University of Pennsylvania ScholarlyCommons IRCS Technical Reports Series Institute for Research in Cognitive Science December 1996 Computational Methods for Realistic Image Synthesis Min-Zhi Shao University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/ircs_reports Shao, Min-Zhi, "Computational Methods for Realistic Image Synthesis" (1996). IRCS Technical Reports Series. 113. https://repository.upenn.edu/ircs_reports/113 University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-96-35. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ircs_reports/113 For more information, please contact [email protected]. Computational Methods for Realistic Image Synthesis Abstract In this thesis, we investigate the computational methods for both diffuse and general reflections in realistic image synthesis and propose two new approaches: the overrelaxation solution and the Bernstein polynomial solution. One of the major concerns with the radiosity method is its expensive computing time and memory requirements. In this thesis, we analyze the convergence behavior of the progressive refinement adiosityr method and propose two overrelaxation algorithms: the gathering and shooting solution and the positive overshooting solution. We modify the conventional shooting method to make the optimal use of the visibility information computed in each iteration. Based on a concise record of the history of the unshot light energy distribution, a solid convergence speed-up is achieved. Though a great effort has been made to extend the radiosity method to accommodate general non- diffuse reflection, the current algorithms are still quite limited to simple environment settings. In this thesis, we propose using the piecewise spherical Bernstein basis functions over a geodesic triangulation to represent the radiance function. The representation is intrinsic to the unit sphere, and can be efficiently stored, evaluated, and subdivided by the numerically stable de Casteljau algorithm. We demonstrate that the computation of other fundamental radiometric quantities such as vector irradiance and reflected radiance can be reduced to the integration of the piecewise spherical Bernstein basis functions. A novel geometric integration algorithm based on adaptive domain subdivision is presented for the Bernstein- B´ezier polynomials over a geodesic triangle on the unit sphere. Comments University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-96-35. This thesis or dissertation is available at ScholarlyCommons: https://repository.upenn.edu/ircs_reports/113 COMPUTATIONAL METHODS FOR REALISTIC IMAGE SYNTHESIS Min-Zhi Shao A Dissertation in Computer and Information Science Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 1996 Dr. Norman I. Badler Supervisor of Dissertation Dr. Mark Steedman Graduate Group Chairperson c Copyright 1996 by Min-Zhi Shao To my grandmother, Jing-Mei Zhou, and my parents, Ya-Mei Zhong and Jian Shao Acknowledgments I owe a great deal to my advisor, Professor Norman Badler, for his guidance, support, and patience. Norm has provided an excellent environment in which to work. Without his open mind and flexibility, this work would never have been completed. I want to thank the other members of my doctoral committee: Professors David Ebert, Jean Gallier, Sampath Kannan, and Dimitris Metaxas, for their encouragement and helpful comments. I would like to take this opportunity to thank the faculties in the Computer and Information Science and the Mathematics Departments. In particular, I thank Professors Ruzena Bajcsy, Herman Gluck, Jerry Kazdan, Michael Palis, Sanguthevar Rajasekaran, and Eero Simoncelli for their fascinating lectures. My sincere thanks also extend to the past and present members of the Graphics Laboratory. Their friendships and kind helps made my life at Penn much easier and truly memorable. Finally, I wish to express my deep gratitude to my parents for giving me the strength to pursue my dreams, and for their sacrifices to provide me the best education possible, and to my grandmother for her quiet love in all these years. This thesis is dedicated to them. ABSTRACT Computational Methods for Realistic Image Synthesis Min-Zhi Shao Supervised by Dr. Norman I. Badler In this thesis, we investigate the computational methods for both diffuse and general reflections in realistic image synthesis and propose two new ap- proaches: the overrelaxation solution