Geometry Images Xianfeng Gu Steven Gortler Hugues Hoppe

Geometry Images Xianfeng Gu Steven Gortler Hugues Hoppe Harvard University Microsoft Research

Irregular meshes Vertex 1 y 1 z 1 Vertex 2 y 2 z 2 … Face 2 1 3 Face 4 2 3 …

Texture mapping Vertex 1 y 1 z 1 s 1 t 1 Vertex 2 y 2 z 2 s 2 t 2 … Face 2 1 3 Face 4 2 3 … t normal map s

Complicated rendering process Vertex 1 y 1 z 1 s 1 t 1 Vertex 2 y 2 z 2 s 2 t 2 … Face 2 1 3 Face 4 2 3 … random access! ~40 M Δ/sec

Semi-regular representations [Eck et al 1995] [Lee et al 1998] [Khodakovsky 2000] [Guskov et al 2000] … irregular vertex indices only semi-regular

Geometry Image completely regular sampling geometry image 257 x 257; 12 bits/channel 3 D geometry

Basic idea cut parametrize demo

Basic idea cut sample

Basic idea cut store render [r, g, b] = [x, y, z]

How to cut ? sphere in 3 D 2 D surface disk

How to cut ? sphere in 3 D l 2 D surface disk Genus-0 surface any tree of edges

How to cut ? torus (genus 1) l Genus-g surface 2 g generator loops minimum

Surface cutting algorithm (1) Find topologically-sufficient cut: 2 g loops [Dey and Schipper 1995] [Erickson and Har-Peled 2002] (2) Allow better parametrization: additional cut paths [Sheffer 2002]

Step 1: Find topologically-sufficient cut (a) retract 2 -simplices (b) retract 1 -simplices

Results of Step 1 genus 6 genus 3 genus 0

Step 2: Augment cut l l Make the cut pass through “extrema” (note: not local phenomena). Approach: parametrize and look for “bad” areas.

Step 2: Augment cut …iterate while parametrization improves

Results of Steps 1 & 2 genus 1 genus 0

Parametrize boundary a a’ a’ a Constraints: n n cut-path mates identical length endpoints at grid points no cracks

Parametrize interior n Geometric-stretch metric – minimizes undersampling [Sander et al 2001] – optimizes point-sampled approx. [Sander et al 2002]

Stretch parametrization Previous metrics (Floater, harmonic, uniform, …)

Sample geometry image

Rendering (65 x 65 geometry image)

Rendering with attributes geometry image 2572 x 12 b/ch normal-map image 5122 x 8 b/ch rendering

Advantages for hardware rendering l Regular sampling no vertex indices. l Unified parametrization no texture coordinates. Raster-scan traversal of source data: geometry & attribute samples in lockstep. Summary: compact, regular, no indirection

Normal-Mapped Demo geometry image 129 x 129; 12 b/ch demo normal map 512 x 512; 8 b/ch

Results 257 x 257 normal-map 512 x 512

Results 257 x 257 color image 512 x 512

Mip-mapping 257 x 257 129 x 129 65 x 65 boundary constraints set for size 65 x 65

Hierarchical culling view-frustum culling geometry image backface culling normal-map image

Compression Image wavelet-coder 295 KB 1. 5 KB + topological sideband (12 B) fused cut

Compression results 295 KB 1. 5 KB 3 KB 12 KB 49 KB

Rate distortion

Some artifacts aliasing anisotropic sampling

Summary l l Simple rendering: compact, no indirection, raster-scan stream. Mipmapped geometry Hierarchical culling Compressible

Future work l Better cutting algorithms l Feature-sensitive remeshing l Tangent-frame compression l Bilinear and bicubic rendering l Build hardware

Pre-shaded Demo geometry image 129 x 129; 12 b/ch demo color map 512 x 512; 8 b/ch