Institutionen för teknik och naturvetenskap

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1 Institutionen för teknik och naturvetenskap Department of Science and Technology Examensarbete Accurate BRDF Modelling for Wide Angle Scattering Examensarbete utfört i Datorgrafik vid Tekniska högskolan vid Linköpings universitet av Tanaboon Tongbuasirilai LiTH-ITN-EX--13/066--SE Norrköping 2013 Department of Science and Technology Linköpings universitet SE Norrköping, Sweden Linköpings tekniska högskola Linköpings universitet, Campus Norrköping Norrköping

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3 Accurate BRDF Modelling for Wide Angle Scattering Examensarbete utfört i Datorgrafik vid Tekniska högskolan vid Linköpings universitet av Tanaboon Tongbuasirilai LiTH-ITN-EX--13/066--SE Handledare: Examinator: Joel Kronander itn, Linköpings universitet Jonas Unger itn, Linköpings universitet Norrköping, 30 oktober 2013

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5 Avdelning, Institution Division, Department Media and Information Technology Department of Science and Technology SE Norrköping Datum Date Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ISBN ISRN LiTH-ITN-EX--YY/NNNN--SE Serietitel och serienummer Title of series, numbering ISSN URL för elektronisk version [ Titel Title Accurate BRDF Modelling for Wide-Angle Scattering Accurate BRDF Modelling for Wide Angle Scattering Författare Author Tanaboon Tongbuasirilai Sammanfattning Abstract In this thesis, a modified BRDF model for wide-angle scattering is presented. The proposed model is developed from empirical observations of several BRDF models. The model is an extention of the classical microfacet models. By replacing the two cosines of elevation angles with functions and exponent parameters, our model is able to give a special characteristic which we have not found in any other BRDF models. The characteristic at wide-angle scattering can be, for example, seen on the polyethylene material. In addition, our proposed model can greatly improve relative error from the reference model. The average relative error improvement is about 20 percent for a cosine weighted error metric,e 1, and 10 percent for a logarithmic error metric, E 2,. Moreover, we also introduce a new optimization approach for the proposed terms. This approach can do optimization so that our proposed model gives at least an equivalent error to the reference model. Nyckelord Keywords brdf, microfacet, wide-angle scattering

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7 Abstract In this thesis, a modified BRDF model for wide-angle scattering is presented. The proposed model is developed from empirical observations of several BRDF models. The model is an extention of the classical microfacet models. By replacing the two cosines of elevation angles with functions and exponent parameters, our model is able to give a special characteristic which we have not found in any other BRDF models. The characteristic at wide-angle scattering can be, for example, seen on the polyethylene material. In addition, our proposed model can greatly improve relative error from the reference model. The average relative error improvement is about 20 percent for a cosine weighted error metric,e 1, and 10 percent for a logarithmic error metric, E 2,. Moreover, we also introduce a new optimization approach for the proposed terms. This approach can do optimization so that our proposed model gives at least an equivalent error to the reference model. iii

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9 Acknowledgments I would like to express my deep gratitude to the examiner and supervisor, Dr. Jonas Unger and Joel Kronander, for valuable discussion, constructive suggestions and useful comments of this work. I would like to thank to the technicians of the laboratory for their assist in offering me resources of this work, such as computer, software and access to the laboratory. Finally, I wish to thank my family for their encouragement throughout my study. Norrköping, October 2013 Tanaboon Tongbuasirilai v

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11 Contents Notation ix 1 Introduction Radiometry Reflectance Types of Surfaces Types of Reflection Types of Models Rendering Background Lambertian reflectance model Phong and Blinn-Phong reflectance model Cook-Torrance BRDF model Neumann BRDF model Ward reflectance model An anisotropic Phong light reflection model BRDF parameter fitting Parameter fitting in an anisotropic BRDF ABC model The Proposed BRDF Model Variations of cosine of elevation angles Proposed model Data Fitting Parameter fitting for the comparison Parameter fitting for the proposed model Result Comparison Double fitting Rendering results vii

12 viii CONTENTS 6 Conclusions and Future Work 33 A Data Table 37 Bibliography 47

13 Notation Notation ω i ω o f r (ω i, ω o ) f rs (ω i, ω o ) f rd (ω i, ω o ) R V H N k d k s F() A.B φ θ θ i θ o θ h Meaning incoming solid angle outgoing solid angle Bidirectional reflectance distribution function Bidirectional reflectance distribution function modelling the specular part Bidirectional reflectance distribution function modelling the diffuse part Reflection vector Viewing vector Halfway vector Surface normal vector diffuse coefficient specular coefficient fresnel reflectance function dot product between vector A and B azimuth angle elevation angle elevation angle of incoming light elevation angle of outgoing light elevation angle of halfway vector ix

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15 1 Introduction Image synthesis is the process of generating images from digital representations of a scene. When both a geometric representation and a material appearance are defined, an image can be generated by applying image synthesis techniques to these representations. Geometric data defines the shape of an object, such as size and location, in the scene. Another representation is material appearance. The appearance of materials is also necessary for rendering realistic objects, as it describes how light is reflected at surfaces Defining the appearance of materials needs the understandings of several disciplines such as the human visual system, physics, and mathematics. Naturally, the way in which humans see objects in the real world is described by interactions between light and matter. Light emitted from light sources reaches materials in the scene where it is reflected and finally observed by the viewer. Each material has different properties, therefore light can interact with materials in many different ways causing various effects in the real world. The basic material properties are reflection and refraction. These properties relate directly to radiometric quantities, such as irradiance, radiosity and radiance. These terms describe light interactions on material surfaces and can be used to derive general reflectance functions. 1.1 Radiometry Radiometry is the study of physical light measurement. There are four fundamental quantities involved in the image synthesis. 1. Flux, Φ, describes the total flow of energy at surface per unit time, expressed as W att(w )(J oule/ sec). 1