and the Bernstein polynomial solution. One of the major concerns with the radiosity method is its expensive com- puting time and memory requirements. In this thesis, we analyze the con- vergence behavior of the progressive refinement radiosity method and propose two overrelaxation algorithms: the gathering and shooting solution and the positive overshooting solution. We modify the conventional shooting method to make the optimal use of the visibility information computed in each iteration. Based on a concise record of the history of the unshot light energy distribution, a solid convergence speed-up is achieved. Though a great effort has been made to extend the radiosity method to accommodate general non-diffuse reflection, the current algorithms are still quite limited to simple environment settings. In this thesis, we propose using the piecewise spherical Bernstein basis functions over a geodesic triangulation to represent the radiance function. The representation is intrinsic to the unit sphere, and can be efficiently stored, evaluated, and subdivided by the numer- ically stable de Casteljau algorithm. We demonstrate that the computation of other fundamental radiometric quantities such as vector irradiance and re- flected radiance can be reduced to the integration of the piecewise spherical Bernstein basis functions. A novel geometric integration algorithm based on adaptive domain subdivision is presented for the Bernstein-Bezier´ polynomials over a geodesic triangle on the unit sphere. Contents 1 Introduction 1 1.1 Background and Contribution 1 1.2 Thesis Outline 4 2 Radiative Transfer 6 2.1 Radiometric Concepts 6 2.2 Reflectance Models 9 2.2.1 Ideal Diffuse Reflectance Model 9 2.2.2 Ideal Specular Reflectance Model 10 2.2.3 Torrance-Sparrow Reflectance Model 11 2.2.4 Beckmann-Spizzichino Reflectance Model 13 2.3 Radiance Equation 14 3 Radiosity Method 17 3.1 Classical Method 17 3.2 Galerkin Method 21 3.3 Iterative Solution 25 4 Overrelaxation Radiosity Solution 28 4.1 Gathering Iteration 28 4.2 Shooting Iteration 29 4.3 Gathering and Shooting Iteration 35 vi 4.4 Positive Overshooting Iteration 42 4.5 Results 44 5 Bernstein Polynomial Solution 54 5.1 Spherical Representations 54 5.2 Bernstein Polynomials 57 5.2.1 Planar Triangle 57 5.2.2 Spherical Triangle 61 5.3 Radiative Transfer Computations 63 5.3.1 Radiance Representation 63 5.3.2 Vector Irradiance 65 5.3.3 Reflected Radiance 66 5.4 Spherical Integration 68 5.5 Results 72 6 Conclusions 85 Bibliography 90 vii List of Tables 4.1 Percentages of patch shooting times in different iteration stages for the test environment. 33 4.2 The computing time statistics for the first test environment (unit: minute). 52 4.3 The computing time statistics for the second test environment (unit: minute). 52 viii List of Figures 2.1 Local geometry of the incident and reflected radiant flux. 7 x 2.2 Geometry of radiance leaving point x and arriving at point . 15 3.1 Geometry of form factor. 19 4.1 A test environment: from left to right and top to bottom, 25, 100, 200, 400, 800, 2,000, 5,000 iterations, and the converged solution. 31 4.2 Curves of the sorted patch vertex radiosity accuracy in differ- ent iteration stages for the test environment. The vertical axis shows the ratio of the intermediate and converged radiosity. The horizontal axis shows all the patch vertices sorted in increasing ratio (thus the curves are all monotonically non-decreasing). The curve family represents the increasing accuracy of the algorithm as the number of iterations (as labeled on each curve) increases. 32 4.3 Gathering and shooting interaction between the current shoot- j ing patch i and any other patch in the environment. 40 4.4 The first test environment. 46 4.5 Conventional shooting method. 46 4.6 Gathering and shooting method. 47 4.7 Overshooting methods. 48 4.8 The second test environment. 49 4.9 Conventional shooting method. 49 4.10 Gathering and shooting method. 50 ix 4.11 Overshooting methods. 51 5.1 Planar barycentric coordinates. 58 5.2 Gaussian quadrature formulas for the triangle. 60 5.3 Spherical barycentric coordinates. 61 5.4 Planar barycentric subdivision. 69 5.5 Spherical barycentric subdivision. 70 5.6 Spherical monomial approximations of function f 1 for 2 3 3 0 4 . 77 5.7 Spherical monomial approximations of function f 1 for 11 13 24 24 0 16 . 80 5.8 An example of Phong reflections with Lambertian sources. 83 5.9