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Abstract

In this paper we propose a new physically plausible, anisotropic Bidirectional Reflectance Distribution Function (BRDF) for fitting and for importance sampling in global illumination rendering. We demonstrate that the new model is better in data fitting than existing BRDF models. We also describe efficient schemes for sampling the proposed anisotropic BRDF model. Furthermore, we test it on a GPU-based real-time rendering algorithm and show that material design can be done with this anisotropic BRDF model effectively. We also show that the new model has effective real-time rendering performance.
Keywords: BRDF representation; Anisotropic reflection model; Monte Carlo rendering; BRDF data fitting

1.   Introduction

Global illumination algorithms solve the rendering equation [1] (see Table 1 for notation):

 (1)

where  is the Bidirectional Reflectance Distribution Function (BRDF) [2]. In the rest of the paper, we ignore the notation of the point in the BRDF.
Physically plausible BRDFs obey both Helmholtz reciprocity and energy conservation laws. Reciprocity means that the incoming and outgoing directions may be exchanged:

(2)

Energy conservation states that the total reflected power cannot exceed the incident power:

(3)

Parameters of BRDFs can be obtained by fitting to measured data. As real BRDFs are always reciprocal and energy conserving, accurate fitting requires physically plausible BRDF models. In addition, in Monte Carlo rendering algorithms, BRDF models are best used with their sampling formulas that result in random directions distributed proportionally to the cosine weighted BRDFs.

This paper proposes physically plausible, analytical BRDF models with the aim of accurate data fitting, efficient importance sampling and effective real-time rendering. The proposed models are based on halfway vector representation and use the standard Beckmann distribution as a normalized microfacet distribution. We demonstrate that they can represent both isotropic and anisotropic materials more effectively than existing BRDF models. We also show thatWard’s [3] importance sampling formulas are good for our proposed BRDF models and are more efficient than Lawrence et al. [4] factorization method. Furthermore, we demonstrate that our proposed BRDF models are applicable for Monte Carlo rendering algorithms, and material tuning can be done with a few parameters. Finally, we show that our new models not only are well-suited for batch use, but they map well onto modern GPUs with a performance advantage of around 20% over previous methods.

Table 1: Notation used throughout this paper.

2.   Related work

The first microfacet based BRDF model was proposed by Torrance and Sparrow [5], which was improved by Cook and Torrance [6]. Although these microfacet based BRDF models are effective in modeling many real materials, they cannot represent anisotropic materials, and do not have efficient importance sampling formulas. Ward’s [3] and Duer’s [7] BRDF models are simplifications of the Cook-Torrance model. Their microfacet distribution functions are not normalized and are not physically plausible. However, they can represent both isotropic and anisotropic materials and have efficient importance sampling formulas.
Neumann et al. [8] analyzed energy conservation of the some analytic models, such as Phong [9], Blinn-Phong [10], Cook-Torrance [6], Ward [3] models and they proposed a normalization factor which can be seen as shadowing/masking term to make these models energy conserving. Ashikhmin and Shirley [11] proposed an anisotropic variation of the Phong [9] reflection model, which was further developed in [12]. Ashikhmin et al. [13] also introduced a general framework for creating microfacet based BRDF models. We note that the Ashikhmin-Shirley [11] microfacet distribution function is not strictly normalized (so this distribution is not physically plausible), since they ignored the  term when calculating the integral in Equation 7. The correctly normalized form of their microfacet distribution function can be found in [14].
Edwards et al. [15] use halfway vector disks for BRDF modeling. Their model is energy conserving and has efficient sampling formulas but it is not physically plausible since it is not reciprocal. Lawrence et al. [4] introduced a new sampling method using nonnegative matrix factorizations (NMF) and demonstrated its effectiveness through comparing to analytical BRDF models such as Lafortune et al. [16].
Ngan et al. [17] experimentally examined analytical BRDF models. Sun et al. [18], Krivánek and Colbert [19] recently demonstrated the possibility of BRDF editing with analytical BRDF models. At the same time, they introduced test beds for analytical BRDF models as well.
3.   Microfacet distribution function
In our analytical anisotropic BRDF model, we use the following normalized microfacet distribution function which is physically plausible:

(4) 
    
(5)

For isotropic materials, , thus we get the following microfacet distribution:
(6)

which is the standard Beckmann distribution function [6].
Our normalized microfacet distribution function obeys the following equation

