Nicolas Dalloz

Serge Mazauric

Thierry Fournel

Mathieu Hébert

How to design a recto-verso print displaying different images in various everyday-life lighting conditions

Introduction

Books, magazines and most documents are made of paper sheets printed in recto-verso mode. Each page is expected to be viewed in reflection mode; the paper opacity is maximized in order to prevent that various pages are seen simultaneously, which would alter the readability of the content of each page. In contrast, in the present study, we do want to see simultaneously different pages in order to display in transmittance mode content that is not visible on the individual pages in reflection mode. Original visual effects can thus be produced, whose potential for graphical arts has not been exploited yet. The probable reason for this is that no automated color management method is available today for these types of "multiview prints", and their design is not intuitive: the images displayed are different from the digital layouts that are effectively printed. The graphical designer must therefore predict how the light fluxes meeting the recto and verso halftone colors are physically combined, an impossible task without prediction tools.

The multiview imaging concept is well known through the example of lenticular images, varying according to the angle of observation [START_REF] Baar | Image ghosting reduction in lenticular relief prints[END_REF]. Other techniques based on special inks or supports can reveal hidden patterns or display different images by varying the illumination or observation conditions: they use the specularity of metallic inks (colors are different in the specular and off-specular directions [START_REF] Hersch | Reproducing color images with embedded metallic patterns[END_REF][START_REF] Babaei | Color Reproduction ofMetallic-Ink Images[END_REF][START_REF] Pjanic | Color Imaging and Pattern Hiding on a Metallic Substrate[END_REF]), the anisotropy of line-halftones color on metal supports (colors vary according to the azimuth angle of observation [START_REF] Pjanic | Color changing effects with anisotropic halftone prints on metal[END_REF]), or fluorescent inks (colors depend on the UV power of the light source [START_REF] Rossier | Hiding patterns with daylight fluorescent inks[END_REF]). In contrast with these techniques, the one that we present here does not need any special material, source or tool: it can be obtained with standard office paper and general use digital printer.

The spectral reflectance and transmittance model for rectoverso halftone prints necessary to compute the multiview images is presented in Section 2. It is an improved version of the DPRT model presented in Ref. [START_REF] Mazauric | 2-flux transfer matrix model for predicting the reflectance and transmittance of duplex halftone prints[END_REF], applicable to symmetrical printing supports, which have same optical properties on their two sides. The number of color patches needed for the model's calibration is considerably reduced in this case. Once the model is calibrated, it is possible to compute the amounts of inks to print on the recto and verso sides in order to display the targeted colors in the considered illumination modes.

The first prototypes of multiview recto-verso prints, computed with the original DPRT model, were presented in Ref. [START_REF] Mazauric | Model-based Design of Recto-Verso Prints Displaying Different Images According to the Illuminated Face[END_REF]. Different patterns were displayed in reflection and transmission modes. In this paper, we extend the principles of multiview images to illumination conditions allowing simultaneous illumination of the two sides, therefore combining the reflection and transmission modes.

The difficulty of designing multiview images by recto-verso printing comes from the limited number of pairs of colors that we can display in the two illumination modes selected. For example, if yellow ink is printed on the recto side, to display a yellow color in the reflectance mode, it is impossible to display a saturated blue color in transmission mode. Likewise, if the color on the recto side is dark, the color in transmission mode is necessarily dark too. The number of displayable pairs of colors is even more limited with the illumination conditions that we address in this paper, where both reflected and transmitted fluxes add. These issues are presented in Section 3, as well as the crucial questions of color matching and registration between the recto and verso layouts.

In Section 4, we present some concepts of visual effects that can be produced: binary multiview images, color-to-grey images and texture contrast images. Section 5 draws the conclusions.

Spectral reflectance-transmittance prediction model for recto-verso prints

A spectral reflectance and transmittance prediction model, called DPRT model [START_REF] Mazauric | 2-flux transfer matrix model for predicting the reflectance and transmittance of duplex halftone prints[END_REF], has been recently proposed for printing supports, possibly non-symmetric (i.e., having different optical properties on the recto and the verso sides), printed with halftone colors. This model relies on a discrete 2-flux model for layered materials [START_REF] Hébert | Assessing the capacity of twoflux models to predict the spectral properties of layered materials[END_REF], based on flux transfer matrices. It is somehow an extension of the Clapper-Yule reflection model for multi-ink halftone colors [START_REF] Hersch | Base models for color halftone reproduction[END_REF], even though the Neugebauer primaries are not characterized by an intrinsic spectral transmittance but considered together with the paper as forming an "inked paper" characterized by a flux transfer matrix. The DPRT model can predict simultaneously, for any pair of halftone colors printed on the recto and verso sides, the transfer factors indicated on Figure 1, namely spectral reflectances R(λ) and R'(λ), and spectral transmittances T(λ) and T'(λ), on the recto and verso sides respectively. After several tests carried out with inkjet printer on supercalendered paper and simple office paper, the DPRT model showed its capacity to provide fairly accurate color predictions for rectoverso halftone prints in both reflection and transmission modes [START_REF] Mazauric | 2-flux transfer matrix model for predicting the reflectance and transmittance of duplex halftone prints[END_REF]. However, the number of color patches to print and the number of measurements to carry out for its calibration is high in comparison to the 44 reflectance measurements needed to calibrate the classical Clapper-Yule model for CMY halftone colors when the dot gain is calibrated according to the method proposed by Hersch and Crété [START_REF] Hersch | Improving the Yule-Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions[END_REF][START_REF] Hébert | Review of spectral reflectance prediction models for halftone prints: calibration, prediction and performance[END_REF]: in the DPRT model for CMY halftones, the spectral properties of the inked paper are computed from 64 recto-verso color patches to be measured in reflectance and transmission modes on both recto and verso sides (256 measurements), and the curves assessing dot gain in the halftones, still computed according to the method by Hersch and Crété, need 36 recto-only color patches to be measured in reflectance and transmission modes (72 measurements). This total of 328 measurements, mandatory when the printing support is non-symmetric, can be considerably reduced if the support is symmetric. This model with faster calibration has been called "Double-Layer Reflectance and Transmittance" (DLRT) model. We propose to present it below, after a short recap about the flux transfer matrices that we use.

Flux transfer matrices

Flux transfer matrices have been recently introduced in the color reproduction domain in order to ease the description of lightmatter interactions in the case of planar layered materials [START_REF] Hébert | Two-flux and multiflux matrix models for colored surfaces[END_REF]. They relate the incoming and outgoing fluxes at each of the different layers and interfaces within the material. Each component in the material, i.e., layer and interface (labeled k), is represented by a transfer matrix of the form:

1 1 k k k k k k k k R R T T R R T ′ - ⎛ ⎞ = ⎜ ⎟ ′ ′ - ⎝ ⎠ M , (1)
where k R , k R′ , k T , k T ′ represents the four transfer factors of the component, defined in a similar manner as in Figure 1. With spectral transfer factors, a transfer matrix is defined for each waveband.

