Area‐Preserving Parameterizations for Spherical Ellipses

Abstract

We present new methods for uniformly sampling the solid angle subtended by a disk. To achieve this, we devise two novel area-preserving mappings from the unit square [0, 1] 2 to a spherical ellipse (i.e. the projection of the disk onto the unit sphere). These mappings allow for low-variance stratified sampling of direct illumination from disk-shaped light sources. We discuss how to efficiently incorporate our methods into a production renderer and demonstrate the quality of our maps, showing significantly lower variance than previous work.

Stratified Sampling of Projected Spherical Caps

“…Gamito [Gam16] proposed a method for cylindrical and disk luminaires based on bounding their solid angle by simpler shapes followed by rejection sampling. Recently, Guillén et al [GUnK*17] derived analytical parameterizations of the solid angle of disks.…”

Area-Preserving Parameterizations for Spherical Ellipses

“…The second allows for very efficient sampling, introducing minimal error. In-depth implementation details can be found in [7]. Our low-distorsion mapping, and its associated sampling procedures, are currently implemented in the Arnold renderer [4]; a commercial software for movie production that recently received the Technical Award of the Academy (i.e.…”

“…The lowdistortion mapping has the best properties for its use in Monte Carlo, since it is both efficient to evaluate and preserves the discrepancy properties of the sampling sequence. The complete mathematical details can be consulted on the main document [7].…”

“…In contrast, our method allow for a explicit stratification of the disk's solid angle projection. Our work has been published in Computer Graphics Forum [7], and presented at the Eurographics Symposium on Rendering 2017.…”

Real‐Time Rendering of Wave‐Optical Effects on Scratched Surfaces

“…The graphics community therefore has invested major efforts to develope efficient approximate methods for shading with extended light sources. In opposition to standard brute‐force path tracing where luminaires are sampled, many techniques have been studied to efficiently sample area sources of different geometry [SWZ96]; in particular spherical triangles [Arv95, 5.2], spherical rectangles [UnFK13], and spherical ellipses [GUnK*17]. When applied to strongly specular material models with high angular bandwidth, these integration strategies require a high number of samples for converged, noise‐free images, making them unsuitable for most real‐time rendering purposes.…”