3D Geometry Coding using Mixture Models and the Estimation Quantization Algorithm

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3D Geometry Coding usingMixture Models andthe
Estimation Quantization Algorithm

• Sridhar Lavu
• Masters Defense
• Electrical Computer Engineering

DSP Group
Rice University
September 2002
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3D Surfaces

• Video games
• Animations - Bugs Life, Toy Story 2
• 3D object modeling - CAD
• e-commerce

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3D Surfaces

• Geometry, color, texture
• 3D scanning
• Polygon meshes
• Problem - large data sets
• Geometry compression

100,000 triangles
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Contribution

• 3D geometry coder
• Multilevel representation
• Normal meshes
• EQ algorithm
• Estimation-Quantization (EQ)
• Local context information
• RD optimization

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Related Work

• Zerotree coder for the wavelet coefficients of
  normal meshes
• RD optimization based quantization algorithm for
  the wavelet coefficients of meshes

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Outline

• 3D surface data
• Multilevel representation
• Normal meshes
• Wavelet transform
• EQ algorithm
• Error metrics
• Results

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3D geometry data

• Geometry
• Polygon meshes
• Geometry connectivity

Geometry 0.0 0.0 0.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0
1.0 0.0 0.5 0.5 1.0
Connectivity 0 1 2 2 3 1 0 1 4 1 2 4 2 3 4 3 0 4
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Multilevel Representations
Original
Coarse
Multilevel triangular meshes
Original ? Normal meshes
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Normal meshes

• Multilevel representation
• Base mesh
• Successively refine the mesh
• Subdivision

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Subdivision

• Linear subdivision
• Butterfly subdivision
• Loop subdivision

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Butterfly Subdivision
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Normal Meshes

• Predict b and n
• Find intersection
• Store offset
• 1 number per vertex

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Wavelet Transforms

• Irregular data
• Lifting scheme predict and update
• Subdivision predict step
• Wavelet transforms
• Butterfly wavelet transform
• Loop wavelet transform

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Wavelet Transforms and Normal Meshes
Wavelet coefficients
Non-normal vertices
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Related Work - Zerotree
Zerotrees
Mesh zerotree coding
Mesh zerotree
Zerotree coding
EQ coding
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Review

• Multilevel representations for meshes
• Normal meshes
• Wavelet transforms
• Subdivision
• Lifting
• Related work - ZT based algorithm
• Contribution EQ based algorithm

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3D EQ Coder

• Local context information
• Model for wavelet coefficients
• Generalized Gaussian distribution
• EQ Algorithm
• Estimate Step
• Quantize Step
• RD Optimization

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Wavelet Coefficient Model

• Generalized Gaussian distribution

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Wavelet Coefficient Model

• Generalized Gaussian (GGD)
• ? Shape Fixed at each level
• ? Variance Local neighborhood
• ? Mean Zero

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EQ Algorithm

• Scan the vertices
• Estimate, quantize, encode
• Estimate step - variance
• Local neighborhood
• Causal neighborhood
• Quantized neighbors
• Quantize step
• Deadzone quantizer
• RD optimization

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EQ Algorithm (cont.)

• RD optimization
• Rate -log(probability)
• Distortion MSE of coefficients
• Entropy coding
• Arithmetic coder

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Normal vs. Tangential

• Smooth surfaces
• Global error contribution
• Normal Higher
• Tangential Lower
• Precision
• Normal Higher Lower l
• Tangential Lower Higher l
• Most tangential components are zero
• Single quantizer per level

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Neighborhood
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Ordering - Base Triangles
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Ordering - Vertices
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Summary of EQ Algorithm

• Pick l
• Determine ordering
• Ordering of base triangles
• Ordering inside each base triangle
• Local causal neighborhood
• Estimate s
• Quantize using RD optimization
• Normal vs. tangential

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Performance Measure

• Error metrics
• MSE ?
• Hausdorff distance
• Min, max, mean, mean squared

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Results

• Metric - PSNR
• Bits-per-vertex (bpv)
• Reconstructed mesh vs. original mesh
• Metro and MeshDev software tools

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Results - EQ vs. ZT
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Results EQ vs. ZT(Lifted Butterfly)
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Results - EQ vs. ZT(Loop Wavelets)
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Results (Bounds)

• Upper bound
• Complete context
• Lower bound
• No context

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Summary

• Multilevel representations
• Normal meshes
• Wavelet transforms
• GGD model
• Local context based coder
• EQ vs. ZT

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Conclusion Future Work

• Conclusions
• GGD model EQ algorithm
• 0.5 1 dB gain
• Future work
• Vertex based error for RD optimization
• New algorithms
• Space-Frequency quantization (SFQ)

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Scaling Coefficients andConnectivity

• Scaling coefficients
• Vertices of base mesh
• Uniform quantization
• Connectivity
• Semi-regular connectivity
• Base mesh connectivity
• TG Coder (lossless)

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Lifting (Predict, Update)
Forward
Inverse
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Lifting - Haar

• Split
• Predict
• Update

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Loop Wavelet Transform
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Causal Neighborhoods
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EQ Unpredictable sets

• Empty causal neighborhood
• Zero s estimate
• Classify as unpredictable (U) set
• Model U set as zero-mean GGD
• Use a single s and n for U set

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EQ Threshold step

• Iteration of E and Q steps
• First iteration
• Threshold coefficients
• Partition U and P sets
• Estimate s and n
• Use estimates in next iteration

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Normal Predictable Set
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Normal Unpredictable Set
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Tangential Set
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Hausdorff Distance
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Mesh Zerotree Coding
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Results Venus PSNR
BPV 0.25 0.5 1.0 2.0 3.0 4.0
EQ lifted BW 63.7 68.6 74.2 79.2 81.7 83.2
ZT lifted BW 63.0 68.2 73.7 78.9 81.7 81.9
EQ unlifted BW 63.5 68.6 74.1 78.9 81.4 83.0
ZT unlifted BW 62.4 67.8 73.0 78.4 81.2 81.5
EQ Loop Wavelet 60.0 65.3 71.3 76.4 79.4 81.4
ZT Loop Wavelet 60.9 66.1 71.8 77.1 79.7 79.7
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Results Rabbit PSNR
BPV 0.25 0.5 1.0 2.0 2.5 3.0
EQ lifted BW 70.3 75.7 80.9 84.2 85.1 85.6
ZT lifted BW 69.3 75.1 80.9 84.0 84.1 84.1
EQ unlifted BW 70.0 75.3 80.6 84.0 85.0 85.5
EQ unlifted BW 68.7 74.7 80.4 83.6 83.6 83.6