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

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.