniques alone are insucient to address the lack of training data. As a result, the development of segmentation methods that can eectively handle limited amounts of data remains a crucial area of research. The applicability of the most eective segmentation methods to actual industrial problems is dependent on overcoming this challenge. This can be done in two manners: rst, by pursuing the development of segmentation techniques that require minimal training data; second, by relying on data augmentation and image generation techniques to obtain suciently large training datasets for the task at hand. This part of the manuscript presents my main contributions to these two lines of research. My main contribution to the development of segmentation techniques is related to the work conducted during the PhD thesis of Kaïwen Chang between 2016 and 2019, which I co-directed along with Jesús Angulo at the Centre for Mathematical Morphology [Chang, 2019]. During the course of this PhD thesis, we developed an algorithm based on the Eikonal equation to generate the superpixel segmentation of a

  (a) FMS (w 0 = 3, w 1 = 0) (b) FMS (w 0 = 1, w 1 = 0) (c) FMS (w 0 = 3, w 1 = 3) (d) SLIC (m=10) (e) ERGC (f ) ISF (α=0.5, β=12)

Figure 4 . 1 :

 41 Figure 4.1: Illustration of the superpixel segmentation on an image of the BSDS500, for K = 200 superpixels [Chang and Figliuzzi, 2020].

Figure 4 .

 4 Figure 4.2 displays the recall, undersegmentation error, compactness and density metrics of FMS, SLIC, ISF and ERGC averaged over the 500 images of the BSDS500.

  Figure 4.2: Comparison between SLIC, ERGC, IFS and FMS algorithms on BSDS500 [Chang and Figliuzzi, 2020].

  the distance function to the solution of the Eikonal equation: Proposition 2. Let G be an undirected, weighted graph. Then, the gradient of the distance function d(•, ∂G) satises the Eikonal equation ||∇t(v)|| = 1

  Figure 5.1: Segmentation results obtained with the Eikonal and normalized cut algorithms. The boundary recall are 0.64 and 0.63, respectively and the boundary precision are 0.33 and 0.29. The nal partition comprises 70 segments [Chang, 2019].

  transfer learning techniques unsuitable to overcome the lack of training data. As a consequence, developing eective segmentation methods that can handle limited annotated data is a critical research area. I describe in this chapter research work conducted in collaboration with David Paulovics (Student, Institut de Physique de Nice), Dr. Frédéric Blanc (Researcher, Institut de Physique de Nice) and Théo Dumont (Student, Mines Paris, PSL University) that seek to train a neural network for performing the segmentation of images obtained during rheology experiments based on a dataset of training images entirely synthesized by a morphological model. 6.1 Context Image processing techniques are crucial for interpreting the outcomes of rheometry experiments on non-Brownian suspensions. The two most signicant quantities for characterizing suspension properties are the concentration elds and the viscosity. Traditionally, these are determined by recording images of the suspension particles during ow at regular time intervals. Figure 6.1 illustrates the principle of a recording device, where a at laser sheet illuminates a transparent suspension and excites the uorescence of a dye dissolved in the liquid. A camera perpendicular to the laser plane captures the uorescent light. An image obtained from this device is displayed in Figure 6.1. In this image, the spherical particles of the suspension appear as black disks.

Figure 6

 6 Figure 6.2: Left: experimental image of the suspension. Right: synthetic image of the suspension constructed with our procedure [Paulovics et al., 2023].

  Figure 6.3: Segmentation results obtained for image #3 (a) with the CAN network (b), K-means (c), Otsu thresholding (d) and adaptive thresholding (e). Correct detections (tp) are displayed in blue, false positives (fp) in yellow and false negatives (fn) in green [Paulovics et al., 2023].

Figure 6 . 5 :

 65 Figure 6.5: Average over the 5 images of Tab. 6.2 of the segmentation metrics (recall, precision, IoU) for the CAN, K-means, Otsu and adaptive thresholding algorithms [Paulovics et al., 2023].

