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International audience; Optical characterization and appearance prediction of translucent materials is needed in several fields of engineering such as computer graphics, dental restorations or 3D printing technologies. In the case of strongly diffusing materials, flux transfer models like the Kubelka-Munk model (2-flux) or 4-flux model have been successfully used to this aim for decades. However, they lead to inaccurate prediction of the color variations of translucent objects of different thicknesses. Indeed, as they assume Lambertian fluxes at any depth within the material and in particular at the bordering interfaces, they fail to model the internal reflectance at the interfaces, penalizing the accuracy of the optical parameter extraction. The aim of the paper is to investigate the impact of translucency on light angular distribution and corresponding internal reflectances, by the mean of the radiative transfer equation, which describes more rigorously the impact of the scattering on the light propagation. It turns out that the light angular distribution at the bordering interfaces, assumed to be flat, is far from being Lambertian, and the internal reflectance may vary a lot according to the layer's thickness, refractive index, scattering and absorption coefficients. This work not only enables to better understand the impact of scattering within a translucent layer but also invites to revisit the well-known Saunderson correction used in 2-or 4-flux models.

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International audience; In many commercial instruments for measuring reflectance, the area illuminated on the measured object is identical to the area from which light is collected. This configuration is suitable for strongly scattering materials such as paper, but issues arise with translucent materials, because a portion of the incident light spreads around the illuminated area by subsurface transport and escapes the detection system. This phenomenon, referred to as edge loss, yields erroneous, underestimated reflectance measurements. In the case of colored and opalescent materials, the impact of edge loss on the measured reflectance varies with the wavelength, which is a significant issue for spectrophotometer and colorimeter users. In the present study, we investigate the edge-loss phenomenon with an emphasis on human skin measurement. In particular, we use a mathematical model to estimate the PSF of translucent materials, relying on the diffusion approximation of the radiative transfer theory, to predict edge-loss measurement error. We use this model to discuss the suitability of several commercial spectrophotometers to accurately measure the translucent materials of various optical properties and show that not all devices can adapt to all translucent materials.

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13 pags., 2 figs., 1 tab., The bidirectional reflectance distribution function (BRDF) and the bidirectional scattering - surface reflectance distribution function (BSSRDF), which relate radiance at the surface to irradiance and radiant flux, respectively, are regarded as the most fundamental scattering quantities used to determine the reflectance of objects. However, for materials where the optical radiation is transmitted under the surface, this radiance depends not only on irradiance and radiant flux, but also on the size of the irradiated area of the surface. This article provides insight into such dependence under the special condition in which the radiance is evaluated within the irradiated area and, consequently, is produced by both the insurface reflection and the subsurface scattering, in contrast to the situation in which the radiance is evaluated at non-irradiated areas and only subsurface scattering contributes. By explicitly considering both contributions, two other scattering quantities are defined: one that accounts exclusively for the insurface reflection and the other that accounts for subsurface scattering. In this regard, these quantities might be considered more fundamental than the BRDF and the BSSRDF, although they are coincident with these two functions apart from the above-mentioned special condition and for materials with negligible subsurface scattering. In this work, the relevance of the proposed scattering quantities is supported by experimental data, practical considerations are given for measuring them, and their relation to the bidirectional transmittance distribution function (BTDF) is discussed., This investigation was carried out within the scope of EMPIR project 18SIB03, “New quantities for the measurement of appearance” (BxDiff). EMPIR is jointly funded by the EMPIR participating countries within EURAMET and the European Union. Some of the authors (CSIC) are would like to acknowledge project PGC2018-096470-B-I00 BISCAT (MCIU/AEI/FEDER,UE). We are grateful to Covestro Deutschland AG for providing with translucent samples.

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The color of a surface structured at the mesoscopic scale differs from the one of a flat surface of the same material because of the light interreflections taking place in the concavities of the surface, as well as the shadowing effect. The color variation depends not only on the surface topology but also on the spectral reflectance of the material, its matte or glossy finishing, and the angular distribution of the incident light. For an accurate prediction of the radiance perceived from each point of the object by an observer or a camera, we must take into account comprehensively the multiple paths of light which can be reflected, scattered or absorbed by the material and its surface. In this paper, we focus on the light reflection component due to the material-air interface, in the special case of a surface structured with parallel, periodical, specular V-shaped ridges, illuminated either by collimated light from any direction of the hemisphere, or by diffuse light. Thanks to an analytical model, we compute the radiance reflected in every direction of the hemisphere by accounting for the different interreflections, according to the angular reflectance of the panels and the aperture angle of the cavity. We can then deduce the apparent reflectance of the cavity when viewed from a large distance.

