Geometry Image Xianfeng Gu Steven Gortler Hugues Hoppe

Geometry Image Xianfeng Gu, Steven Gortler, Hugues Hoppe SIGGRAPH 2002 Present by Pin Ren Feb 13, 2003

Irregular Triangle 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 … random access! t random access! s normal map

Irregular Regular, How? l Previous work: [Eck et al 1995] [Lee et al 1998] [Khodakovsky 2000] [Guskov et al 2000] … Remesh into Semi-Regular Connectivity

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

Basic idea cut parametrize

Basic idea cut sample

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

Creation of Geometry Image l How can we get the Geometry Image? – Cut M into M’ which has the topology of a disk – Parameterize: piecewise linear map from domain unit square D to M’ – Resample it at D’s grid points l Key Points: – Good Cut – Good Parameterization l Approach: Combine those two goals together!

Surface cutting algorithm (1) Find topologically-sufficient cut: For genus g: 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 Make the cut pass through “extrema” (note: not local phenomena). l Approach: parametrize and look for “bad” areas.

Step 2: Augment cut …iterate while parametrization improves

Parameterize Methods l Boundary – – – l To avoid Crack: constraints apply To avoid degeneracy: more constraints Minor adjustments for better result Interior – – Geometric-Stretch metric Other metric: Floater …

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

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

Sampling

Rendering Span each quad of samples with two triangles.

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

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

Advantages Regular Sampling – no vertex indices l Unified Parameterization – no texture coord. l Directly Mip-mapping, l l Rendering process is done in SCAN ORDER! – Much simpler than traditional rendering process – Inherently natural for hardware acceleration.

Compression l Completely regular sample means: – Can take full advantages of off-the-shelf image compression codes. Image Wavelets Coder: 295 KB 1. 5 KB plus 12 B sideband

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

Limitations l Higher l Since genus can be problematic it is based on sampling approach, – it does suffer from artifacts – Has difficulty to capture sharp surface features.

Summary l Geometry Image is a novel method to represent geometries in a completely regular and simple way. l It has some very valuable advantages over traditional triangular meshes. l May Inspire new hardware rendering tech. l Based on sampling, may not be able to capture all the details

All pictures credit to the original Siggraph 02 presentation slides

More Pics 1 257 x 257 normal-map 512 x 512

More Pics 2 257 x 257 color image 512 x 512

More Pics 3 – artifacts aliasing anisotropic sampling

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