A Fast LBG Codebook Training Algorithm A Fast LBG Codebook Training Algorithm for Vector Quantizationfor Vector Quantization

Presented by Presented by 蔡進義蔡進義

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MotivationMotivation A fast A fast codebook-training algorithmcodebook-training algorithm based on based on

LBG algorithm.LBG algorithm.

To reduce the computational cost in the To reduce the computational cost in the codebook training processes.codebook training processes.

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OutlineOutline IntroductionIntroduction

Previous WorksPrevious Works

Proposed MethodProposed Method

Some ExperimentsSome Experiments

Discussions and ConclusionsDiscussions and Conclusions

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Image Compression techniquesImage Compression techniques Block truncation codingBlock truncation coding

Transform codingTransform coding

Hybrid codingHybrid coding

Vector quantizationVector quantization Simple structure and low bit rateSimple structure and low bit rate

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VQ schemeVQ scheme The VQ scheme can be divided into three parts:The VQ scheme can be divided into three parts:

Codebook generationCodebook generation Encoding procedureEncoding procedure Decoding procedureDecoding procedure

encoding decoding

CodebookCodebook CodebookCodebook

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Codebook GenerationCodebook Generation The most important task for VQ scheme is to The most important task for VQ scheme is to

design a good codebook.design a good codebook. LBGLBG (Linde-Buzo-Gray) algorithm / (Linde-Buzo-Gray) algorithm / LloydLloyd

clustering algorithmclustering algorithm

The LBG algorithm is an iterative procedure.The LBG algorithm is an iterative procedure.cb0

cb1

cbn

…

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Euclidean DistanceEuclidean Distance The dimensionality of vector = k (= w*h)The dimensionality of vector = k (= w*h)

An input vector x = (xAn input vector x = (x11, x, x22, …, x, …, xkk))

A codeword yA codeword yii = (y = (yi1i1, y, yi2i2, …, y, …, yikik))

The Euclidean distance between x and yThe Euclidean distance between x and yii

k

jijjii yxyxyxd

1

22)(),(

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Codebook GenerationCodebook Generation

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Codebook generationCodebook generation

Training Images Training set

.

.

.

01...

N-1N

VQ Codebook TrainingVQ Codebook Training

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Codebook generationCodebook generation

Training set

Codebook initiation

Initial codebook

.

.

.

01...

254255

01...

N-1N

VQ Codebook TrainingVQ Codebook Training

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ImageIndex table

Vector Quantization Encoder

wh

Image compression technique

VQ Encoding ProcedureVQ Encoding Procedure

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ImageIndex table

Vector Quantization Decoder

wh

Image compression technique

VQ Decoding ProcedureVQ Decoding Procedure

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Codebook searchCodebook search To reduce the computational cost for the To reduce the computational cost for the

segmentation procedure in the LBG segmentation procedure in the LBG algorithm, many fast algorithms for algorithm, many fast algorithms for codebook searchcodebook search have been developed. have been developed. Partial Distortion Search (PDS)Partial Distortion Search (PDS) Mean-distance-ordered Partial Codebook Search Mean-distance-ordered Partial Codebook Search

(MPS)(MPS) Integral Projection Mean-sorted Partial Search Integral Projection Mean-sorted Partial Search

(IPMPS)(IPMPS)

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OutlineOutline IntroductionIntroduction

Previous WorksPrevious Works

Proposed MethodProposed Method

Some ExperimentsSome Experiments

Discussions and ConclusionsDiscussions and Conclusions

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GoalGoal To reduce the computation cost in finding To reduce the computation cost in finding

the closest codeword in the codebook.the closest codeword in the codebook. PDSPDS MPSMPS IPMPSIPMPS

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PartialPartial distortion search (PDS) distortion search (PDS)

Closest codeword searchClosest codeword search

If If the minimal distancethe minimal distance of each input vector could of each input vector could not be found early, the PDS method can just reduce not be found early, the PDS method can just reduce little computation time.little computation time.

s

iijj dyx

1min

2)(

(a0, a1, a2, …, a15) input vector

(b0, b1, b3, …, b15) codeword

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Mean-distance-Mean-distance-ordered ordered Partial Codebook Search Algorithm Partial Codebook Search Algorithm (MPS)(MPS)

The Squared Euclidean Distance (SED)The Squared Euclidean Distance (SED)

The Squared Mean Distance (SMD)The Squared Mean Distance (SMD)

The minimal SED codeword is usually in the The minimal SED codeword is usually in the neighborhood of the minimal SMD codeword.neighborhood of the minimal SMD codeword.

m

iijjiE yxYXd

1

2)(),(

m

i

m

iijjiM yxYXd

1

2

1

)(),(

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Mean-distance-Mean-distance-ordered ordered Partial Codebook Search Algorithm Partial Codebook Search Algorithm (MPS)(MPS)

),(),( min iME YXdYXdm

SMD

SED

reject

),(),( iEiM YXdmYXd ),(),( min iME YXdYXdm

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Integral ProjectionIntegral Projection Mean-sorted Partial Search Algorithm Mean-sorted Partial Search Algorithm (IPMPS)(IPMPS)

Based on Based on multiple distortion measuresmultiple distortion measures with with different levels of computational complexity.different levels of computational complexity.

