RateDistortion Optimization for Geometry Compression of Triangular Meshes

1
Rate-Distortion Optimization for Geometry
Compression of Triangular Meshes

• Frédéric Payan

PhD Thesis
Supervisor Marc Antonini
I3S laboratory - CReATIVe Research
Group Université de Nice - Sophia
Antipolis Sophia Antipolis - FRANCE
2
Motivations

• Goal
• propose an efficient compression algorithm for
  highly detailed triangular meshes
• Objectives
• High compression ratio
• Rate-Quality Optimization
• Multiresolution approach
• Fast algorithm

3
Summary

• Background
• Distortion criterion for multiresolution meshes
• Optimization of the Rate-Distorsion trade-off
• Experimental results
• Conclusions and perpectives

4
Summary
I. Background

• Background
• Triangular Meshes
• Remeshing
• Multiresolution analysis
• Compression
• Bit allocation

5
Triangular Meshes
I. Background

• 3D modeling
• Applications
• Medecine
• CAD
• Map modeling
• Games
• Cinema
• Etc.

6
Irregular meshes
I. Background

• valence different of 6
• gt 2 informations
• Geometry (vertices)
• Connectivity (edges)

4 neighbors
5 neighbors
9 neighbors
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Examples
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I. Background
Irregular meshes (2)

• Multiresolution Analysis
• Without connectivity modification gt wavelet
  transform for irregular meshes (S.Valette et
  R.Prost, 2004)
• A mesh is only one instance of the surface
  geometry gt Remeshing
• goal regular and uniform geometry sampling

gt Considered solution Semi-regular remeshing
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Summary
I. Background

• Background
• Triangular Meshes
• Remeshing
• Multiresolution analysis
• Compression
• Bit allocation

10
Semi-regular remeshing
I. Background
Coarse mesh
Original mesh
Subdivised mesh (1)
Finest semi-regular version
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Semi-regular remesher
I. Background

• MAPS (A. Lee et al. , 1998)
• Coarse mesh (geometryconnectivity)
• N sets of 3D details (geometry) gt 3 floating
  numbers
• Normal Meshes (I. Guskov et al., 2000)
• Coarse mesh (geometryconnectivity)
• N sets of 3D details (geometry) gt 1 floating
  number

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Normal Meshes
I. Background

• Known direction normal at the surface

gt More compact representation
13
Summary
I. Background

• Background
• Triangular Meshes
• Remeshing
• Multiresolution analysis
• Compression
• Bit allocation

14
Multiresolution analysis
I. Background

Details
Details
Details
Details

• Multiresolution Representation
• Low frequency (LF) mesh (geometry topology)
• N sets of wavelet coefficients (3D vectors)
  (geometry)

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Summary
I. Background

• Background
• Triangular Meshes
• Remeshing
• Multiresolution analysis
• Compression
• Bit allocation

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Compression
I. Background

• Objective reduce the information quantity
  useful for representing numerical data
• 2 approachs Lossy or lossless compression
• High compression ratii
• gt Lossy compression

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Compression scheme
I. Background
Wavelet coefficients
Semi-regular
Entropy Coding
Q
1010
Transform
Remeshing
Optimize the Rate-Distortion (RD)tradeoff
Preprocessing
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Summary
I. Background

• Background
• Triangular Meshes
• Remeshing
• Multiresolution analysis
• Compression
• Bit allocation

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Bit allocation goal
I. Background

• Optimization of the tradeoff between bitstream
  size and reconstruction quality
• minimize D(R)
• or
• minimize R(D)

D
R
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Bit allocation and meshes
I. Background

• Related Works (geometry compression)
• Zerotree coding
• PGC Progressive Geometry Compression (A.
  Khodakovsky et al., 2OOO)
• NMC
• Normal Mesh Compression ( A. Khodakovsky et I.
  Guskov, 2002).
• gt Stop coding when bitstream given size is
  reached.
• Estimation-quantization (EQ) coding
• MSEC Geometry Compression of Normal Meshes
  Using Rate-Distortion Algorithms (S. Lavu et al.,
  2003)
• gt Local RD optimization.

