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Main results of physics are observed energy conservation, reciprocity rule, microfacet theory and numerous phenomena involved in light re ection are accounted for, in a physically plausible way incoherent and coherent re ection, spectrum modi cations, anisotropy, self-shadowing, multiple surface and subsurface re ection, di erences between homogeneous and heterogeneous materials. Keywords : Physically-Based Rendering, Bidirectional Re ectance Distribution Function, Optimization 1 Introduction Computation of a re ectance model is the heart of every rendering method because it provides the illumination of objects in the scene, and therefore the color of pixels in the image.

Re ectance models currently in use can be divided in two main families : empirical models and theoretical ones. Empirical models 12, 3, 6] are computationally e cient but are lacking of physical validity energy conservation law, for instance and thus do not provide plausible values of energy or intensity. In fact, they are generally only used to create bright spots on surfaces in order to add some tridimensional information which helps to understand the image.

Therefore they are limited to applications where good-looking pictures are su cient computer generated imagery for movies or commercials. On the other side, theoretical models 5, 9, 20] involve higher computational costs but provide quantitative values that have shown to be in good adequation with experimental data. Therefore they are well adapted to applications for which physically-based rendering is essential simulation for lighting industry or architecture.

This paper proposes a kind of intermediary model between empirical and theoretical models. In Section 2, simple notations are presented and used to reformulate several existing re ectance models.

Section 3 focuses on some unsatisfactory points that can be found in these models. In Section 4, a general purpose optimization technique is detailled and several low-cost alternatives to expensive terms involved in existing models are proposed.

Finally, Section 5 presents the new re ectance model, which uses that optimization technique to combine several interesting features of previously disjointed work into an inexpensive formulation well-suited to computer graphics.

V 0 is the incident radiance reaching point P from direction? V 0 R P; V; V 0 is the BRDF of the surface at point P between directions V and V 0 V is the set of possible directions for incident light ie hemisphere above the surface dV 0 is a di erential solid angle surrounding direction V 0 V , V 0 and N are unit vectors so N V 0 is the cosine of the angle between N and V 0 The re ected radiance is the integral, for all possible directions, of incident radiances scaled by the BRDF and the projected solid angle.

It should be noted that such a formulation is well adapted to rendering algorithms radiosity, path tracing, two pass methods that perform e ectively a true integration and for which the solid angle information is available. Other rendering algorithms direct illumination methods, ray tracing consider only a discrete sum of light contributions, and therefore are unable to provide close simulation of real phenomena as required by physically-based rendering.

Equation 1 is a monochromatic equation expressed for a given wavelength. In the present paper, we use the following notation convention : every term that is function of the wavelength will be subscripted by. The BRDF has got two important properties that result directly from physics of light 2].

The limit case perfectly di use surfaces or Lambert surfaces is obtained when the BRDF becomes a constant function ie the light is equally re ected in every direction. Specular surfaces : The light is re ected only in a small area around the mirror direction. The limit case perfectly specular surfaces or Fresnel surfaces is obtained when the BRDF becomes a Dirac function ie the light is re ected in one single direction. Figure 1 presents the notations that will be used to formulate BRDF models throughout the paper.

A complete review of all the models that have been proposed in the litterature is out of scope here the interested reader may refer to 17] where such a survey has been done , we will focus especially on the three theoretical models that have been used to de ne our new model. In that model, a surface is supposed to be composed of so-called microfacets which are small smooth planar elements.

Only microfacets for which the normal vector is in direction H see Figure 1 contribute to the re ection between V and V 0. Therefore, it implies a condition on the slope distribution function 2] : 2. Coherent re ection has been used for years in computer graphics it is the fundamental principle of recursive ray tracing but only for perfectly specular surfaces; He et al.

Ward has presented a simple model 20] in which the rotational symmetry of isotropic BRDF is replaced by an elliptical asymmetry of varying excentricity : 1 t2? By examinating existing re ectance models, one can nd several points that appear somewhat unsatisfactory. For instance, the BRDF is formulated as a linear combination with constant weights between a di use part and a specular one.