  , B. Figliuzzi, T. Dumont and F. Blanc. A supervised algorithm entirely trained on a synthetic dataset to segment granular suspension images. Preprint, 2023. R. Alais, P. Dokládal, A. Erginay, B. Figliuzzi, E. Decencière. Fast macula de- tection and application to retinal image quality assessment. Biomedical Signal Processing and Control, 55, 101567, 2020.

  nations of the endmembers present in the observed surface weighted by their respective abundances. The linear mixing model has a straightforward physical interpretation: it corresponds to situations where each incoming light ray only interacts with a single endmember before reaching the sensor. A large corpus of methods based upon geometrical or statistical approaches have been considered to solve the problem of linear unmixing, which is now relatively well understood [Bioucas-Dias et al., 2012]. Still, a number of problems remain open: Endmembers variability. The spectral signatures of the elementary materials often exhibit intrinsic variability within the observations. The issue of endmembers variability is still an active ongoing research topic in HU. Two approaches are classically employed in the literature for dealing with endmembers variability:

  applications are very dierent to the ones classically occurring in remote sensing, which foster the development of dedicated signal processing methods to process them. From a broader perspective, the development of hyperspectral sensors for applications in chemistry and materials science raises novel questions on a theoretical level: the mixing models encountered in these applications are often strongly non-linear, the endmember variability is enhanced and the observations are often highly mixed. This topic is the last axis of research that I wish to explore in the coming years, in connection with the thematic of image processing for the materials engineering. My main contributions to this research eld to day is the PhD thesis of Tarek Zenati, which I co-supervised along with Jesús Angulo (Center for Mathematical Morphology, Mines Paris, PSL University) between 2019 and 2023. During this PhD thesis, we worked on the development of algorithmic methods enabling the detection and the chemical characterization of selective oxides that can form on steel sheet surfaces during their manufacturing[Zenati et al., 2022b,c]. The PhD thesis was funded by ArcelorMittal Research through the CIFRE program. I summarize in chapter 7 the methodology that we developed.

Figure 7

 7 Figure 7.2: Spectrum recovery in the presence of a single layer of oxide (left: SiO 2 , 120 nm; right: Mn-Al-O, 183 nm) [Zenati et al., 2023].

Figure 7 . 3 :

 73 Figure 7.3: Reectance recovery in the presence of a multiple layers of oxides (left:

  progressively eroding the surface of the samples, provided us with information on the atomic species across the depth of the samples. The experiments allowed us to obtain K = 5 oxidized samples identied by the references 19 and 22 for the HP1180 thermal cycles (with respective dew points of -40 • C and -20 • C) and 5, 9 and 11 for the DP980 thermal cycle (with respective dew points of -50 • C, -35 • C and -5 • C). Overall, we can notice that the thermal cycle HP1180 favors the obtaining of a mixture of Mn 2 SiO 4

Figure 7

 7 Figure 7.4: Recovery of the observations for sample 19[Zenati et al., 2023].

Figure 7 . 5 :

 75 Figure 7.5: Scanning electron microscopy image of a slice of sample 19. The covering layer of oxides appears in black on the image [Zenati et al., 2023].

Figure 7

 7 Figure 7.6: Recovery of the observations for sample 5 [Zenati et al., 2023].

Figure 7 . 7 :

 77 Figure 7.7: Scanning electron microscopy image of a slice of sample 5 [Zenati et al., 2023].

  , B. Figliuzzi and S. H. Ham. Surface oxides characterization based on hyperspectral observations . Chemometrics and Intelligent Laboratory Systems, in review, 2023. T. Zenati, B. Figliuzzi and S. H. Ham. Characterization of surface oxides from hyperspectral measurements. In Workshop on Hyperspectral Image and Signal Processing : Evolution in Remote Sensing (WHISPERS) (pp. 1-5). IEEE. 2022. T. Zenati, B. Figliuzzi and S. H.