(7)

This equation means that  is a heighfield distribution, i.e. the total projected area of all the microfacet faces should be equal to the base area [14] (if ).
4.   Energy conservation and reciprocity
If a microfacet distribution function is described with the halfway vector, then the BRDF model which uses it may satisfy the following equation to fulfill energy conservation [15]:

(8)

Unfortunately, this upper bound does not obey reciprocity. Because of that, as Neumann at al. [8] proposed, we have to make a slight modification:

(9)

This equation produces a physically plausible BRDF model if the microfacet distribution function is normalized. However, in the course of our experimental analysis on measured BRDF data set, we did not get impressive fitting results with these normalization factors. Thus, we rather propose to use following BRDF model in data fitting:

(10)

To show why Equation 10 is better than Equation 9 for data fitting, we first plot the albedo functions of measured BRDF data. The albedo function depends on outgoing vector and according to the energy conservation law, its values should not be greater than 1 (Equation 3).

We selected three typical materials from Matusik et al. [20] data set for plotting albedo functions: blue-metallic-paint, nickel, and yellow-matte-plastic. While blue-metallic-paint and nickel are semi-glossy and glossy materials, respectively, yellow-matte-plastic is a specular material and its specularity sharply increases near grazing angles (see Figure 1). Matusik et al. [20] data set contains measurement noise especially at the grazing angles [15, 4]. The plotted albedo functions are also in Figure 1. When the outgoing angle increases, the albedo values of blue-metallic-paint and nickel decreases, but yellow-matte-plastic behaves differently. Its albedo values increase when the outgoing angle increases and then sharply decreases.
Figure 2 shows the albedo functions of Equation 9 for different parameters. Note that when the outgoing angle increases, albedo values decrease. With this behavior, diffuse, semi-glossy and glossy materials such as blue-metallic-paint and nickel can be represented well, but this formulation is not suitable for representing specular and highly specular materials such as yellow-matte-plastic. On the other hand, if we plot the albedo functions of Equation 10 for different  andparameters, we get the plots of Figure 3 that can mimic many kinds of data. Ifis set near to 0, diffuse, semi-glossy and glossy materials such as blue-metallic-paint and nickel can be represented well. Furthermore, if is given a bigger value, specular and highly specular materials such as yellow-matte-plastic can be defined. If equals to 0, this representation is exactly physically plausible for all parameters. On the other hand, if is not equal to 0, the combination of model parameter values determines the physical plausibility (see Figure 3).

Figure 2: Albedo functions of the proposed physically plausible BRDF model (see Equation 9) for different m values. Fresnel term is chosen as f0 = 1:0.



Figure 3: Albedo functions of the proposed data fitting BRDF model (see Equation 10) for different m and a values. If m and a are correctly chosen, this model is physically plausible as well. Fresnel term is chosen as f0 = 1:0.

5.   The new anisotropic BRDF model and its fitting

A general BRDF model consists of diffuse and specular terms. We develop three different versions of our new anisotropic model. The simple version is the sum of a pure Lambertian term and a single specular lobe:

(11)

where  is the diffuse albedo of the material,  is specular reflectivity and  is the Fresnel function of the material, which can be cheaply approximated by Schlick’s [21] formula:

(12)

    Based on the observation that the diffuse reflection is due to non-direct photon collisions, the diffuse and specular parts can be coupled using the Fresnel function describing ideal mirror reflections:

(13)

Finally, we also consider the multiple-specular lobe version of the proposed model. Using more than one specular lobe, we can represent the surface roughness on multiple scales, which reduces the fitting error. We also use different Fresnel terms for each specular lobe, which reduces the fitting error as well. In this way, we can model mixture materials like a car paint. The multiple-specular lobe model is:

(14) 

                                                                                               

5.1.   Fitting the simple model

We fitted the proposed simple model to 100 isotropic and 4 anisotropic materials taken from [20], using Ngan’s [17] fitting procedure and the  error metric. First, we assumed a single specular lobe.
    In our analytical BRDF model,  and  are the diffuse and specular reflectivities and are linear parameters. On the other hand, ; ; and  are specular lobe parameters and are non-linear parameters. To produce plausible results, we should include constraints; ;  and  in the parameter estimation procedure based on MATLAB FMINCON function.
Brushed aluminum, yellow satin, purple satin and red velvet are four anisotropic materials. Although Ngan et al. [17] underlined that only brushed aluminum and yellow satin are suitable for analytic modeling, we wanted to see the ability of data fitting performance of our analytical BRDF model even for these materials. In Table 2, there are estimated parameters of our model. In Figure 4, renderings of these materials using our anisotropic model can be seen.

