On the validity of two-flux and four-flux models for light scattering in translucent layers: angular distribution of internally reflected light at the interfaces.

Optical characterization and appearance prediction of translucent materials are required in many fields of engineering such as computer graphics, dental restorations or 3D printing technologies. In the case of strongly scattering materials, flux transfer models like the Kubelka-Munk model (2-flux) or the Maheu's 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 rely on the assumption of lambertian fluxes at any depth within the material, they fail to model the internal reflectance at the interfaces, penalizing the accuracy of the optical parameter extraction. The aim of this 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 scattering on light propagation. It turns out that the light angular distribution at the bordering interfaces is often far from being lambertian, and that the internal reflectance may vary significantly according to the layer's thickness, refractive index, scattering and absorption coefficients and scattering anisotropy. This work enables to better understand the impact of scattering within a translucent layer and also invites to revisit the well-known Saunderson correction used in 2- or 4-flux models.

Bronzing effect: predicting the color shine of printed surfaces by estimating the refractive index of the inks.

This paper investigates the optical phenomenon responsible for the colored shine that sometimes appears at the surface of ink layers in the specular direction, often called "bronzing" or "gloss differential." The prediction of this shine effect relies on the Fresnel formulas of the air/ink interface. The complex refractive index of the ink must therefore be determined, which is made difficult because of the roughness of inked printing supports. We propose a generic method that can be applied to any ink, without any prior knowledge of its composition or the printing substrate. In order to reduce light scattering, a solid colored area is printed with the studied ink on a glossy paper previously printed with black ink. By ellipsometry, we determine the effective refractive index of the sample. The intrinsic complex refractive index of the ink can then be extracted by modeling the optical response of the inked surface with a set of Gaussian oscillators, among which one of them approaches residual scattering. With this data, we could proceed to a fine colorimetric analysis of the bronzing color of some cyan, magenta, and yellow inks. In particular, we show that this gloss color is slightly shifted from the complementary of the ink's usual color in diffuse reflection.

Photometric properties of piles of glass plates: retrospective.

Stacked glass plates have discreetly accompanied the understanding of light since the origins of modern optics. They were studied by Bouguer, Lambert, Brewster, Arago, Stokes, Rayleigh, and many others, whose successive works progressively refined the predictive formulas of the reflectance and transmittance of piles of glass plates as a function of the number of plates and the angle of incidence by considering the decay of light flux by absorption, the multiple reflections between plates, the change in the degrees of polarization, and the possible interferential effects. Through this history of ideas about the optical properties of piles of glass plates, up to the mathematical formalisms from only a few years ago, we show that these successive works, and their subsequent errors and corrections, are inseparable from the evolution of the quality of the glass available each time, in particular its absorptance and its transparency, which strongly influence the quantities and the degree of polarization of the reflected and transmitted beams.

Instructive errors of Bouguer, Lambert, and Arago in the first determinations of angular reflectances on flat surfaces: discussion.

The first photometric measurements performed in the eighteenth century were based on brightness matching between two illuminated surfaces. In 1760, Bouguer and Lambert proposed the first methods to measure the angular reflectance of a flat surface, and Arago proposed a third one in the mid-nineteenth century. These pioneering experiments provided rather good estimates of the values we can predict or measure much more accurately today, considering that the human visual system was the only available light detector at that time. We show that the errors made in their measurements come not only from experimental uncertainties but also from incomplete knowledge of the physical properties of light, leading to incorrect assumptions in their models. The main errors are (i) the fact that light is totally reflected at grazing incidence, (ii) the glass plates they used were not perfectly clear, and (iii) light is partially polarized after transmission across the surface. By highlighting the impact of these three errors, we can better understand the state of knowledge in optics at that time and question our current practices in radiometric measurements and calculations.

Evaluating edge loss in the reflectance measurement of translucent materials.

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.

Design of rough microgeometries for numerical simulation of material appearance.

