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

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

SummarySummary

BackgroundBackground

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

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

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

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)

SummarySummary

BackgroundBackground

MSE approximation for semiregular meshesMSE approximation for semiregular meshes

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

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

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

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

,, #

#

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

SummarySummary

BackgroundBackground

MSE approximation for semiregular meshesMSE approximation for semiregular meshes

Experimental resultsExperimental results

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

Curves PSNR-Bitrate for our Curves PSNR-Bitrate for our MAPS MAPS CoderCoder

Curves PSNR-Bitrate for the Curves PSNR-Bitrate for the Normal Normal CoderCoder

SummarySummary

BackgroundBackground

MSE approximation for semiregular meshesMSE approximation for semiregular meshes

Experimental resultsExperimental results

ConclusionConclusion

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)

This is the end….This is the end….

My homepage: My homepage:

http://www.i3s.unice.fr/~fpayan/http://www.i3s.unice.fr/~fpayan/

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

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

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

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

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

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)

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