16 2 1 Introduction 2. Irradiance, E, describes the incident flux on a surface per unit area, expressed as E = dφ da (W att/m2 ). 3. Radiant exitance, M or B, is the exitant flux on a surface per unit area, expressed as M = B = dφ da (W att/m2 ). 4. Radiance, L, is flux per unit projected area per unit solid angle, expressed d as L = 2 Φ dωdacosθ (W att/(steradian.m2 )) From the definitions above and together with light transport theory[4], the relationships describing scattering can be derived. One important relationships is the Bidirectional Reflectance Distribution Function(BRDF). 1.2 Reflectance BRDF is a function describing how incident light reflects from a surface point of materials[14]. The BRDF is a four dimensional function, where θ i, φ i, θ o andφ o denote the spherical coordinates for the incoming and outgoing radiance[19]. One variable pair is an incident light direction to a surface and another variable pair is an exitant light direction from the surface. A pair of spherical coordinate variables defines a solid angle which is a measure on a 3D unit sphere and measured in steradians[4]. A BRDF can be written as the proportion of radiance and irradiance as follows. f r (ω i ω o ) = f r (θ i, φ i, θ o, φ o ) = dl o(ω o ) de i (ω i ) (1.1) where L o, the radiance, is the amount of flux leaving a surface area, and E i, the irradiance, is the amount of flux incident on the surface area. Two important properties which physically based BRDFs have to obey are reciprocity and energy conservation[14],[19]. Reciprocity indicates that when the incident and exitant directions are swapped, the BRDF remains unchanged. This property can be expressed mathematically as f r (ω i ω o ) = f r (ω o ω i ) (1.2) Another important property is energy conservation. When light interacts with a surface, reflected energy must be less than or equal to the energy of incident light. f r cos(θ o )dω o 1 (1.3) Ω

17 1.2 Reflectance 3 (a) diffuse distribution. (b) specular distribution (c) glossy distribution. (d) retro reflective distribution. Figure 1.1: Examples of the four different scattering types [14]. For decades, BRDFs have been developed and they can be categorized in many types of models such as analytical models, physically plausible models[11], datadriven models[10] and parametric models[9]. However, there are material properties in which we have to consider when designing BRDF model. Such properties are types of surface and reflection distribution property Types of Surfaces Surface materials can be characterized into four broad groups, diffuse, specular, glossy or retro-reflective[14]. Diffuse surfaces reflect light equally in all directions. Examples of near perfect diffuse surfaces are dull chalkboards and matte paint. Specular or perfect specular surfaces reflect light at the same angle of incident light. For some specular surfaces, light can also be transmitted through the specular surfaces with different angle of incident light. This phenomenon is called refraction. In this case, the Fresnel equations[4] can be applied for describing both reflection and refraction. Many real-world objects do not have a perfectly diffuse or specular reflectance, instead a glossy appearance. These surfaces are a combination between diffuse and specular parts[4]. Retro-reflective surfaces are mostly used along roads and street signs. These surfaces reflect light mostly back towards the incident direction. Since each type of surfaces has different reflection properties, it also affects the BRDF models used to describe the material. Figure 1.1 illustrates light distributions on each material surface.

18 4 1 Introduction Figure 1.2: On the left, the sphere is an isotropic material. The right sphere shows anisotropic reflection Types of Reflection The reflection distribution property also influences BRDF models. Each material can be either isotropic or anisotropic. Most BRDF models have been developed from isotropic materials. Isotropic materials reflect the same amount of light when incident and exitant directions are rotated around the normal axis on the surface. This kind of reflection can be found on most objects[14]. For anisotropic reflection the amount of reflected light is different when incoming and outgoing directions are rotated. Figure 1.2 illustrates the effects of the reflection distribution property. 1.3 Types of Models BRDF models, called reflectance models, can be divided in two broad groups, analytical models and empirical models. Analytical models are constructed from assumptions. Such assumptions can be used to derive mathematical model expressions. On the other hand, an empirical model is based on observation. By conducting experiments, the models can then be formulated by fitting the model to the data. One can also use the approach of data fitting to find best fit parameters in analytical models from measurement. Empirical models also include data-driven model and parametric models. 1.4 Rendering Modern image synthesis techniques render images by solving the rendering equation[6]

