A Fast LBG Codebook Training Algorithm
for Vector Quantization
Presented by 蔡進義
Motivation


A fast codebook-training algorithm based on LBG
algorithm.
To reduce the computational cost in the codebook
training processes.
2
Outline

Introduction

Previous Works

Proposed Method

Some Experiments

Discussions and Conclusions
3
Image Compression techniques

Block truncation coding

Transform coding

Hybrid coding

Vector quantization

Simple structure and low bit rate
4
VQ scheme

The VQ scheme can be divided into three parts:



Codebook generation
Encoding procedure
Decoding procedure
encoding
decoding
Codebook
Codebook
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Codebook Generation

The most important task for VQ scheme is to
design a good codebook.


LBG (Linde-Buzo-Gray) algorithm / Lloyd clustering
algorithm
The LBG algorithm is an iterative procedure.
cb0
cb1
…
cbn
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Euclidean Distance

The dimensionality of vector = k (= w*h)

An input vector x = (x1, x2, …, xk)

A codeword yi = (yi1, yi2, …, yik)

The Euclidean distance between x and yi
d ( x, yi )  x  yi
2
k
  ( x j  yij )
2
j 1
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Codebook Generation
8
VQ Codebook Training

Codebook generation
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N-1
N
Training Images
Training set
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VQ Codebook Training

Codebook generation
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N-1
N
Training set
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Initial codebook
Codebook initiation
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VQ Encoding Procedure
Image compression technique
h
w
Image
Index table
Vector Quantization Encoder
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VQ Decoding Procedure
Image compression technique
h
w
Image
Index table
Vector Quantization Decoder
12
Codebook search

To reduce the computational cost for the
segmentation procedure in the LBG algorithm,
many fast algorithms for codebook search have
been developed.



Partial Distortion Search (PDS)
Mean-distance-ordered Partial Codebook Search (MPS)
Integral Projection Mean-sorted Partial Search (IPMPS)
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Outline

Introduction

Previous Works

Proposed Method

Some Experiments

Discussions and Conclusions
15
Goal

To reduce the computation cost in finding the
closest codeword in the codebook.



PDS
MPS
IPMPS
16
Partial distortion search (PDS)

Closest codeword search
(a0, a1, a2, …, a15) input vector
s
 ( x j  yij )  d min
2
i 1
(b0, b1, b3, …, b15) codeword

If the minimal distance of each input vector could not be
found early, the PDS method can just reduce little
computation time.
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Mean-distance-ordered Partial Codebook Search Algorithm
(MPS)

The Squared Euclidean Distance (SED)
m
d E ( X , Yi )   ( x j  yij )
2
i 1

The Squared Mean Distance (SMD)
m
m
d M ( X , Yi )  ( x j   yij )
i 1

2
i 1
The minimal SED codeword is usually in the
neighborhood of the minimal SMD codeword.
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Mean-distance-ordered Partial Codebook Search Algorithm
(MPS)
m  d E ( X , Ymin )  d MM ((X
X,,Y
Yii))  m  d E ( X , Yi )
SMD
m  d E ( X , Ymin )  d M ( X , Yi )
reject
SED
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Integral Projection Mean-sorted Partial Search Algorithm
(IPMPS)


Based on multiple distortion measures with different
levels of computational complexity.
Three kinds of integral projections:
n
VPx ( j )   X (i, j ),1  j  n
i 1
n
HPx ( j )   X (i, j ),1  i  n
j 1
n
MPx  
i 1
n
 X (i, j )
j 1
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Integral Projection Mean-sorted Partial Search Algorithm
(IPMPS)

Three distortion measures:
d M ( X , Yi )  ( MPX  MPYi )
2
n  d min  d M ( X , Yi )  n  d E ( X , Yi )
2
2
n
dV ( X , Yi )   (VPX (k )  VPYi (k ))
2
k 1
n
d H ( X , Yi )   ( HPX (k )  HPYi (k ))
2
k 1
Test conditions
For each codeword Yi
d M ( X , Yi )  n  d E ( X , Yi )
2
dV ( X , Yi )  n  d E ( X , Yi )
d H ( X , Yi )  n  d E ( X , Yi )
n  d min  d M ( X , Yi )
2
n  d min  dV ( X , Yi )
n  d min  d H ( X , Yi )
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Outline

Introduction

Previous Works

Proposed Method

Some Experiments

Discussions and Conclusions
22
Generalized Integral Projection Model (GIP)

To reduce the computational cost


MPS and IPMPS
IPMPS employs the concept of integral projection
to reject further codeword in search.
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Generalized Integral Projection Model (GIP)
1.
Initially, choose one possible projection map of the pair
(p, q).

2.
3.
p segments with q pixels in each segment
For each input vector, compute the projection PX(k) of
these p segments.
The distortion measure corresponding to this projection
map is defined as:
p
d ( p ,q ) ( X , Yi )   ( PX (k )  PYi (k ))
2
k 1
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Generalized Integral Projection Model (GIP)
4.
For each codeword, the following inequality can be
easily proven true
d ( p ,q ) ( X , Yi )  q  d E ( X , Yi )
5.
The test condition for this projection map can be
constructed.
q  d min  d ( q , p ) ( X , Yi )
pair(p, q)
possible projection map
test condition
m!
( p!)( q!)
p
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Segment maps
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Fast LBG Algorithm
1.
2.
3.
Initially, select a set of test conditions by
repeatedly applying the GIP model with different
projection maps of the desired pair (p, q).
Sort the current codebook by the mean values of
the codewords.
For each vector, find the corresponding closest
codeword.
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Fast LBG Algorithm
4.
Record the index of the closest codeword for each
training vector.
5.
Update each codeword
6.
Overall averaged distortion
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Outline

Introduction

Previous Works

Proposed Method

Some Experiments

Discussions and Conclusions
29
Experiment Methods 512*512 image
LBG
PDS
MPS
30
Experiment Results
the property of
the training set
FLBG-1a
FLBG-1b
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Outline

Introduction

Previous Works

Proposed Method

Some Experiments

Discussions and Conclusions
32
Conclusions




A generalized integral projection model is developed to
produce the test conditions for the speedup of the search
process for the VQ codebook design.
To use these test conditions to eliminate the need of
calculating the squared Euclidean distance.
The property of image
By choosing proper sets of test conditions for different
training sets, a great deal of computation cost can be
reduced.
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