Consecutive components can also be represented together by one transfer matrix, obtained as the product of the transfer matrices representing the individual components. The matrix multiplication order reproduces the stacking order of the components. For example, the transfer matrix P representing the stack of three diffusing layers featured in Figure 2, respectively represented by the transfer matrices 1 2 , , M M and 3 , M is given by:

1 2 3 = ⋅ ⋅ P M M M .
(2)

Printing support

The printing support is a sheet of strongly scattering medium, e.g., paper, having same optical properties on both sides. It is assumed to have an effective refractive index of 1.5, with flat interfaces with air. It has been shown in Ref. [START_REF] Hébert | Extending the Clapper-Yule model to rough printing supports[END_REF] that rough interfaces do not significantly modify the optical effects of the airpaper interfaces in this case. The paper is modeled by the three component shown in Figure 3: a diffusing paper layer bordered by two interfaces. The diffusing paper layer, considered without its interfaces with air, have a reflectance ρ(λ) and a transmittance τ(λ). It is represented by a transfer matrix M defined for each wavelength as:

2 2 1 ρ 1 τ ρ τ ρ - ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ - ⎝ ⎠ M (4)
The reflectances and transmittances of the interfaces depend on the measuring geometry. In the present study, measurements were done with the X-rite Color i7 spectrophotometer relying on the di:8° geometry in reflection mode (illumination with perfectly diffuse light over the hemisphere, and observation at 8° from the normal of the sample by including the specular reflection component), and the d:0° geometry in transmission mode (illumination with perfectly diffuse light over the hemisphere, and observation in the normal of the sample). The corresponding transfer factors are given in Figure 2. For these geometries and the refractive index value 1.5, the transfer matrix representing the interface at the recto side is:

R 1 R 1 T 1 T 1 R 2 R 2 T 2 T 2 R 3 R 3 T 3 T 3 ′ ′ Layer 2
1 1.111 0.667 1 0.044 0.403 d s in out s d in r r t t r r t - - ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ - ⎝ ⎠ ⎝ ⎠ F , (5)
and the one representing the interface at the verso side is:

1 1.111 0.044 1 0.667 0.403 s d i n o u t s d out r r t t r r t - - ⎛ ⎞ ⎛ ⎞ ′ = = ⎜ ⎟ ⎜ ⎟ - ⎝ ⎠ ⎝ ⎠ F . ( 6
)
Once the spectral reflectance R(λ) and transmittance T(λ) of the paper sheet are measured, we can compute a transfer matrix P representing the paper sheet with its interfaces:

2 2 1 1 R T R T R - ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ - ⎝ ⎠ P , (7)
then a transfer matrix M representing the paper layer without the interfaces, thanks to the following matrix equation:

1 1 - - ′ = ⋅ ⋅ M F P F . ( 8
)
The reduction of needed calibration measurements comes from the idea of subdividing the paper layer into two identical sublayers, both represented by the transfer matrix:

1/2 1 = A M . ( 9
)

Solid colors

In CMY printing, one has eight Neugebauer primaries: white (no ink), cyan, magenta, yellow, red (magenta and yellow), green (cyan and yellow), blue (cyan and magenta) and black (cyan, magenta and yellow).

Let us print the recto-side of the paper with a solid color (100% surface coverage) corresponding to one of seven Neugebauer primaries containing at least one ink, labeled 2,...,8 i = (i = 1 denotes the "white" primary, i.e., the unprinted paper). We measure the four transfer factors

( ) ( ) ( ) , , , i i i R T R′ λ λ λ and
( ) i T ′ λ . The matrix Pi1 representing this primary (printed on the recto side only) is given for each wavelength by:

1 1 1 i i i i i i i i R R T T R R T ′ - ⎛ ⎞ = ⎜ ⎟ ′ ′ - ⎝ ⎠ P . ( 10
)
We can deduce the matrix Mi1 representing the inked paper without interfaces:

1 1 1 1 i i - - ′ = ⋅ ⋅ M F P F , (11)
where matrices F and F' have same meaning as in Eq. ( 8).

As for the unprinted paper, we subdivide the inked paper into two sublayers (see Figure 4), one for the recto side containing the inks, represented by a matrix i A , and one for the verso side, similar to the sublayers present in the unprinted paper, therefore represented by the matrix 1 A given by Eq. ( 9). We thus have:

1 1 1 1 1 1 / 2 1 i i i - - - - = ⋅ ′ = ⋅ ⋅ ⋅ A M A F P F M ( 12
)
Matrix Ai is to be computed for each wavelength, and for each of the seven Neugebauer primaries containing at least one ink. The matrix i A for the Neugebauer primary "white" (no ink) is simply the matrix 1

A given by Eq. ( 9). The transfer factors of the inked sublayer are computed from the entries of

{ } i i j p = A
by similar equations as Eq. ( 3):

( )

21
′ = = - ′ = = A ( 13
)
Figure 4. Flux transfers within a paper sheet printed with a solid primary on the recto side.

Recto-verso halftone colors

For a halftone color printed on the recto side of the paper, with effective surface coverages i a for the eight Neugebauer primaries i, we also consider the inked paper without interfaces, and more precisely the sublayer on the recto side which contains the inks (the other sublayer being unprinted paper, therefore represented by the matrix 1

A introduced previously). The four transfer factors of this recto-side sublayer are functions of their equivalent ρ , ρ , τ , i i i ′ and τ i ′ for the eight Neugabeur primaries considered separately, given by Eq. ( 13):

8 8 { } { } 1 1 8 8 { } { } 1 1 ρ ρ ρ ρ τ τ τ τ i i i i a ii a ii i i a ii a ii i i a a a a = = = = ′ ′ = = ′ ′ = = ∑ ∑ ∑ ∑ (14)
The recto-side sublayer is represented by the transfer matrix:

{ } { } { } { } { } { } { } { } 1 ρ 1 ρ τ τ ρ ρ τ i i i i i i i i a a a a a a a a ′ - ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ′ ′ - ⎝ ⎠ A . ( 15
)
If a halftone color is printed on the verso side, with effective surface coverages i a′ for the eight Neugebauer primaries i, the verso-side sublayer has the following transfer factors, again functions of the ones given by Eq. ( 13) thanks to the symmetry of the printing setup:

r s ρ ρ t in τ τ r d t out recto sublayer ρ ρ τ τ verso sublayer Air Interface F A i A 1 P F′ r d r s ′ ′ ′ ′ ′ Air Interface t in t out { , i a } i a′ Recto Verso 8 { } { } 1 1 8 8 { } { } 1 1 ρ ρ ρ ρ τ τ τ τ i i i i a ii a ii i i a ii a ii i i a a a a ′ ′ = = ′ ′ = = ′ ′ ′ ′ = = ′ ′ ′′ = = ∑ ∑ ∑ ∑ (16)
The verso-side sublayer is represented by the transfer matrix:

{ } { } { } { } { } { } { } { } 1 ρ 1 ρ τ τ ρ ρ τ i i i i i i i i a a a a a a a a ′ ′ ′ ′ ′ ′ ′ ′ - ⎛ ⎞ ′ = ⎜ ⎟ ⎜ ⎟ ′ ′ ′ ′ - ⎝ ⎠ A . ( 17
)
Finally, the transfer matrix representing the recto-verso print, whose structure is shown in Figure 5, is obtained by multiplying the transfer matrices of the recto-side interface, the recto-side sublayer, the verso-side sublayer and the verso-side interface:

{ } { } { } , i i i i a a a a ′ ′ ′ ′ = ⋅ ⋅ ⋅ P F A A F (18)
The global transfer factors of the recto-verso print are deduced

from { } { } , i i ij a a p ′ = P
by similar equations as Eq. (3). On each side, the conversion of nominal surface coverages (c, m, y) for the three inks cyan, magenta and yellow, into effective surface coverages for the eight Neugebaeur primaries, relies on twelve nominal-to-effective surface coverage functions according to the method recommended by Hersch and Crété [START_REF] Hersch | Improving the Yule-Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions[END_REF], and on the Demichel equations (see details in Ref. [START_REF] Hébert | Review of spectral reflectance prediction models for halftone prints: calibration, prediction and performance[END_REF]).

