Abstract

The four-flux model is a method to solve light radiative-transfer problems in planar, possibly multilayer structures. The light fluxes are modeled as two collimated and two diffuse beams propagating forward and backward perpendicularly to the layer stack. In the present contribution, we develop a four-flux model relying on a matrix formalism to determine the reflectance and transmittance factors of stacks of components by knowing those of each individual component. This model is also extended to generate the bidirectional scattering distribution function of the stack by considering an incoming collimated flux in any direction and by taking into account the directionality of the diffuse fluxes exiting from the material at the border components of the stack. The model is applied to opaque Lambertian backgrounds with flat or rough interfaces for which analytical expressions of the BSDF are obtained.

© 2015 Optical Society of America

1. INTRODUCTION

Predicting the visual appearance of objects by carrying out an acceptable computation effort is often a challenge because of the variety of materials and surface finishes and the complexity of the multiple optical phenomena occurring within the material layers. Accurate prediction is however, crucial for the digital design of objects. It requires modeling the multiple reflections and scattering of light at the interfaces and within the turbid media composing the material. A radiometric approach consists in writing the balance for the absorbed and scattered fluxes by introducing the properties of single scattering. In the general tridimensional case, multiple scattering is mostly solved by Monte Carlo methods where each individual photon event is described with probabilities. For specific material structures, especially stacks of planar layers, one can solve the radiative transfer equation (RTE) [1]. This integro-differential equation takes into account both the spatial position and the orientations of the incident and scattered fluxes. However, as it is complex to solve, many resolution techniques and approximations have been suggested. One common approximation is not to take into account explicitly lateral scattering within the material. Therefore, the equation depends only on one spatial dimension, namely the depth z within the stack. Regarding the angular distribution of light, it can be discretized into N-annular solid angles. This N-flux model was first proposed by Mudgett and Richards [2] in the case of azimuthally isotropic scattering and generalized by Stamnes et al. [3] under the so-called discrete ordinate method. N can exceed 20 [2] but small N values present the advantage of simple expressions for the reflectance and the transmittance factors. For N=2, only two hemispherical fluxes with constant radiance propagate toward positive and negative z. In that case, the RTE has analytical solutions, which are well known as the Kubelka–Munk formulas [4,5]. This two-flux model was also extended to determine the diffuse reflectances and transmittances of stacks of scattering layers [6]. Whereas the Kubelka–Munk model is the result of the continuous integration of the RTE, the Kubelka 1954 model can be interpreted as the corresponding discrete summation. The correspondences between the continuous and discrete two-flux approaches have already been discussed [7,8].

However, considering two diffuse fluxes is not possible when the incident light is collimated and a part of it becomes diffuse. The four-flux model with two additional collimated fluxes propagating perpendicularly to the planar layers toward positive and negative z directions improves the reflectance and transmittance predictions in case of collimated illumination. Resolutions of the RTE according to the four-flux approach with various boundary conditions were proposed by Beasley et al. [9], Mudgett and Richards [2], and Ishimaru [10]. The formulation proposed by Maheu et al. [11,12] became the main reference when compared with exact calculations for specific cases [13] or after comparisons with Monte Carlo simulations [14]. In their formulation Maheu et al. introduced an average path-length parameter that can take values from 1 for a collimated flux to 2 for a perfect isotropic radiation. They also introduce a forward scattering ratio. The determination of these parameters was discussed by several authors [1520]. The four-flux model can be expressed by using a matrix formalism as suggested by Rozé et al. in the case of multilayer [21]. Recent formulations [22,23] also enable predicting interface effects. The four-flux model can be used for various scattering systems as illustrated by recent publications [2427]. However, even if the four-flux approach is much easier to use than more elaborate models, the simplicity of the two-flux approximation is still often preferred. For this reason intermediate models between two flux and four flux have also been proposed [2830].

In the present study, we use the four-flux model without specifically focusing on the resolution of the RTE. In Section 2 we adopt a matrix formalism to calculate the reflectance and transmittance factors of a superposition of optical components (interfaces and propagating media). This approach can be seen as an extension of the Kubelka model [6] to four fluxes. The main contribution of this study, presented in Section 3, consists in adapting the four-flux matrix model in order to generate families of bidirectional scattering distribution factors (BSDF) by considering an incident collimated flux in any direction and directional diffuse fluxes exiting the material. The bidirectional transfer matrices are described for the particular cases of highly scattering (Lambertian) and of nonscattering (transparent) components in Section 4. We consider flat or rough dielectric interfaces in Section 5. The complete method is finally presented in Section 6 to determine the BSDF in the special cases where a flat or a rough interface is at the top of an opaque Lambertian background.

2. FOUR-FLUX MATRIX MODEL

The four-flux model considers a parallel planar structure of material and therefore reduces the radiative-transfer equation to a problem with one spatial dimension. It can be presented as a special case of the N-flux model [2] where the radiation field for each position in the stack of layers is composed of two collimated beams Ic and Jc and two isotropic diffuse beams Id and Jd. The fluxes propagate perpendicularly to the plane, forward (Ic and Id) and backward (Jc and Jd).

A multilayer material can be described as a succession of interfaces and media. Each component of the stack, interface or medium, gives rise to flux transfers: front side reflectance r, back side reflectance r, forward transmittance t, and backward transmittance t. They can be collimated-to-collimated (label cc), diffuse-to-diffuse (label dd), or collimated-to-diffuse (label cd) transfers. Figure 1 represents the flux transfers for a stack of two components.

 figure: Fig. 1.

Fig. 1. Flux transfers between two components represented by thin arrows. Bold arrows correspond to fluxes.

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Let us consider the component labeled k. The fluxes labeled by superscripts k and k1 are related according to the following equations where, for the sake of writing simplicity, we omit the label k in the transfer factors:

{Jck1=rccIck1+tccJckIck=tccIck1+rccJckJdk1=rcdIck1+tcdJck+rddIdk1+tddJdkIdk=tcdIck1+rcdJck+tddIdk1+rddJdk.
It is easy to verify that these equations express the relationships shown in Fig. 1.

A. Matrix Formulation

The system of Eq. (1) can be presented under two possible matrix equations. The first matrix equation used to solve the RTE [31] in a multi-angle approach is shown in Appendix A. We consider here the second matrix equation, which focuses on the transfer nature (collimated-to-collimated, diffuse-to-diffuse, and collimated-to-diffuse) and is therefore specific to the four-flux approach,

(rcc100tcc000rcd0rdd1tcd0tdd0)(Ick1Jck1Idk1Jdk1)=(0tcc001rcc000tcd0tdd0rcd1rdd)(IckJckIdkJdk).
The matrix on the left-hand side can be inverted if tcctdd0. By left-multiplying both members of Eq. (2) with the inverse of the left-most matrix, we obtain the following equation exhibiting the transfer matrix of the considered component, which is written for convenience under a 2×2 block form:
(Ick1Jck1Idk1Jdk1)=(Mcc02,2McdMdd)(IckJckIdkJdk),
where each block is a 2×2 matrix. The two blocks on the diagonal, corresponding to collimated-to-collimated transfers (xx=cc) and diffuse-to-diffuse transfers (xx=dd) are
Mxx=1txx(1rxxrxxtxxtxxrxxrxx),
and the left-bottom block corresponding to collimated-to-diffuse transfers (cd) is
Mcd=1tcctdd(tcdrcctcdrcdtccrcdtddrddtcdtcc(tcdtddrcdrdd)rcc(rcdtddrddtcd)).
From a given transfer matrix, the reflectances and transmittances of the component (or stack of components) can be obtained provided Mcc(1,1)0 and Mdd(1,1)0. Let us express, for example, the front-side reflectance and the front-to-back transmittance for the different types of transfers. For collimated-to-collimated (xx=cc) and diffuse-to-diffuse (xx=dd) transfers, they are given by
txx=1Mxx(1,1)andrxx=Mxx(2,1)Mxx(1,1),
and, for collimated-to-diffuse transfers, they are given by
tcd=Mcd(1,1)Mcc(1,1)Mdd(1,1)andrcd=Mcd(2,1)Mcc(1,1)Mcd(1,1)Mdd(2,1)Mcc(1,1)Mdd(1,1).

