Anisotropic spherical Gaussians

Abstract

1 SG, 9.2%, 180 fps 3 SGs, 6.0% , 76 fps 5 SGs, 3.4 %, 55 fps 7 SGs, 2.0% , 36 fps 9 SGs, 1.1% , 27 fps 1 ASG 11 SGs, 0.54% , 22 fps 13 SGs, 0.26% , 19 fps 15 SGs, 0.11% , 17 fps 1 ASG, 0.10% , 125 fps reference Figure 1: Comparison of the SG (Spherical Gaussian) based approximation with the ASG (Anisotropic Spherical Gaussian) based approximation in rendering a highly anisotropic metal dish, under an environment light and two local lights. The BRDF of the metal dish is approximated by different number of ASGs or SGs in different images. Notice the superior property of ASGs over SGs. The result generated by 1 ASG already matches the path-traced reference well (with a L 2 error of 0.10%), and achieves a high framerate of 125 fps, while, to achieve a similar quality, more than 10 SGs are required, but with much lower framerates (19 fps for 13 SGs or 17 fps for 15 SGs). The L 2 error and the framerates for each configuration are also given in the corresponding subtitle. AbstractWe present a novel anisotropic Spherical Gaussian (ASG) function, built upon the Bingham distribution [Bingham 1974], which is much more effective and efficient in representing anisotropic spherical functions than Spherical Gaussians (SGs). In addition to retaining many desired properties of SGs, ASGs are also rotationally invariant and capable of representing all-frequency signals. To further strengthen the properties of ASGs, we have derived approximate closed-form solutions for their integral, product and convolution operators, whose errors are nearly negligible, as validated by quantitative analysis. Supported by all these operators, ASGs can be adapted in existing SG-based applications to enhance their scalability in handling anisotropic effects. To demonstrate the accuracy and efficiency of ASGs in practice, we have applied ASGs in two important SG-based rendering applications and the experimental results clearly reveal the merits of ASGs.

On-line learning of parametric mixture models for light transport simulation

“…The Kent distribution [1982] is a directional distribution that does support anisotropy, but becomes bi-modal for a range of parameters and is numerically unstable. An anisotropic spherical function [Xu et al 2013] derived from the Bingham distribution [1974] was developed for interactive rendering. However, at the moment, there is no known analytical sampling procedure.…”

A Survey of BSDF Measurements and Representations

“…Noisy, irregular and sparse reflectance and transmittance measurements can be represented by analytical BRDF models [7][8][9][10][11][12][13][14][15][16][17], BTDF models [1,6,18] or BSDF models [1,6,18]. However, this kind of irregular and sparse measurements cannot be represented by any datadriven material models.…”

Modified filtered importance sampling for virtual spherical Gaussian lights

“…Spherical Gaussians [18,19] are often used for approximating the rendering of various types of materials under environment maps or area lights [20][21][22][23][24]. This is because they have closed-form solutions for the integral, product, and product integral, which are fundamental operations to evaluate rendering integrals.…”

“…This paper approximates the incoming radiance using spherical Gaussians for the analytical approximation of the rendering integral [19,22].…”

“…For simplicity, anisotropic spherical Gaussians are used only for the first bounce which is more visually important than the second bounce. In addition, the product integral of a spherical Gaussian and anisotropic spherical Gaussian [19] has a reasonable computation cost. An anisotropic spherical Gaussian is defined aś G(ω, ξ x , ξ y , ξ z , η x , η y ) …”