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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. [ABSTRACT FROM AUTHOR]

This paper is turned to the advanced Monte Carlo methods for realistic image creation. It offers a new stratified approach for solving the rendering equation. We consider the numerical solution of the rendering equation by separation of integration domain. The hemispherical integration domain is symmetrically separated into 16 parts. First 8 sub-domains are equal size of orthogonal spherical triangles. They are symmetric each to other and grouped with a common vertex around the normal vector to the surface. The hemispherical integration domain is completed with more 8 sub-domains of equal size spherical quadrangles, also symmetric each to other. All sub-domains have fixed vertices and computable parameters. The bijections of unit square into an orthogonal spherical triangle and into a spherical quadrangle are derived and used to generate sampling points. Then, the symmetric sampling scheme is applied to generate the sampling points distributed over the hemispherical integration domain. The necessary transformations are made and the stratified Monte Carlo estimator is presented. The rate of convergence is obtained and one can see that the algorithm is of super-convergent type. [ABSTRACT FROM AUTHOR]