Estimation-Quantization Geometry Coding using Normal Meshes Sridhar Lavu Hyeokho Choi Richard Baraniuk Rice University

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

3D Mesh Representation Mesh representation Problem Goal 3D scan Point clouds Polygon mesh Problem Massive data size Michelangelo’s statue of David: > billion triangles Connectivity 0 1 2 2 3 1 0 1 4 1 2 4 2 3 4 3 0 4 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 Goal Compression

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 Quantized coefficients Estimate sigmai2 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 MSE Metro 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