The justi cation usually given by the authors is that, for a large class of materials, di use and specular components come from di erent physical phenomena, and thus they may have di erent colors. One classical example is a plastic surface on which light can be re ected either by the uncolored substrat in a coherent way ie surface re ection is specular or by the colored pigments beneath the surface in an incoherent way ie subsurface re ection is di use 5].

But, as noticed by Shirley 15], such a linear combination with constant weights is incorrect because proportions of di use and specular components are usually not constant but function of the incident angle. Taking the example of a varnished wood oor see Figure 2 , one can see that, according to the Fresnel law, for large incident angles most light is re ected specularly by the varnish, whereas for small incident angles, most light penetrates the varnish before beeing re ected di usely by the wood.

For such materials metals, for instance there is rather a kind of continuum between perfect di use and perfect specular behaviours see Figure 3 according to the roughness of the surface. Therefore a linear combination with constant weights is inadequate again. Figure 3 : Continuum between di use and specular for surface re ection Another weak point in existing models appears when light reaches or leaves a rough surface where self-obstruction from one microfacet to another occurs.

Usually, a geometrical attenuation coe cient G in Equation 8 is used as a multiplicative factor to express the ratio of light which is not subject to that obstruction. But in real life, the remainder of the light ie 1? G is re ected in other directions and not simply blocked. Currently, none of the existing re ectance models does correctly account for that reemission of self-obstructed light. As said, empirical re ectance models are inexpensive but their lack of physical validity prevents their use in any physically-based rendering system.

On the other hand, theoretical models are physically accurate but involve complex mathematical expressions which are computationally expensive and preclude hardware implementations. Moreover, when including such a re ectance model in an image synthesis software, the error generated by other stages of the rendering pipeline tessellation for geometrical modeling, spectral sampling for optical modeling, directional sampling for global illumination, interpolation at almost every steps does usually totally cancel the bene t of greater accuracy.

In other words, there is no need to compute BRDF at a precision of 0. A possible solution could be to replace expensive formulas in theoretical re ectance models by some well-chosen low-cost alternative functions.

In Section 4, we present a new technique that enables to nd such approximations. One classical optimization technique which has been applied several times in computer graphics to speed up an algorithm that involves the computation of a complex function is to store many sample values of the function in a table, and compute missing values by interpolation usually linear or cubic.

Implementing a whole theoretical re ectance model which such a technique would require numerous tables in order to account for various surface properties and illuminating conditions which means high memory costs and di culties to switch to hardware implementations 10, 4, 21]. To overcome this limitation, a possible solution is to use piece-wise Taylor approximants.

But creating large ranges of values where the approximation is accurate implies to use many pieces, for which continuity in their derivatives cannot always be insured. Another classical technique which exists since the beginning of the century and has been applied to numerous scienti c elds is to use Pade approximants 1] in which a rational fraction is generated according to the Taylor expansion of the function.

Compared to pure polynomial approximations, Pade approximants have usually a much better accuracy when leaving the neighbourhood of the origin.

Piece-wise Pade approximants have also been proposed but rarely used in practice, because insuring continuity of the derivatives becomes almost impossible. The previous approximation methods, which deal all with Taylor expansions, have got two strong limitations. First, the Taylor expansion of the function has to be known, this is not always possible even with numerical techniques. Second, speci c properties of the function are generally not preserved.

We propose here another method that we simply call rational fraction approximation. This method di ers from the Pade approximation technique by the fact that we do not use Taylor expansions to nd the coe cients of the numerator and denominator polynomials. The idea is to study the function that we want to approximate, in order to nd what we call kernel conditions.

A kernel condition can be any intrinsic characteristic of the function : value at a given point either of the function or of one of its derivative, integral or di erential equation it obeys to The detection and the choice of kernel properties can be done in several ways, by using its mathematical de nition, by picking some of its remarkable values or even by plotting the function and examinating the graph.