Three kinds of integral projections:Three kinds of integral projections:

nijiXjHPn

jx

1),,()(1

n

j

n

ix jiXMP

11

),(

njjiXjVPn

ix

1),,()(1

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Integral Projection Mean-sorted Partial Search Algorithm Integral Projection Mean-sorted Partial Search Algorithm (IPMPS)(IPMPS)

Three distortion measures:Three distortion measures:2)(),(

iYXiM MPMPYXd

n

kYXiH kHPkHPYXdi

1

2))()((),(

n

kYXiV kVPkVPYXdi

1

2))()((),(

),(),( 2iEiM YXdnYXd

),(),( iEiV YXdnYXd

),(),( iEiH YXdnYXd

For each codeword Yi ),(min2

iM YXddn

),(min iV YXddn

),(min iH YXddn

),(),( 2min

2iEiM YXdnYXddn

Test conditions

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OutlineOutline IntroductionIntroduction

Previous WorksPrevious Works

Proposed MethodProposed Method

Some ExperimentsSome Experiments

Discussions and ConclusionsDiscussions and Conclusions

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Generalized Integral Projection Model (GIP)Generalized Integral Projection Model (GIP)

To reduce the computational costTo reduce the computational cost MPS and IPMPSMPS and IPMPS

IPMPS employs the concept of integral IPMPS employs the concept of integral projection to reject further codeword in projection to reject further codeword in search.search.

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Generalized Integral Projection Model (GIP)Generalized Integral Projection Model (GIP)

1.1. Initially, choose one possible projection map of Initially, choose one possible projection map of the pair (p, q). the pair (p, q).

p segments with q pixels in each segmentp segments with q pixels in each segment

2.2. For each input vector, compute the For each input vector, compute the projection projection PPXX(k) of these p segments.(k) of these p segments.

3.3. The The distortion measuredistortion measure corresponding to this corresponding to this projection map is defined as:projection map is defined as:

p

kYXiqp kPkPYXdi

1

2),( ))()((),(

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Generalized Integral Projection Model (GIP)Generalized Integral Projection Model (GIP)

4.4. For each codeword, the following inequality can For each codeword, the following inequality can be easily proven truebe easily proven true

5.5. The test condition for this projection map can be The test condition for this projection map can be constructed.constructed.

),(),(),( iEiqp YXdqYXd

),(),(min ipq YXddq

pair(p, q) test conditionpossible projection map

pqp

m

)!)(!(

!

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Segment mapsSegment maps

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Fast LBG AlgorithmFast LBG Algorithm1.1. Initially, select Initially, select a set of test conditionsa set of test conditions by by

repeatedly applying the repeatedly applying the GIP modelGIP model with with different projection maps of the desired pair different projection maps of the desired pair (p, q).(p, q).

2.2. Sort the current codebook by the mean Sort the current codebook by the mean values of the codewords.values of the codewords.

3.3. For each vector, find the corresponding For each vector, find the corresponding closest codeword.closest codeword.

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Fast LBG AlgorithmFast LBG Algorithm4.4. Record the index of the closest codeword Record the index of the closest codeword

for each training vector.for each training vector.

5.5. Update each codewordUpdate each codeword

6.6. Overall averaged distortionOverall averaged distortion

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OutlineOutline IntroductionIntroduction

Previous WorksPrevious Works

Proposed MethodProposed Method

Some ExperimentsSome Experiments

Discussions and ConclusionsDiscussions and Conclusions

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Experiment Methods Experiment Methods 512*512 image512*512 image

LBGPDSMPS

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Experiment ResultsExperiment Results

the property of the

training set

FLBG-1aFLBG-1b

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OutlineOutline IntroductionIntroduction

Previous WorksPrevious Works

Proposed MethodProposed Method

Some ExperimentsSome Experiments

Discussions and ConclusionsDiscussions and Conclusions

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ConclusionsConclusions A A generalized integral projectiongeneralized integral projection model is model is

developed to produce the test conditions for the developed to produce the test conditions for the speedup of the search process for the VQ codebook speedup of the search process for the VQ codebook design.design.

To use these To use these test conditionstest conditions to eliminate the need of to eliminate the need of calculating the squared Euclidean distance.calculating the squared Euclidean distance.

The property of imageThe property of image

By choosing proper sets of test conditions for By choosing proper sets of test conditions for different training sets, a great deal of computation different training sets, a great deal of computation cost can be reduced.cost can be reduced.