The calibration of the model needs 44 CMY patches printed on the recto only, the same as for reflectance-only models [START_REF] Hébert | Review of spectral reflectance prediction models for halftone prints: calibration, prediction and performance[END_REF]: 8 solid color patches for the 8 Neugebauer primaries, whose spectral reflectances and transmittances are measured on both the recto and verso sides, and 36 halftone patches required to compute the twelve nominal-to-effective ink surface coverage functions, whose spectral reflectance and transmittance are measured on the recto side. This makes a total of 104 measurements, therefore lower than the 328 measurements needed by the DPRT model [START_REF] Mazauric | 2-flux transfer matrix model for predicting the reflectance and transmittance of duplex halftone prints[END_REF].

Prediction accuracy

The model was tested on a few tens of different recto-verso halftone samples printed with the Xerox Phaser 6500 electrophotographic printer and a standard 80 g/m² office paper. The prediction accuracy is assessed by the average CIELAB ΔE94 value computed between the measured and predicted spectra, by converting them first into CIE 1931 XYZ tristimulus values, calculated with a D65 illuminant, and then into CIE 1976 L*a*b* color coordinates [START_REF] Sharma | Digital Color Imaging Handbook[END_REF], using as white references a perfectly white reflector (unit reflectance) or transmitter (unit transmittance). The average ΔE94 values obtained in pure reflectance and pure transmission modes are shown in Table 1. By way of comparison, we also give the average ΔE94 values obtained for the same set of samples by two other models: the Duplex Clapper-Yule model [START_REF] Mazauric | 2-flux transfer matrix model for predicting the reflectance and transmittance of duplex halftone prints[END_REF], which is a direct extension of the multi-ink Clapper-Yule model to recto-verso halftone prints, and the Yule-Nielsen model used in reflection mode [START_REF] Viggiano | The color of halftone tints[END_REF][START_REF] Hersch | Improving the Yule-Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions[END_REF] or in transmission mode [START_REF] Hébert | Yule-Nielsen based recto-verso color halftone transmittance prediction model[END_REF].

The DLRT model is slightly less accurate than the other two models in reflection mode, but is it noticeably more accurate in transmission mode, which is the key point for the visual effects that we show in the next section.

Doubled paper

Without any additional calibration measurement, the model is able to predict the reflectance and transmittance of recto-verso colors on a doubled paper sheet (e.g. a paper sheet folded in two), by simply replacing Eq. ( 18) with the following one:

{ } { } { } , i i i i a a a a ′ ′ ′ ′ ′ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ P F A A F F A A F (19)

Inverse approach: displaying different images under different illumination conditions

Once the model is calibrated and its prediction accuracy has been verified, we can use it in an inversed mode to produce multiview prints, i.e., prints displaying different images according to the illumination and observation configurations. For example, we can illuminate the print on the observer's side (reflection mode) or the opposite side (transmission mode).

The first prototypes of multiview recto-verso prints were presented in Ref. [START_REF] Mazauric | Model-based Design of Recto-Verso Prints Displaying Different Images According to the Illuminated Face[END_REF]. Different patterns were displayed in reflection mode (only the recto appears) and in transmission mode (the recto and verso images are mixed). Pure transmission mode means that only the verso side is illuminated, while the recto side receives no light. This illumination mode is rather constraining because it requires to be in a very dark room: if ambient light slightly illuminates the recto side, the displayed colors are not the targeted ones.

Illumination modes

In order to allow a certain amount of ambient light in the transmission mode, we define illumination conditions where the print is simultaneously illuminated on both sides, with spectral irradiances having similar spectral distributions but different magnitudes. These irradiances can be written ( ) ( )

1 i E -α
λ for the recto side and ( ) i E α λ for the verso side, where α is a number between 0 and 1. The observer, located in front of the recto side, perceives a spectral radiance defined as follows if we assume that the print is nearly Lambertian:

r s ρ ρ t in τ τ r d t out recto sublayer ρ ρ τ τ verso sublayer Air Interface F A A P F′ Recto r d r s ′ ′ ′ ′ ′ Air Interface t in t out Verso { } i a′ { } i a { } i a { } i a { } i a { } i a { , i a { } i a′ { } i a′ { } i a′ { } i a′ } i a′ ( ) ( ) ( ) ( ) ( ) ( ) 1 1 T i L R E T E α ′ λ = -α λ λ + α λ λ ⎡ ⎤ ⎣ ⎦ π (20)
where

( ) R λ and
( )

T ′ λ denote respectively the recto-side reflectance and the verso-to-recto transmittance of the print.

In the next sections, we will consider two values for α: • α = 0, which corresponds to the pure reflection mode. The radiance perceived by the observer, depends only on the amount of inks printed on the recto side, is:

( ) ( ) ( ) 1 R i L R E λ = λ λ π (21)
• α = 0.9, which means that the irradiance on the verso is 9 times the irradiance on the recto. This configuration corresponds to the lighting conditions of a window with overcast sky on the one side, and ambient light in a small room without artificial lighting on the other side. This mode is referred to as "α-transmission mode".

Multiview recto-verso prints: main issues

The ink quantities to print on the recto and verso sides are obtained by performing a brute-force optimization in order to obtain the targeted CIE 1976 L*a*b* values, or the targeted spectral radiance, in the two selected illumination modes. We generally compute first the ink surface coverages on the recto side yielding the reflection-mode image, then the ones on the verso side so as to obtain the targeted image in α-transmission mode.

However, the computation of multiview images remains delicate, mainly because the CMY images that are transferred onto paper are not the ones that we want to display. The CMY layouts to print on the recto and the verso sides are interdependent. This is a rather unusual configuration in graphical arts, where the digital layout is generally similar to the physical image that we will see once printed. Moreover, the graphic designer must cope with the issues listed below.

Conditional color gamut

The ink surface coverages on the recto side determine the set of colors that can be displayed in α-transmission mode, which is referred to as conditional gamut. The of a conditional gamut depends on the halftone color on the recto side, as well as the transmittance of the paper. The standard office paper that we use has a transmittance around 15%, whereas its reflectance is around 80%. With the α value of 0.9 that we considered, the radiances reflected and transmitted by the paper have comparable orders of magnitude; the coloration due to the inks on the recto side therefore tends to predominate on the coloration due to the inks on the verso, thus limiting the gamut displayable in α-transmission mode. Figure 6 shows an example of conditional gamuts for the αtransmission mode corresponding to two colors printed on the recto side. These conditional gamuts have been generated by predicting the spectral radiances perceived in α-transmission mode for each of the colors selected on the recto side, represented by certain surface coverages for the cyan magenta and yellow inks, and by considering 1000 halftone colors on the verso side, randomly selected. The spectral radiances are then converted into CIE 1931 XYZ tristimulus values, calculated with a D65 illuminant, and then into CIE 1976 L*a*b* color coordinates using as white reference the spectral radiance of the unprinted paper in this illumination mode.

Color matching

It is possible that areas expected to display one color in αtransmission mode must display different colors in reflection mode. The ink surface coverages therefore vary over these areas, on both the recto and verso sides. To be sure that uniform color is displayed in α-transmission mode, we must ascertain that the spectral radiances ( ) T L α λ issued from all the pixels in these areas are metameric.

In practice, since we use the same three inks on the recto and verso sides, we can reproduce almost spectrally the radiance ( ) T L α λ in all areas where the targeted color is reproducible. We thus have a spectral matching. The rms deviation between the different spectral radiance suffices to assess their similarity. We can also use a color difference metric, for example the CIE ΔE94.