This gives a system of n equations and n unknowns where n is the number of conditions. For a non-polarized electromagnetic wave, its formulation is 15] : 2 2 1 a? One di culty that precludes a general use of F in every rendering environment comes from the fact that n and k are seldom known.

Some experimental values exist 11] but usually one can only nd a single value n and k for a wavelength in the middle of the visible spectrum. But even so, the computation of the Fresnel factor remains expensive and further optimization should be possible. By examinating Figure 5, one can see that the shape of the curves does not vary very much according to the kind of material.

Therefore, a step further could be to make F u only dependent on f. The formulation proposed by Smith 18] introduced in the computer graphics eld by 9] is not subjected to these restrictions and has been experimentally validated. Despite its complicated form, the shape of the function see Figure 6 is quite simple.

To characterize it, we choose the following kernel conditions : 8 4. Moreover, compared to others, this formulation depends only on the rms slope m of the microfacets and does not introduce any arbitrary constant : t2? When the surface is smoother small values for m microfacet normals H come closer to the average normal N. And for perfectly smooth surfaces m is null D t becomes a Dirac function see Figure 7.

The normalization condition Equation 7 is an obvious kernel condition for D t. And nally, D 1 gives a last kernel condition : Z1 1 8t 2 0; 1? Therefore, we propose : m3 21 8t 2 1? The curves drawn in polar coordinates are shown on Figure 7. The rational fraction approximation scheme enables to speed-up the computation of re ectance models but does not provide a solution for the other unsatisfactory points discussed in Section 3.

With regards to that discussion, an appealing BRDF model should include the following features : Main results of physics Energy Conservation Law, Helmholtz Reciprocity Rule, Microfacet Theory should be ful lled to enable physically-based rendering A continuum between Lambert and Fresnel surfaces should be provided A distinction between homogeneous and heterogeneous materials should be made Both isotropic and anisotropic behaviours should be accounted for Only a small number of simple and meaningful parameters should control the model Only expressions with low computational costs should be used A new model which includes all these features is presented here and can be viewed as an intermediary model between empirism and theory.

First, the role of every parameter can be understood intuitively and therefore easily de ned by a non-specialist user. Second, the parameters can also be assigned according to experimental data 11]. When only geometrical optics is involved an hypothesis made by almost every rendering technique , the spectral and the directional behaviours of the BRDF can be separated ie rays are re ected in the same direction, whatever their wavelength in two multiplicative factors S and D.

Indeed, a usual technique for a Monte-Carlo process is to generate a stochastic importance sampling to improve the convergence 7]. G v0 expresses the ratio of re ected resp. But, as discussed in Section 3, we want to account for reemission of self-obstructed light ie 1? Similarly, because coherent re ection is not considered, Equation 25 cannot provide a Fresnel re ector even with a Dirac slope distribution function.

The weights a; b; c could be speci ed directly by the user, but it would represent three additional parameters per material. We propose rather a physically plausible automatic scheme which provides a quadratic interpolation between the three fundamental behaviours of a surface, according to the roughness factor.

In order to show various illumination e ects incidence angles ranging from grazing to normal and varying either fast or slow a simple test scene similar to 9] has been chosen. Every cylinder on the pictures has been rendered individually at a x resolution using Monte-Carlo ray-tracing.

Top row resp. To achieve a better understanding of the behaviour of the new model, only direct illumination from a single light source put at the view point is shown. In order to exhibit anisotropy, the cylinder is made of brushed metal, having concentric circular scratches on its top and parallel horizontal scratches on its body.

Top row of Picture 2 is similar to the bottom row of Picture 1, but indirect specular illumination from the surrounding environment is shown this time. Finally, bottom row of Picture 2 illustrates the bene t of accounting for the Fresnel law in the spectral factor.

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