MSE Approximation for model-based compression of multiresolution semiregular meshes Frederic Payan,...
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![Page 1: MSE Approximation for model-based compression of multiresolution semiregular meshes Frederic Payan, Marc Antonini I3S laboratory - CReATIVe Research Group.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d575503460f94a36110/html5/thumbnails/1.jpg)
MSE ApproximationMSE Approximationfor model-based compression of for model-based compression of
multiresolution semiregular meshesmultiresolution semiregular meshes
Frederic Payan, Marc AntoniniFrederic Payan, Marc Antonini
I3S laboratory - CReATIVe Research GroupUniversite de Nice Sophia Antipolis - FRANCE
13th European Conference on Signal Processing, Antalya, Turkey, 2OO5
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DWT: discrete wavelet transformDWT: discrete wavelet transform
Q: quantizationQ: quantization
MotivationsMotivationsDesign an efficient wavelet-based lossy compression Design an efficient wavelet-based lossy compression method for the geometry of semiregular meshesmethod for the geometry of semiregular meshes
Q Entropy Coding
DWT 1010…
Wavelet coefficientsSemiregular mesh
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SummarySummary
BackgroundBackground
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Wavelet transformWavelet transform
N-level multiresolution decompositionN-level multiresolution decomposition:: low frequencylow frequency (LF) mesh: (LF) mesh: connectivity + geometryconnectivity + geometry N sets of wavelet coefficients (N sets of wavelet coefficients (3D vectors3D vectors): ): geometrygeometry
…
Details Details Details Details
I. Background
One-level decomposition
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Compression: principleCompression: principle
Compression Compression Optimization of the rate-Optimization of the rate-distortion (RD) tradeoffdistortion (RD) tradeoff
I. Background
R
D
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
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Proposed bit allocationProposed bit allocationfind the set of find the set of optimal quantization stepsoptimal quantization steps that that minimizes the total distortionminimizes the total distortion at one at one user-given target bitrate .user-given target bitrate .
Distortion criterion: Distortion criterion: Mean Square ErrorMean Square Error
targetconstraintwith minimize
RqRqD
T
T
*q
1#
0
2
2ˆ
#
1 SR
jjjSRT vv
SRMSED
semiregular semiregular verticesvertices
Quantized Quantized verticesverticesNumber of verticesNumber of vertices
I. Background
TDtargetR
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Problem statementProblem statement
I. Background
In order to speed the allocation process up, howexpressing MSEsr directly from the quantizationerrors of each coefficient subband?
1.1. The distortion is measured on The distortion is measured on the vertices the vertices (Euclidean Space)(Euclidean Space)
2.2. The quantization is done on the The quantization is done on the coefficient coefficient subbands (Transformed space)subbands (Transformed space)
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SummarySummary
BackgroundBackground
MSE approximation for semiregular meshesMSE approximation for semiregular meshes
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Previous worksPrevious works
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
II. MSE approximation for semiregular meshes
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Triangular sampling:Triangular sampling:
Principle of a wavelet coder/decoder for meshesPrinciple of a wavelet coder/decoder for meshes
MSE approximation for meshesMSE approximation for meshes
II. MSE approximation for semiregular meshes
M
D
Q D
D
+M^
Q
Dh3
h0
g3
g0
s0
s3
s0
s3
^
^
0 0 0
00
0
LF coset (0)
n1
n2
HF coset 1
1 1
1
HF coset 22
2
2
HF coset 3
3
3
3
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Method: global stepsMethod: global steps
II. MSE approximation for semiregular meshes
We follow aWe follow a deterministic approach deterministic approach
quantization error quantization error additive noiseadditive noise
We exploitWe exploit the polyphase notations the polyphase notations
3
2
1
0
s
s
s
s
GM
3,32,31,30,3
3,22,21,20,2
3,12,11,10,1
3,02,01,00,0
GGGG
GGGG
GGGG
GGGG
G
Polyphase notation of the synthesis filters
withwith
the polyphase components
iii ss ˆ
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SolutionSolution
II. MSE approximation for semiregular meshes
withwith
3
0jiiSR MSEwMSE
3
0
2,#
#
jji
ii
d
GSR
SRw
Zk
k
MSE of the coset i
withwith
1
0
3
1,,0,10,1
N
i lliliNNSR MSEWMSEWMSE
lili
li wwSR
SRW 0
,, #
#
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Model-based algorithmModel-based algorithm
Probability density Function of the coordinate sets:Generalized Gaussian Distribution (GGD)
=> Model-based algorithm
Complexity : 12 operations / semiregular vertexExample : 0.4 second (PIII 512 Mb Ram)
=> Fast allocation process
II. MSE approximation for semiregular meshes
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SummarySummary
BackgroundBackground
MSE approximation for semiregular meshesMSE approximation for semiregular meshes
Experimental resultsExperimental results
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SimulationsSimulations
Two versions of our algorithm are proposed:Two versions of our algorithm are proposed:1.1. for for MAPSMAPS meshes meshes + + Lifted butterflyLifted butterfly scheme scheme
2.2. for for NormalNormal meshesmeshes + + Unlifted butterflyUnlifted butterfly scheme scheme
Comparison with the zerotree codersComparison with the zerotree coders PGCPGC (for (for MAPS meshesMAPS meshes) and ) and NMCNMC (for (for Normal meshes)Normal meshes)
Comparison criterion: PSNR based on the Comparison criterion: PSNR based on the Hausdorff Hausdorff distancedistance (computed with (computed with MESHMESH))
sd
peakPSNR 10log20
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Curves PSNR-Bitrate for our Curves PSNR-Bitrate for our MAPS MAPS CoderCoder
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Curves PSNR-Bitrate for the Curves PSNR-Bitrate for the Normal Normal CoderCoder
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SummarySummary
BackgroundBackground
MSE approximation for semiregular meshesMSE approximation for semiregular meshes
Experimental resultsExperimental results
ConclusionConclusion
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ConclusionsConclusions
Contribution: Contribution: derivation of an derivation of an MSE approximation for the geometry of MSE approximation for the geometry of semiregular meshessemiregular meshes
Interest: Interest: fast model-based bit allocationfast model-based bit allocation optimizing the quality of optimizing the quality of the quantized meshthe quantized mesh
V. Conclusions and perspectives
An efficient compression method An efficient compression method for semiregular meshesfor semiregular meshes outperforming outperformingthe state of the art zerotree methodsthe state of the art zerotree methods
(up to 3.5 dB)(up to 3.5 dB)
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This is the end….This is the end….