  The rst research axis detailed in this manuscript concerns the development of morphological models allowing to reproduce the microstructure of heterogeneous materials. My research perspectives on the subject are essentially applicative. In the short term, as part of several research projects carried out in collaboration with an industrial partner, we are seeking to develop a morphological model making it possible to reproduce the geometric distribution of melanosomes in the different layers of the epidermis. Coupled with optical simulation tools, this model aims to study the inuence of a certain number of biological parameters of the skin, such as the density of melanosomes or their distribution in the form of clusters, on the optical

  By adopting the point of view of mathematical morphology, we can generate a Poisson point process in a domain Ω of R d by exploring Ω with a compact structuring element K and implanting a random number of points following the probability law (2.9) at each location.Boolean model Most classical models of stochastic geometry rely on Poisson pointprocesses. An archetypal model is for instance the Boolean model. The Boolean model is a grain model which is obtained by implanting random primary grains A ′ on the N germs (x n ) 1≤n≤N of a Poisson points process P with intensity θ. Note that primary grains can possibly overlap. The resulting set A is

  To simulate the interstices between the particles of the same aggregate, we use a random Johnson-Mehl tessellations restricted to each aggregate, or connected component of the rst scale of the microstructure. For each connected aggregate A of the rst scale of the simulation, we consider n germs and we simulate a random number ζ n according to an exponential law with some mean k. The cells of the tessellation are then dened by

  Johnson-Mehl tessellation on a domain Ω, let us consider the specic version of the Eikonal equation. Given a Poisson point process P with realizations (x i ) i=1,...,N and corresponding germination times (τ i ) i=1,...,N , we consider the problem:

Table 3

 3

		1	1.	0.45	0.5
	Θ	0.13	0.85	0.45	0.5
	Std. Dev.	0.03	0.25	0.05	0.04
	Min. Val.	0.05	0.5	0.2	0.15
	Max. val.	0.2	2.	0.9	1.05

.2: Optimal parameters found by the Metropolis-Hastings algorithm for the Cox-Boolean model of disks.

  Hence, to facilitate the comparison between the Eikonal and Ncut approaches, for each image, we computed the normalized cut segmentation with the same number of segments as the one yielded by the Eikonal algorithm.

		Eikonal	NCut
	Boundary recall	0.76 ± 0.09 0.74 ± 0.08
	Boundary precision	0.36 ± 0.12 0.31 ± 0.11
	Seg. covering	0.32 ± 0.12 0.23 ± 0.09
	Number of segments	66 ± 10	63 ± 9

Table 5.1: Results of the Eikonal and of the normalized cut algorithm on the test images of the BSD.

  and a leaky rectier linear unit (leaky ReLU) non-linear activation function Φ. The depth d of all hidden convolutional layers is kept xed in the CAN architecture.

	Layer L (s) for s =	1	2	3	4	5	6	7
	Input channels	3	24	24	24	24	24	24
	Output channels	24	24	24	24	24	24	2
	kernel size

Table 6 .

 6

1: Architecture of the Context Aggregation Network (CAN). The total number of trainable parameters for this architecture is 27162

Table 7 .

 7

		Path Lasso	Group Lasso
		Single oxide	Multiple oxides	Single oxide	Multiple oxides
	Recall	1	0.917	0.992	0.885
	Precision	0.956	0.910	0.954	0.924
	Thickness error	2.6 nm	10.7 nm	2.6 nm	6.6 nm

1: Detection metrics

Table 7 .

 7 Theseestimations are consistent with observations of the sample conducted on SEM images, which conrm the presence of a covering oxide layer with a thickness ranging from 100 to 150 nm. They are also in good agreement with the XPS measurements conducted at the surface and the GD-OES measurements conducted across the depth of sample, which indicates the presence of aluminum oxides near the surface.

		Mixture 1		Mixture 2			Mixture 3
		Sub.	MnO	Sub.	Mn-Si-O	SiO2	Sub.	Mn-Al-O	Al2O3
	v f	-	-	0.27	0.69	0.04	0	0	1
	α	-			0.89			0.11
	h	-			139 nm			190 nm

2: Estimation results obtained for sample 19.