Moreover, being given the three ink surface coverages on the recto side, the ink surface coverage permitting to display a given reproducible color in α-transmission mode have unique values. This is not true with more than three inks: ink selection strategies should be used, as the Under Color Removal (UCR) methods for black ink management in CMYK printing [START_REF] Kipphan | of Print Media[END_REF].

Registration between the recto and verso images

The color matching quality does not only depend on the accuracy of the prediction model and the relevance of the color metric used to match the colors of neighboring pixels, but also on the registration between the recto and verso images, which is often a challenging issue in printing. Imperfect registration can create ghost images in the α-transmission mode. The human visual system is capable of perceiving lines or contours thinner than 100 µm, especially when they are well contrasted and surrounded by a uniform background. This length is much smaller than the mechanical accuracy of most duplex printers. This is why in the example presented here, we printed the recto and verso images on different paper sheets and superposed them manually. The printing support, in this case, is a doubled paper sheet. The model, calibrated with recto-only color patches printed on a single sheet, allows predicting the reflectance and transmittance of printed doubled papers sheets as explained at the end of Section 2. The advantage of doubling the thickness of the support is that its point spread function in transmission mode increased, thus limiting the visibility of slightly misregistrated contours. However, the doubled paper being more opaque (its transmittance is less than 10%), the conditional gamuts in α-transmission mode are consequently reduced.

Concepts of multiview images

Ideal multiview print would display any color image in one illumination mode, and any other image in the other illumination mode. However, due to the limitations explained in the previous section, this kind of print cannot be achieved by recto-verso printing. In this section, we propose some concepts of images that can be produced: the display of different binary images, the display of one image with different colors (e.g. multicolor in one mode, and grey in the other mode), or the revelation in transmission mode of uniformly colored patterns in a textured background. ,

T R T T L X L L Y L X L L Y α α α α ′ ′ ≡ ≡ ≡ ′ ′ ≡ ≡ ≡ a aa aa a aa aa , , , , , , . R T

R

L a denotes the spectral radiance observed in reflection mode when a halftone with ink surface coverages a are printed on the recto, ( ) , T L α ′ a a the radiance observed in α-transmission mode when halftones with ink surface coverages a and a' are printed on the recto side, respectively the verso side, and symbol " X ≡ " means "yields the perceived color X".

In order to be sure that the four pairs of colors ( )

, u v
X Y can be reproduced, we can verify that the colors v Y belong to the conditional gamuts in α-transmission mode for the selected ink surface coverages u

x on the recto side. We can also just check that the color distance ΔE94 between the radiances ( )

1 1, , T u L α ′ a a and ( ) 2 2,
,

T u L α ′ a a
is small, hopefully below 1 unit. In practice, the ink surface coverages 1 a , 2 a , 1,1 ′ a and 1,2 ′ a can be chosen, and the ink surface coverages 2,1 ′ a and 2,2 ′ a computed by using the DLRT prediction model in inversed mode in order to satisfy conditions (22) at best.

Notice that the displayed image in α-transmission mode may be composed of large uniform areas. The registration between the recto and the verso images is therefore critical.

The print shown in Figure 7 displays a text in reflection mode which disappears by color matching in α-transmission mode, while another text, printed in black on the verso side, appears.

Contrast in the binary images

The two colors 1 X and 2 X selected on the recto side must generate conditional gamuts in α-transmission mode that partially meet each other in order to be sure that two colors 1 Y and 2 Y can be found in their intersection. If we want a good contrast in reflection mode, colors 1 X and 2 X must be far from each other in the color space, but the intersection of their conditional gamuts may be small, or even empty. This is illustrated by Figure 9, which shows a digital layout for the recto side (displayed in reflectance mode) containing eight colors i X , as well as the conditional gamuts that would be obtained in α-transmission mode if each of these eight colors was printed on a doubled office paper. We see that the conditional gamuts associated with the darkest and brightest colors are disjoint. Therefore, the colors 1 X and 2 X should be sufficiently close to each other in order to generate conditional gamuts with non-empty intersection. Moreover, the intersection should be large enough to select well distinct colors 1 Y and 2 Y and obtain a sufficient contrast in α-transmission mode.

A good contrast therefore implies a poor contrast in αtransmission mode (or even the impossibility to display a binary image), and reciprocally a good contrast in α-transmission mode means that the two colors on the recto side are close to each other, therefore that the contrast in reflectance mode is poor. There is a trade-off to find in order to have enough contrast in the two modes.

In practice, optimal contrast is reached when the total amounts of inks on the recto and verso sides (i.e., c m y c m y ′ ′ ′ + + + + + ) are as different as possible between the different areas. This total amount of inks has a strong influence on the lightness of the colors, while the human visual systems seems to me more sensible to lightness variations than hue or chroma variations in an image [START_REF] Fairchild | Color Appearance Models[END_REF].

Color-to-grey images

A second type of multi-view image, after binary images, consists in displaying a color image in reflection mode and an achromatic version of this image in transmission mode.

In order to simplify the explanations of the principles, we propose to consider a quantized image containing eight colors, like the ones shown in Figure 9. Each of these eight colors, printed on the recto side with their respective ink surface coverages ( )

, , i i i i c m y = a
, will be turned into an achromatic color on the grey axis, characterized by values a* = b* = 0 and a given lightness value L* in the CIE 1976 L*a*b* space. The lightness of the grey color is expected to be similar to the one of the original colors.

Color-to-grey effect can be achieved only if the eight conditional gamuts, associated with the eight halftone colors printed on the recto side, meet the grey axis. We select in each conditional gamut a reproducible lightness value, as close as possible to the lightness of the original color. The ink surface coverages on the verso side are computed in order to obtain these achromatic colors.

We can see from Figure 9 that the Mona Lisa image is not adapted to the color-to-grey effect: no conditional gamut meets the grey axis (ordinate axis in the right-most graph). The Eiffel Tower image is more adapted, even though the two conditional gamuts associated with the darkest halftones on the recto do not meet the grey axis. In this case, we select the color with lowest chroma.

The grey-to-level print computed from the Eiffel Tower image is shown in Figure 10. We observe, as expected, that the dark areas remain slightly chromatic in α-transmission mode.

Texture contrast images

A third type of multiview image that can be produced with recto-verso prints relies on the perceived contrast between uniform and textured areas in an image, permitting to distinguish patterns. In the following example, shown in Figure 11, the two images printed on the resto and verso sides are a tessellation of eight colors, distributed randomly. The colors on the recto side are arbitrarily chosen, making sure that the corresponding conditional gamuts in α-transmission mode have a non-empty intersection. In this intersection, a color is selected, which will fill the patterns to reveal in α-transmission mode. The eight halftone colors on the verso side are then computed in order to obtain this α-transmission color. In the pattern, each selected color on the recto side faces the computed color on the verso side. Outside the pattern, the colored squares are rearranged in order to obtain a color texture. If the rearrangement is random, since no information is contained in the recto-and versoimages and a message is revealed in α-transmission mode, we have a visual cryptography scheme similar to the one based on printed films presented in Ref. [START_REF] Machizaud | Two-out-of-two color matching based visual cryptography schemes[END_REF], extending to color the principles introduced by Naor and Shamir with black and white images [START_REF] Naor | Visual cryptography[END_REF]. We can also use custom rearrangement in order to increase the perceived texture outside the message, and thus maximize the texture contrast between the message and the background. We can consider the obtained print as an example of stenographic image.