B. Matrix Multiplication

In order to obtain the transfer matrix representing a stack of components, the components’ individual transfer matrices are multiplied by respecting the stacking order of the components. With two components characterized by matrices M1 and M2, from front to back, the transfer matrix of the two components together is

M1M2=(Mcc1Mcc202,2Mcd1Mcc2+Mdd1Mcd2Mdd1Mdd2).
Applying the 2×2 matrix multiplication Mxx1Mxx2 and the formulas Eq. (6) yields, for collimated-to-collimated (xx=cc) or diffuse-to-diffuse (xx=dd) transfers, the global transmittances and reflectances of the two-component stack,
txx=txx1txx21rxx1rxx2andrxx=rxx1+rxx2txx1txx11rxx1rxx2.
Applying the 2×2 matrix operation Mcd1Mcc2+Mdd1Mcd2 and the Eqs. (7) yields for collimated-to-diffuse transfers,
tcd=tcd1tdd21rdd1rdd2+tcc1tcd21rcc1rcc2+tcc1tdd2(rcd1rcc2+rdd1rcd2)(1rcc1rcc2)(1rdd1rdd2),
rcd=rcd1+tcd1rdd2tdd11rdd1rdd2+tcd1rcc2tcc11rcc1rcc2+tcc1tdd1(rcd2+rdd2rcd1rcc2)(1rcc1rcc2)(1rdd1rdd2).
If either collimated-only or diffuse-only fluxes are considered, the model becomes the two-flux model described by its corresponding 2×2 matrices (either Mcc or Mdd). Similar relations, as Eq. (9), were derived by Stokes [32] in order to predict the specular reflectances and regular transmittances of stacks of glass plates. They were also used later by Kubelka [6] to predict the diffuse reflectances and transmittances of stacks of strongly scattering layers.

Given the complex expressions of the collimated-to-diffuse reflectance and transmittance [Eqs. (10) and (11)], even with two components only, the matrix formalism is much more convenient. However, the matrix computation is valid only when the following condition is satisfied for each component:

tccitddi0.
For example, with an opaque component this condition cannot be satisfied. In these cases the matrix calculations are first performed with the literal expressions of these transmittances. They are then set to zero at the very final step of the calculation.

3. EXTENSION FOUR-FLUX MATRIX MODEL TO EXPRESS THE BSDF OF COMPONENT STACKS

In its original expression [11], the four-flux model assumes collimated and isotropic hemispherical diffuse fluxes propagating perpendicular to the stack of layers. In order to define BSDF models from the four-flux matrix method presented in Section 2, some adaptations are needed. The BSDF expresses the bidirectional reflectance and transmittance distribution functions (BRDF and BTDF), for which the incident illumination is assumed to be a unique collimated flux in any direction i of the upper hemisphere not only at the normal incidence. Moreover, the diffuse fluxes exiting the material from the border components of the stack are not necessarily assumed Lambertian and can therefore depend on the output direction o of the upper hemisphere for BRDF and of the lower hemisphere for BTDF.

The radiometric definitions and relations used in this section are detailed in the literature, for example in [33,34].

A. BSDF Configuration

According to the definition of the BSDF, the incident illumination is assumed to be a collimated flux in the incident direction i. There is no incident diffuse flux or upward incident flux. In the case of the system represented in Fig. 1, this means that Id0=Jc2=Jd2=0. The scattered light is captured in every direction o. Figure 2 explains the notations.

 figure: Fig. 2.

Fig. 2. Useful notations for defining the BSDF.

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B. Collimated Fluxes Related to the Incident Direction

The collimated incident beam is defined for a freely chosen orientation i within the upper hemisphere. The collimated fluxes after multiple reflections and transmissions remain within the incident plane. Knowing the different refractive indices, the directions of the downward fluxes are defined from the incident direction i and by Snell’s refraction law. The direction of each upward flux is deduced from the direction of the corresponding downward flux according to Snell’s reflection law.

C. Directional Diffuse Output Transfers

In the original four-flux model, the diffuse fluxes are assumed to be hemispherical with angle-independent radiance (i.e., Lambertian). In our approach we make an exception for the diffuse fluxes exiting the material from the first or from the last component of the stack. Therefore, the reflectance and transmittance factors to be chosen depend on the position of the component in the stack. Directional light that exits the stack of components is diffused according to the BRDF and the BTDF of the bordering components. Table 1 defines the collimated-to-diffuse and diffuse-to-diffuse transfer factors to be used according to the definitions by Nicodemus et al. [33]. For the sake of simplicity, we specify vector i only when the incident light is collimated and/or vector o only when light is captured in one direction (see Tables 1 and 2).

Tables Icon

Table 1. Collimated-to-Diffuse and Diffuse-to-Diffuse Reflectance and Transmittance Factors of a Component According to its Position in the Component Stacka

Tables Icon

Table 2. Expressions of the Different Reflectance or Transmittance Factors in Function of the BRDF or BTDF f, where 2πfcosθdω=θ=0π/2φ=0π/2fcosθsinθdθdφ

By knowing the BRDF and the BTDF of a component, we can define its reflectance or transmittance factors according to the formulas given by Table 2 [33]. For a stack of components, according to Table 1 and after the matrix multiplications, one can deduce the BRDF fr(i,o) and BTDF ft(i,o) of the stack

fr(i,o)=rcd(i,o)πandft(i,o)=tcd(i,o)π,
where the bidirectional reflectance and transmittance factors, rcd(i,o) and tcd(i,o), are given by Eq. (7).

Moreover, in the case where part of the incident collimated flux remains collimated after exiting the border components, the specular reflectance rcc(i) and the regular transmittance tcc(i), given by Eq. (6) with xx=cc, have to be added to the BSDF formally by using Dirac delta functions [35].

In contrast with more elaborated models based on the multi-angle scattering approach [36], we assume that the diffuse fluxes lose their directionality within the stack. In many systems this limitation has a weak influence when at least one component of the stack is sufficiently scattering to make the assumption acceptable. But even when it is not the case, the directionality of the collimated fluxes is preserved within the stack, and the directionality of the diffuse fluxes is conserved at the extreme components of the stack. A more restrictive limitation, but intrinsic to the four-flux approach, is the fact that the angular spreading of the collimated fluxes is not rendered.

4. TRANSFER MATRICES FOR LAMBERTIAN AND NONSCATTERING COMPONENTS

Among the optical components, the Lambertian scattering as well as the nonscattering components are interesting limit cases. Their presence in a stack enables important simplifications of the prediction method, especially when the border components of the stack are nonscattering.

A. Lambertian Component

For a Lambertian component, light is uniformly scattered over the hemisphere independent of the orientation of the incident light. It is worth noting that it is an ideal case [37]. Consequently, an incident collimated light is entirely transformed into diffuse light and tcc=rcc=0 [Fig. 3(a)]. However, to fulfill the condition, Eq. (12), we artificially define a transmittance tcc, which we will set to zero hereafter, and the following matrix Mcc:

Mcc=1tcc(1000).
The assumption of a Lambertian component implies that the reflectance and transmittance factors are identical for a collimated or a diffuse incident light flux and for every output direction o,
{rcd(i,o)=rcd(i)=rddtcd(i,o)=tcd(i)=tdd(o)=tddand{rcd(i)=rddtcd(i,o)=tcd(i)=tdd(o)=tdd.
Therefore, the transfer matrix for a Lambertian component is independent of its position in the stack according to Table 1. Assuming tcctdd0, the collimated-to-diffuse transfer matrix [Eq. (5)] can be expressed as
Mcd=(1tccrddtddrddtcctddtddrddrddtdd).
For example, we can define a perfectly Lambertian rough interface or a perfectly Lambertian medium. In the last case, the medium is generally considered as symmetrical and
{rcd=rdd=rcd=rddtcd=tdd=tcd=tdd.
We can check that the superposition of Lambertian components is also Lambertian as its transfer matrix verifies Eqs. (14) and (15).

 figure: Fig. 3.

Fig. 3. Flux transfers (a) for a Lambertian component, (b) for a nonscattering component.

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B. Nonscattering Component

Without scattering, the collimated fluxes cannot be transferred into diffuse fluxes [Fig. 3(b)]. Therefore,

Mcd=02,2.
For example, a nonscattering component can be a perfect plane interface or a nonscattering medium. In the last case, the light cannot be reflected by such a medium (rcc=rcc=rdd=rdd=0) and the matrices Mcc and Mdd are then diagonal.

For the special case of a nonscattering component, the diffuse-to-diffuse reflectances or transmittances are the integrals of the collimated beams in all directions of the upper or lower hemisphere, and can then be directly expressed from the corresponding collimated-to-collimated transfer factors

hdd=1πθi=0π/2φi=02πhcc(i)cosθisinθidθidφi=θi=0π/2hcc(i)sin2θidθi,
where h is r, r, t or t.

