Estimation-Quantization Geometry Coding

using Normal Meshes

Sridhar LavuHyeokho Choi Richard Baraniuk

Rice University

3D Surfaces

Applications– Video games– Animations– 3D Object modeling– e-commerce

3D Mesh Representation• Mesh representation

– 3D scan– Point clouds– Polygon mesh

Geometry0.0 0.0 0.01.0 0.0 0.01.0 1.0 0.00.0 1.0 0.00.5 0.5 1.0

Connectivity0 1 22 3 10 1 41 2 42 3 43 0 4• Goal

– Compression

• Problem – Massive data size– Michelangelo’s statue

of David: > billion triangles

Multiscale Representation

• Regular or semi-regular meshes

• Connectivity Base mesh connectivity

Wavelet Transform

• Prediction residuals– Wavelet transform– 3D coefficients (x,y,z)

Normal Meshes

• Normal mesh representation• 3D (x,y,z) 1D normal coefficient

Wavelet Coefficients• Normal wavelet coefficients• Tangential wavelet coefficients

• Goal – Model + Encode

Wavelet Coefficient Model

• Statistical model for normal mesh wavelet coefficients

• Expectation-Quantization model[Lopresto, Orchard, Ramchandran], DCC 1997

• ni ~ N(0,sigmai2)

• sigmai2 = local variance

– large rough region– small smooth region

Details

• Causal neighborhood– Estimate sigmai

2

• Quantized coefficients– Modified model– Generalized Gaussian density– Fixed shape at each scale– Estimate variance for each vertex

Vertex Scanning Order• Each scale• Each base triangle

Vertex Neighborhood

Estimate-Quantization Steps

• Estimate step– Shape parameter– Variance parameter

• R-D optimized quantize step– Rate = - log probability– Distortion = MSE of coefficient– Pick a lambda R-D operating point

• Entropy code– Arithmetic coder

Summary

Error Metrics• Different surfaces

– Original mesh surface– Normal re-meshing– EQ algorithm coded mesh

• MSE

• Metro– “average distance between two

meshes”– Hausdorff distance

PSNR Plots• 0.5 – 1dB gain over

zero-tree coder [Guskov, Vidimce, Sweldens, Schroder], SIGGRAPH 2000

• Similar results with other data sets

Conclusions• 0.5 – 1dB gain

– Over state-of-the-art mesh zerotree coder

• 3D surfaces much easier to compress than 2D images– Very smooth (continuous)– Worst case: sharp crease

• Future research– More appropriate distortion metrics in

normal mesh wavelet domain