The color of a surface structured at the mesoscopic scale differs from the one of a flat surface of the same material because of the light inter-reflections taking place in the concavities of the surface, as well as shadowing effects. The color variation arises not only in scattering materials, but also in the absence of scattering, e.g., in metals and clear dielectrics, just as a consequence of multiple specular reflections between neighboring flat facets of the surface. In this paper, we investigate such color variation in the case of an infinitely long V-shaped groove, having in mind the visual appearance of a surface composed of many structures of that sort, all parallel and identical. We develop a full model of multiple specular reflections, accounting for the ray position and orientation and the polarization effects occurring at each reflection. We compare that situation with two approximate models, more usual and easier to compute, where light is assumed to remain unpolarized all along, or where the $p$p- and $s$s-polarized components are treated separately. Spectral reflectances were predicted for various materials and angles of cavities, under diffuse illumination. In most cases, the three models predict very similar bi-hemispherical reflectances, but the hemispherical-directional reflectances can vary noticeably in certain observation directions. This study might help achieve a more physically realistic rendering of dielectric or metallic ridged surfaces in computer graphics.

International audience; The different areas of a concave object illuminate each other by a multiple light reflection process, called interreflections, depending on the geometries of the object and the lighting. For an accurate prediction of the radiance perceived from each point of the object by an observer or a camera, an interreflection model is necessary, taking into account the optical properties and the shape of the object, the orientation(s) of the incident light which can produce shadows, and the infinite number of light bounces between the different points of the object. The present paper focusses on the irradiance of two adjacent planar panels (V-cavity) illuminated by collimated light from any direction of the hemisphere, or by diffuse light. According to the reflectance of the material and the angle of the cavity, the loss of irradiance near the fold due to the shadowing effect is partly compensated by the gain in radiance due to the interreflections.

International audience; The optical and visual characterization of translucent materials according to their shape and thickness is still a challenge for scientific research due to the difficulty to describe and predict in a simple manner the angular and spectral distributions of the reflected and transmitted light. In this paper, the light reflection and transmission properties of slightly scattering polymer sheets stacked with each other are studied. The light that is regularly reflected and transmitted by the material, propagated without scattering, as well as the light which is scattered out of the regular directions, are predicted using only a few measurements from a commercial reflectance/transmittance spectrophotometer and an extended two-flux model of the radiative transfer. The proposed methodology, validated by experiments, can be applied by laboratories equipped with an integrating sphere device which is usually used in the context of strongly diffusing rather than translucent materials. A discussion states the potential and the limitations of the proposed methodology.

Strongly scattering supports coated with thick transparent medium display a bright halo with a characteristic ring shape when illuminated in one point by a thin pencil of light. The halo, whose size is related to the coating thickness, is due to the Fresnel internal reflections of the light scattered by the diffusing support at the coating-air interface. The angular distribution of the reflected light strongly varies over the halo according to the distance from the point initially illuminated, a fact that cannot be observed when a large area of the surface is illuminated as in usual reflectance and bidirectional reflectance distribution function measurements. By considering a Lambertian background and a transparent layer on top of it, both of them being possibly absorbing, we develop a bidirectional subsurface scattering reflectance distribution function model, based on analytical equations and matrix numerical computation, which enables a detailed description of the spatial and angular distribution of the scattered light including the multiple reflections between the background and the coating-air interface. Some applications in which this subsurface scattering phenomenon can be an issue are addressed, such as the reflectance measurement, which can be undervalued when the geometry is not adapted to the coating thickness, or the impact of the phenomenon on heterogeneously colored surfaces such as coated or laminated halftone prints.