It is worth noting that the transfer matrix for the superposition of nonscattering components verifies Eq. (18). The resulting component is therefore also nonscattering. The reflectances and transmittances can be determined by operating independently with the 2×2 transfer matrices Mcc and Mdd. However, the diffuse reflectances and transmittances obtained by diffuse-to-diffuse transfer matrix multiplication [Eqs. (9) with xx=dd] are crude approximations for nonscattering components. One needs to calculate first the collimated-to-collimated reflectances and transmittances [Eqs. (9) with xx=cc] and integrate them over the hemisphere [Eq. (19)].

C. Scattering Components Surrounded by Nonscattering Components

When several components are superposed, sub-stacks of nonscattering components are first regrouped and their corresponding transfer matrices are determined. The expressions of their diffuse-to-diffuse transmittance factors depend on if they are boundary component or not according to Table 1. In the case that the nonscattering sub-stack is the first component, the hemispherical-directional tdd(o) is

tdd(o)=tcc(o)/n2,
where n is the refractive-index ratio between the initial and final media.

If it is the last component of the stack, the hemispherical-directional tdd(o) is

tdd(o)=n2tcc(o).
The factor 1/n2 in Eq. (20), respectively, the factor n2 in Eq. (21), is related to the conservation of the optical extent and takes into account the extension of the light beam toward a less refractive medium, respectively, the contraction of the light beam toward a more refractive medium [34].

A nonscattering component presents the advantage to preserve the bidirectional reflectance factor, respectively, transmittance factor, when it is the first or the last component. Let us consider a nonscattering component (index 1) on any scattering component (index 2) as represented in Fig. 4(a). By using Eq. (11) and the properties of the nonscattering component 1 [Eqs. (18) and (20)], the resulting bidirectional reflectance factor rcd(i,o) can be expressed in terms of the one of the component 2, rcd2(i2,o2):

rcd(i,o)=tcc1(i)tcc1(o)n2rcd2(i2,o2)(1rcc1(i)rcc2(i))(1rdd1rdd2),
where the directions i2 and o2 are related to the directions, respectively, i and o according to Snell’s refraction law.

 figure: Fig. 4.

Fig. 4. (a) Bidirectional reflectance factor rcd(i,o) of a nonscattering component (index 1) on a scattering component (index 2). (b) Bidirectional transmittance factor tcd(i,o) of a scattering component (index 1) on a nonscattering component (index 2).

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With similar considerations, we can obtain the bidirectional transmittance factor for any scattering component (index 1) on a nonscattering component (index 2) as represented in Fig. 4(b):

tcd(i,o)=tcc2(o)n2(tcd1(i,o1)1rdd1rdd2+tcc1(i)rcd1(i)rcc2(i)(1rcc1(i)rcc2(i))(1rdd1rdd2)),
where the direction o1 is related to the outgoing direction o according to Snell’s refraction law.

5. TRANSFER MATRICES FOR FLAT AND ROUGH INTERFACES

As the first and the last components mainly influence the BRDF and BTDF of a layered material, the scattering responses of these two components must be analyzed carefully. These border components are most often interfaces. We present the corresponding four-flux matrices for a flat or a rough interface between two dielectric media labeled 0 and 1, with respective refractive indices n0 and n1. The relative refractive index of the interface is denoted as n=n1/n0.

A. Flat Interface

A flat interface is a nonscattering component, which does not enable any collimated-to-diffuse light transfer. Therefore, the 2×2 matrix Mcd is a zero matrix. The collimated-to-collimated transfers are given by the Fresnel formulas as functions of the incident direction i. By calling rcc=R01(i) and tcc=T01(i), and by considering

{nsin(θi1)=sin(θi)tcc=T01(i)=1R01(i)rcc=R10(i1)=R01(i)tcc=T10(i1)=1R01(i),
the collimated-to-collimated transfer matrix can be written according to Eqs. (5) and (24),
Mcc=1T01(i)(1R01(i)R01(i)12R01(i)).
Regarding the diffuse-to-diffuse transfer matrix, the bihemispherical reflectance factor rdd is obtained by the angular integration of Eq. (19) [38],
rdd=r01=θi=0π/2R01(i)sin2θidθi.
The other bihemispherical reflectance and transmittance factors can then be easily deduced from rdd by the relations
{tdd=t01=1r01tdd=t10=t01/n2rdd=r10=1t10
These bihemispherical factors only depend on the refractive index ratio n and can be expressed analytically [39].

According to Eqs. (20) and (21), if the flat interface is the first or the last component of the stack, the hemispherical-directional transmittance factors can be calculated as

tdd(o)=T01(o)/n2,
tdd(o)=n2T01(o).

B. Rough Interface

Any model describing the BSDF of a rough interface can be used in the four-flux model presented in this paper. We adopt the micro-facet model described by Walter et al. [35]. The roughness parameter σ becomes an additional index for all transfer factors. We assume that the incident collimated flux is completely converted into diffuse fluxes. Hence, the collimated-to-collimated transfers are assumed to be zero [Eq. (14)]. The bidirectional reflectance factor is defined as

r01σ(i,o)=πR01(i,hr)Dσ(hr)G(i,o,hr)4(i·n)(o·n),
and the bidirectional transmittance factor as
t01σ(i,o)=π|i·ht||o·ht|(i·n)(o·n)n12T01(i,ht)Dσ(ht)G(i,o,ht)(n0(i·ht)+n1(o·ht))2,
where Dσ is the distribution of the micro-facet normals, G is a shadowing masking term, and the directions hr and ht are defined as hr=(i+o)/i+o and ht=(n0i+n1o)/n0i+n1o.

All entries of the transfer matrix are reflectance and transmittance factors that can be calculated according to the position of the component within the stack (Table 1) and to the angular distribution of the incident flux on both faces (relations of Table 2). For example, the following equation gives the expression of the bihemispherical reflectance factor r01σ as a function of the directional-hemispherical reflectance factor r01σ(i):

r01σ=θi=0π/2r01σ(i)sin2θidθi.
Appendix B shows the calculated directional-hemispherical and the bihemispherical factors of a rough interface for various roughness parameter values and presents a way to preserve the energy at the interface.

6. FLAT OR ROUGH INTERFACE ON AN OPAQUE LAMBERTIAN BACKGROUND

As case studies, we consider a flat or a rough interface on the top of an opaque Lambertian background [Figs. 5(a) and 5(b)]. These systems involve only two components and the matrix calculations result in compact analytical relations.

 figure: Fig. 5.

Fig. 5. (a) Perfectly flat interface on a Lambertian background, (b) micro-facet rough interface on a Lambertian background, (c) distribution of interfaced Lambertian micro-facets.

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The calculation consists in multiplying the transfer matrix of the interface described in Section 5 by the transfer matrix of the opaque Lambertian background defined in Section 4.B. As a Lambertian component, the substrate matrix must verify Eqs. (14) and (16). The background diffuse reflectance is denoted as ρ=rdd=rcd. Although the substrate is opaque, we artificially assume that tcc0 and tdd0 [Eq. (12)] while tcd=tcd=tcc=tdd=0 and rcd=rcc=rdd=0. The transfer matrices of the opaque Lambertian background can then be written as

Mcc=(1tcc000),Mdd=(1tdd0ρtdd0)andMcd=(00ρtcc0).

A. Flat Interface on a Lambertian Background

As the first components of the stack, the transfer matrices for the plane interface are [Eqs. (25)–(28)]

Mcc=(1T01(i)R01(i)T01(i)R01(i)T01(i)12R01(i)T01(i)),Mdd=(1t01r10t01r01t01t01T01(o)/n2r01r10t01)andMcd=02,2.
The resulting transfer matrix is obtained by multiplying the interface matrix Eq. (34) with the background matrix Eq. (33). We obtain the following matrices for the stack of the two components:
Mcc=(1tccT01(i)0R01(i)tccT01(i)0),Mdd=(1ρr10t01tdd0r01+ρ(t01T01(o)/n2r01r10)t01tdd0)andMcd=(r10ρt01tcc0ρt01T01(o)/n2r01r10t01tcc0).
From these matrices, using Eqs. (6) and (7), we can deduce the transfer factors of the interfaced background. Then by setting tcc=tdd=0 (these latter were artificially maintained nonzero), we obtain that the transmittances are zero as expected since the material is opaque. The overall collimated-to-collimated reflectance corresponds to the specular reflectance R01(i) of the flat interface. Finally, the collimated-to-diffuse reflectance rcd of the interfaced background enables deducing the BRDF without its specular component thanks to Eq. (13):
fr(i,o)=1πn2T01(i)T01(o)ρ(1r10ρ).
This analytical relation was first obtained by Elias et al. [40] and can be interpreted as a bidirectional extension of the earlier spectral reflectance model by Williams and Clapper [41] for gelatin-based photographic color prints assuming a non-absorbing gelatin layer. The Saunderson correction [42] deals with the same system (flat interface on a Lambertian background) but in the more basic two-flux approach for diffuse light beams. The bidirectional calculation can be easily extended to the case of a stack of nonscattering components instead of a single flat interface either by using the four-flux matrix formalism or by replacing the interface regular transmittances T01 and internal diffuse reflectance r10 of Eq. (36) by the equivalent factors of the nonscattering multilayer [43]. The model can be also generalized to a non-Lambertian background by using Eq. (22), which allows calculating the BRDF of a flat interface on the top of any substrate whose bidirectional reflectance factor is known.