Table 2: Estimated parameters of our Anisotropic analytical BRDF model on four anisotropic materials which are from Matusik


Figure 4: (a) Brushed aluminum, (b) Purple satin, (c) Redvelvet and (d) Yellow satin are reconstructed using our Anisotropic BRDF model. For the model parameters and L2 error.

We also choose six analytical anisotropic BRDF models for comparing to our model: Ashikhmin-Shirley (A&S) [11], Ashikhmin-Premože (A&P) [12], Edwards et al. [15], Lafortune et al. [16], Ward [3], and Ward-Duer [7]. We applied the same fitting procedure to estimate model parameters of these models. Table 3 summarizes the errors of these models. In Figure 5, we represent brushed aluminum with some of these anisotropic BRDF models. As it can be seen, our anisotropic BRDF model has better approximation capability than others.

Table 3: The L2 errors of seven analytical anisotropic models to four anisotropic data which are from Matusik et al. [MPBM03] data set.


Figure 5: Brushed aluminum is reconstructed using (a) the Ashikhmin-Shirley, (b) the Ashikhmin-Premože, (c) the Edwards et al., (d) the Ward, (e) the Ward-Duer and (f) our Anisotropic BRDF model. For the model parameters and L2 errors, see Table 2 and Table 3, respectively.

In addition, we choose six analytic isotropic BRDF models for comparing to the isotropic case of our BRDF model: Ashikhmin-Shirley [11], Blinn-Phong [10], Cook-Torrance [6], Lafortune et al. [16], Ward [3] andWard-Duer [7].We selected 30 materials from Matusik’s data set randomly and applied the same fitting procedure to estimate model parameters of these models. In Figure 6, there are the logarithmic  errors of these models. Our BRDF model gives the lowest errors on 25 materials of these 30 materials. Consequently, its approximation capability is effective for isotropic materials as well.


Figure 6: L2 errors (logarithmic scale) of seven analytic models to 30 isotropic BRDF data which are from Matusik et al. [MPBM03] data set. The BRDFs are sorted in the errors of the Lafortune model (Black) for visualization purpose. One specular lobe is used for all models.

In Figure 7, the polar plots in the incidence plane for four models are shown. Ashikhmin-Shirley [11], Cook-Torrance [6] and Ward [3] models have , and  terms, respectively, and they act as shadowing/masking terms. However, they are not optimized for one material and do not vary from material to material. Our proposed anisotropic BRDF model optimizes this shadowing/masking term with an additional parameter. In this way, it more closely matches the measured BRDF data than existing BRDF models.


Figure 7: Polar plots of various fitted models against the measured BRDF (black dashed lines) of "dark-specular-fabric" in the incidence plane. Cubic root applied for visualization purpose. The Ashikhmin-Shirley L2 error is 0.0042, the Cook-Torrance L2 error is 0.0036, the Ward L2 error is 0.0132 and our Anisotropic model L2 error is 0.0027 for this material.

5.2.   Multiple specular lobes

We choose six analytical BRDF models: Ashikhmin-Shirley [11], Blinn-Phong [10], Cook-Torrance [6], Ward [3], Ward-Duer [7] and our proposed model. We fitted them to randomly selected 30 materials form Matusik’s data set, and used the two specular lobe versions of each model. In Figure 8, there are the logarithmic  errors of these models. Our BRDF model gives the lowest errors on 28 materials out of 30 materials. Although the addition of specular lobes decreases the errors and increases the visual quality, its effectiveness varies from model to model. In these 30 materials, one extra specular lobe reduces the error by 4.00% in the Blinn-Phong model, 7.00% in the Ward model, 11.41% in the Ward-Duer model, 20.72% in the Ashikhmin-Shirley model, 21.42% in the Cook-Torrance model and by 27.62% in our proposed model. Consequently, its approximation capability is even more effective when it is used with multiple lobes.

Figure 8: L2 errors (logarithmic scale) of six analytic models to 30 isotropic BRDF data which are from Matusik et al. [MPBM03] data set. The BRDFs are sorted in the errors of the Ward-Duer model (Magenta) for visualization purpose. Two specular lobes are used for all models.