Microfacet-based material appearance models are commonly considered as a physical plausible representation of matter-light interaction. With such models, the microgeometry of a surface element is defined by a statistical distribution of microfacets. The mathematical formulation ensures physical plausibility, such as energy conservation and reciprocity. Many authors have addressed microfacet bidirectional scattering distribution function (BSDF) representations, with various normal distribution functions (NDFs) and their relationship with shadowing and masking, or the effects due to multiple light scattering on the microgeometry. However, an extensive study on how an actual microgeometry drives material appearance still is missing. This question is a key issue for inverse design and manufacturing. This paper contributes to filling this gap by proposing a complete pipeline composed of a microgeometry generation process and numerical lighting simulation. From any input NDF, our method generates a controlled and structured microgeometry, integrated within numerical light scattering simulation. Reflected light is gathered using a virtual goniophotometer. From a given set of parameters, we use our pipeline to study the impact of microgeometry structures on material light scattering in the case of rough surfaces. The obtained results are discussed and compared with already existing approaches when they exist in the pipeline.

Modeling, measuring, and using BRDFs: significant French contributions.

The scattering of light by a surface is described by the bidirectional reflectance distribution function (BRDF). Unfortunately, this function cannot be straightforwardly acquired or modeled. French researchers have proposed interesting contributions to the field, with several models and accurate experimental systems. For instance, the National Metrological Institute (LNE-CNAM) has implemented the best angular resolution goniospectrophotometer (0.015°). Modeling the BRDF has also been deeply studied in France, especially with the microfacet theory in recent years, a better understanding of the shadowing-masking function, new general distribution functions, visible normals, interfaced Lambertian microfacets, and analysis concerning light multiple reflections. This paper presents the state of the art regarding some significant French contributions in these fields.

Angular reflectance model for ridged specular surfaces, with comprehensive calculation of inter-reflections and polarization.

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.

Halo and subsurface scattering in the transparent coating on top of a diffusing material.

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.

Rendering Rough Opaque Materials with Interfaced Lambertian Microfacets.

Specular microfacet distributions have been successfully employed by many authors for representing glossiness of materials. They are generally combined with a Lambertian term to account for the colored aspect. These representations make use of the Fresnel reflectance factor at the interface, but the transmission factor at the interface should also be managed. One solution is to employ a multi-layered model with a single layer for the rough interface, which requires a numerical simulation for handling the multiple reflections of light between the substrate and the interface. In this paper, we propose rather to use a representation corresponding to a Fresnel interface lying on a Lambertian substrate, for which the multiple reflections of light between the interface and the substrate can be expressed analytically. With this interfaced Lambertian model, we show how Fresnel transmission affects the material appearance for flat and rough surfaces with isotropic and anisotropic distributions, that produce light backscattering effects. We also propose a methodology for using such materials in any physically based Monte Carlo rendering system, as well as an approximate representation, suitable for GPU applications or measured data fitting. Our approach generalizes several previous models, including flat Lambertian materials as well as specular and Lambertian microfacets. Our results illustrate the wide range of materials that can be rendered with this representation.

Spectral and Color Changes of Ag/TiO2 Photochromic Films Deposited on Diffusing Paper and Transparent Flexible Plastic Substrates.

Giving paper and polymer photochromic properties under laser irradiation is challenging due to the low resistance of these materials to heat, their flexibility, and their possibly irregular structure. However, we could successfully deposit TiO2/Ag/TiO2 layers stacking on flexible white glossy paper and transparent polyethylene terephalate (PET) substrates using a reactive magnetron sputtering technique, and tailor coloration changes after laser irradiation, alternating visible and ultraviolet (UV) wavelengths. The sample colors are characterized by a panel of chromas depending on the irradiation conditions. We demonstrate that these chroma changes are due to morphological changes of Ag nanoparticles (NPs) after visible laser irradiation of the colored as-deposited sample. The process exhibits a good reversibility after subsequent UV irradiation due to the growth of new metallic Ag NPs. The colors displayed in diffuse reflection by the paper samples are more saturated than the ones displayed in regular transmission by PET samples. We demonstrate the efficiency of the photochromic process on such support by printing high resolution patterns exhibiting different colors depending on the observation conditions.

Experimental study on merits of virtual cleaning of paintings with aged varnish.

To assess the accuracy of virtual cleaning of Old Master paintings (i.e. digital removal of discolored varnishes), a physical model was developed and experimentally tested using reflectance imaging spectroscopy on three paintings undergoing conservation treatment. The model predicts the reflectance spectra of the painting without varnish or after application of a new varnish from the reflectances of the painting with the aged varnish, given the absorption of the aged varnish and the scattering terms. The resulting color differences between the painting actually and virtually cleaned can approach the perceivable limit. Residual discrepancies are ascribable to spatial variations in the characteristics of the aged varnish (scattering, optical thickness) and the exposed painting (surface roughness).