B. Rough Interface on a Lambertian Background

By using the notations introduced in Section 5, as well as Eqs. (4) and (5), the transfer matrices for the rough interface as the first component of the stack

Mcc=(1Tcc000),Mdd=(1t01σr10σt01σr01σt01σt01σt10σ(o)r01σr10σt01σ)andMcd=(t01σ(i)Tcct01σr10σ(i)t01σt01σr01σ(i,o)t01σ(i)r01σTcct01σt10σ(i,o)t01σr10σ(i)r01σTcct01σ),
where Tcc is the collimated-to-collimated transmittance of the rough surface, first assumed to be nonzero.

After multiplying the rough-interface matrix, Eq. (37), by the opaque Lambertian background matrix, Eq. (33), we obtain

Mcc=(1Tcctcc000),Mdd=(1r10σρtddt01σ0r01σ+ρ(t01σt10σ(o)r01σr10σ)tddt01σ0)andMcd=(t01σ(i)Tcctcct01σr10σρtcct01σ0t01σr01σ(i,o)t01σ(i)r01σTcctcct01σ+ρt01σt10σ(o)r01σr10σtcct01σ0).
From these matrices, using Eqs. (7) and (13) and by setting Tcc=tcc=tdd=0 (artificially maintained nonzero during the calculations), we can deduce the corresponding BRDF with no additional specular term:
fr(i,o)=1π(r01σ(i,o)+t01σ(i)t10σ(o)ρ1r10σρ).
The first term of Eq. (39) is the BRDF of the rough interface (single scattering). The second term is due to the multiple reflections between the inner face of the rough interface and the Lambertian background. As far as we could see in the literature, such an analytical expression for this system has never been published. However, as explained in Appendix B, usual micro-facet models strongly underestimate the internal diffuse reflectance r10σ. To compensate for this energy loss at the interface, we assume this reflectance to be independent of the roughness [see r10σ1t10σ in Fig. 8(b)]. Consequently, we apply Eq. (39) by replacing r10σ by the internal diffuse reflectance r10 of a flat interface.

C. Interfaced Lambertian Facets

It is interesting to compare the configuration described by Eq. (39) and Fig. 5(b) with the one described by Fig. 5(c), which was developed in a previous work [44], with the same micro-facet slope distribution. The corresponding BRDF can be written as

fr(i,o)=r01σ(i,o)π+ρπ(1r10ρ)1n2(i·n)(o·n)2πT01(i,m)T01(o,m)Dσ(m)G(i,o,m)(i·m)(o·m)dωm,
where the integral sums up the radiances related to every micro-facet normal m in the hemisphere.

It can be first noted that both models are equivalent for a flat interface (σ=0) with the expression given in Eq. (36). Moreover, the first term r01σ(i,o), due to surface scattering, is similar for both Eqs. (39) and (40). It corresponds to a Cook–Torrance-like specular lobe [45] whose expression is given in Eq. (30). We therefore focus the comparison on the second term of Eqs. (39) and (40), due to volume scattering.

When a Lambertian background has a flat interface, its volume BRDF trends rapidly toward zero at grazing incident angles (see Fig. 6 for σ=0). This effect is reduced when considering a rough interface superposed over the Lambertian background [Fig. 6(a)]. The resulting volume BRDF tends to the one of a Lambertian reflector when the roughness increases. This BRDF is azimuthally isotropic, which is not the case for the distribution of interfaced Lambertian facets [Fig. 6(b)]. The variation of the volume BRDF depends on the observation angle. It decreases when the observation is from the part of the hemisphere containing the specular direction [θo>0 in Fig. 6(b)] but it increases for grazing angles toward the backscattering direction [θo<0 in Fig. 6(b)]. This difference between both models is striking even when there is no refractive index change (n=1) between media. In that specific case, Fig. 5(b) [Eq. (39)] is equivalent to the flat Lambert background while Fig. 5(c) [Eq. (40)] corresponds to a distribution of Lambertian micro-facets and is therefore equivalent to an Oren–Nayar-like model [46].

 figure: Fig. 6.

Fig. 6. Volume BRDF in the incident plane, assuming a Beckmann distribution for Dσ and the corresponding Smith shadowing masking function for G [47,48], with θi=60° (backscattering direction at θo=60° and specular direction at θo=60°), n=1.5 and different roughness parameters σ. (a) Rough interface on a Lambertian background [second term of Eq. (39) with r10σ=r10], (b) distribution of interfaced Lambertian facets [second term of Eq. (40)].

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7. CONCLUSION

The methodology presented in this paper opens new perspectives to solve radiative transfer problems. We adapt the four-flux model by describing a material as a stack of discrete components. We introduce a 4×4 transfer matrix that describes each component of the stack, collimated-to-collimated, diffuse-to-diffuse, and collimated-to-diffuse flux transfers both in transmittance and reflectance. Stacks of components are built by carrying out the corresponding matrix multiplications. Special matrices describe a Lambertian or a nonscattering component, a flat or a rough interface, and a border component given by its BRDF and BTDF. By construction, the 4×4 transfer matrix can be reduced to 2×2 matrices, and the two-flux models for either collimated-only or diffuse-only beams are special cases of this four-flux model. The use of four-flux is justified as soon as at least one component induces collimated-to-diffuse light transfers, and particularly for translucent materials where the two-flux models fail. We also extend the four-flux to obtain the BSDF of multilayer systems. A restriction of the model is the directionality loss of the diffuse fluxes within the layer stack. However, in most common cases, this limitation has a negligible impact because the model accounts for the directionality of the collimated fluxes at every position within the stack and for the directionality of the diffuse fluxes for the most external scattering components of the stack. Another restriction is that the four-flux approach does not describe the progressive angular broadening of the collimated beams. However, the method is easy to use and offers compact matrix expressions and quick computations. For computer graphics, it allows us to generate families of virtual BSDF for a superposition of components knowing the BSDF of each component.

We apply the method for a flat and a rough interface on an opaque Lambertian background. Although systems with only two components are relatively basic, they offer a wide variety of physical-based reflectance models from matte to glossy materials. We intend to apply this methodology to more complex systems where the compact matrix formalism will be an attractive feature.

APPENDIX A: ANOTHER TRANSFER MATRIX

The transfer matrix chosen in the paper focuses on the nature of the light transfers (labeled cc, dd, or cd). After rearrangement of the flux order for the vectors, another matrix relation alternative to Eq. (2) can be obtained:

(tcc000tcdtdd00rcc010rcdrdd01)(Ick1Idk1Jck1Jdk1)=(10rcc001rcdrdd00tcc000tcdtdd)(IckIdkJckJdk).
By introducing the 2×2 matrices T, R, T, and R, the matrix relation can be rewritten by block
(T02,2R12,2)(Ick1Idk1Jck1Jdk1)=(12,2R02,2T)(IckIdkJckJdk).
By assuming tcctdd0, the matrix of the left side is inverted and left multiplied to the matrix of the right side:
(Ick1Idk1Jck1Jdk1)=M(IckIdkJckJdk),withM=(T102,2RT112,2)(12,2R02,2T)=(T1T1RRT1TRT1R)=(M11M12M21M22).
Given the 2×2 block matrices Mij of the transfer matrix, the following matrix relations can be obtained:
{R=M21M111T=M111R=M111M12T=M12M21M111M12.
This transfer-matrix formalism was classically used to solve the radiative transfer equation with N×N block matrices [31]. Let us note that both matrix expressions [Eqs. (2) and (A1)] are related by permutation matrices.

