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Examining the Distribution of Sampling Point Sets on Sphere for Monte Carlo Image Rendering

AIP Conference Proceedings 2010年第1期1281卷 2103-2106页

作者： A. A. Penzov I. T. Dimov N. M. Mitev G. I. Sirakova L. Szirmay‐Kalos aDepartment of Parallel Algorithms Institute for Information and Communication Technologies Bulgarian Academy of Sciences Acad. G. Bonchev Str. bl. 25 A 1113 Sofia Bulgaria bBudapest University of Technology and Economics Magyar Tudósok krt. 2 H‐1117 Budapest Hungary

This paper presents a study of several non‐uniform and uniform techniques for sampling of unit sphere and gives an comparative analysis on uniform sampling techniques. Each sampling technique generates point sets distributed on sphere. We are interested to examine the quality of their distributions, which could impact on the Monte Carlo image creation. Generalized discrepancy is designed as a measure for equidistribution of point sets on spherical sampling patterns as well as applied to analyze sampling techniques used in rendering. We generate sphere sampling patterns using various random number generators and Halton low discrepancy sequence. The generalized discrepancy is computed for increasing number of points for each sampling pattern. Finally, the results for different uniform sampling techniques are studied and analyzed by comparison of the uniformity of distributions.

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Monte Carlo Estimator for Image Creation with Symmetric Sampling of Phong BRDF Model

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AIP Conference Proceedings 2010年第1期1281卷 2095-2098页

作者： Anton A. Penzov Ivan Dimov Gergana Sirakova Vladimir Koylazov aDepartment of Parallel Algorithms Institute for Information and Communication Technologies Bulgarian Academy of Sciences Acad. G. Bonchev Str. bl. 25 A 1113 Sofia Bulgaria bChaos Software Alexander Malinov Blvd. bl. 33 entr. B. 1729 Sofia Bulgaria

The paper is directed to the advanced rendering techniques for realistic image creation. We construct and offer a new Monte Carlo estimator for numerical solution of the rendering equation based on Phong BRDF (Bidirectional Reflectance Distribution Function) model. We consider the kernel of Phong rendering equation and present the Monte Carlo solution by a sum of two independent integrals, one for diffuse and one for specular part respectively. The diffuse integral equation is solved by applying Combined Uniform Separation of integration domain to achieve variance reduction. The hemispherical integration domain is symmetrically separated into 8 sub‐domains of equal orthogonal spherical triangles and 8 sub‐domains of equal spherical quadrangles. All spherical triangles, spherical quadrangles respectively are symmetric each to other as well as have fixed vertices and computable parameters. The symmetric sampling scheme is applied to generate the sampling points and solve the diffuse integral equation. The integration domain of specular integral equation is approximated by conical solid angle of most important region of interest. The normal vector of the conical solid angle is the direction of ideally reflected by the surface viewing vector. We show that the conical solid angle can be successfully approximated with rotation of all orthogonal spherical triangles sub‐domains, constructed at solving the diffuse integral equation, being easy to reuse the sampling points.

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A Quasi Monte Carlo Solution of the Rendering Equation by Uniform Quadrangle Separation

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AIP Conference Proceedings 2009年第1期1148卷 184-188页

作者： Anton A. Penzov Ivan T. Dimov Vladimir N. Koylazov aDepartment of Parallel Algorithms Institute for Parallel Processing Bulgarian Academy of Sciences Acad. G. Bonchev Str. bl. 25 A 1113 Sofia Bulgaria bACET Centre University of Reading Whiteknights PO Box 217 Reading RG6 6AH UK cChaos Software Alexander Malinov Blvd. bl. 33 entr. B. 1729 Sofia Bulgaria

The paper is directed to the advanced Quasi Monte Carlo methods for realistic image synthesis. We propose and consider a new Quasi Monte Carlo solution of the rendering equation by uniform quadrangle separation of integration domain. The hemispherical integration domain is uniformly separated into 12 equal size and symmetric sub‐domains. Each sub‐domain represents a solid angle, subtended by spherical quadrangle, very similar by form to plane unit square. Any spherical quadrangle has fixed vertices and computable parameters. A bijection of unit square into spherical quadrangle is find and the symmetric sampling scheme is applied to generate the sampling points uniformly distributed over hemispherical integration domain. Then, we apply the stratified Quasi Monte Carlo integration method for solving the rendering equation. The estimate of the rate of convergence is obtained. We prove the superiority of the proposed Quasi Monte Carlo solution of the rendering equation for arbitrary dimension of the sampling points. The uniform separation leads to convergence improvement of the Plain (Crude) Quasi Monte Carlo method.

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Quasi Monte Carlo Approach with Uniform Separation for Solving of the Rendering Equation

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AIP Conference Proceedings 2008年第1期1046卷 114-118页

作者： Anton A. Penzov Ivan T. Dimov Stanislava S. Stoilova aDepartment of Parallel Algorithms Institute for Parallel Processing Bulgarian Academy of Sciences Acad. G. Bonchev Str. bl. 25 A 1113 Sofia Bulgaria bACET Centre University of Reading Whiteknights P.O. Box 217 Reading RG6 6AH UK cDepartment of Computational Mathematics Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev Str. bl. 8 1113 Sofia Bulgaria

This paper is directed to the advanced parallel Quasi Monte Carlo (QMC) methods for realistic image synthesis. We propose and consider a new QMC approach for solving the rendering equation with uniform separation. First, we apply the symmetry property for uniform separation of the hemispherical integration domain into 24 equal sub‐domains of solid angles, subtended by orthogonal spherical triangles with fixed vertices and computable parameters. Uniform separation allows to apply parallel sampling scheme for numerical integration. Finally, we apply the stratified QMC integration method for solving the rendering equation. The superiority our QMC approach is proved.

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