Slide nén dữ liệu transform coding PJEG

Idea of Transform Coding
1) Transform the input pixels x0,x2,...,xN-1 into
coefficients c0,c1,...,cN-1 (real values):

CSE 592
Data Compression

– The coefficients should have the property that most of
them are near zero.
– Therefore most of the “energy” is compacted into a few
coefficients.

Fall 2006

Lecture 14:
Transform Coding
JPEG

2) Scalar quantize the coefficient:
– This is bit allocation!
– Important coefficients should have more quantization
levels (== represented with more accuracy).

3) Entropy encode the quantized values.
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Decoding

Block Diagram of Transform Coding

1) Entropy decode the quantized values.

Encoder
input

x

2) Compute approximate coefficients c’0,c’1,...,c’N-1
from the quantized values.
3) Inverse transform c’0,c’1,...,c’N-1 to x’0,x’1,...,x’N-1
which is (we hope) a good approximation of the
original x0,x1,...,xN-1.

coefficients

c

transform

symbols

entropy
decoding

s

symbols

quantization
bit allocation

bit stream

s

entropy
coding

output

coefficients

decode
symbols

c’

b

inverse
transform

x’

Decoder
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Mathematical Properties of Transforms

Why Coefficients?
A c=x
T

 a 00 L

 M
 a 0,N -1 L

• Linear Transformation:
– Defined by a real N x N matrix A = (aij)

 a 00 L a 0,N-1   x 0   c 0 


M   M  =  M 
 M
aN-1,0 L aN-1,N-1   xN-1  c N-1 

a N -1,0   c 0   x 0 

M   M  =  M 
a N -1, N -1  c N -1   x N -1 

 a N -1,0 
 a 00 
 x0 







=
+
+
c
c
M
L
M

 N −1  M 

 0
a N -1,N -1 
 a 0,N -1 
 x N -1 

• Orthonormality: A −1 = A T (transpose)

basis vectors

coefficients

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Why Orthonomality?

Example (with Compaction)
1 1 1 


2 1 − 1
A 2 = I ⇒ A −1 = A

A=

• The energy of the data equals the energy of
the coefficients:
N−1

∑c
i=0

2
i

A T = A = A −1

= c T c = (Ax)T (Ax)
N−1

= (x T A T )(Ax) = x T (A T A)x = x T x = ∑ x i

orthonormal

1 1 1  b  2b


  = 
2 1 − 1 b  0 

2

i =0

compaction

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Discrete Cosine Transform (1D)


aij = 



1
i=0
N

2
(2 j + 1)iπ
cos
N
2N

Basis Vectors (1D)
1

1

0.5

0.5
0

0

i >0

0

1

2

0

3

-0.5

-0.5

-1

-1

1

row 0

2

3

row 1

N = 4:
.5
.5
.5

 .5
 .65328 .270598 - .270598 - .65328 

A=

 .5
- .5

- .5
.5


.270598 - .65328 .65328 - .270598

1

1

0.5

0.5
0

0
0

1

0

3

1

2

3

-1

-1

row 3

row 2

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Decomposition in Terms of Basis
Vectors

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Block Transform
Image

.5
 .653281 
 .5 
 .270598 
 x0 
.5
 .270598 
−.5
 
 - .653281
 c + 
c +  c + 

c =  x1 
0
1
2
3
.5
− .270598
−.5
 x2 
 .653281 
 

 

 


.270598
−
.5
.653281
.5
 

 



 x3 

DC coefficient

2

-0.5

-0.5

Each 8x8 block is
individually coded

AC coefficients

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2-Dimensional Block Transform

8x8 DCT Basis Vectors

Transform

Block of pixels X

a00
a
10
A=
a20

a30

x00 x01 x02 x03
x10 x11 x12 x13
x20 x21 x22 x23
x30 x31 x32 x33

a01 a02
a11 a12
a21 a22
a31 a32

a03 
a13 

a23 

a33 

N−1

Transform rows rij = ∑ a kj xik
k =0
N −1

N−1

N−1

N −1 N −1

m= 0

m =0

k =0

m= 0 k = 0

Transform columns c ij = ∑ aimrmj = ∑ a im ∑ akj x mk = ∑
Summary: C = AXA

∑a

a x mk

im kj

T

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Importance of Coefficients

Quantization Tables

• The DC coefficient is (in general) the most important.
• The AC coefficients (in general) become less
important as they are farther from the DC coefficient.
• One example bit allocation (bits per coefficient):

• For an nxn block we construct an nxn matrix Q:
– Qij indicates how large the quantization interval is for
coefficient cij.

