MSE Approximation for model-based compression of multiresolution semiregular meshes

1. MSE Approximationfor model-based compression of multiresolution semiregular meshes Frederic Payan, Marc Antonini I3S laboratory - CReATIVe Research Group Universite de Nice Sophia Antipolis - FRANCE 13th European Conference on Signal Processing, Antalya, Turkey, 2OO5

2. Semiregular mesh Wavelet coefficients EntropyCoding Q 1010… DWT Motivations Design an efficient wavelet-based lossy compression method for the geometry of semiregular meshes • DWT: discrete wavelet transform • Q: quantization

3. Summary • Background

4. I. Background Wavelet transform N-level multiresolution decomposition: • low frequency (LF) mesh: connectivity + geometry • N sets of wavelet coefficients (3D vectors): geometry … Details Details Details Details One-level decomposition

5. I. Background Compression: principle D • Compression  Optimization of the rate-distortion (RD) tradeoff R • Multiresolution datahow dispatching pertinently the bits across the subbands in order to obtain the highest quality for the reconstructed mesh? => Solution: bit allocation process

6. I. Background Proposed bit allocation • find the set of optimal quantization steps that minimizes the total distortion at one user-given target bitrate . • Distortion criterion: Mean Square Error semiregular vertices Quantized vertices Number of vertices

7. I. Background Problem statement • The distortion is measured on the vertices (Euclidean Space) • The quantization is done on the coefficient subbands (Transformed space) In order to speed the allocation process up, how expressing MSEsr directly from the quantizationerrors of each coefficient subband?

8. Summary • Background • MSE approximation for semiregular meshes

9. II. MSE approximation for semiregular meshes Previous works • The MSE of data quantized by a wavelet coder can be approximated by a weighted sumof the MSE of each subband • The weights depend on the coefficients of the synthesis filters But… shown only for data sampled on square grids and not for the mesh geometry! • Challenge: develop an MSE approximation for a data sampled on a triangular grid

10. II. MSE approximation for semiregular meshes ^ s0 s0 h0 g0 Q D D ^ M M + ^ s3 s3 g3 h3 Q D D n2 0 2 3 0 0 1 2 3 2 3 0 0 n1 0 1 1 MSE approximation for meshes • Triangular sampling: • Principle of a wavelet coder/decoder for meshes LF coset (0) HF coset 1 HF coset 2 HF coset 3

11. II. MSE approximation for semiregular meshes Method: global steps • We follow a deterministic approach • quantization error  additive noise • We exploit the polyphase notations with Polyphase notation of the synthesis filters the polyphase components

12. II. MSE approximation for semiregular meshes Solution • For a one-level decomposition • MSE approximation for a N-level decomposition with MSE of the coset i with

13. II. MSE approximation for semiregular meshes Model-based algorithm Probability density Function of the coordinate sets:Generalized Gaussian Distribution (GGD) => Model-based algorithm Complexity : 12 operations / semiregular vertex Example : 0.4 second (PIII 512 Mb Ram) => Fast allocation process

14. Summary • Background • MSE approximation for semiregular meshes • Experimental results

15. Simulations • Two versions of our algorithm are proposed: • for MAPS meshes + Lifted butterfly scheme • for Normalmeshes + Unlifted butterfly scheme • Comparison with the zerotree codersPGC (for MAPS meshes) and NMC (for Normal meshes) • Comparison criterion: PSNR based on the Hausdorff distance (computed with MESH)

16. Curves PSNR-Bitrate for our MAPS Coder

17. Curves PSNR-Bitrate for the Normal Coder

18. Summary • Background • MSE approximation for semiregular meshes • Experimental results • Conclusion

19. V. Conclusions and perspectives Conclusions Contribution: derivation of an MSE approximation for the geometry of semiregular meshes Interest: fast model-based bit allocation optimizing the quality of the quantized mesh An efficient compression method for semiregular meshes outperformingthe state of the art zerotree methods(up to 3.5 dB)

20. This is the end…. My homepage: http://www.i3s.unice.fr/~fpayan/

21. II. MSE approximation for semiregular meshes MSE approximation for meshes Proposed MSE approximation is well-adapted for the lifting schemes because the polyphase components of such transforms depend on only the prediction and update operators

22. IV. Experimental results Geometrical comparison NMC (62.86 dB) Proposed algorithm (65.35 dB) Bitrate = 0.71 bits/iv

23. III.Optimization of the Rate-Distorsion trade-off MSE of one subband i MSE relative to the tangential components MSE relative to the normal components

24. III.Optimization of the Rate-Distorsion trade-off Optimization of the Rate-Distorsion trade-off • Objective : find the quantization steps that maximize the quality of the reconstructed mesh • Scalar quantization (less complex than VQ) • 3D Coefficients => data structuring?

25. III.Optimization of the Rate-Distorsion trade-off Constraint relative to the bitrate Distortion How solving the problem? • Find the quantization steps and lambda that minimize the following lagrangian criterion: • Method: => first order conditions

26. III.Optimization of the Rate-Distorsion trade-off Solution • Need to solve(2N + 4) equations with (2N + 4) unknowns PDF of the component sets:Generalized Gaussian Distribution (GGD)=> model-based algorithm (C. Parisot, 2003)

27. III.Optimization of the Rate-Distorsion trade-off Model-based algorithm compute the variance and α for each subband compute the bitratesfor each subband λ Target bitratereached? Look-up tables new λ compute the quantizationstep of each subband