APPENDIX B: ENERGY PRESERVATION FOR A MICRO-FACET ROUGH INTERFACE

We consider a rough interface between two media with n=1.5 whose bidirectional transfer factors are given by the Eqs. (30) and (31). The function Dσ is the Beckmann distribution and the corresponding shadowing masking function G is the one described by Smith [47] and generalized by Bourlier et al. [48]. Figure 7 shows the different directional-hemispherical factors. As expected, the rougher the surface is, the more Lambertian these resulting factors are. The differences with the flat interface (σ=0) are more pronounced at grazing incident angles from the less to the more refractive medium [Fig. 7(a)]. The differences are much more important in the opposite direction due to the effect of total reflection for a large part of the incident hemisphere [Fig. 7(b)].

 figure: Fig. 7.

Fig. 7. Directional-hemispherical reflectance (solid lines) or transmittance (dashed lines) factors of rough interfaces as a function of the incident angle θi for different roughness parameters σ with n=1.5 (a) from medium 0 to medium 1, (b) from medium 1 to medium 0.

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From relations similar to Eq. (32), the different bihemispherical factors can be calculated and are represented in Fig. 8 in terms of the roughness parameter. The conservation of energy should give r+t=1. This is however, not the case because the micro-facet models do not account for interactions with multiple facets. The energy loss increases steadily with the roughness parameter. It is relatively weak from medium 0 to medium 1 (3% loss for σ=0.6). It has a larger impact on the diffuse transmittance t01σ than on the diffuse reflectance r01σ [Fig. 8(a)]. From medium 1 to medium 0, the loss is much more important (20% loss for σ=0.6). Due to the total reflections, it mainly impacts the diffuse reflectance r10σ [Fig. 8(b)]. Let us note that because of energy loss, r01σ continuously decreases with the roughness parameter while t10σ is almost constant.

 figure: Fig. 8.

Fig. 8. Bihemispherical factors of a rough interface in terms of the roughness parameter σ with n=1.5 (a) from medium 0 to medium 1, (b) from medium 1 to medium 0.

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As a physical description of the interactions with multiple facets seems to be very complex, these effects are mostly ignored or empirically corrected. For example, Jakob et al. [36] suggest to reintroduce the energy loss as a diffuse radiation in reflection and transmission so that the energy is conserved. With the same idea, we suggest to first calculate r01σ, respectively, t10σ, and then deduce t01σ=1r01σ, respectively, r10σ=1t10σ, to preserve energy conservation. The internal diffuse reflectance r10σ can then be considered as independent of the roughness and can be approximated by the internal diffuse reflectance r10 of a flat interface.

Acknowledgment

L. Simonot thanks the Poitou-Charentes region for founding a visiting fellowship at the EPFL, Laboratoire des Systèmes Périphériques, from June to August 2015.

REFERENCES

1. S. Chandrasekhar, Radiative Transfer (Dover, 1960).

2. P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971). [CrossRef]  

3. K. Stamnes, S. Chee Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media,” Appl. Opt. 27, 2502–2510 (1988). [CrossRef]  

4. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

5. P. Kubelka, “New contributions to the optics of intensely light-scattering material, part I,” J. Opt. Soc. Am. 38, 448–457 (1948). [CrossRef]  

6. P. Kubelka, “New contributions to the optics of intensely light-scattering material, part II. Non-homogeneous layers,” J. Opt. Soc. Am. 44, 330–334 (1954). [CrossRef]  

7. M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech. 2007, P02018 (2007). [CrossRef]  

8. M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008). [CrossRef]  

9. J. K. Beasley, J. T. Atkins, and F. W. Billmeyer, “Scattering and absorption in turbid media,” in Electromagnetic Scattering, R. L. Rowell and R. S. Stein, eds. (Gordon and Breach, 1967), pp. 765–785.

10. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

11. B. Maheu, J. N. Le Toulouzan, and G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984). [CrossRef]  

12. B. Maheu and G. Gouesbet, “Four-flux models to solve the scattering transfer equation. Special cases,” Appl. Opt. 25, 1122–1228 (1986). [CrossRef]  

13. G. A. Niklasson, “Comparison between four flux theory and multiple scattering theory,” Appl. Opt. 26, 4034–4036 (1987). [CrossRef]  

14. B. Maheu, J. P. Briton, and G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple and complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989). [CrossRef]  

15. Y. P. Wang, S. W. Zheng, and K. F. Ren, “Four-flux model with adjusted average crossing parameter to solve the scattering transfer equation,” Appl. Opt. 28, 24–26 (1989). [CrossRef]  

16. W. E. Vargas and G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997). [CrossRef]  

17. W. E. Vargas and G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997). [CrossRef]  

18. W. E. Vargas, “Generalized four-flux radiative transfer model average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 37, 2615–2623 (1998). [CrossRef]  

19. C. A. Arancibia-Bulnes and J. C. Ruiz-Suarez, “Average path-length parameter of diffuse light in scattering media,” Appl. Opt. 38, 1877–1883 (1999). [CrossRef]  

20. C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001). [CrossRef]  

21. C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001). [CrossRef]  

22. N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009). [CrossRef]  

23. M. Hébert, R. D. Hersch, and P. Emmel, “Fundamentals of optics and radiometry for color reproduction,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1021–1077.

24. F. El Haber, X. Rocquefelte, C. Andraud, B. Amrani, S. Jobic, O. Chauvet, and G. Froyer, “Prediction of the transparency in the visible range of x-ray absorbing nanocomposites built upon the assembly of LaF3 or LaPO4 nanoparticles with poly(methyl methacrylate),” J. Opt. Soc. Am. B 29, 305–311 (2012). [CrossRef]  

25. S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013). [CrossRef]  

26. K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014). [CrossRef]  

27. L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014). [CrossRef]  

28. J. W. Ryde, “The scattering of light by turbid media—Part 1,” Proc. Roy. Soc. (London) A131, 451–464 (1931).

29. H. Pauli and D. Eitel, “Comparison of different theoretical models of multiple scattering for pigmented media,” Colour 73, 423–426 (1973).

30. A. B. Murphy, “Modified Kubelka–Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D 39, 3571–3581 (2006). [CrossRef]  

31. R. Aronson and D. L. Yarmush, “Transfer-matrix method for gamma-ray and neutron penetration,” J. Math. Phys. 7, 221–237 (1966). [CrossRef]  

32. G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860). [CrossRef]  

33. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

34. M. Hébert and P. Emmel, “Two-flux and multiflux matrix model for color surface,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1233–1277.

35. B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

36. W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

37. A. Kienle and F. Forschum, “250 years Lambert surface: does it really exist,” Opt. Express 19, 3881–3889 (2011). [CrossRef]  

38. D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).

39. R. Molenaar, J. J. ten Bosch, and J. R. Zijp, “Determination of Kubelka–Munk scattering and absorption coefficient,” Appl. Opt. 38, 2068–2077 (1999). [CrossRef]  

40. M. Elias, L. Simonot, and M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001). [CrossRef]  

41. F. C. Williams and F. R. Clapper, “Multiple internal reflections in photographic color prints,” J. Opt. Soc. Am. 43, 595–597 (1953). [CrossRef]  

42. J. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942). [CrossRef]  

43. L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006). [CrossRef]  

44. L. Simonot, “A photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009). [CrossRef]  

45. R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH ‘81 Proceedings of the 8th Annual Conference on Computer Graphics and Interactive Techniques (1981), Vol. 15, pp. 307–316. [CrossRef]  

46. M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995). [CrossRef]  

47. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967). [CrossRef]  

48. C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002). [CrossRef]  

References

  • View by:

