MSE%20Approximation%20for%20model-based%20compression%20of%20multiresolution%20semiregular%20meshes

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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
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Motivations

• Design an efficient wavelet-based lossy
  compression method for the geometry of
  semiregular meshes

• DWT discrete wavelet transform
• Q quantization

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Summary

• Background

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Wavelet transform
I. Background

Details
Details
Details
Details
One-level decomposition

• N-level multiresolution decomposition
• low frequency (LF) mesh connectivity geometry
• N sets of wavelet coefficients (3D vectors)
  geometry

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Compression principle
I. Background
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?
• gt Solution bit allocation process

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Proposed bit allocation
I. Background

• 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
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Problem statement
I. Background

1. The distortion is measured on the vertices
   (Euclidean Space)
2. 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?
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Summary

• Background
• MSE approximation for semiregular meshes

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Previous works
II. MSE approximation for semiregular meshes

• The MSE of data quantized by a wavelet coder can
  be approximated by a weighted sum of 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
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II. MSE approximation for semiregular meshes
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
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Method global steps
II. MSE approximation for semiregular meshes

• We follow a deterministic approach
• quantization error ? additive noise
• We exploit the polyphase notations

with
Polyphase notation of the synthesis filters
the polyphase components
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Solution
II. MSE approximation for semiregular meshes

• For a one-level decomposition
• MSE approximation for a N-level decomposition

with
MSE of the coset i
with
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Model-based algorithm
II. MSE approximation for semiregular meshes

• Probability density Function of the coordinate
  setsGeneralized Gaussian Distribution (GGD)

• gt Model-based algorithm
• Complexity 12 operations / semiregular vertex
• Example 0.4 second (PIII 512 Mb Ram)
• gt Fast allocation process

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Summary

• Background
• MSE approximation for semiregular meshes
• Experimental results

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Simulations

• Two versions of our algorithm are proposed
• for MAPS meshes Lifted butterfly scheme
• for Normal meshes Unlifted butterfly scheme
• Comparison with the zerotree coders PGC (for MAPS
  meshes) and NMC (for Normal meshes)
• Comparison criterion PSNR based on the Hausdorff
  distance (computed with MESH)

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Curves PSNR-Bitrate for our MAPS Coder
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Curves PSNR-Bitrate for the Normal Coder
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Summary

• Background
• MSE approximation for semiregular meshes
• Experimental results
• Conclusion

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Conclusions
V. Conclusions and perspectives

• 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)
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This is the end.

• My homepage
• http//www.i3s.unice.fr/fpayan/

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MSE approximation for meshes
II. MSE approximation for semiregular 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

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Geometrical comparison
IV. Experimental results
NMC (62.86 dB)
Proposed algorithm (65.35 dB)
Bitrate 0.71 bits/iv
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MSE of one subband i
III.Optimization of the Rate-Distorsion trade-off
MSE relative to the tangential components
MSE relative to the normal components
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Optimization of the Rate-Distorsion trade-off
III.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 gt data structuring?

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How solving the problem?
III.Optimization of the Rate-Distorsion trade-off

• Find the quantization steps and lambda that
  minimize the following lagrangian criterion
• Method
• gt first order conditions

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Solution
III.Optimization of the Rate-Distorsion trade-off

• Need to solve (2N 4) equations with (2N 4)
  unknowns

PDF of the component setsGeneralized Gaussian
Distribution (GGD)gt model-based algorithm (C.
Parisot, 2003)
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Model-based algorithm
III.Optimization of the Rate-Distorsion trade-off
compute the variance and a for each subband
compute the bitratesfor each subband
?
Target bitratereached?
Look-up tables
new ?
compute the quantizationstep of each subband