19 1.4 Rendering 5 Figure 1.3: Figure shows the vectors and the notations which are relevant to the rendering equation. L r (x ω o ) = L e (x ω o ) + f r (x, ω i ω o )L(x ω i )cos(n, ω i )dω (1.4) where L r (x ω o ) is the radiance reflecting from a surface point x to the ω o direction. L e (x ω o ) is the emitted radiance from the surface point x. The integral part describes the reflected radiance, over the hemisphere above the surface point x, reflecting from the surface point x to the viewer. Figure 1.3 shows the notations and the related vectors. The rendering equation is an integral equation which has no closed form of solution. One way to solve the rendering equation is using Monte Carlo methods[4]. Monte Carlo methods estimate the values of integrals by using random sampling techniques. Let I be the solution of the integral equation,i = f (x)dx. One can estimate the solution as follows. Î = 1 N f (Xi ) p(x i ), X i p(x) (1.5) where p(x) is a probability density function (pdf) for random sampling variables. X is the sampling variable and N is the number of samples. In the simplest case, p(x) is chosen as the uniform distribution resulting in a crude Monte Carlo estimator. However, the crude Monte Carlo method has a major drawback. The method has a slow convergence rate which decreases at the rate of O( 1 N )[14]. This means that more samples are needed to reduce the error, the variance, of the method. The error of the estimated solution, in the rendering, causes noisy images. Thus the number of samples needs to be large in order to improve the quality of the images for the rendering.

20 6 1 Introduction Figure 1.4: The left image shows a noisy visual result from the Monte Carlo method. The right image shows improved image quality by choosing suitable importance sampling function p(x). Due to the high variance of the Monte Carlo estimator, variance reduction techniques are needed. For our work, the importance sampling technique is used. Importance sampling is a technique where the random variables are sampled by the probability density function p(x) closed to the function f (x). Figure 1.4 shows the problem of solving the rendering equation by the Monte Carlo method and improving images quality by importance sampling. In this report, we are presenting an improvement that can be applied to the class of microfacet BRDF models. We introduce functions and exponent parameters on the two cosine functions of elevation angles derived physically from the Cook- Torrance model[3]. Our proposed model is a parametric model and developed empirically. The model is an alternative approach for BRDF models based on the microfacet theory. This work was inspired by recently proposed BRDF models especially the model of Kurt et al. [7]. In our work, we consider the ABC Model [9] as a reference model to demonstrate that our proposed model can improve numerical error from parameter fitting and BRDF scattering plots. Moreover, we show that our new parameters are able to simulate characteristics which, to our knowledge, no other microfacet model can simulate. The characteristics can be found on wide-angle scattering, particularly on near grazing angles. By varying the new parameters, one can shrink or extend the distribution tails. The next chapter includes a review of some of the most common BRDF models, beginning with the Lambertian reflectance model followed by analytical models and empirical models. In Chapter 3, we describe our proposed model. In Chapter 4, we also propose an optimization approach which improves the numerical error. Chapter 5 shows comparisons both in numerics and graphics as well as discussion. Finally, we present the conclusion of the work and our future work.

21 2 Background Many previous BRDF models have been developed based on both empirical and analytical approaches for using in computer graphics. Most early BRDF models in computer graphics are analytical models and are thus based on assumptions and theories from physics. Recently, the tools for measuring reflectance data were created since real data needs to be used in order to evaluate the models. The development for material measurement tools has been an important part of the development of BRDF models. The more accurate the data is, the more accurate BRDF models can be developed. Currently, the most efficient and accurate data used in the computer graphics research is from Matusik et al.[10]. The database called MERL(Mitsubishi Electric Research Laboratories) consists of 100 of isotropic materials. The MERL database has attracted a lot of attention and has helped researchers develop more accurate and efficient empirical models than previous analytical models [7], [9]. Although empirical models are more accurate than analytical models, analytical models are still necessary as a fundamental model for developing empirical models. Before going into complex models, a simple model will be shown. This section begins with a description of a diffuse model, the Lambertian reflectance, then some empirical and analytical models will be described. Finally, the ABC model, which is the reference model for this study, is explained in detail. 2.1 Lambertian reflectance model Materials that exhibit the Lambertian reflectance are called diffuse reflectors[4]. They reflect light equally in all directions over the hemisphere. Thus, this reflectance model can be expressed in a simple function as follows : 7

22 8 2 Background Figure 2.1: Illustration of incoming and outgoing vectors and its halfway vector. f r (ω i, ω o ) = ρ d π (2.1) where ρ d [0, 1] for energy conservation. The Lambertian model is important for modeling glossy surfaces. Since glossy surfaces exhibit both diffuse and specular reflection, many glossy surface models include the Lambertian model, as a part of the diffuse reflection, and the varied specular term, based on the observation or assumptions. 2.2 Phong and Blinn-Phong reflectance model The models being described in this section are focused only on the specular part, i.e. reflection varies as a function of light direction and viewing angles. The original Phong model and Blinn-Phong models[15] [2] use the cosine relationship between angles along with an exponent parameter. f rs (ω i, ω o ) = C(R.V ) n f rs (ω i, ω o ) = C(N.H) n (2.2a) (2.2b) where equation 2.2a is the Phong model and 2.2b the is Blinn-Phong model. The Phong model makes use of angle between perfect reflection direction, R, and viewing direction, V. Instead, the Blinn-Phong model requires surface normal, N, and the halfway vector, H. The halfway vector can be calculated as H = V +R V +R. Both models have a specular coefficient, C, and an exponent parameter, n. The models obey the reciprocal property which is important when used in backward ray tracing. However, when Phong shaders were developed, based on the BRDF definition, they were not reciprocal. Therefore, Lewis[8] proposed a formulation of energy-conserving and reciprocal phong shaders.