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Proposed bit allocation
I. Background

• Low computational complexity
• Improve the quantization process
• Maximize the quality of the reconstructed
  meshaccording to a given target bitrate

gt Which distortion criterion for evaluating the
losses?
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Summary

• Background
• Distortion criterion for multiresolution meshes
• Optimization of the Rate-Distorsion trade-off
• Experimental results
• Conclusions and perpectives

23
Coding/Decoding
II. Distortion criterion for multiresolution
meshes
Semi-regular
Entropy coding
Q
1010
Transform
Remeshing
Target bitrate or distortion
Preprocessing
Entropy Decoding
Q
Quantized semi-regular
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Considered distorsion criterion
II. Distortion criterion for multiresolution
meshes

• MSE due to quantization of the semi-regular mesh

Number of vertices
semi-regular vertices
quantized semi-regular vertices
MSE for one subband
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Related works
II. Distortion criterion for multiresolution
meshes

• K.Park and R.Haddad (1995)
• M-channel scheme
• quantization model noise plus gain
• B.Usevitch (1996)
• quantization model additive noise
• N decomposition levels
• Sampled on square grids

Filter bank
Problem - non adapted for lifting scheme !
- usable for any sampling grid ?
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Lifting scheme for meshes
II. Distortion criterion for multiresolution
meshes

• 3 prédiction operators P
• gt wavelet coefficients
• 3 update operators U
• gt LF mesh
• Triangular grid gt 4 channels

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Triangulaire sampling
II. Distortion criterion for multiresolution
meshes

• 1 triangular grid gt 4 cosets

LF subband (0)
HF subband 1
HF subband 2
HF subband 3
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4-channel lifting scheme analysis
II. Distortion criterion for multiresolution
meshes
LF
-P1
U1
HF 1
split
-P2
U2

HF 2
Semi-regular mesh
-P3
U3
HF 3

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4-channel lifting scheme synthesis
II. Distortion criterion for multiresolution
meshes
LF
P
-U
HF 1
P
-U
Merge

HF 2
Semi-regular mesh
P
-U

HF 3
gt Derivation of the MSE of the quantized
meshaccording to the quantization error of each
4 subband
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II. Distortion criterion for multiresolution
meshes
Proposed Method

• Input signal
• Quantization error model  additive noise 
• S is one realization of a stationar and ergodic
  random process gt deterministic quantity
• gt MSE of the input signal

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Proposed Method Hypothesis
II. Distortion criterion for multiresolution
meshes

• Uncorrelated error in each subband
• Subband errors mutually uncorrelated

Synthesis filter energy
Quantization error energy
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Proposed Method principle
II. Distortion criterion for multiresolution
meshes

• Synthesis filter energy
• Polyphase components of the filters
• Cauchy theorem
• Quantization error energy
• Uncorrelated error in each subband

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Proposed Method solution
II. Distortion criterion for multiresolution
meshes
For 1 decomposition level
MSE of the subband i
Weights relative to the non-orthogonal filters
with
Polyphase component
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polyphase representation
II. Distortion criterion for multiresolution
meshes

• Lifting scheme
• gt Polyphase components depend on only the
  prediction and update opérators

New formulation gt can be applied easily to
lifting scheme
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Proposed Method solution
II. Distortion criterion for multiresolution
meshes
For N decomposition levels
avec
et
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Outline
II. Distortion criterion for multiresolution
meshes

• This formulation can be applied to lifting scheme
• Global formulation of the weights for any
• Grid and related subsampling
• number of channels M
• Number of decomposition levels N

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Experimental Results
II. Distortion criterion for multiresolution
meshes
gt PSNR Gain up to 3.5 dB
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II. Distortion criterion for multiresolution
meshes
Visual impact
Original
Without the weights
With the weights
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Coding/Decoding
II. Distortion criterion for multiresolution
meshes
Semi-regular
Entropy coding
Q
1010
Transform
Remeshing
Target bitrate or distortion
Preprocessing
Entropy Decoding
Q
Quantized semi-regular
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MSE and irregular mesh
II. Distortion criterion for multiresolution
meshes