Conclusions

We introduced in the present paper a model based on flux transfer matrices capable of predicting the spectral reflectance and transmittance of recto-verso halftone prints by means of 104 measurements for its calibration, a rather reasonable number in comparison with previous models. The accuracy of the model is sufficiently good to address the inverse approach, which consists in computing the ink surface coverages in the recto-side and verso-side halftone colors in order to display targeted images under different illumination conditions. We thus obtain multiview prints. The illumination modes considered in our examples are the reflectance mode, where only the recto side is illuminated, and a so-called αtransmission mode, where the print receives 10% of the total irradiance on the recto side, and 90% on the verso side, the observer being in front of the recto side. The design of multiview images is made difficult by the fact that the number of pairs of colors displayable by one recto-verso color patch in the two modes is rather limited. The main issues are discussed in this paper, and different strategies permitting to design prints with interesting visual effects are proposed: the first one is based on binary color images, the second one transforms a color image into an achromatic version of this image, and the third one is based on contrast texture images, where a uniformly colored pattern surrounded by a multicolor texture can be distinguished.

In the future, it would be interesting to develop a multi-gamut mapping algorithm capable of computing automatically the ink surface coverages on the recto and verso sides in order to display any pair of color images, taking into account the non-reproducible colors and trying to modify the two original images until all their colors are reproducible in the considered illumination configurations. This is a challenging task, but a tool like this seems necessary if we want that graphical designer exploit the potential of multiview recto-verso images in graphical applications.

Figure 1 .

 1 Figure 1. Transfer factors of a recto-verso print.

Figure 2 .

 2 Figure 2. Flux transfers within a stack of three diffusing layers.

Figure 3 .

 3 Figure 3. Flux transfers within a paper sheet.

Figure 5 .

 5 Figure 5. Flux transfers within a recto-verso halftone print.

  The obtained points in the CIE1976 L*a*b* space are represented on the (a*,b*)-and (L*,C*)-planes, where C* denotes the chroma coordinate given by

Figure 6 .

 6 Figure 6. Conditional color gamuts in α-transmission mode that would be obtained by printing the halftone colors (c, m, y) = (0.1, 0.1, 0.1) (blue dots) and (0.2, 0.1, 0.1) (red dots) on a doubled office paper (electrophotographic printing). The conditional gamuts are represented in the (a*,b*)-and (L*,C*)planes of the CIE 1976 L*a*b* color space.

Figure 7 .

 7 Figure 7. Multiview recto-verso print based on binary images. Top row: digital layouts for (a) the recto side, (b) the verso side. Bottom row: Pictures of the print in (c) reflection mode and (d) in 0.9-transmission mode.

Figure 8 .

 8 Figure 8. Multiview recto-verso print based on binary images, representing the Eiffel tower in reflection mode, and Mona Lisa in α-transmission mode. Top row: binary images to be displayed (a) in reflection mode, (b) in 0.9transmission mode; middle row: digital layouts for (c) the recto side, (d) the verso side, bottom row: (e) Picture of the print in 0.9-transmission mode.

Figure 9 .

 9 Figure 9. Quantized images of (a) the Eiffel Tower, and (b) Mona Lisa, containing each one eight colors, and corresponding conditional gamuts in α-transmission mode that would be obtained if each of these eight colors was printed on the recto side of a double office paper (electrophotographic printing). The conditional gamuts are represented by 1000 points randomly selected and plotted in the (a*,b*)-and (L*,C*)-planes of the CIE1976 L*a*b* color space with the corresponding color among the eight colors.

Figure 10 :

 10 Figure 10: Pictures of a multiview recto-verso print presenting a color-to-grey effect: (a) image displayed in reflection mode, (b) image displayed in 0.9transmission mode.

Figure 11 :

 11 Figure 11: Pictures of a steganographic print based on texture contrast: (a) recto-side image and (b) verso-side image displayed in reflection mode, (c) image displayed in 0.9-transmission mode.

Table 1 .

 1 Compared prediction accuracy between three models*

	Reflectance	Transmittance

*Average ΔE 94 value computed between predicted and measured spectra

Acknowledgement

This work was supported by the French National Research Agency (ANR) within the program "Investissements d'Avenir" (ANR-11-IDEX-0007), in the framework of the LABEX MANUTECH-SISE (ANR-10-LABX-0075) of Université de Lyon.

The authors would also like to thank Dorian Saint-Pierre, Louis Duveau and Baptiste Langlois for their participation to the production of samples.

Binary images

The easiest design option for multiview prints is based on binary images, thus displaying two colors in each illumination mode. Two colors, denoted as 1 X and 2 X , are selected to be displayed in reflection mode. They are specified by the ink surface coverages on the recto side, ( )

Y and 2 Y , are also selected for the α-transmission mode. They are reproduced by four different halftones, with ink surface coverages denoted as

, ,

, ( , ) {1, 2} u v ∈ , corresponding to areas where color u X is expected in reflection mode and color v Y is expected in α-transmission mode. These ink surface coverages must satisfy the following conditions:


Pierre Seroul

Mathieu Hébert

Marie Cherel

Romain Vernet

Raphael Clerc

Matthieu Jomier

Model-based skin pigment cartography by high-resolution hyperspectral imaging

The determination of local components in human skin from in-vivo spectral reflectance measurements is crucial for medical applications, especially for aiding the diagnostic of skin diseases. Hyper-spectral imaging is a convenient technique since one spectrum is acquired in each pixel of the image, and by inverting a light scattering model, we can retrieve the concentrations of skin components in each pixel. The good performance of the method presented in this paper comes from both the imaging system and the model. The hyper-spectral camera that we conceived uses polarizing filters in order to remove gloss effects generated by the stratum corneum; it provides a high resolution image (1120 × 900 pixels), with a thin spectral sampling of 10 nm over the visible spectrum. The acquisition time of 2 seconds is short enough to prevent movement effects of the imaged area, which is usually the main issue in hyperspectral imaging. The model relies on a two-layer model for the skin, and the Kubelka-Munk theory with Saunderson correction for the light reflection. An optimization method enables computing, in less than one hour, several skin parameters in each of the million of pixels. These parameters (blood, melanin and bilirubin volume fractions, oxygen saturation…) are then displayed under the form of density images. Different skin structures, such as veins, blood capillaries, hematoma or pigmented spots, can be highlighted. The deviation between the measured spectrum and the one computed from the fitted parameters is evaluated in each pixel.

Introduction

Skin chromophores and their influence on skin color perception are a major concern in dermatology and cosmetics in order to evaluate pigmentation issues [START_REF] Preece | Spectral filter optimization for the recovery of parameters which describe human skin[END_REF][START_REF] Cotton | A noninvasive skin imaging system[END_REF][START_REF] Jolivot | Skin Parameter Map Retrieval from a Dedicated Multispectral Imaging System Applied to Dermatology/Cosmetology[END_REF]. Non-invasive imaging systems of the skin are used in a wide range of clinical applications in the last decades, in particular in order to get information on pigmentation disorders: melanin index, port wine vein, vitiligo, erythema [START_REF] Nilsson | Tissue Viability Imaging for Assessment of Skin Erythema and Blanching[END_REF][START_REF] Jung | Characterization of Port Wine Stain Skin Erythema and Melanin Content Using Cross-Polarized Diffuse Reflectance Imaging[END_REF]… Since pigmentation determines the spectral reflectance of the skin, the best way to observe it is to make images in different spectral bands, either in three wide spectral bands as in classical RGB imaging [START_REF] Liu | Melanin and Hemoglobin Identification for Skin Disease Analysis[END_REF][START_REF] Tsumura | Independent-component analysis of skin color image[END_REF], or in more than 10 short spectral bands as in multispectral imaging [START_REF] Elbaum | Computer-Aided Melanoma Diagnosis[END_REF][START_REF] Tsumura | Mapping pigmentation in human skin from a multi-channel visible spectrum image by inverse optical scattering technique[END_REF]. There already exist several multispectral imaging instruments for skin observation on the marketplace (e.g. SIAscope from Astron Clinica Limited, or MelaFind from Mela Sciences, …) generally dedicated to specific measurements. In the present study, we used the hyperspectral camera that we developed and presented in Section 2, SpectraCam®, whose spectral and spatial resolutions are higher than the aforementioned ones, thus permitting more accurate skin analysis.