23 2.3 Cook-Torrance BRDF model 9 f r (ω i, ω o ) = k d + k s f rs (ω i, ω o ) (2.3) By bounding diffuse and specular coefficients,k d and k s, to 1, k d + ks = 1, the shaders are energy conserving and they are also reciprocal. For more detail about the shaders, please refer to [8]. One major drawback of the models occurs when near grazing angles are rendered since the cosine value is close to 0 when the angle is large. Thus, for great incident angles, the reflectance is quite dark. This leads to non-realistic effects. 2.3 Cook-Torrance BRDF model This model [3] is widely used in computer graphics since it can simulate many types of surfaces ranging from diffuse to specular surfaces. It is assumed that material surfaces are groups of v-shaped microfacets. When light interacts with the microfacets it then reflects specularly,in the mirror direction. The model is in the following form. f r (ω i, ω o ) = D(ω h)f(ω o )G(ω o, ω i ) 4cosθ i cosθ o (2.4) where D is the statistical distribution of microfacets, and G is the geometrical attenuation term. G is in the form of min(1, 2(N.H)(N.V ), 2(N.H)(N.L) ) (V.H) (V.H) Normally the Beckman Distribution is used as the distribution, D, which is expressed as D = exp( 1 tanα m 2 cos 4 α m )2 [3]. 2.4 Neumann BRDF model Neumann et al.[11] proposed a basic BRDF model derived from mathematical modeling. The basic BRDF is defined by using the relations of projected area of the viewing vector and incoming light vector on a unit disk. The model also obeys two important properties for BRDFs which are reciprocity and energy conservation. Moreover, it is easy to define the sample distributions used for importance sampling. In this work, the terms, m(l, V ) and ξ(m), are important for the visual results in isotropic and anisotropic reflection. m(l, V ) is an arbitrary metric between two unit projected vectors. For isotropic reflection, m(l, V ) is defined as a Euclidean distance, thus it corresponds to a circle shaped reflection. m(l, V ) = sqrt(( B A (z.u))2 + (z.v) 2 ) (2.5) When it is defined as elliptic norm as the equation above, the shape of the re-

24 10 2 Background Figure 2.2: The figure shows scattering plot from the Phong and stretching Phong model. The left scattering curve demonstrates the draw back of the Phong model. The right image is the result of the stretching Phong model. This figure is from [12]. flection is elliptic which results in the anisotropic reflection. ξ(m) is a highlight profile function which affects to p(r) probability density function used in importance sampling. The authors presented the use of an exponential function for the highlight profile function, ξ(m) = e sm where m [0, 2]. The models were tested on many materials such as metals, plastics and polishing. However, they did not compare the results with other conventional models or measured data. Moreover they also proposed a compact metallic BRDF model. The scattering in the greater angles of the Phong model, however, creates a problem. When the reflection angle moves closer to 90 degrees, the Phong model produces a darker reflectance which is unrealistic, see figure 2.2. Neumann et al.[12] then proposed a correction term. This term can be called a stretching term since it can stretch reflectance value when incident angles are large. When this term is applied to the Phong model, reciprocity still exists. The correction term is 1 max(cosθ i,cosθ o ). Using this correction term with the Phong[15] or Blinn-Phong model[2] allows us to describe the reflectance as : cos n α f r (ω i, ω o ) = C max(cosθ i, cosθ o ) cos n α f r (ω i, ω o ) = C max(cosθ i, cosθ o ) p (2.6a) (2.6b) where α is the angles between L and V vectors or N and H vectors depending on which the Phong or the Blinn-Phong models are used. For equation 2.6b, we can vary the parameter p, p [0, 1]. If p = 0, it is the reciprocal Phong model. If p = 1, it leads to new models for glossy and specular surfaces proposed in the paper. This correction term can also be applied to the Ward model.

25 2.5 Ward reflectance model Ward reflectance model In the original paper, Ward[18] proposed a measuring tool and BRDF model. The tool was developed based on image based technique by using a half-silvered plastic hemisphere in order to reflect light to a camera. However, there is an important limitation when capturing reflectance data near grazing angles. The proposed model is constructed by using Gaussian distributions. The geometric attenuation term and the Fresnel factor are widely used for microfacet based models. However, in the Ward model, the geometric attenuation term and Fresnel factor 1 are omitted and replaced by a normalization factor,. The Ward BRDF model 4πm is described by : 2 f r (ω i, ω o ) = k d π + k 1 e ( tan2 θ h s cosθi cosθ o 4πm 2 (2.7) where m is the standard deviation of the surface slope. m 2 ) This model is reciprocal because of the symmetry in the equation. Moreover, Ward claimed that the normalization factor controls energy conservation if it is chosen in a proper way. A slightly modified version of this model can also account for anisotropic materials. f r (ω i, ω o ) = k e d π + k 1 ( tan 2 θ h ( cos2φ + sin2 φ )) m 2 x m 2 y s (2.8) cosθi cosθ o 4πm x m y where m x, m y are the standard deviation of the surface slope in x and y direction respectively. An interesting property of the Ward model is that the term in the exponent of the Gaussian function can control anisotropic reflection. Although the Ward model can visualize both isotropic and anisotropic materials, there is still a problem at near grazing angles. Neumann et al. [12] then proposed a new model based on the stretching term. f r (ω i, ω o ) = k d π + k 1 s max(cosθ i, cosθ o ) e ( tan2 θ h m 2 ) 4πm 2 (2.9) 2.6 An anisotropic Phong light reflection model The Ashikhmin-Shirley model[1] was developed based on six constraints. The model was developed for any surface material and met the following requirements :

