Sampling and Variance Analysis for Monte Carlo Integration in Spherical Domain
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Date
2015-09
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Abstract
This dissertation introduces a theoretical framework to study different sampling
patterns in the spherical domain and their effects in the evaluation of global illumination
integrals. Evaluating illumination (light transport) is one of the most essential aspect in image
synthesis to achieve realism which involves solving multi-dimensional space integrals. Monte
Carlo based numerical integration schemes are heavily employed to solve these high
dimensional integrals. One of the most important aspect of any numerical integration method
is sampling. The way samples are distributed on an integration domain can greatly affect the
final result. For example, in images, the effects of various sampling patterns appears in
the form of either structural artifacts or completely unstructured noise. In many cases, we may get completely false (biased) results due to the sampling pattern used in integration.
The distribution of sampling patterns can be characterized using their Fourier power spectra.
It is also possible to use the Fourier power spectrum as input, to generate the corresponding
sample distribution. This further allows spectral control over the sample distributions.
Since this spectral control allows tailoring new sampling patterns directly from the input
Fourier power spectrum, it can be used to improve error in integration.
However, a direct relation between the error in Monte Carlo integration and the sampling
power spectrum is missing. In this work, we propose a variance formulation, that establishes
a direct link between the variance in Monte Carlo integration and the power
spectra of both the sampling pattern and the integrand involved.
To derive our closed-form variance formulation, we use the notion of homogeneous sample
distributions that allows expression of error in Monte Carlo integration, only in the form of
variance. Based on our variance formulation, we develop an analysis tool that can be used to
derive theoretical variance convergence rates of various state-of-the-art sampling patterns.
Our analysis give insights to design principles that can be used to tailor new sampling
patterns based on the integrand.
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Citation
@phdthesis{singh:tel-01217082, TITLE = {{Sampling and Variance Analysis for Monte Carlo Integration in Spherical Domain}}, AUTHOR = {Singh, Gurprit}, URL = {https://hal.archives-ouvertes.fr/tel-01217082}, SCHOOL = {{R3AM}}, YEAR = {2015}, MONTH = Sep, KEYWORDS = { Stochastic Sampling ; Global Illumination ; Monte Carlo Integration ; Int{\'e}gration Monte Carlo}, TYPE = {Theses}, PDF = {https://hal.archives-ouvertes.fr/tel-01217082/file/phdthesis.pdf}, HAL_ID = {tel-01217082}, HAL_VERSION = {v1}, }