Representing Spherical Functions with Rhombic Dodecahedron

Equal-Area Spherical Maps for Compute Graphics
Liang Wan

Spherical map is the foundation for many applications in computer graphics, such as environment mapping, precomputed radiance transfer, radiosity and image-based rendering. For these applications that involve intensive computation on spherical domain, it is generally desirable to employ equal-area and uniform spherical maps as the underlying parameterization. However, equal-area spherical maps are less exploited in graphics.

In this dissertation, we first introduce an equal-area spherical map, HEALPix, which is borrowed from astrophysics. Its associated sample pattern is uniformly distributed over the sphere surface. Then we discuss its application in environment mapping. Although the HEALPix representation is more balanced than the traditional cubemap, it cannot utilize the built-in hardware operations like cubemap, and the mipmapping construction is more complicated. Therefore, its rendering speed is not comparable to that of the cubemap. This motivates us to invent a new six-face spherical map, called isocube. Unlike cubemap, isocube is an equal-area mapping, i.e., each texel spans the same area and is equally important. Due to the six-face representation, isocube can fit nicely into the cubemap hardware and hence can fully exploit the hardware operations tailored for cubemap. In addition, its mapping computation only involves a small overhead. Therefore the look-up speed for isocube is very fast.

Next we discuss another application of HEALPix map, the dynamic environment sequence sampling. Previous work only considers the static case where the environment map is approximated by finite directional light sources. In the dynamic case, the individual regeneration of samples for each frame may introduce abrupt changes in the rendering animation. To handle this temporal inconsistency problem, we propose spherical q$^2$-tree based on HEALPix map. The adaptiveness of the quadtree can suppress the abrupt changes between consecutive frames, and hence a rather smooth rendering can be produced. This method, however, generates sample pattern independently for each environment frame, and therefore it may still cause unexpected jumps in some situations. To fully utilize the temporal coherence in the sequence, we present a global sampling approach which treats the dynamic environment sequence as a 3D volume and performs sampling in the volume. The volumetric sampling adapts the sample number according to each frame and restricts the temporal/spatial changes within small subdivided volumes, and hence ensures a smooth sampling sequence. Within the framework, we present a volumetric importance metric and develop a binary-quad tree to perform stratification. Experimental results show that our volumetric sampling method can generate light samples with a better temporal consistency.