8 7 5 3 2 1 0 0
7
5
3
2
1
0
0

5
3
2
1
0
0
0

3
2
1
0
0
0
0

2
1
0
0
0
0
0

1
0
0
0
0
0
0

0
0
0
0
0
0

0

0
0
0
0
0
0
0

0
0
0
0
0
0
0

• Encode cij with the label:
compression
55 bits for 64
pixels = .86 bpp

• Decode sij to:

c'ij = sijQij.
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Example Quantization

c' = 2 ⋅ 24 = 48

• c= 54.2, Q = 6

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Example Quantization Table (JPEG)

• c = 54.2, Q = 24 s =  54.2 + 0.5 = 2

 24

• c = 54.2, Q = 12


 c ij
sij =  + 0.5

 Qij


 54.2
s=
+ 0.5 = 5

 12
c' = 5 ⋅12 = 60

 54.2
s=

+ 0.5 = 9

 6
c' = 9 ⋅ 6 = 54

16
12

11
12

10
14

16
19

24
26

40
58

51
60

61
55

14

14
18
24
49
72

13
17
33
35
64
92

16
22
37
55
78
95

24
29
56
64
87
98

40
51
68

81
103
112

57
87
109
104
121
100

69
80
103
113
120
103

56
62
77
92
101
99

Increase the bit rate = halve the size of quantization intervals.
Decrease the bit rate = double the size of quantization intervals.
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JPEG (1987)

Block Transmission

• Let P = [pij], 0 < i,j < N be an image with 0 < pij < 256.
• Center the pixels around zero:

• DC coefficient:
– DC coefficients don’t change much from block to
neighboring block. Hence, their labels change
even less.
– Predictive coding with differences is used to code
the DC label.

– xij = pij - 128

• Code 8x8 blocks of X using the DCT.
• Choose a quantization table:
– The table depends on the desired bit rate and is built into
JPEG.

• AC coefficients:

• Quantize the coefficients according to the quantization

table:

– Do a zig-zag coding.

– The quantization symbols can be positive or negative.

• Transmit the labels (in a coded way) for each block.
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Zig-Zag Coding

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Example Block of Labels

• AC labels are coded in zig-zag order:
– Usually use a special entropy coding to take advantage of
the ordering of the bit allocation (quantization).

-8

2

0

0

0

0

0

0

0

0

0

0

0

0

0

3

1

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Coding order of AC labels
2 –8 3 0 0 0 0 1 1 0 0 1 0 0 .....
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Coding Labels

Coding AC Label Sequence

• Categories of labels
–
–
–
–

1
2
3
4

• A symbol has three parts (Z,C,B)

{0}
{-1, 1}
{-3,-2,2,3}
{-7,-6,-5,-4,4,5 6 7}

– Z for number of zeros preceding a label 0 < Z < 15
– C for the category of the of the label
– B for a C-1 bit number for the actual label

• End of Block symbol (EOB) means the rest of
the block is zeros. EOB = (0,0,-)
• Example: 2 –8 3 0 0 0 0 1 1 0 0 1 0 0 .....

• Label is indicated by two numbers Cat,B

• Examples label C,B
0
2
-4

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1
3, 2
4, 3

(0,3,2)(0,5,7)(0,3,3)(4,2,1)(0,2,1)(2,2,1)(0,0,-)

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Coding AC Label Sequence
• Z,C have a prefix code
• B is a C-1 bit number
0

0
1
Z 2

3

1010

1

Notes on Transform Coding
• Video Coding:

Partial prefix
code table

C

2

– MPEG uses DCT.
– H.263 and H.261 use DCT.

3

00

01

100

1100

11011

11110001

1110

11111001

1111110111

111010

111110111

111111110101

• Audio Coding:
– MP3 = MPEG 1 Layer 3 uses DCT.

• Alternative Transforms:
– Lapped transforms remove some of the blocking
artifacts.
– Wavelet transforms do not need to use blocks at
all.

(0,3,2) (0,5,7) (0,3,3) (4,2,1) (0,2,1) (2,2,1) (0,0,-)
100 10 11010 0111 100 11 1111111000 1 01 1 11111001 1 1010
46 bits representing 64 pixels = .72 bpp

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