26 12 2 Background 1. Physically plausible 2. Anisotropy 3. Intuitive parameters 4. Fresnel behavior 5. Non-Lambertian diffuse term 6. Supporting Monte Carlo method At the time, no model could satisfy all these properties. Then the authors selected a model from [12], shown in equation (N.H) n f rs (ω i, ω o ) = C max(cosθ i, cosθ o ) F(cosθ h) (2.10) By inputing a Fresnel term in the equation 2.10, this model can satisfy items 1,3,4 and 6. After that, the exponent n is adjusted in a similar way as for the Ward model. Thus, the model can exhibit anisotropic reflection. f rs (ω i, ω o ) = C (N.H)n u cos 2 φ+n v sin 2 φ F(cosθ max(cosθ i, cosθ o ) h ) (2.11) To satisfy the energy-conserving property, the model must be in the following form. f rs (ω i, ω o ) = (nu + 1)(n v + 1) 8π (N.H) n u cos 2 φ+n v sin 2 φ (cosθ h )max(cosθ i, cosθ o ) F(cosθ h) (2.12) The equations above come with the Fresnel term,f(cosθ h ). This term can be approximated by using Schlick s approximation [16]. F(cosθ h ) = f + (1 f )(1 (cosθ h )) 5 (2.13) where f is the material reflectance for the normal incidence. For non-lambertian reflection, the authors derived a closed form BRDF to get the diffuse term. This term also satisfies energy conservation and reciprocity. f rd (ω i, ω o ) = 23k d 23π (1 f )(1 (1 cosθ i 2 )5 )(1 (1 cosθ o 2 )5 ) (2.14) Then, the combination between equation 2.12 and 2.14 is the Ashikhmin-Shirley BRDF model, a model which satisfies all six properties.

27 2.7 BRDF parameter fitting 13 f r = f rd + f rs (2.15) 2.7 BRDF parameter fitting Ngan et al.[13] conducted an experiment where they investigated many analytical BRDF models using a data-fitting approach. They used an optimization technique for minimizing the mean squared error to find the best specular lobe parameters for each analytical model. By using a single specular lobe, then analytic models with Fresnel effects, such as the Cook-Torrance model[3] and the Ashikhmin-Shirley model[1], perform lower errors than the models without the Fresnel term. When additional lobes are added, the error is also reduced. Another important finding is the use of the halfway vector to model the microfacet distributions. They found that H.N is a better representation than V.R to model specular reflections. Ngan et al. also proposed an acquisition system for anisotropic materials. The acquisition system is an image based system based on cylindrical objects with 3 degrees of freedom. Moreover, four anisotropic materials were tested against anisotropic BRDF models and the proposed acquisition system. However only two of them can be reconstructed by using the data-fitting approach. The fabric objects are complex and difficult to model because of their thread arrangement. 2.8 Parameter fitting in an anisotropic BRDF Kurt et al.[7] proposed new anisotropic models based on the Beckman distribution function. They proposed three different BRDF models ie. the single specular lobe model, the ideal mirror reflection model and the multiple specular lobe model. The paper shows how the distribution can be applied to all proposed BRDF models. The proposed models were modified from the original Cook- Torrance model by removing the shadowing and masking term and adding an exponent on the denominator as follows. f r (ω i, ω o ) = k d π + k s F(cosθ h )D(h) 4(cosθ h )(cosθ i cosθ o ) α f r (ω i, ω o ) = k d(1 F(cosθ h )) π + k s F(cosθ h )D(h) 4(cosθ h )(cosθ i cosθ o ) α f r (ω i, ω o ) = k lobes d π + k sl F l (cosθ h )D l (h) 4(cosθ h )(cosθ i cosθ o ) α l l=1 (2.16a) (2.16b) (2.16c) where equation 2.16a, 2.16b and 2.16c are the single specular lobe model, the ideal mirror reflection model and the multiple specular lobe model respectively. The distribution, D(h), is as follows.

28 14 2 Background D(h) = 1 πm x m y cos 4 θ h e tan2θh( cos 2 φ h m 2 x + sin2 φ h m 2 ) y (2.17) For m x = m y, we will get a distribution identical to Beckman distribution function. One interesting point in the proposed models is the α parameter. In the microfacet model, the geometric attenuation term is quite important to model rough surfaces[17]. All proposed models do not have the geometric attenuation term. However, the results shown in the paper still had lower errors than the Cook-Torrance model which has the geometric attenuation term. Thus, the exponent parameter seems to have an impact on BRDF. 2.9 ABC model Recently, Löw et al.[9] developed two new BRDF models based on a thorough study of the MERL database. They found a symmetric relationship in vectors, the viewing and reflection vectors, projected onto the unit disk. Also, they found that most of the scattering curves resemble an inverse-power-law shape. This means that Gaussian or Phong-like functions cannot model this behavior accurately. From these two conclusions, they proposed two new BRDF models, the "smooth surface" model and a microfacet model both based on a modified ABC function S(f ), inspired by rayleigh-rice theory from optically smooth surfaces. S(f ) = A (1 + Bf 2 ) C (2.18) The ABC function was employed to both smooth surface BRDF and microfacet BRDF with different parameter f as follows. f r (ω i, ω o ) = k d π + F(cosθ h)s( D P ) f r (ω i, ω o ) = k d π + F(cosθ h)g(ω i, ω o )S( 1 N.H) cosθ i cosθ o (2.19a) (2.19b) where D P is the projected deviation vector, D P = V P R P. The smooth surface model is inspired by Rayleigh-Rice theory for smooth surfaces but the parameters dependent on the wavelength are removed. The ABC microfacet model is a modification of the Cook-Torrance model and uses the ABC function as the distribution function. The fitting process was done by using a least squares approach. Two different error metrics were used in the fitting process : the cosine weighted function and the logarithmic function. The experiment results show that the inverse-power-

