Marques, RicardoBouville, ChristianBouatouch, Kadi2019-04-302019-04-302019Marques R, Bouville C, Bouatouch K. Spectral analysis of quadrature rules and fourier truncation-based methods applied to shading integrals. IEEE Trans Vis Comput Graph. 2019 Apr 26:1-14. DOI: 10.1109/TVCG.2019.29134181077-2626http://hdl.handle.net/10230/37154We 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. Finally, 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 by a set of rendering experiments.application/pdfeng© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. http://dx.doi.org/10.1109/TVCG.2019.2913418info:eu-repo/semantics/articlehttp://dx.doi.org/10.1109/TVCG.2019.2913418Rendering equationSpectral analysisMonte Carlo methodsSpherical harmonics decompositioninfo:eu-repo/semantics/openAccess