  1. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  2. P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
    [Crossref]
  3. K. Stamnes, S. Chee Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media,” Appl. Opt. 27, 2502–2510 (1988).
    [Crossref]
  4. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).
  5. P. Kubelka, “New contributions to the optics of intensely light-scattering material, part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [Crossref]
  6. P. Kubelka, “New contributions to the optics of intensely light-scattering material, part II. Non-homogeneous layers,” J. Opt. Soc. Am. 44, 330–334 (1954).
    [Crossref]
  7. M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech. 2007, P02018 (2007).
    [Crossref]
  8. M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
    [Crossref]
  9. J. K. Beasley, J. T. Atkins, and F. W. Billmeyer, “Scattering and absorption in turbid media,” in Electromagnetic Scattering, R. L. Rowell and R. S. Stein, eds. (Gordon and Breach, 1967), pp. 765–785.
  10. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  11. B. Maheu, J. N. Le Toulouzan, and G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
    [Crossref]
  12. B. Maheu and G. Gouesbet, “Four-flux models to solve the scattering transfer equation. Special cases,” Appl. Opt. 25, 1122–1228 (1986).
    [Crossref]
  13. G. A. Niklasson, “Comparison between four flux theory and multiple scattering theory,” Appl. Opt. 26, 4034–4036 (1987).
    [Crossref]
  14. B. Maheu, J. P. Briton, and G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple and complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
    [Crossref]
  15. Y. P. Wang, S. W. Zheng, and K. F. Ren, “Four-flux model with adjusted average crossing parameter to solve the scattering transfer equation,” Appl. Opt. 28, 24–26 (1989).
    [Crossref]
  16. W. E. Vargas and G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
    [Crossref]
  17. W. E. Vargas and G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997).
    [Crossref]
  18. W. E. Vargas, “Generalized four-flux radiative transfer model average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 37, 2615–2623 (1998).
    [Crossref]
  19. C. A. Arancibia-Bulnes and J. C. Ruiz-Suarez, “Average path-length parameter of diffuse light in scattering media,” Appl. Opt. 38, 1877–1883 (1999).
    [Crossref]
  20. C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
    [Crossref]
  21. C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
    [Crossref]
  22. N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
    [Crossref]
  23. M. Hébert, R. D. Hersch, and P. Emmel, “Fundamentals of optics and radiometry for color reproduction,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1021–1077.
  24. F. El Haber, X. Rocquefelte, C. Andraud, B. Amrani, S. Jobic, O. Chauvet, and G. Froyer, “Prediction of the transparency in the visible range of x-ray absorbing nanocomposites built upon the assembly of LaF3 or LaPO4 nanoparticles with poly(methyl methacrylate),” J. Opt. Soc. Am. B 29, 305–311 (2012).
    [Crossref]
  25. S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
    [Crossref]
  26. K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
    [Crossref]
  27. L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
    [Crossref]
  28. J. W. Ryde, “The scattering of light by turbid media—Part 1,” Proc. Roy. Soc. (London) A131, 451–464 (1931).
  29. H. Pauli and D. Eitel, “Comparison of different theoretical models of multiple scattering for pigmented media,” Colour 73, 423–426 (1973).
  30. A. B. Murphy, “Modified Kubelka–Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D 39, 3571–3581 (2006).
    [Crossref]
  31. R. Aronson and D. L. Yarmush, “Transfer-matrix method for gamma-ray and neutron penetration,” J. Math. Phys. 7, 221–237 (1966).
    [Crossref]
  32. G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860).
    [Crossref]
  33. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).
  34. M. Hébert and P. Emmel, “Two-flux and multiflux matrix model for color surface,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1233–1277.
  35. B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.
  36. W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).
  37. A. Kienle and F. Forschum, “250 years Lambert surface: does it really exist,” Opt. Express 19, 3881–3889 (2011).
    [Crossref]
  38. D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).
  39. R. Molenaar, J. J. ten Bosch, and J. R. Zijp, “Determination of Kubelka–Munk scattering and absorption coefficient,” Appl. Opt. 38, 2068–2077 (1999).
    [Crossref]
  40. M. Elias, L. Simonot, and M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
    [Crossref]
  41. F. C. Williams and F. R. Clapper, “Multiple internal reflections in photographic color prints,” J. Opt. Soc. Am. 43, 595–597 (1953).
    [Crossref]
  42. J. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
    [Crossref]
  43. L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
    [Crossref]
  44. L. Simonot, “A photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009).
    [Crossref]
  45. R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH ‘81 Proceedings of the 8th Annual Conference on Computer Graphics and Interactive Techniques (1981), Vol. 15, pp. 307–316.
    [Crossref]
  46. M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
    [Crossref]
  47. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
    [Crossref]
  48. C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
    [Crossref]

2014 (3)

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

2013 (1)

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

2012 (1)

2011 (1)

2009 (2)

N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
[Crossref]

L. Simonot, “A photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009).
[Crossref]

2008 (1)

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[Crossref]

2007 (1)

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech. 2007, P02018 (2007).
[Crossref]

2006 (2)

A. B. Murphy, “Modified Kubelka–Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D 39, 3571–3581 (2006).
[Crossref]

L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
[Crossref]

2002 (1)

C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
[Crossref]

2001 (3)

M. Elias, L. Simonot, and M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[Crossref]

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

1999 (2)

1998 (1)

1997 (2)

1995 (1)

M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[Crossref]

1989 (2)

1988 (1)

1987 (1)

1986 (1)

1984 (1)

1977 (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

1973 (1)

H. Pauli and D. Eitel, “Comparison of different theoretical models of multiple scattering for pigmented media,” Colour 73, 423–426 (1973).

1971 (1)

1967 (1)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[Crossref]

1966 (1)

R. Aronson and D. L. Yarmush, “Transfer-matrix method for gamma-ray and neutron penetration,” J. Math. Phys. 7, 221–237 (1966).
[Crossref]

1954 (1)

1953 (1)

1948 (1)

1942 (2)

J. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
[Crossref]

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).

1931 (2)

J. W. Ryde, “The scattering of light by turbid media—Part 1,” Proc. Roy. Soc. (London) A131, 451–464 (1931).

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

1860 (1)

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860).
[Crossref]

Ala-Nissila, T.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Aloui, F.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Amrani, B.

Andraud, C.

Arancibia-Bulnes, C. A.

Aronson, R.

R. Aronson and D. L. Yarmush, “Transfer-matrix method for gamma-ray and neutron penetration,” J. Math. Phys. 7, 221–237 (1966).
[Crossref]

Atkins, J. T.

J. K. Beasley, J. T. Atkins, and F. W. Billmeyer, “Scattering and absorption in turbid media,” in Electromagnetic Scattering, R. L. Rowell and R. S. Stein, eds. (Gordon and Breach, 1967), pp. 765–785.

Bayou, S.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Beasley, J. K.

J. K. Beasley, J. T. Atkins, and F. W. Billmeyer, “Scattering and absorption in turbid media,” in Electromagnetic Scattering, R. L. Rowell and R. S. Stein, eds. (Gordon and Breach, 1967), pp. 765–785.

Becker, J.-M.

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[Crossref]

Berginc, G.

C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
[Crossref]

Billmeyer, F. W.

J. K. Beasley, J. T. Atkins, and F. W. Billmeyer, “Scattering and absorption in turbid media,” in Electromagnetic Scattering, R. L. Rowell and R. S. Stein, eds. (Gordon and Breach, 1967), pp. 765–785.

Bourlier, C.

C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
[Crossref]

Briton, J. P.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Chauvet, O.

Chee Tsay, S.

Clapper, F. R.

Cook, R. L.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH ‘81 Proceedings of the 8th Annual Conference on Computer Graphics and Interactive Techniques (1981), Vol. 15, pp. 307–316.
[Crossref]

d’Eon, E.

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Dong, N.

N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
[Crossref]

Eitel, D.

H. Pauli and D. Eitel, “Comparison of different theoretical models of multiple scattering for pigmented media,” Colour 73, 423–426 (1973).

El Haber, F.

Eldridge, J. I.

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

Elias, M.

M. Elias, L. Simonot, and M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[Crossref]

Emmel, P.

M. Hébert and P. Emmel, “Two-flux and multiflux matrix model for color surface,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1233–1277.

M. Hébert, R. D. Hersch, and P. Emmel, “Fundamentals of optics and radiometry for color reproduction,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1021–1077.

Forschum, F.

Froyer, G.

Ge, J.

N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
[Crossref]

Ginsber, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Girasole, T.

C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

Gouesbet, G.

Granqvist, C. G.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Gréhan, G.

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

Guo, S. M.

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

Hébert, M.

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[Crossref]

L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
[Crossref]

M. Hébert and P. Emmel, “Two-flux and multiflux matrix model for color surface,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1233–1277.

M. Hébert, R. D. Hersch, and P. Emmel, “Fundamentals of optics and radiometry for color reproduction,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1021–1077.

Hersch, R. D.

L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
[Crossref]

M. Hébert, R. D. Hersch, and P. Emmel, “Fundamentals of optics and radiometry for color reproduction,” in Handbook of Digital Imaging, M. Kriss, ed. (Wiley, 2015), pp. 1021–1077.

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Jakob, O.

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Jakob, W.

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Jayaweera, K.

Jobic, S.

Judd, D. B.

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).

Kienle, A.

Kubelka, P.

Laaksonen, K.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Le Toulouzan, J. N.

Lebaudy, P.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Lecamp, L.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Li, H.

B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

Li, S.-Y.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Maheu, B.

Marschner, S.