29 2.9 ABC model 15 law shape of the reflectance distribution can describe measured real-world materials with significantly higher accuracy compared to e.g. the Cook-Torrance and Ashikmin-Shirley models.

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31 3 The Proposed BRDF Model The previous chapter presented an overview of existing BRDF models. Based on this study, we have developed a new BRDF model which is the main result of this thesis. In this chapter, we start by introducing a problem statement. We then explain our new model. Realistic rendering is one of the most crucial techniques used in many computer graphics related industries such as film and game industries. The objective for this image synthesis is to simulate real-world objects into a scene to make audiences have realistic experiences. The BRDF model, thus, plays an important role for realistic rendering. Some BRDF models are easy to use but cannot simulate objects realistically. Some were developed based on physical theory. Even though BRDF models have been developed for decades, they have still not achieved image quality comparable to the level of measured BRDF data. Measured BRDF data gives rendering results close to real- world objects. However, there are many drawbacks for using the measured BRDF data such as storage and time consuming. These drawbacks make the rendering technique quite hard to apply to industrial projects. So much research has been done for BRDF models in order to discover the more accurate BRDF models. One observation in which we made during the review of research in BRDF models is the variations of denominators, which are two cosine functions of elevation angles. In the previous chapter, many models have been described and most of them are equipped with cosθ i or cosθ o or both as the denominator terms. Each model has a different exponent weight of the denominators and different functions for the cosine terms. This observation can be summarized as follows. Ward [18] proposed a model by weighting both cosθ i and cosθ o with a square root. This means that both terms are weighted with same amount and are equally 17

32 18 3 The Proposed BRDF Model Table 3.1: BRDF models and its variations Model its denominator Ward cosθi cosθ o Neumann et al. max(cosθ i, cosθ o ) p Ashikhmin and Shirley max(cosθ i, cosθ o ) Kurt et al. (cosθ i cosθ o ) α important to the BRDF model. Later, Neumann et al. [12] considered a different idea that first originated from the Phong model. They suggested a stretching term, max(cosθ i, cosθ o ), by applying the stretching term as a denominator. The stretching term can deal with a problem of reflection angle close to 90 degrees, see figure 2.2. This term can also replace as a denominator term in Ward model. The ability of the stretching term assisted Ashikhmin and Shirley [1] to develop a better BRDF model. They developed an anisotropic model based on the model from Neumann et al. [12]. This model has gained a good reputation due to its efficiency in simulating anisotropic materials. Finally, Kurt et al. [7] developed an anisotropic BRDF model based on the Beckman distribution. They imposed a new parameter for the cosine functions. Instead of weighting cosθ o and cosθ i with the square root, Kurt et al. changed the square root to a parameter, α, which it influenced the reflection value close to 90 degrees. They claimed that this parameter can simulate many types of materials. Moreover, the results also show that the model performs better than the Cook-Torrance model. In summary, Table 3.1 shows BRDF models and the variations used in each model. The observation above leads to an idea that a BRDF model should have other variations which would give a more accurate BRDF model than the above. So, in this work, we proposed a modification of the BRDF model by using the new variation. One fundamental model which we used for this study is the microfacet model. Since this model naturally has cosθ i and cosθ o terms as denominators and is efficient in rendering realistic objects, then the microfacet model is a suitable model to modify based on our observation. In reference to the microfacet model, one must think what the distribution function for the model would be. In the previous chapter, information is provided about the ABC model which is more efficient than the Cook-Torrance model. Thus, the ABC-BRDF model is used as a reference model for the study. 3.1 Variations of cosine of elevation angles Up to this point, some variations have been presented. In this section, two new variations are introduced and tested with some other variations for this study. One property which we must keep in mind is the reciprocity. Since physically plausible models must hold this property, the variations should not lead the model to break this law. Here are the chosen variations for this study.