Moreover, we choose six BRDF representations for visually comparing to the isotropic case of our BRDF model: Ashikhmin-Shirley [11], Cook-Torrance [6], Edwards et al. [15], Lawrence et al. [4], Ward [3] and Ward-Duer [7]. We selected three materials (blue-metallic-paint, nickel and yellow-matte-plastic) from Matusik’s data set and applied the same fitting procedure to estimate model parameters of the analytical models. To increase fitting quality, we used three specular lobes for the analytical models. We used the same scene configuration for rendering as Edwards et al. [15] used in their work. Figure 9 shows renderings of the Princeton scene [15]. The table was rendered with the approximation of measured nickel in this scene. Since the viewer looks at the table from grazing angles, it is distinctive for approximation capability. We also calculated the PSNR (Peak Signal-to-Noise Ratio) values for each color channel and obtained their averages [22]. The PSNR values and difference images between the reference image and rendered images are also presented in Figure 9. Results indicate that our proposed BRDF model most closely matches the measured BRDF data. Furthermore, while our proposed BRDF model requires only few parameters to represent materials, Lawrence et al. [4] required 176KB, 120KB and 384KB for representing blue-metallic-paint, nickel and yellow-matte-plastic, respectively.


Figure 9: Visual comparisons of well-known BRDF representations on the Princeton scene. (a) Reference image was rendered using measured BRDF data. (b), (c), (d), (e), (f), (g), (h) were rendered using the Ashikhmin-Shirley, the Cook-Torrance, the Edwards et al., the Lawrence et al. factored BRDF, the Ward, the Ward-Duer and our Anisotropic BRDF model, respectively. While (a), (b), (c), (d), (f), (g) and (h) were rendered at 262144 samples/pixel, (e) was rendered at 4096 samples/pixel. Insets show a difference between the reference image and the rendered image and darker portions in these difference images mean higher disparity. Below each image we also report (PSNR value).

Furthermore, we fitted isotropic case of our model to another BRDF data set which was measured by Rump et al. [23]. This data set includes three metallic car paints: specular blue, green-blue flip-flop and silver metallic. When we fitted analytical models to this data set, we used Rump et al.’s [23] fitting procedure and error metric. As can be seen from Figure 10, our model represents metallic car paints better than the Cook-Torrance BRDF model.

Figure 10: (a) Specular blue, (b) Green-blue flip-flop, (c) Silver metallic are reconstructed using the Cook-Torrance BRDF model. (d) Specular blue, (e) Green-blue flip-flop, (f) Silver metallic are reconstructed using our Anisotropic BRDF model. Below each image we also report (Error value).

6.   Importance sampling

In Monte Carlo rendering algorithms the outgoing radiance is obtained as:

(15)  

In above equation, probability density function  should mimic  for efficient importance sampling.
Let  and  be two canonical uniform random variables in the range; . Importance sampling equations for our anisotropic BRDF model are:

(16)  
(17)

With the help of Equation 16 and Equation 17, we can calculate the halfway vector:

(18)

After that, we can find the sampling direction with the following formula:

(19)

Probability density function for our anisotropic BRDF model is:

(20)

where is from Equation 4. For the derivation of sampling formulas, see Walter’s [24] notes.
The ratio of the product of the BRDF and cosine of the orientation angle and the sampling density (see Equation 15) will be the weight of the sample. For our anisotropic BRDF model, weighting function is the following:
(20)

The ratio of the product of the BRDF and cosine of the orientation angle and the sampling density (see Equation 15) will be the weight of the sample. For our anisotropic BRDF model, weighting function is the following:

(21)

Actually, the sampling procedure of the our anisotropic model is the same as that of the Ward’s [3] model. However, our weighting function is different, which makes our model have different importance sampling efficiency. If our model did not contain the Fresnel term, the importance sampling efficiency would be better than that of Ward’s [3] BRDF model.
Furthermore, we compared sampling efficiency of our anisotropic model to Edwards et al. [15] sampling method, Lawrence et al. [4] factorization method, and Ward-Duer [7] sampling method. We used the same scene configuration as in Figure 9. Visual comparison of sampling quality is shown in Figure 11. Figure 11 also includes closeup views of some important regions, such as the teapot handle. As Figure 11 demonstrates, we obtain better sampling efficiency than Lawrence et al. [4] factored representation. As a consequence of this, our anisotropic BRDF model can be used both for representing and importance sampling the BRDFs.