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Marschner, S. R.

B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

Menu, M.

M. Elias, L. Simonot, and M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[Crossref]

Molenaar, R.

Mouzali, M.

S. Bayou, M. Mouzali, F. Aloui, L. Lecamp, and P. Lebaudy, “Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers,” Polym. J. 45, 863–870 (2013).
[Crossref]

Mudgett, P. S.

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

Murphy, A. B.

A. B. Murphy, “Modified Kubelka–Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D 39, 3571–3581 (2006).
[Crossref]

Nayar, S. K.

M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[Crossref]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Nieminen, R. M.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Niklasson, G. A.

Niklasson, G. A. C.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Oren, M.

M. Oren and S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[Crossref]

Pauli, H.

H. Pauli and D. Eitel, “Comparison of different theoretical models of multiple scattering for pigmented media,” Colour 73, 423–426 (1973).

Puisto, S. R.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Ren, K. F.

Richards, L. W.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsber, and T. Limperis, “Geometrical consideration and nomenclature for reflectance,” J. Res. Natl. Bur. Stand. 160, 1–52 (1977).

Rocquefelte, X.

Rostedt, N. K. J.

K. Laaksonen, S.-Y. Li, S. R. Puisto, N. K. J. Rostedt, T. Ala-Nissila, C. G. Granqvist, R. M. Nieminen, and G. A. C. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: Conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Sol. Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

Rozé, C.

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

Ruiz-Suarez, J. C.

Ryde, J. W.

J. W. Ryde, “The scattering of light by turbid media—Part 1,” Proc. Roy. Soc. (London) A131, 451–464 (1931).

Saillard, J.

C. Bourlier, G. Berginc, and J. Saillard, “One and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations,” IEEE Trans. Antennas Propag. 50, 312–324 (2002).
[Crossref]

Saunderson, J. L.

Simon, K.

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech. 2007, P02018 (2007).
[Crossref]

Simonot, L.

Smith, B. G.

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[Crossref]

Stamnes, K.

Stokes, G. G.

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860).
[Crossref]

Tafforin, A. G.

C. Rozé, T. Girasole, and A. G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

ten Bosch, J. J.

Torrance, K. E.

B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in SIGGRAPH ‘81 Proceedings of the 8th Annual Conference on Computer Graphics and Interactive Techniques (1981), Vol. 15, pp. 307–316.
[Crossref]

Vargas, W. E.

Vöge, M.

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech. 2007, P02018 (2007).
[Crossref]

Walter, B.

B. Walter, S. R. Marschner, H. Li, and K. E. Torrance, “Microfacet models for refraction through rough surfaces,” in Proceeding of Eurographics Symposium on Rendering (2007), pp. 195–206.

Wang, L.

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

Wang, Y. P.

Williams, F. C.

Wiscombe, W.

Yarmush, D. L.

R. Aronson and D. L. Yarmush, “Transfer-matrix method for gamma-ray and neutron penetration,” J. Math. Phys. 7, 221–237 (1966).
[Crossref]

Zhang, Y.

N. Dong, J. Ge, and Y. Zhang, “Four-flux Kubelka–Munk model of the light reflectance for printing of rough surface,” Proc. SPIE 7241, 72411I (2009).
[Crossref]

Zheng, S. W.

Zijp, J. R.

ACM Trans. Graph. (1)

W. Jakob, E. d’Eon, O. Jakob, and S. Marschner, ”A comprehensive framework for rendering layered materials, “ ACM Trans. Graph. 33, 1–12 (2014).

Acta Mater. (1)

L. Wang, J. I. Eldridge, and S. M. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

Appl. Opt. (12)

R. Molenaar, J. J. ten Bosch, and J. R. Zijp, “Determination of Kubelka–Munk scattering and absorption coefficient,” Appl. Opt. 38, 2068–2077 (1999).
[Crossref]

P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[Crossref]

K. Stamnes, S. Chee Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media,” Appl. Opt. 27, 2502–2510 (1988).
[Crossref]

B. Maheu, J. N. Le Toulouzan, and G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
[Crossref]

B. Maheu and G. Gouesbet, “Four-flux models to solve the scattering transfer equation. Special cases,” Appl. Opt. 25, 1122–1228 (1986).
[Crossref]

G. A. Niklasson, “Comparison between four flux theory and multiple scattering theory,” Appl. Opt. 26, 4034–4036 (1987).
[Crossref]

B. Maheu, J. P. Briton, and G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple and complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
[Crossref]

Y. P. Wang, S. W. Zheng, and K. F. Ren, “Four-flux model with adjusted average crossing parameter to solve the scattering transfer equation,” Appl. Opt. 28, 24–26 (1989).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[Crossref]

W. E. Vargas, “Generalized four-flux radiative transfer model average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 37, 2615–2623 (1998).
[Crossref]

C. A. Arancibia-Bulnes and J. C. Ruiz-Suarez, “Average path-length parameter of diffuse light in scattering media,” Appl. Opt. 38, 1877–1883 (1999).
[Crossref]

L. Simonot, “A photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009).
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Atmos. Environ. (1)

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Figures (8)

Fig. 1.
Fig. 1. Flux transfers between two components represented by thin arrows. Bold arrows correspond to fluxes.
Fig. 2.
Fig. 2. Useful notations for defining the BSDF.
Fig. 3.
Fig. 3. Flux transfers (a) for a Lambertian component, (b) for a nonscattering component.
Fig. 4.
Fig. 4. (a) Bidirectional reflectance factor r cd ( i , o ) of a nonscattering component (index 1) on a scattering component (index 2). (b) Bidirectional transmittance factor t cd ( i , o ) of a scattering component (index 1) on a nonscattering component (index 2).
Fig. 5.
Fig. 5. (a) Perfectly flat interface on a Lambertian background, (b) micro-facet rough interface on a Lambertian background, (c) distribution of interfaced Lambertian micro-facets.
Fig. 6.
Fig. 6. Volume BRDF in the incident plane, assuming a Beckmann distribution for D σ and the corresponding Smith shadowing masking function for G [47,48], with θ i = 60 ° (backscattering direction at θ o = 60 ° and specular direction at θ o = 60 ° ), n = 1.5 and different roughness parameters σ. (a) Rough interface on a Lambertian background [second term of Eq. (39) with r 10 σ = r 10 ], (b) distribution of interfaced Lambertian facets [second term of Eq. (40)].
Fig. 7.
Fig. 7. Directional-hemispherical reflectance (solid lines) or transmittance (dashed lines) factors of rough interfaces as a function of the incident angle θ i for different roughness parameters σ with n = 1.5 (a) from medium 0 to medium 1, (b) from medium 1 to medium 0.
Fig. 8.
Fig. 8. Bihemispherical factors of a rough interface in terms of the roughness parameter σ with n = 1.5 (a) from medium 0 to medium 1, (b) from medium 1 to medium 0.

Tables (2)

Tables Icon

Table 1. Collimated-to-Diffuse and Diffuse-to-Diffuse Reflectance and Transmittance Factors of a Component According to its Position in the Component Stack a

Tables Icon

Table 2. Expressions of the Different Reflectance or Transmittance Factors in Function of the BRDF or BTDF f , where 2 π f cos θ d ω = θ = 0 π / 2 φ = 0 π / 2 f cos θ sin θ d θ d φ

Equations (44)

