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We propose a theoretical framework, based on the theory of Sobolev spaces, that allows for a comprehensive analysis of quadrature rules for integration over the sphere. We apply this framework to the case of shading integrals in order to predict and analyze the performances of quadrature methods. We show that the spectral distribution of the quadrature error depends not only on the samples set size, distribution and weights, but also on the BRDF and the integrand smoothness. The proposed spectral analysis of quadrature error allows for a better understanding of how the above different factors interact. We also extend our analysis to the case of Fourier truncation-based techniques applied to the shading integral, so as to find the smallest spherical/hemispherical harmonics degree L (truncation) that entails a targeted integration error. This application is very beneficial to global illumination methods such as Precomputed Radiance Transfer and Radiance Caching. Our proposed framework is the first to allow a direct theoretical comparison between quadrature- and truncation-based methods applied to the shading integral. This enables, for example, to determine the spherical harmonics degree L which corresponds to a quadrature-based integration with N samples. Our theoretical findings are validated through a set of rendering experiments.
This article was published in the following journal.
Name: IEEE transactions on visualization and computer graphics
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Analysis based on the mathematical function first formulated by Jean-Baptiste-Joseph Fourier in 1807. The function, known as the Fourier transform, describes the sinusoidal pattern of any fluctuating pattern in the physical world in terms of its amplitude and its phase. It has broad applications in biomedicine, e.g., analysis of the x-ray crystallography data pivotal in identifying the double helical nature of DNA and in analysis of other molecules, including viruses, and the modified back-projection algorithm universally used in computerized tomography imaging, etc. (From Segen, The Dictionary of Modern Medicine, 1992)
A method of chemical analysis based on the detection of characteristic radionuclides following a nuclear bombardment. It is also known as radioactivity analysis. (McGraw-Hill Dictionary of Scientific and Technical Terms, 4th ed)
Methods for performing three-dimensional measurement and motion analysis using stereoscopic radiographs.
A class of natural contraceptive methods in which SEXUAL ABSTINENCE is practiced a few days before and after the estimated day of ovulation, during the fertile phase. Methods for determining the fertile period or OVULATION DETECTION are based on various physiological indicators, such as circulating hormones, changes in cervical mucus (CERVIX MUCUS), and the basal body temperature.
A repetitive and persistent pattern of behavior in which the basic rights of others or major age-appropriate societal norms or rules are violated. These behaviors include aggressive conduct that causes or threatens physical harm to other people or animals, nonaggressive conduct that causes property loss or damage, deceitfulness or theft, and serious violations of rules. The onset is before age 18. (From DSM-IV, 1994)