Figure 11: The Princeton scene with global illumination was rendered using a path tracing algorithm for visual comparison of sampling efficiency. Paths up to five bounces long were included. While (a), (b), (c) and (d) were rendered using the Edwards et al., the Lawrence et al., the Ward-Duer and our Anisotropic model at 64 samples/pixel, respectively, (e), (f), (g) and (h) were rendered using the Edwards et al., the Lawrence et al., the Ward-Duer and our Anisotropic model at 4096 samples/pixel, respectively. The bottom rows show closeups of the highlighted regions.

7.   Real-Time rendering implementation

We tested and compared our anisotropic analytical model in a GPU-based real-time rendering algorithm [KC08]. In real-time rendering implementation, we used Fresnel blended version of the our BRDF model (see Equation 13). As we mentioned before, when , our BRDF model is used to represent isotropic materials (see Equation 6). In Figures 12 and 13, the isotropic and anisotropic cases of our BRDF model can be seen, respectively.
Moreover, the rendering frame rates for varying environment maps are shown in Figure 14. Although Figure 14 does not contain FPS values for the Blinn-Phong model, we found average of 34.575 FPS for it. As can be seen from Figure 14, our anisotropic BRDF model has effective real-time rendering performance. These results were produced on a consumer-level PC with Intel Pentium 4 3.00GHz processor, 2GB RAM and a Nvidia GeForce 6600 GT 256MB graphics card.
Finally, material design can be done with our anisotropic model interactively (see Figure 15).

Figure 12: Various objects rendered with isotropic version of our analytic BRDF model. For all scenes, m = mx = my = 0:001; f0 = 0:5;a = 0:25; kd = [0;0;0]; ks = [1:0;1:0;1:0].




Figure 13: A sphere rendered with anisotropic version of our analytic BRDF model. For both scenes, kd = [0:15;0:15;0:15], ks = [0:85;0:85;0:85], f0 = 0:75 and a = 0:2.



Figure 14: Plot of the real-time rendering performance with respect to the number of frame per second (FPS) for a given environments when rendering a sphere at a resolution of 550550. The 40 samples were used during rendering.


Figure 15: Material design can be done with analytical BRDF models on the fly. This is true for our proposed Anisotropic model as well.

8.   Conclusions and future work

In this work, we have provided a comprehensive review of analytical BRDF models and proposed new physically plausible anisotropic BRDF models. Our proposed models are based on halfway vectors, have normalized microfacet distribution function, have efficient importance sampling procedure and efficient performance in both data fitting and real-time rendering. We also tested them in a real-time rendering algorithm and have shown that material editing can be done dynamically.
As a future work, we would like to use our physically plausible models in Bidirectional Scattering Distribution Function (BSDF) which equals to sum of BRDF and Bidirectional Transmittance Distribution Function (BTDF).

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About the Authors:

Murat Kurt

was born in Izmir, Turkey, in 1981. He received B.Sc. degree in the civil engineering from Engineering Department, Dokuz Eylul University in 2002 and M.Sc. degree in computer science from International Computer Institute, Ege University in 2007. He has been a Research Assistant with the International Computer Institute since 2005. His research interests are algorithms, computer graphics with focuses on BRDF models, appearance modeling, global illumination algorithms, photorealistic rendering and digital image processing with focuses on image registration, wavelets, LPR. He is also currently a Ph.D. student in the International Computer Institute, Ege University.

László Szirmay-Kalos

heads the Computer Graphics Group and the Department of Control Engineering and Information Technology at Budapest University of Technology and Economics. His research interests include Monte Carlo global illumination algorithms, distributed high-performance visualization, and GPU implementation. Szirmay-Kalos published more than 200 papers on these topics. He is a Eurographics Fellow. Contact him at szirmay@iit.bme.hu.



Jaroslav Křivánek

is a postdoctoral research fellow at Cornell University. Prior to joining Cornell in summer 2008, he was a researcher and later an assistant professor at the Czech Technical University in Prague. He received his Ph.D. degree from Universite de Rennes 1 and Czech Technical University in December 2005, and his M.Sc. degree, also from Czech Tech, in March 2001. In 2003, he was a research associate at University of Central Florida. His research interest include various aspects of rendering, mostly focusing on global illumination, user interfaces for relighting, and real-time rendering.