The concentration and localization of chromophores and photophores in the skin cannot be measured directly by optical device, but they can be deduced from light signal captured by the imaging system, by using a spectral reflectance model for the skin taking into account the spectral absorbance of the chromophores [START_REF] Stamatas | Non-invasive measurements of skin pigmentation in situ[END_REF]. Two main approaches are usually followed in the literature: Monte-Carlo simulation, and 2-flux models. The Monte-Carlo method consists in a stochastic stimulation of light paths in the skin with multiple interactions with scatterers [START_REF] Preece | Spectral filter optimization for the recovery of parameters which describe human skin[END_REF][START_REF] Meglinski | Computer simulation of the skin reflectance spectra[END_REF][START_REF] Nishidate | Estimation of melanin and hemoglobin in skin tissue using multiple regression analysis aided by monte carlo simulation[END_REF][START_REF] Shi | Multispectral method for skin imaging development and validation[END_REF][START_REF] Hirose | Nonlinear estimation of chromophore concentrations, shading and surface reflectance from five band images[END_REF]. The drawback of this method is that it cannot be easily reversed in order to deduce skin parameters from measured spectra [START_REF] Zhang | Determination of human skin optical properties from spectrophotometric measurements based on optimization by genetic algorithms[END_REF]: several minutes are needed for treating one spectrum, even with fastened methods such as the adaptive method proposed by Spanier [START_REF] Spanier | Geometrically convergent learning algorithms for global solutions of transport problems[END_REF]. The two-flux models rely on a physical model of light propagation which can be seen as a special solution of the radiative transfer equation, standing in case of strong scattering. The Kubelka-Munk model predicts the reflectance and transmittance of one homogeneous layer as a function of its thickness [START_REF] Kubelka | New contributions to the optics of intensely light-scattering material, part I[END_REF], and the Kubelka compositional model can predict the reflectance and transmittance of stacks of homogenous layers [START_REF] Kubelka | New contributions to the optics of intensely light-scattering material, part II: nonhomogeneous layers[END_REF]. Two-flux models have been used in a wide range of applications, included skin analysis [START_REF] Jolivot | Skin Parameter Map Retrieval from a Dedicated Multispectral Imaging System Applied to Dermatology/Cosmetology[END_REF][START_REF] Anderson | The optics of human skin[END_REF][START_REF] Krishnaswamy | A biophysicallybased spectral model of light interaction with human skin[END_REF][START_REF] Igarashi | The appearance of human skin: A survey[END_REF], most of the time without Saunderson correction [START_REF] Saunderson | Calculation of the color pigmented plastics[END_REF], which means that the optical effect of the air-skin interface is not taken into account. More advanced models describe more thinly the scattering of light by the skin tissues, especially the angular distribution of the scattered light, thanks to methods based on the radiative transfer theory with a multilayer model [START_REF] Zhang | Determination of human skin optical properties from spectrophotometric measurements based on optimization by genetic algorithms[END_REF][START_REF] Prahl | Light transport in tissues[END_REF][START_REF] Magnain | Skin color modeling using the radiative transfer function solved by the auxiliary function method[END_REF]. This approach is physically more rigorous but difficult to use in practice due to the needed computation time for computing the parameters from bi-directional spectral reflectance measurements. The number of parameters should be also restricted in order to guarantee uniqueness of the solution in the optimization process [START_REF] Zonios | Modeling diffuse reflectance from semi-infinite turbid media: application to the study of skin optical properties[END_REF].

The method that we present in this paper relies on the Kubelka-Munk model with Saunderson correction (Section 3), by considering the simplified model for skin tissues presented in Section 4. The inversed model, fastened by an optimized algorithm, enables computing rather fast five skin parameters: epidermis depth, melanin volume fraction, blood volume fraction, oxygen saturation and bilirubin volume fraction. These parameters can be displayed under the form of density maps [START_REF] Jolivot | Skin Parameter Map Retrieval from a Dedicated Multispectral Imaging System Applied to Dermatology/Cosmetology[END_REF]. The estimation error between simulated and real spectrum is computed using least square and spectral angle similarity (SAS) [START_REF] Yuhas | Discrimination among semi-arid landscape endmembers using the spectral angle mapper (sam) algorithm[END_REF]. In order to verify the performance of the method, acquisitions were made on ten Caucasian skins to correlate results reported by previous works [START_REF] Jacques | Skin optics[END_REF][START_REF] Huzaira | Topographic Variations in Normal Skin, as Viewed by In Vivo Reflectance Confocal Microscopy[END_REF] and pigment maps were compared to literature [START_REF] Stamatas | In vivo measurement of skin erythema and pigmentation: New means of implementation of diffuse reflectance spectroscopy with a commercial instrument[END_REF] (Section 6).

Hyperspectral imaging system

The imaging system that we used is a modular device for hyperspectral imaging, SpectraCam®, developed by Newtone Technologies, France. In its standard mode, it can capture up to 31 images in different thin wavebands of the visible spectrum (400-700 nm), every 10 nm. For specific clinical applications, it can also capture one image every 1 nm through a spectral band of 10 nm. The surface is illuminated at approximately 8° from its normal by four large visible band power LEDs symmetrically disposed around optical axis (see Figure 1). The acquisition time for each waveband is selected in order to prevent saturation while ensuring good signal-to-noise ratio. The total acquisition time for the 31 images is 2.5 seconds, which is appreciable for measurements on alive surfaces: the risk of movement is rather low, and the images corresponding to the different wavelengths are most often perfectly registered with each other.

Figure 1 -SpectraCam (left) and its schematic (right). Light emitted by LEDs is transmitted through the polarizers (P.) and reflected by the skin. A tunable filter (TF) let pass a specific wavelength and is used as analyzer. The resulting image is captured by a monochromatic camera.

In order to avoid artifacts due to the specular reflection of light by the skin surface (gloss), the system integrates two polarization filters, one for polarizing the incident light, another one for the reflected light. This system uses the fact that when the incident light is polarized, the light reflected by the surface of a dielectric medium has similar polarization as the incident light, whereas the light scattering into the diffusing medium is unpolarized. Thus, when the second polarizing filter is parallel to the first one (Parallel-Polarization mode, hereinafter denoted as PP), it transmits both the specularly and diffusely reflected light components. When it is perpendicular to the first filter (Cross-Polarization mode, CP), it transmits only the diffused light, without the specularly reflected component (see Figure 2).

For an analysis of the pigment composition of the skin, the CP mode is preferable since information is deduced from the spectral distribution of the diffusely reflected light. The PP mode is useful for an analysis of the skin surface structure, highlighted by the specular reflectance. Note that the specular reflected light can be computed by subtracting the CP-image to the-PP image. The SpectraCam® device also enables analyzing fluorescence emission by photophores thanks to a UV-A lamp emitting around 365 nm. 31 images are captured corresponding to the same wavelengths as for the chromophores analysis. When the UV lamp is on and the white light off, the measured spectrum corresponds to the fluorescence emittance. This mode is not used in the present study.