International audience; A classical way of coloring a surface in order to create a still image is the application of a colored coating. The more recent digital printing systems enable depositing thick coatings or successive ink layers. The color rendering of the surface depends on the optical properties of the coated materials (optical index, spectral scattering and absorption coefficients) and their thickness. In order to predict its spectral reflectance as a function of these parameters, the so-called two-flux approach is to be tested in first since the model is simple and relies on analytical equations. It has a good chance to provide accurate predictions for coatings made of solid layers of strongly scattering or nonscattering media, or even complex stratified coatings obtained by stacking nonsymmetrical components such as printed films. The generalized Kubelka-Munk model summarized in this paper enables treating all these configurations with a unified mathematical formalism. But it has limitations and may provide poor color predictions for certain types of layered materials. We therefore propose a simple method based on parameters of the model to check the precision of the two-flux model for a given type of coating.

International audience; We introduce a model allowing convenient calculation of the spectral reflectance and transmittance of duplex prints. It is based on flux transfer matrices and enables retrieving classical Kubelka-Munk formulas, as well as extended formulas for non-symmetric layers. By making different assumptions on the flux transfers, we obtain two predictive models for the duplex halftone prints: The "duplex Clapper-Yule model" which is an extension of the classical Clapper-Yule model, and the "duplex primary reflectance-transmittance model". The two models can be calibrated from either reflectance or transmittance measurements; only the second model can be calibrated from both measurements, thus giving optimal accuracy for both reflectance and transmittance predictions. The conceptual differences between the two models are deeply analyzed, as well as their advantages and drawbacks in terms of calibration. According to the test carried out in this study with paper printed in inkjet, their predictive performances are good provided appropriate calibration options are selected.

International audience; Leaf reflectance of monocotyledons generally displays a strong azimuthal anisotropy due to the longitudinal orientation of the veins. The Cook and Torrance (CT) bidirectional reflectance distribution function model was adapted to account for this distinctive feature. The resulting ACT (Anisotropic Cook and Torrance) model is based on the decomposition of the roughness parameter into two perpendicular components. It is evaluated on sorghum (Sorghum halepense) and wheat (Triticum durum) leaf BRF (Bidirectional Reflectance Factor) measurements acquired using a conoscope system. Results show that the ACT model fits the measurements better than azimuthally isotropic surface models: the root mean square error computed over all the BRF measurements for both leaves decreases from ≈0.06 for the Lambertian model to ≈0.04 for the CT model and down to ≈0.03 for the ACT model. The adjusted value of the refraction index is plausible (n ≈ 1.32) for both leaves while the retrieved roughness values perpendicular to the veins (sorghum = 0.56; wheat = 0.46) is about two times larger than that parallel to the veins (sorghum = 0.27; wheat = 0.18). Nonetheless, the observed residual discrepancies between the ACT model simulations and the measurements may be explained mainly by the Lambertian assumption of the volume scattering.

International audience; The color rendering of superposed coloring components is often an issue either to predict or to simulate the appearance of colored surfaces. In graphical software, for example, transparence options are available to display different layouts on top of each other. With two colored layers, tuning the transparency of the top layer enables transitioning continuously from the color of this top layer to the color of the bottom layer. However, these options based very often only on additive color mixing offer limited transitions between two base colors. It would be advantageous to introduce more advanced options, providing, for example, realistic renderings of superposed paint layers. This is the aim of the present study, where simple models are proposed to create intermediate configurations between additive and subtractive color mixings. These models rely on the spectral power distribution of a finite set of primaries with given proportions. They may be extended to RGB color reflective or transmissive systems if the red, green, and blue wavebands do not overlap each other. An additional parameter is introduced to tune the proportions of additive and subtractive mixings, each type of mixing being based on its set of primaries. Various simulations of color mixings are presented, illustrating the possibilities offered by this model in addition to those permitted by the purely additive and subtractive mixings.

The definition of BRDF as a ratio of radiance to irradiance assumes that the geometrical optics framework applies, implicitly meaning that spatial coherence and diffraction of light have no significant effect in the reflection process. However, recent applications of BRDF push at increasing the angular resolution and thus at reducing the solid angles for illumination and collection. Therefore speckle, an optical effect inherent to the stochastic nature of scattering objects, becomes apparent. We suggest that BRDF should be redefined as the statistical average over that effect.