33 3.2 Proposed model max(a, B) α 2. (AB) α 3. min(a, B) α 4. max(a, B) α min(a, B) β where A, B are cosθ i and cosθ o. max(x, y), min(x, y) are functions to find maximum and minimum values between x, y. α, β are arbitrary exponents. For the first two variations, we know their effects on the reflectance curve. The first one is from Neumann et al.[12]. The second one is from Kurt et al.[7]. The third variation is the counterpart of the max(a, B) α, min(a, B) α, which still keeps the ABC model reciprocal. If the model uses either max(a, B) α or min(a, B) α as the denominator, either the value of A or B will influence the computation of the model for a pair of incident and exitant angles. The model will lose the information of one of the two cosine terms. Thus, we introduced max(a, B) α min(a, B) β in order to keep both cosine terms in computation. This variation gives us characteristics from both maximum and minimum functions. Moreover, our variation keeps the reciprocal property of the BRDF model and shows flexibility over the second choice in which we can weight both cosθ i and cosθ o with different exponents. 3.2 Proposed model Due to the flexibility of max(a, B) α min(a, B) β, we then proposed the following BRDF model based on this choice of variation. Thus, our model can be expressed as follows. f r (ω i, ω o ) = k d π + F(cosθ h )G(ω i, ω o )S( 1 N.H) max(cosθ i, cosθ o ) α min(cosθ i, cosθ o ) β (3.1) For this study, we experimented and made comparisons of our proposed model to the rest of the variations. Other choices of the variations are expressed as follows. f r (ω i, ω o ) = k d π + F(cosθ h)g(ω i, ω o )S( 1 N.H) max(cosθ i, cosθ o ) α f r (ω i, ω o ) = k d π + F(cosθ h)g(ω i, ω o )S( 1 N.H) min(cosθ i, cosθ o ) α f r (ω i, ω o ) = k d π + F(cosθ h)g(ω i, ω o )S( 1 N.H) (cosθ i cosθ o ) α (3.2a) (3.2b) (3.2c) G term is the same as used in Cook-Torrance model. F(cosθ h ) is the Fresnel equation for unpolarized incident light.

34 20 3 The Proposed BRDF Model F(c) = 1 (g c) 2 { } [c(g + c) 1]2 2 (g + c) [c(g c) 1] 2 (3.3) where c = cosθ h and g 2 = η 2 + c 2 1. To evaluate the performance of our model, several comparisons have been made. The evaluation was designed to compare the errors between the reference model and all four models stated in this chapter. The error differences were to evaluate the efficiency of each model. The optimization process must be carried out in order to find the best fit parameters and error of each model. In the next chapter, the optimization process is explained. The two error metrics, used in the optimization, were proposed by Löw et al. [9]. In addition, we also introduced an optimization approach for improving error based on our new model.

35 4 Data Fitting To evaluate BRDF models, measured data is necessary for evaluation. The most widely used measured data for evaluating BRDF model to date is the MERL database. The database consists of 100 different material samples. BRDF models have been developed based on theories or empirical observations. However, when they are used to simulate realistic materials, model parameters must be known. To find the parameter values, the optimization is employed to find the best parameters in which the error between measured data and the BRDF model is minimized. In this study, we followed the data fitting method proposed by Löw et al.[9]. They proposed two different error metrics E 1 and E 2. E 1 is an error metric which is weighted by the cosine function. E 2 is a logarithmic error metric. Löw et al. concluded that parameters fit by the logarithmic error metric, the E 2, provide better rendering results. However the E 1 is still important because it is a common error metric. The two error metrics are expressed as follows. E m (p) = (gm jkl (p) ĝm jkl ) 2 sinθo k φo θ l o k θ j i j k l (4.1a) 21

36 22 4 Data Fitting Let p be a set of parameters and m be 1 or 2, g m (p) is formulated as follows. g 1 (ω i, ω o ; p) = cosθ i f r (ω i, ω o ; p) gˆ 1 (ω i, ω o ) = cosθ ifr ˆ (ω i, ω o ) g 2 (ω i, ω o ; p) = ln(1 + cosθ i f r (ω i, ω o ; p)) gˆ 2 (ω i, ω o ) = ln(1 + cosθ i f r (ω i, ω o )) (4.2a) (4.2b) (4.2c) (4.2d) g is reflection data from any BRDF model, ĝ is measured data. The equations above are definitions of error metrics. The optimization approach is to minimize the error, E m (p), and the result is a set of parameters, p, and its corresponding error. In this study, the least squares was used in the fitting process. The MATLAB command for the fitting is lsqcurvef it. This command is provided in the Optimization toolbox. The algorithm by default is trust region reflective, so the boundary values of the parameters must be set. The parameters for all BRDF models are k d, A, B, C, η, α, β. 4.1 Parameter fitting for the comparison The previous chapter has stated that comparisons must be made in order to evaluate the efficiency of the four BRDF models. The comparison is then performed by doing the optimization for all BRDF models, including the reference model. Due to the algorithm used, boundary values must be set. The test is performed for all 100 materials in MERL database. All parameters are fit together once. Once the fitting process has been done, the error differences of each material corresponding to each BRDF model were written in a table of comparison. In our work, we evaluated the efficiency by two schemes, Best performance and Non improvement schemes. The best performance scheme is to count the number of materials where the error difference of the variation has the lowest error difference values. For example, a material had 4 error differences from each variation. If the lowest error difference falls into our model, our model will be increased by one for the best performance scheme. Thus, if the variation has the highest value, the variation is the most efficient model. The non improvement scheme is to count the number of materials where the negative error differences are counted. For example, a material had 4 error differences. If negative error differences are found in max(a, B) α and min(a, B) α, then both max(a, B) α and min(a, B) α are increased by one for the non improvement scheme. Thus, the lower the counter is, the better the performance will be. Table 4.1 shows an example of comparison results.