• Quality of the reconstructed mesh
• Reference irregular mesh
• Used metric
• geometrical distance between two surfaces
• the surface-to-surface distance (s2s)

gt Is the MSE suitable to control the quality?
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Quality of the reconstructed mesh
II. Distortion criterion for multiresolution
meshes

• Forward distance
• distance between one point and one surface

Quantized mesh (semi-regular)
Input mesh (irregular)
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Simplifying approximations
II. Distortion criterion for multiresolution
meshes

• Normal meshes
• gt infinitesimal remeshing error
• gt uniform and regular geometry sampling
• Highly detailed meshes
• gt densely sampled geometry

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Hypothesis asymptotical case
II. Distortion criterion for multiresolution
meshes
gt

• Preservation of the LF subbands
• gt normal orientations slightly modified gt
  errors lie in the normal direction (normal
  meshes)

n
n
n
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Proposed heuristic
II. Distortion criterion for multiresolution
meshes
Asymptotical case normal meshes
gt MSE suitable criterion to controlthe
quality of the reconstructed mesh
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Summary

• Background
• Distortion criterion for multiresolution meshes
• Optimization of the Rate-Distorsion trade-off
• Experimental results
• Conclusions and perpectives

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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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Local frames
III.Optimization of the Rate-Distorsion trade-off

• Normal at the surface z-axis of the local frame
• gt Coefficient
• Tangential components (x and y-coordinates)
• Normal components (z-coordinates)

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Histogram of the polar angle
III.Optimization of the Rate-Distorsion trade-off

• Local frame

?
0
90
180
gt Components treated separately (2 scalar
subbands)
gt Most of coefficients have only normal
components
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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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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
?
Complexity 12 operations / semi-regular Example
0.4 second (PIII 512 Mb Ram) gt Fast process.
Target bitratereached?
Look-up tables
new ?
compute the quantizationstep of each subband
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Summary

• Background
• Distortion criterion for multiresolution meshes
• Optimization of the Rate-Distorsion trade-off
• Experimental results
• Conclusions and perpectives

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Compression scheme
IV. Experimental results
UnliftedButterfly
SQ
3D-CbAC
Normalmeshes
Context-based Bitplane Arithmetic Coder
(EBCOT-like)
Preprocessing
Touma-Gotsman coder
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Visual results
IV. Experimental results
CR 226 0.82 bits/iv
CR 9000.2 bits/iv
Input mesh(irregular)
CR 832.2 bits/iv
Compression ratio
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Comparison
IV. Experimental results

• Quality criterion
• State-of-the-art methods
• NMC (Normal meshes Butterfly NL zerotree)
• EQMC (Normal meshes Butterfly NL EQ)
• PGC (MAPS Loop)

Bounding box diagonal
s2s between the irregular input mesh and the
quantized semi-regular one
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PSNR-bitrate curve Rabbit
IV. Experimental results
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PSNR-bitrate curve Feline
IV. Experimental results
gt PSNR Gain up to 7.5 dB
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PSNR-bitrate curve Horse
IV. Experimental results
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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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Summary

• Background
• Distortion criterion for multiresolution meshes
• Optimization of the Rate-Distorsion trade-off
• Experimental results
• Conclusions and perpectives

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Conclusions
V. Conclusions and perspectives

• New shape compression method
• Contributions
• Weighted MSE suitable distortion criterion
• Original formulation of the weights (suitable in
  case of lifting scheme)
• Bit alllocation of low computational complexity
  that optimizes the quality of a quantized mesh.
• An original Context-based Bitplane Arithmetic
  Coder

gt Better results thanthe state-of-the-art
methods.
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Perspectives
V. Conclusions and perspectives

• Take into account some visual properties the
  human eye appreciates (local curvature, volume,
  smoothness) Reference Z.Karni and C.Gotsman,
  2000
• Algorithm for huge meshes on the flow
  compressionReference A. Elkefi et al., 2004

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End