Table 7 .

 7

		Mixture 1		Mixture 2			Mixture 3
		Sub.	MnO	Sub.	Mn-Si-O	SiO2	Sub.	Mn-Al-O	Al2O3
	v f 0.37 0.63 0.13	0.40	0.47	0.14	0.43	0.43
	α	0.44		0.32			0.24
	h	182 nm		183 nm			87 nm

3: Estimation results obtained for sample 5.

  Ham. A supervised approach for the detection of surface oxides from hyperspectral measurements. In Workshop on Hyperspectral Image and Signal Processing : Evolution in Remote Sensing (WHISPERS) (pp. 1-5). 2022. T. Zenati, B. Figliuzzi and S. H. Ham. Surface Oxide Detection and Characterization Using Sparse Unmixing on Hyperspectral Images. In International Con-ference on Image Analysis and Processing (pp. 291-302). Springer. 2022.B. Figliuzzi, S. Velasco-Forero, M. Bilodeau, J. Angulo. A bayesian approach to linear unmixing in the presence of highly mixed spectra. In International Conference on Advanced Concepts for Intelligent Vision Systems(pp. 263-274).

	Springer, 2016.
	Chapter 8
	Conclusion and perspectives

I have presented in this manuscript research work that I have carried out since 2013

 et al., 1975;[START_REF] Snook | Dynamics of shear-induced migration of spherical particles in oscillatory pipe ow[END_REF]. It is usually necessary to properly parameterize these algorithms and to adjust the parameterization depending on the image being processed, which makes the use of these algorithms relatively dicult in practice.

To overcome these diculties, we developed an image processing algorithm based on a convolutional network [START_REF] Chen | Fast image processing with fully-convolutional networks[END_REF], which brings an advantage over traditional image processing techniques by alleviating the need of updating the algorithm parameterization for each novel image. However, like all supervised learning algorithms, convolutional networks require a dataset of annotated experimental images to be trained.

Constructing the ground truth is a time consuming task, which renders the use of supervised algorithms dicult in a lot of problems related to physical applications. In addition, the annotation process can be error prone. The main originality of our approach is that we entirely trained the network on a series of synthetic images generated with morphological models rather than on images of real experiments.

The lack of annotated data has long been identied as a critical issue that prevents the use of state-of-the-art supervised algorithms in many image processing problems. In particular, annotating images obtained during physical experiments is often expensive, which triggers interest in alternative methods where ground truth images are generated in a synthetic manner. The development of such methods is increasingly being studied in the literature [START_REF] Barisin | Methods for segmenting cracks in 3d images of concrete: A comparison based on semi-synthetic images[END_REF][START_REF] Jahanian | Generative models as a data source for multiview representation learning[END_REF][START_REF] Nagy | Automatic detection of marine litter: A general framework to leverage synthetic data[END_REF][START_REF] Ravuri | Classication accuracy score for conditional generative models[END_REF]. In [START_REF] Besnier | This dataset does not exist: training models from generated images[END_REF], a generative adversarial network is for instance used to generate a dataset of images similar to those of ImageNet. These generated images are then used to train a classication network. In [START_REF] Baradad | Learning to see by looking at noise[END_REF], the authors investigate image generation models that produce images from simple random processes. These generated images are subsequently used as training data for a visual representation learner. Finally, in [START_REF] Barisin | Methods for segmenting cracks in 3d images of concrete: A comparison based on semi-synthetic images[END_REF], morphological models are used to generate synthetic 3D images of cracks in concrete that are used to train a supervised segmentation model. -2019: L'Oréal R&I Development of a morphological model to describe the microstructure of a colloidal medium in order to enhance its optical properties.

-2016: Yokohama Rubber Co. -2017: Thermal-uid engineering 2 nd year course of the civil engineering program at Mines Paris. I was participating to the teaching of the