Vector Quantization

2

outline

Introduction Two measurement : quality of image and bit rat

e Advantages of Vector Quantization over Scalar

Quantization The Linde-Buzo-Gray Algorithm Cell-split algorithm Standard VQ encoding Summary

3

Two measurement Quality of image

M

iji

N

jij

M

i

N

jji

xx

x

SNR

1

2^

1

1 1

^

)(

MSEPSNR

2

10

255log10

M

i

N

jjiij xx

NMMSE

1 1

2^

)()*

1(

4

Two measurement (conti)

The amount of compression will be described in terms of the rate, which will be measured in bits per sample. Suppose we have a codebook of size k, and the input vector is of dimension L. We need to use bits to specify which of the code-vectors was selected. The rate for an L-dimensional vector quantizer with a codebook of size K is .

L

k2log

k2log

5

VQ IntroductionEncoder Decoder Reconstruction

Find closest code-vector

Table lookup

Source output

UnblockGroup into vectors

6

Vector Quantization encoding

VQ was first proposed by Gray in 1984. First, construct codebook which is composed of

codevector. For one vector being encoding, find the nearest

vector in codebook. (determined by Euclidean distance)

Replace the vector by the index in codebook. When decoding, find the vector corresponding b

y the index in codebook.

7

Two important issue

Codevectors in codebook need representative. It affect the quality of de-compressed image a lot. So, how to build a good codebook is an important issue.

Euclidean distance is time-consuming. How to fasten to search for the nearest vector is also an important issue.

8

LBG Algorithm

Proposed by Linde, Buzo, Gray The basic idea is to divide a group of vecto

r. To find a most representative vector from one group. Then gather the vectors to form a codebook.

9

LBG Algorithm1. Divide image into blocks. Then we can view on

e block as k-dimension vector. Ex: block: 4x4 , consider 512x512 image, which can be divided into blocks. Each block can be viewed 16 dimension vector.

2. Arbitrarily choose initial codebook.3. Set these initial codebook as centroids. Other

vectors are grouped. Vectors are in the same group when they have the same nearest centroid.

4. Again, to find new centroids for every group. Get a new codebooks. Repeat 2,3 steps until the centroids of every group converge.

1638444

512512

10

Standard VQ encoding For one vector to be encoding, compute the Eucl

idean distance with every codevectors in codebook, and find the codevector with smallest Euclidean distance.To encode Codebook i codewords

),...,,( 21 kxxxx },...,,{ 21 mYYYY

),...,,( 21 ikiii YYYY

2

11

2

1

)(min)(

k

vlvv

m

l

k

vcvv YxYx

11

Cell split method(細胞分裂法 )

),..,,( 111 xxxX

),..,,( 111 yyyY ),..,,( 111 zzzZ ii xy ii xy

For Y, after grouping, find new centorid

For Z, after grouping, find new centorid

ii

k

yp

pppP

'

),..,,( 21

ii

k

yp

qqqQ

'

),..,,( 21

ii

k

Zr

rrrR

'

),..,,( 21

ii

k

Zs

sssS

'

),..,,( 21

12

algorithm

1. Divide image into blocks. Choose a block (k-dimension) X=(x1, x1,…,x1) as initial vector.

2. Spit X vector into two vector Y=(y1, y1,…,y1) and Z=(z1, z1,…,z1)yi=xi- ,zi=xi+

3. Y and Z are centroids. For all blocks, find the nearest centroid. Re-compute the centroid of blocks and get new centroid Y’ and Z’.

4. Recursively, do Y’ and Z’. Repeat 2 ,3 step. Until find enough number of codevector.

13

14

experienceImage512*512

LBG codebook

Cell spit method =6

Cell spit method =8

Cell spit method =10

PSNR PSNR Gain PSNR Gain PSNR Gain

Girl

Gold

Lena

Pepper

Toys

Tiffancy

average

31.167

29.337

29.080

29.942

28.618

28.268

29.402

31.872 0.705

30.203 0.866

29.927 0.847

30.905 0.963

30.061 1.443

30.669 2.401

30.610 1.204

30.495 0.800

29.456 1.142

28.957 0.621

29.789 1.068

27.861 0.539

30.012 2.338

29.428 1.085

30.562 0.867

29.529 1.218

29.045 0.712

29.828 1.107

28.205 0.873

30.171 2.197

29.557 1.214