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

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Irregular Triangle Meshes Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 Face 2 1 3 Face 4 2 3 … Face 2 1 3 Face 4 2 3 …

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Texture mapping Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 … s1 t1s1 t1s2 t2s2 t2s1 t1s1 t1s2 t2s2 t2 normal map s t Face 2 1 3 Face 4 2 3 … random access!

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

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

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Basic idea cut parametrize

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cut sample

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

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Creation of Geometry Image 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 Key Points: – Good Cut – Good Parameterization Approach: Combine those two goals together!

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

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

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Results of Step 1 genus 6 genus 0 genus 3

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

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Step 2: Augment cut …iterate while parametrization improves

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

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

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

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Sampling

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Rendering Span each quad of samples with two triangles.

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Rendering with Attributes geometry image 257 2 x 12b/ch normal-map image 512 2 x 8b/ch

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Mip-mapping 257x257129x12965x65 boundary constraints set for size 65x65

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

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Compression Completely regular sample means: – Can take full advantages of off-the-shelf image compression codes. Image Wavelets Coder: 295KB 1.5KB plus 12B sideband

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Compression Results 295KB 1.5KB 3KB 12KB 49KB

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

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

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All pictures credit to the original Siggraph02 presentation slides

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More Pics1 257x257 normal-map 512x512

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More Pics2 257x257 color image 512x512

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More Pics3 – artifacts aliasing anisotropic sampling

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

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