The repeatability of the spectral measurement system is similar to spectrophotometers used in color reproduction: the standard deviation over 10 measurements on a same white ceramic, expressed in CIELAB 1976 ΔE*ab, is 0.09 unit with our system, therefore comparable to the Konica Minolta CM-700d instrument (0.04 unit on the same sample) and the X-Rite SP62 instrument (0.05 unit).

Kubelka-Munk model and Saunderson correction

The Kubelka-Munk model describes the propagation of diffuse natural light within a layer of homogenous scattering and absorbing medium, characterized by its thickness h, its linear absorption coefficient K and its linear backscattering coefficient S [START_REF] Kubelka | New contributions to the optics of intensely light-scattering material, part I[END_REF]. The fluxes i and j propagating respectively forwards and backwards within the layer satisfy at every depth z the following system of differential equations

( ) ( ) ( ) ( ) ( ) ( )  = - + +     = - + +   di K S i z Sj z dz dj Si z K S j z dz (1)
By integrating these differential equations [START_REF] Hébert | Fundamentals of Optics and Radiometry for Color Reproduction[END_REF], one obtains the following expressions for the reflectance R and the transmittance T of the layer:

( ) ( ) ( ) = + sinh cosh sinh bSh R b bSh a bSh , (2) and ( ) ( )
= + cosh sinh b T b bSh a bSh , (3) with ( )
= + / a K S S and = - 2 1 b a (4)
As the layer thickness tends to infinity, the reflectance tends to the limit

( ) ∞ + - + = -= 2 K S K K S R a b S (5)
and the transmittance obviously trends to 0.

When the layer is on top of an opaque background with reflectance ρg, the reflectance becomes [START_REF] Hébert | Fundamentals of Optics and Radiometry for Color Reproduction[END_REF] ( )

( ) ( ) ( ) ( ) ( ) - + = - + 1 ρ sinh ρ cosh ρ sinh cosh g g g g a b S h b b S h R a b S h b b S h (6)
Note that since absorption and scattering coefficients generally depend on wavelength, all reflectance and transmittance expressions also depend on wavelength.

Most of the time, the refractive index of the layer is different from the one of the surrounding medium, e.g., air. The reflectance Rg of the layer with background, expressed by Eq. 6, is thus transformed into the following one

= + - 1 in out g S s i g T T R R r r R (7)
where the terms s r , in T and out T are derived from the Fresnel formulas according to the measuring geometry [START_REF] Simonot | Extension of the Williams-Clapper model to stacked nondiffusing colored layers with different refractive indices[END_REF]. This expression generalizes the Saunderson correction for the Kubelka-Munk model for any measuring geometry (in his original paper, Saunderson considered illumination at 45° from the normal of the layer, and observation in the normal of the layer [START_REF] Saunderson | Calculation of the color pigmented plastics[END_REF]). For the 8°:0°geometry, one has

( ) ( ) ( ) = = °= - °= °2 0 8 1 8 0 / s in fresnel fresnel out fresnel r T T R T T n ( 8
)
Where n is the refractive index of the layer and the factor 2 1 / n introduces the fact that the captured radiance has been spread into a larger cone when crossing the interface due to refraction [START_REF] Simonot | Extension of the Williams-Clapper model to stacked nondiffusing colored layers with different refractive indices[END_REF]. Note that 0 s r = because the incident and reflection are different; it is also zero when the specular reflection is removed by polarization filtering. Moreover, it has been shown that in T , out T and i r are almost not modified by eventual roughness of the surface [START_REF] Hébert | Extending the Clapper-Yule model to rough printing supports[END_REF].

Simplified skin model

Human skin has a multilayer structure, each layer having specific optical properties.

Depending on the optical model that is used, the considered number of layers is 2, 3, 5, 7, or even more (Refs. [START_REF] Kim | Light transport in two-layer tissues[END_REF], to [START_REF] Magnain | Skin color modeling using the radiative transfer equation solved by the auxiliary function method: inverse problem[END_REF], respectively). Anderson [START_REF] Anderson | The optics of human skin[END_REF] considers a two-layer system: epidermis, which contains melanin, and dermis, containing collagen fibers as scatterers, and blood and its derivatives as absorbers, among which hemoglobin (Hb), oxyhemoglobin (HbO2), bilirubin, as well as β-carotene. We adopted similar two-layer structure (see Figure 3). We considered the epidermis as a diffusing layer with refractive index n = 1.4, thickness h, absorption coefficient e K and scattering coefficient e S , and the dermis as a diffusing layer with absorption coefficient d K and scattering coefficient d S . The dermis thickness is assumed infinite and its refractive index identical to the one of the epidermis. For this structure where a diffusing layer is on top of a background, Eq. 7 applies. Surprisingly, the optical effect of the air-skin interface (here modeled by the Saunderson correction) is ignored in most works referred to in this paper, except in Ref. [START_REF] Magnain | Skin color modeling using the radiative transfer function solved by the auxiliary function method[END_REF] where specific bounding conditions are used in the auxiliary function method for solving of the radiative transfer equation. Ignoring the reflections and refractions of light at the skin surface is in contradiction with the optical laws and introduces significant error in the skin reflectance model.

For the spectral scattering coefficients of the epidermis and dermis, and the spectral absorption coefficients of the skin pigments used in our model, we referred to Ref. [START_REF] Jacques | Skin optics[END_REF].

The absorption coefficients of the two layers depend on the pigments contained in them and their respective volume fraction: According to the Beer-Lambert-Bouguer law [START_REF] Wyszecki | Color science: Concepts and methods, quantitative data and formulae[END_REF], the spectral absorption coefficient of the absorber mixture is a linear, additive combination of the spectral absorption coefficients of the individual absorbers, weighted by their respective concentration [START_REF] Kortüm | Reflectance Spectroscopy, Principles, Methods, Applications[END_REF] The epidermis contains a baseline with absorption coefficient ). Its thickness is assumed to be infinite.

( ) ( ) ( ) ( ) ( ) 02 02 λ λ λ λ λ d b b H b H b H b H b b i b i K c K c K c K c K = + + + (10)

Model inversion for the computation of the skin parameters

The spectral reflectance of the skin is given by Eq. 7, with ( )

λ g R
given by Eq. 6. In the expression for ( ) λ

λ m Hb Hb bi p m m Hb Hb bi c c c c h p m R R c c c c h R R   =                  , ℎ = argmin , , , , ∑ - (11) { } ( ) ( ) ( ) ( ) 02 02 2 2 , , , , λ λ , , , , arg min arccos λ
The optimization yielding the parameter values relies on the Nelder-Mead method [START_REF] Nelder | A simplex method for function minimization[END_REF].

The fact that the different pigments have different absorption bands insures a unique solution for this optimization process. In order to verify the stability of the obtained values of parameters, we introduced small variations in the hyperspectral image and observed that the variations of the parameter values were low. Moreover, we could verify that a bijective relationship exists between the parameter values and the reflectance spectra, which means that the model can be reversed. This enables attributing a unique set of parameters to every reflectance spectrum of skin. The five parameters are computed for each of the 1 008 000 pixels in less than one hour (3.4 ms per pixel). The computation speed optimization that has been achieved is an advantage for medical applications: it remains important for immediate clinical diagnosis but is acceptable for offline data evaluation in the context of clinical trials.