37 4.2 Parameter fitting for the proposed model 23 Table 4.1: comparison between four variations max(a, B) α min(a, B) α max(a, B) α min(a, B) β (AB) α Best performance Non improvement Figure 4.1: Figure illustrates the double fitting method for our proposed model. Using the parameter results from the first step as initial values for the second step. The α and β parameters for the second step are initialized to Parameter fitting for the proposed model The previous section demonstrates that comparisons are performed by fitting all parameters together. Table 4.1 also shows that our proposed model still has negative error difference value, for 9 materials. Thus, this section provides a method for improving errors based on our proposed model. We proposed a method called double fitting. The fitting method above, applied to the comparisons, does fitting all parameters together. The double fitting method is divided into two steps. First, the fitting is performed in the reference model. This first step gives us all parameters except for α and β. Once we get the parameter values,k d, A, B, C, η, those values are then used as initial values for the next step. The second step has the initial values from the first step except for the initial values of α and β. The initial values of α and β are then given to 1. Figure 4.1 illustrates the process of the double fitting method. This method gave us very interesting results. All error results were equivalent or much improved from the reference model. There were no negative error difference values with the double fitting method. Moreover, relative errors of half database were improved more than 5 percent. In the next chapter, all results and visual outputs are presented. We also discuss improvements to some materials where the rendering results are better than the reference model.

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39 5Result We have provided information about both our proposed model and the data fitting method in the previous chapters. In this chapter, we present the comparisons between the four BRDF models to show the efficiency of our model. Some results are provided to demonstrate improvements and efficiency of the double fitting. Finally, some visual results are shown. 5.1 Comparison To perform the comparisons, we have gathered all error values from the optimization step. The comparisons were made as described in the previous chapter. Both E 1 and E 2 were used for comparisons. Table 5.1 shows the results of the comparisons. The best performance scheme and the non improvement scheme are described in the previous section. The results in table 5.1 show that the max(a, B) α min(a, B) β variation performs the best of all variations because it has the highest number of the best performance scheme and the lowest number of the non improvement scheme. 5.2 Double fitting As shown in table 5.1, even though our proposed model had the smallest number of the non improvement scheme, it still gave us worse errors than the reference model for 9 materials. We used the double fitting approach to improve the optimization errors. For E 1, the double fitting gave the lowest error difference value as x10 7 and the highest value as For E 2, the lowest error difference is x10 6 and the highest error difference is The results 25

40 26 5 Result Table 5.1: The comparisons between four variations Results of the comparisons calculated from cosine weighted error metric, E 1 max(a, B) α min(a, B) α max(a, B) α min(a, B) β (AB) α Best performance Non improvement Results of the comparisons calculated from logarithmic error metric, E 2 max(a, B) α min(a, B) α max(a, B) α min(a, B) β (AB) α Best performance Non improvement Table 5.2: The lowest and highest error differences of each error metric when applying the double fitting approach. E 1 E 2 Lowest x x10 6 Highest showed that there was no negative error difference. Thus, this approach can at least give an equivalent error to the reference model. See table 5.2. Moreover, percentage errors were calculated. The average percentage errors for all 100 materials are shown in table 5.3. There were up to 41 materials which had improvements more than 10% of the error for E 1. For E 2, there were up to 29 materials. In the appendix, we provide raw error data from the test. 5.3 Rendering results We have evaluated our proposed model in numerical approach by comparing errors from the optimization process. In this section, some rendering results are shown. Since the numerical errors or improvements, we have demonstrated in the sections above, cannot represent the efficiency of visual results, then some renderings must be done to show visual improvements. The rendering results were rendered by parameters from the logarithmic error metric, because the error metric gives better visual results than the cosine weighted error metric[9]. Table 5.3: percentage errors of each error metric. E 1 E 2 Average Lowest Highest

41 5.3 Rendering results 27 Figure 5.1: Figure shows rendering results. Left most column shows rendering from the measured data. The middle column shows results from the reference model and the last column is from our proposed model.

42 28 5 Result For black soft plastic and dark blue paint materials, our model can reduce sharpness and some sharp spots so that the materials look more realistic than the reference model. Moreover, the edge of polyethylene material can be best represented by our model. The reference model produces a darker material edge compared to our model which produces a brighter material edge. When we investigated the scattering curves, we found the cause of the sharpness and the darker material edges. Figure 5.2 illustrates the curves of each rendered material. All of the curves are scaled by logarithm. The black soft plastic and the dark blue paint materials, for the measured data, have smooth curves while the purple curves, the reference model, are sharp at the perfect specular angle. This is the cause of the sharpness and sharp spots on the rendered materials. For the polyethylene, our model can generate the scattering curve which spreads out and is close to the measured data. This characteristic of the curve cannot be generated by the reference model. To explain more clearly, figure 5.3 demonstrates how the curve changes when β parameter changes. When β is bigger, the scattering curve can spread the tails out. On the contrary, when β is smaller, the curve tails can be shrunk. The rendering effect of β parameter can be seen in figure 5.4.

43 5.3 Rendering results 29 (a) black soft plastic material reflection plot. (b) dark blue paint material reflection plot (c) polyethylene material reflection plot. Figure 5.2: Figures show scattering curves of materials. The red curve is the measured data. The purple curve is the reference model. The green curve is from our model. The curves are from red channel and scaled by logarithm.

44 30 5 Result (a) β = (b) β = (result from the fitting ). (c) β = Figure 5.3: Figures show the scattering curves of polyethylene materials with different β parameter.

45 Rendering results (a) β = (b) β = (result from the fitting ). (c) β = Figure 5.4: Figures show the rendering of polyethylene materials with different β parameter.