Between additive and subtractive color mixings: intermediate mixing models.

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.

Two-flux transfer matrix model for predicting the reflectance and transmittance of duplex halftone prints.

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 nonsymmetric 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.

Generalization of the geometric description of a light beam in radiometry and photometry.

Radiometric and photometric quantities rely on a geometric description of the beam subtended by a source and a receptor. In this paper, a generalization of this description is proposed as the product of the apparent size of the source times the receptor angular extent, whatever the natures of these elements: point, line, surface, or volume. The obtained flux density per geometric extent expressions are then applied to the determination of the irradiances induced in the near field and far field by a rectilinear source represented as a point source, a line source, and a surface source.

Photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets.

The Lambertian model for diffuse reflection is widely used for the sake of its simplicity. Nevertheless, this model is known to be inaccurate in describing a lot of real-world objects, including those that present a matte surface. To overcome this difficulty, we propose a photometric model where the surfaces are described as a distribution of facets where each facet consists of a flat interface on a Lambertian background. Compared to the Lambertian model, it includes two additional physical parameters: an interface roughness parameter and the ratio between the refractive indices of the background binder and of the upper medium. The Torrance-Sparrow model--distribution of strictly specular facets--and the Oren-Nayar model--distribution of strictly Lambertian facets--appear as special cases.

Spectral prediction model for piles of nonscattering sheets.

The present paper investigates the reflection and transmission properties of piles of nonscattering sheets. Using a spectral prediction model, we perform a detailed analysis of the spectral and color variations induced by variations of the number of superposed sheets, the absorbance of the sheet material, the refractive index of the medium between the sheets, and the reflectance of the background. The spectral prediction model accounts for the multiple reflections and transmissions of light between the interfaces bounding the layers. We describe in detail the procedure for deducing model parameters from measured data. Tests performed with nonscattering plastic sheets demonstrate the excellent accuracy of the predictions. A large set of predicted spectra illustrate the different evolutions of reflected and transmitted spectra as well as the corresponding colors for various types of piles.

Ray scattering model for spherical transparent particles.

We propose a model for the reflectance of a particle medium made of identical, large, spherical, and absorbing particles in a clear binder. A 3D geometrical description of light scattering is developed by relying on the laws of geometrical optics. The amount of light backscattered by a single particle is determined as a function of its absorbance and refractive index. Then, we consider a set of coplanar particles, called a particle sublayer, whose reflectance and transmittance are functions of the particle backscattering ratio and the particle concentration. The reflectance of an infinite particle medium is derived from a description of multiple reflections and transmissions between many superposed particle sublayers. When the binder has a refractive index different from that of air, the medium's reflectance factor accounts for the multiple reflections occurring beneath the air-binder interface as well as for the measuring geometry. The influences of various parameters, such as the refractive indices and the particle absorption coefficient, are examined.

Geometrical considerations in analyzing isotropic or anisotropic surface reflections.

The bidirectional reflectance distribution function (BRDF) represents the evolution of the reflectance with the directions of incidence and observation. Today BRDF measurements are increasingly applied and have become important to the study of the appearance of surfaces. The representation and the analysis of BRDF data are discussed, and the distortions caused by the traditional representation of the BRDF in a Fourier plane are pointed out and illustrated for two theoretical cases: an isotropic surface and a brushed surface. These considerations will help characterize either the specular peak width of an isotropic rough surface or the main directions of the light scattered by an anisotropic rough surface without misinterpretations. Finally, what is believed to be a new space is suggested for the representation of the BRDF, which avoids the geometrical deformations and in numerous cases is more convenient for BRDF analysis.

Extension of the Williams-Clapper model to stacked nondiffusing colored coatings with different refractive indices.

We propose a model for predicting the reflectance and transmittance of multiple stacked nonscattering coloring layers that have different refractive indices. The model relies on the modeling of the reflectance and transmittance of a bounded coloring layer, i.e., a coloring layer and its two interfaces with neighboring media of different refractive indices. This model is then applied to deduce the reflectance of stacked nonscattering layers of different refractive indices superposed with a reflecting diffusing background that has its own refractive index. The classical Williams-Clapper model becomes a special case of the proposed stacked layer model.