My homepage: My homepage:
http://www.i3s.unice.fr/~fpayan/http://www.i3s.unice.fr/~fpayan/
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MSE approximation for meshesMSE approximation for meshes
Proposed MSE approximation is well-adapted for the lifting schemes because the polyphase because the polyphase components of such transforms components of such transforms depend on only depend on only the the prediction and update prediction and update operatorsoperators
3323133
3222122
3121111
321
1
1
1
1
pupupuu
pupupuu
pupupuu
ppp
G
3,32,31,30,3
3,22,21,20,2
3,12,11,10,1
3,02,01,00,0
GGGG
GGGG
GGGG
GGGG
G
II. MSE approximation for semiregular meshes
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Geometrical comparisonGeometrical comparison
NMC NMC (62.86 dB)(62.86 dB)
Proposed algorithmProposed algorithm ( (65.35 dB65.35 dB))
Bitrate = 0.71 bits/iv
IV. Experimental resultsExperimental results
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MSE of one subband MSE of one subband ii
2,1,, iSRiSRJj
jii MSEMSEMSEMSEi
MSE relative to the tangential components
MSE relative to the normal components
III.Optimization of the Rate-Distorsion trade-offIII.Optimization of the Rate-Distorsion trade-off
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Optimization of the Rate-Distorsion Optimization of the Rate-Distorsion trade-offtrade-off
Objective : Objective :
find the quantization steps that find the quantization steps that maximize the quality of maximize the quality of the reconstructed meshthe reconstructed mesh
Scalar quantization Scalar quantization (less complex than VQ)(less complex than VQ)
3D Coefficients => 3D Coefficients => data structuring?data structuring?
targetconstraintwith
minimize
RR
MSE
T
SR
III.Optimization of the Rate-Distorsion trade-offIII.Optimization of the Rate-Distorsion trade-off
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How solving the problem?How solving the problem?
Find the Find the quantization steps and lambdaquantization steps and lambda that minimize the following lagrangian criterion:that minimize the following lagrangian criterion:
Method:Method:=> => first order conditionsfirst order conditions
cible,
0,,,,
0, RqRaqMSEWqJ ji
N
i Jjjiji
JjjijiSR
N
iiji
ii
Distortion Constraint relative to the bitrate
III.Optimization of the Rate-Distorsion trade-offIII.Optimization of the Rate-Distorsion trade-off
![Page 26: MSE Approximation for model-based compression of multiresolution semiregular meshes Frederic Payan, Marc Antonini I3S laboratory - CReATIVe Research Group.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d575503460f94a36110/html5/thumbnails/26.jpg)
SolutionSolution Need to solveNeed to solve
(2N + 4) equations with (2N + 4) unknowns(2N + 4) equations with (2N + 4) unknowns
target0
,,,
,
,,
,,
RqRa
W
a
qR
qMSE
N
i Jjjijiji
i
ji
jiji
jijiSR
i
III.Optimization of the Rate-Distorsion trade-offIII.Optimization of the Rate-Distorsion trade-off
jiq ,
Tji Rq ,
PDF of the component sets:Generalized Gaussian Distribution (GGD)
=> model-based algorithm (C. Parisot, 2003)
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Model-based algorithmModel-based algorithm
compute the variance and compute the variance and αα for each subband for each subband
compute the bitratescompute the bitratesfor each subbandfor each subbandλλ
Target bitrateTarget bitratereached?reached?new new λλ
compute the quantizationcompute the quantizationstep of each subbandstep of each subband
jiR ,
jiq ,
III.Optimization of the Rate-Distorsion trade-offIII.Optimization of the Rate-Distorsion trade-off
Look-up tables