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# Parallel Monte Carlo Approach for Integration of the Rendering Equation

• Ivan T. Dimov
• Anton A. Penzov
• Stanislava S. Stoilova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4310)

## Abstract

This paper is addressed to the numerical solving of the rendering equation in realistic image creation. The rendering equation is integral equation describing the light propagation in a scene accordingly to a given illumination model. The used illumination model determines the kernel of the equation under consideration. Nowadays, widely used are the Monte Carlo methods for solving the rendering equation in order to create photorealistic images.

In this work we consider the Monte Carlo solving of the rendering equation in the context of the parallel sampling scheme for hemisphere. Our aim is to apply this sampling scheme to stratified Monte Carlo integration method for parallel solving of the rendering equation. The domain for integration of the rendering equation is a hemisphere. We divide the hemispherical domain into a number of equal sub-domains of orthogonal spherical triangles. This domain partitioning allows to solve the rendering equation in parallel. It is known that the Neumann series represent the solution of the integral equation as a infinity sum of integrals. We approximate this sum with a desired truncation error (systematic error) receiving the fixed number of iteration. Then the rendering equation is solved iteratively using Monte Carlo approach. At each iteration we solve multi-dimensional integrals using uniform hemisphere partitioning scheme. An estimate of the rate of convergence is obtained using the stratified Monte Carlo method.

This domain partitioning allows easy parallel realization and leads to convergence improvement of the Monte Carlo method. The high performance and Grid computing of the corresponding Monte Carlo scheme are discussed.

## Keywords

Integration Domain Monte Carlo Approach Global Illumination Illumination Model Random Sampling Point
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## Copyright information

© Springer Berlin Heidelberg 2007

## Authors and Affiliations

• Ivan T. Dimov
• 1
• Anton A. Penzov
• 2
• Stanislava S. Stoilova
• 3
1. 1.Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 25 A, 1113 Sofia, Bulgaria and ACET Centre, University of Reading, Whiteknights, PO Box 217, Reading, RG6 6AHUK
2. 2.Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 25 A, 1113 SofiaBulgaria
3. 3.Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 8, 1113 SofiaBulgaria