From the obtained set of parameters, one may compute again the skin reflectance and compare it to the measured ones. This enables assessing the relevance of the parameter values and the accuracy of the model. The deviations may come from the presence of other pigments, non-modeled optical phenomena, or local variations of the optical properties of the skin which are not taken into account in our model. The SAS metric is computed for each pixel of the image and can be also displayed under the form of a density images.

Experimental testing

Clinical experiments were performed onto 10 Caucasian subjects with phototypes I (fair skin) to III (light brown skin) on the Fitzpatrick scale [START_REF] Fitzpatrick | Soleil et peau[END_REF][START_REF] Fitzpatrick | Ultraviolet-induced pigmentary changes: Benefits and hazards[END_REF], 5 males and 5 females between 20 and 50 years old. Seven areas of interest were observed on each subject, located on the right and left inner forearms, on the right and left cheeks and on the forehead. This makes a set of 70 hyperspectral images. The size of each area is 5 cm x 4 cm (1120 × 900 pixels). The SAS value has been shown to be less sensitive than least square error to irradiance variation across the image, especially visible on oxygen saturation maps (Figure 4). Computed over all pixels of the 70 hyperspectral images, the SAS value was 0.044, which is of the same order as the value 0.017 given in Ref. [START_REF] Jolivot | Skin Parameter Map Retrieval from a Dedicated Multispectral Imaging System Applied to Dermatology/Cosmetology[END_REF] where this value can theoretically vary from 0 to П/2. Average skin parameters and their standard deviation, obtained from the 70 hyperspectral images, are given in Table 1. Except from the epidermis thickness that is overestimated compared to literature, all parameters fit in the range detailed in previous works. An originality of our study is the estimation of the bilirubin volume fraction by hyperspectral imaging. We compared our method with the method proposed by Stamatas et al., a reference work in the literature in which spectral data are used to estimated melanin, oxy-hemoglobin and deoxy-hemoglobin [START_REF] Stamatas | In vivo measurement of skin erythema and pigmentation: New means of implementation of diffuse reflectance spectroscopy with a commercial instrument[END_REF]. In both methods, we combined oxy-and deoxy-hemoglobin in order to obtain blood and oxygenation values. Blood map is defined as the sum of oxyand desoxy-hemoglobin concentrations and oxygenation as the ratio between oxyhemoglobin and blood concentrations. method [START_REF] Stamatas | In vivo measurement of skin erythema and pigmentation: New means of implementation of diffuse reflectance spectroscopy with a commercial instrument[END_REF] compared with blood (e), melanin (f) and oxygen (g) maps with the proposed method.

Blood capillaries network is roughly visible using Stamatas & al (Figure 5.b) whereas it is more clearly defined with the proposed method (Figure 5.e). Some pigmented spots, visible in the color image are not identifiable in the Stamatas & al melanin maps (Figure 5.c) whereas they are well defined in the proposed method (Figure 5.f). Furthermore, with the proposed method, some of the spots can be differentiated between the ones visible only on the melanin image and the ones visible both in the melanin and blood images. Finally, with the proposed method, oxygenation map displays clearly veins network (Figure 5.g) whereas they generates artefacts in blood and melanin maps (

Conclusions

The method presented in this paper enables quantitative estimation of pigment concentrations in each point of a given area of human skin, and therefore display of their spatial distribution. The good performance of the method comes from the combination of a high-resolution hyper-spectral imaging system with polarizing filters, and an inverse model based on the two-flux theory (Kubelka-Munk model in a two-layer structure for the skin, with Saunderson correction in order to take into account the internal reflections of light at the skin-air interface). The use of cross-polarization enables removing light reflections of the stratum corneum, which may be an important error factor in chromophores quantification. The model allows retrieving six parameters of Caucasian skin: epidermis depth, melanin volume fraction, blood volume fraction, oxygen saturation and bilirubin volume fraction. The estimation of this latter parameter is an originality of our work compared to the existing works. Images of each of these parameters can be displayed to observe different structures such as veins, blood capillaries, hematoma or pigmented spots. A spectral deviation image is also displayed to highlight structures that do not fit well with the developed skin model. From the tests carried out in this study, we observed good agreement between estimated parameter values with our system and those available in the literature. In the future, we would like

Figure 2 -

 2 Figure 2 -Same skin area observed by the hyperspectral imaging system in crosspolarization mode (left) or parallel-polarization mode (center); subtracting the two images yields the gloss image displayed on the right. Here, hyperspectral images are converted into RGB color images.

  transmittance Te of the epidermis can be computed as a function of its thickness h by using Eqs.2 and 3.The dermis contains the same baseline as the epidermis [absorption coefficient

  reflectance Rd of the dermis can be computed using Eq. 5.

Figure 3 .

 3 Figure 3. Skin model with two layers. Epidermis is composed of baseline (b.) and melanin (m.)and has a thickness h. Dermis is composed of baseline (b.), deoxy-hemoglobin (Hb.), oxyhemoglobin (HbO2) and bilirubin (b.). Its thickness is assumed to be infinite.

2 and

 2 transmittance T of the epidermis are respectively expressed by Eqs. 9). For each pixel of the hyperspectral image, one fits the pigment concentrations and the epidermis thickness as values for which the error between the spectrum ( ) Two errors where implemented as cost function to optimize the model, a classical least square error (Eq. 11) and the Spectral Angle Similarity (SAS) (Eq. 12)[START_REF] Kruse | The Spectral Image Processing System (SIPS)-interactive visualization and analysis of imaging spectrometer[END_REF]:, , ,

Figure 4 .

 4 Figure 4. Color images (a and d), and oxygen saturation images reconstructed using a least square cost function (b and e) and a SAS cost function (c and f)

Figure 5 -

 5 Figure 5 -Color image (a) and blood (b), melanin (c) and oxygen (d) maps implemented using Stamatas & al.method[START_REF] Stamatas | In vivo measurement of skin erythema and pigmentation: New means of implementation of diffuse reflectance spectroscopy with a commercial instrument[END_REF] compared with blood (e), melanin (f) and oxygen (g) maps with the proposed method.

Figure 5 .

 5 b and Figure 5.a) and are not visible in oxygenation maps (Figure 5.d) using the method by Stamatas & al.

Figure 6 .

 6 Figure 6. Color image (a), and density images of blood volume fraction (b), oxygen saturation (c) and melanin volume fraction (d) issued from a hyperspectral image of female eye contour (courtesy of Clarins S.A).

Figure 6

 6 Figure 6 shows another example of observed skin area, with a color image and relevant grey-level images representing density images for three skin parameters: the whole eyelid and eye contour blood irrigation in well described in the blood image (Figure6.b). The dark circle is well defined on the oxygen saturation image (Figure6.d). The melanin image (Figure6.a) shows no specific variations around the eye helping the diagnosis between dark circles due to oxygenation issues and eye contour hyper-pigmentation.

Table 1 .

 1 Skin parameters estimated from 7 areas on 10 Caucasian subjects

	Skin Parameter	Estimated value*	Data from the literature
	Epidermis thickness	41 ± 8 µm	34 ± 4 µm [45]
	Melanin volume fraction	15 ± 5.7%	1.3 -43% [28]
	Blood volume fraction	5.5 ± 3.7%	2 -5% [28]
	Oxygen saturation	50 ± 15%	25-90% [46]
	Bilirubin volume fraction	0.35 ± 0.15%	0.3-1.2 mg/dL [47]**
	*Average over the 10 Caucasian subjects and standard deviation **This concentration of biliburin is measured in serum whereas our measurement is an effective volume
	fraction in the skin