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{ J c k 1 = r cc I c k 1 + t cc J c k I c k = t cc I c k 1 + r cc J c k J d k 1 = r cd I c k 1 + t cd J c k + r dd I d k 1 + t dd J d k I d k = t cd I c k 1 + r cd J c k + t dd I d k 1 + r dd J d k .
( r cc 1 0 0 t cc 0 0 0 r cd 0 r dd 1 t cd 0 t dd 0 ) ( I c k 1 J c k 1 I d k 1 J d k 1 ) = ( 0 t cc 0 0 1 r cc 0 0 0 t cd 0 t dd 0 r cd 1 r dd ) ( I c k J c k I d k J d k ) .
( I c k 1 J c k 1 I d k 1 J d k 1 ) = ( M cc 0 2 , 2 M cd M dd ) ( I c k J c k I d k J d k ) ,
M x x = 1 t x x ( 1 r x x r x x t x x t x x r x x r x x ) ,
M cd = 1 t cc t dd ( t cd r cc t cd r cd t cc r cd t dd r dd t cd t cc ( t cd t dd r cd r dd ) r cc ( r cd t dd r dd t cd ) ) .
t x x = 1 M x x ( 1 , 1 ) and r x x = M x x ( 2 , 1 ) M x x ( 1 , 1 ) ,
t cd = M cd ( 1 , 1 ) M cc ( 1 , 1 ) M dd ( 1 , 1 ) and r cd = M cd ( 2 , 1 ) M cc ( 1 , 1 ) M cd ( 1 , 1 ) M dd ( 2 , 1 ) M cc ( 1 , 1 ) M dd ( 1 , 1 ) .
M 1 M 2 = ( M cc 1 M cc 2 0 2 , 2 M cd 1 M cc 2 + M dd 1 M cd 2 M dd 1 M dd 2 ) .
t x x = t x x 1 t x x 2 1 r x x 1 r x x 2 and r x x = r x x 1 + r x x 2 t x x 1 t x x 1 1 r x x 1 r x x 2 .
t cd = t cd 1 t dd 2 1 r dd 1 r dd 2 + t cc 1 t cd 2 1 r cc 1 r cc 2 + t cc 1 t dd 2 ( r cd 1 r cc 2 + r dd 1 r cd 2 ) ( 1 r cc 1 r cc 2 ) ( 1 r dd 1 r dd 2 ) ,
r cd = r cd 1 + t cd 1 r dd 2 t dd 1 1 r dd 1 r dd 2 + t cd 1 r cc 2 t cc 1 1 r cc 1 r cc 2 + t cc 1 t dd 1 ( r cd 2 + r dd 2 r cd 1 r cc 2 ) ( 1 r cc 1 r cc 2 ) ( 1 r dd 1 r dd 2 ) .
t cci t ddi 0 .
f r ( i , o ) = r cd ( i , o ) π and f t ( i , o ) = t cd ( i , o ) π ,
M cc = 1 t cc ( 1 0 0 0 ) .
{ r cd ( i , o ) = r cd ( i ) = r dd t cd ( i , o ) = t cd ( i ) = t dd ( o ) = t dd and { r cd ( i ) = r dd t cd ( i , o ) = t cd ( i ) = t dd ( o ) = t dd .
M cd = ( 1 t cc r dd t dd r dd t cc t dd t dd r dd r dd t dd ) .
{ r cd = r dd = r cd = r dd t cd = t dd = t cd = t dd .
M cd = 0 2 , 2 .
h dd = 1 π θ i = 0 π / 2 φ i = 0 2 π h cc ( i ) cos θ i sin θ i d θ i d φ i = θ i = 0 π / 2 h cc ( i ) sin 2 θ i d θ i ,
t dd ( o ) = t cc ( o ) / n 2 ,
t dd ( o ) = n 2 t cc ( o ) .
r cd ( i , o ) = t cc 1 ( i ) t cc 1 ( o ) n 2 r cd 2 ( i 2 , o 2 ) ( 1 r cc 1 ( i ) r cc 2 ( i ) ) ( 1 r dd 1 r dd 2 ) ,
t cd ( i , o ) = t cc 2 ( o ) n 2 ( t cd 1 ( i , o 1 ) 1 r dd 1 r dd 2 + t cc 1 ( i ) r cd 1 ( i ) r cc 2 ( i ) ( 1 r cc 1 ( i ) r cc 2 ( i ) ) ( 1 r dd 1 r dd 2 ) ) ,
{ n sin ( θ i 1 ) = sin ( θ i ) t cc = T 01 ( i ) = 1 R 01 ( i ) r cc = R 10 ( i 1 ) = R 01 ( i ) t cc = T 10 ( i 1 ) = 1 R 01 ( i ) ,
M cc = 1 T 01 ( i ) ( 1 R 01 ( i ) R 01 ( i ) 1 2 R 01 ( i ) ) .
r dd = r 01 = θ i = 0 π / 2 R 01 ( i ) sin 2 θ i d θ i .
{ t dd = t 01 = 1 r 01 t dd = t 10 = t 01 / n 2 r dd = r 10 = 1 t 10
t dd ( o ) = T 01 ( o ) / n 2 ,
t dd ( o ) = n 2 T 01 ( o ) .
r 01 σ ( i , o ) = π R 01 ( i , h r ) D σ ( h r ) G ( i , o , h r ) 4 ( i · n ) ( o · n ) ,
t 01 σ ( i , o ) = π | i · h t | | o · h t | ( i · n ) ( o · n ) n 1 2 T 01 ( i , h t ) D σ ( h t ) G ( i , o , h t ) ( n 0 ( i · h t ) + n 1 ( o · h t ) ) 2 ,
r 01 σ = θ i = 0 π / 2 r 01 σ ( i ) sin 2 θ i d θ i .
M cc = ( 1 t cc 0 0 0 ) , M dd = ( 1 t dd 0 ρ t dd 0 ) and M cd = ( 0 0 ρ t cc 0 ) .
M cc = ( 1 T 01 ( i ) R 01 ( i ) T 01 ( i ) R 01 ( i ) T 01 ( i ) 1 2 R 01 ( i ) T 01 ( i ) ) , M dd = ( 1 t 01 r 10 t 01 r 01 t 01 t 01 T 01 ( o ) / n 2 r 01 r 10 t 01 ) and M cd = 0 2 , 2 .
M cc = ( 1 t cc T 01 ( i ) 0 R 01 ( i ) t cc T 01 ( i ) 0 ) , M dd = ( 1 ρ r 10 t 01 t dd 0 r 01 + ρ ( t 01 T 01 ( o ) / n 2 r 01 r 10 ) t 01 t dd 0 ) and M cd = ( r 10 ρ t 01 t cc 0 ρ t 01 T 01 ( o ) / n 2 r 01 r 10 t 01 t cc 0 ) .
f r ( i , o ) = 1 π n 2 T 01 ( i ) T 01 ( o ) ρ ( 1 r 10 ρ ) .
M cc = ( 1 T cc 0 0 0 ) , M dd = ( 1 t 01 σ r 10 σ t 01 σ r 01 σ t 01 σ t 01 σ t 10 σ ( o ) r 01 σ r 10 σ t 01 σ ) and M cd = ( t 01 σ ( i ) T cc t 01 σ r 10 σ ( i ) t 01 σ t 01 σ r 01 σ ( i , o ) t 01 σ ( i ) r 01 σ T cc t 01 σ t 10 σ ( i , o ) t 01 σ r 10 σ ( i ) r 01 σ T cc t 01 σ ) ,
M cc = ( 1 T cc t cc 0 0 0 ) , M dd = ( 1 r 10 σ ρ t dd t 01 σ 0 r 01 σ + ρ ( t 01 σ t 10 σ ( o ) r 01 σ r 10 σ ) t dd t 01 σ 0 ) and M cd = ( t 01 σ ( i ) T cc t cc t 01 σ r 10 σ ρ t cc t 01 σ 0 t 01 σ r 01 σ ( i , o ) t 01 σ ( i ) r 01 σ T cc t cc t 01 σ + ρ t 01 σ t 10 σ ( o ) r 01 σ r 10 σ t cc t 01 σ 0 ) .
f r ( i , o ) = 1 π ( r 01 σ ( i , o ) + t 01 σ ( i ) t 10 σ ( o ) ρ 1 r 10 σ ρ ) .
f r ( i , o ) = r 01 σ ( i , o ) π + ρ π ( 1 r 10 ρ ) 1 n 2 ( i · n ) ( o · n ) 2 π T 01 ( i , m ) T 01 ( o , m ) D σ ( m ) G ( i , o , m ) ( i · m ) ( o · m ) d ω m ,
( t cc 0 0 0 t cd t d d 0 0 r cc 0 1 0 r cd r dd 0 1 ) ( I c k 1 I d k 1 J c k 1 J d k 1 ) = ( 1 0 r cc 0 0 1 r cd r dd 0 0 t cc 0 0 0 t cd t dd ) ( I c k I d k J c k J d k ) .
( T 0 2 , 2 R 1 2 , 2 ) ( I c k 1 I d k 1 J c k 1 J d k 1 ) = ( 1 2 , 2 R 0 2 , 2 T ) ( I c k I d k J c k J d k ) .
( I c k 1 I d k 1 J c k 1 J d k 1 ) = M ( I c k I d k J c k J d k ) , with M = ( T 1 0 2 , 2 RT 1 1 2 , 2 ) ( 1 2 , 2 R 0 2 , 2 T ) = ( T 1 T 1 R RT 1 T RT 1 R ) = ( M 11 M 12 M 21 M 22 ) .
{ R = M 21 M 11 1 T = M 11 1 R = M 11 1 M 12 T = M 12 M 21 M 11 1 M 12 .

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