Wave Letghchgch

The Discrete Wavelet Transformfor Image Compression

Speaker: Jing-De HuangAdvisor: Jian-Jiun Ding

Graduate Institute of Communication EngineeringNational Taiwan University, Taipei, Taiwan, ROC

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Outline

• Subband Coding

• Multiresolution Analysis

• Discrete Wavelet Transform

• The Fast Wavelet Transform

• Wavelet Transforms in Two Dimension

• Image Compression

• Simulation Result

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1. Subband Coding

h0(n)

h1(n)

2

2

2

2

g0(n)

g1(n)

+Analysis Synthesis

1( )y n

0 ( )y n

( )x n ˆ( )x n

1( )H 1( )H

/ 2

Low band High band

0

4

1. Subband Coding

• Cross-modulated

10 0 1 12

10 0 1 12

ˆ ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

X z H z G z H z G z X z

H z G z H z G z X z

ˆ ( ) ( )X z X z

0 0 1 1

0 0 1 1

( ) ( ) ( ) ( ) 0

( ) ( ) ( ) ( ) 2

H z G z H z G z

H z G z H z G z

For finite impulse response (FIR) filters and ignoring the delay

0 1

11 0

( ) ( 1) ( )

( ) ( 1) ( )

n

n

g n h n

g n h n

For error-free reconstruction

FIR synthesis filters are cross-modulated copies of the analysis filters with one (and only one) being sign reversed.

Z-transform :

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1. Subband Coding

• Biorthogonal

(2 ), ( ) ( ) ( ), , {0,1}i jh n k g k i j n i j

0 0 0 1

1 1 1 0

( ), (2 ) ( ), ( ), (2 ) 0

( ), (2 ) ( ), ( ), (2 ) 0

g k h n k n g k h n k

g k h n k n g k h n k

The analysis and synthesis filter impulse responses of all two-band,

real-coefficient, perfect reconstruction filter banks are subject to the biorthogonality constraint

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1. Subband Coding

• Orthonormal – – one solution of biorthogonal– used in the fast wavelet transform– the relationship of the four filter is :

( ), ( 2 ) ( ) ( ), , {0,1}i jg n g n m i j m i j

1 0

1 0

1 2 1 2

( ) ( 1) (2 1 ) , 2K denotes the number of coefficients

( ) is related to ( )

( ) (2 1 ), {0,1}

( ), ( ) is time-reversed versions of ( ), ( ), respectively

n

i i

g n g K n

g n g n

h n g K n i

h n h n g n g n

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2. Multiresolution Analysis

( ) ( )k kk

f x x

• Expansion of a signal f (x) :: real-valued expansion coefficients

( ) : real-valued expansion functionsk

k x

If the expansion is unique, the are called basis functions. ( )k x

The function space of the expansion set :( )k x ( )kk

V span x

*( ), ( ) ( ) ( ) ( ): the dual function of ( )k k k k kx f x x f x dx x x

If is an orthonormal basis for V , then ( )k x ( ) ( )k kx x

If are not orthonormal but are an orthogonal basis for V , then the basis funcitons and their duals are called biorthogonal.

( )k x

0 ,Biorthogonal: ( ), ( )

1 ,j k jk

j kx x

j k

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2. Multiresolution Analysis• Scaling function

/ 2 2, ( ) 2 (2 ), for and ( )j j

j k x x k k x L Z R

The subspace spanned over k for any j : , ( )j j kk

V span x

The scaling functions of any subspace can be built from double-resolution copies of themselves. That is,

where the coefficients are called scaling function coefficients.

( ) ( ) 2 (2 )n

x h n x n

9


• Requirements of scaling function:

1. The scaling function is orthogonal to its integer translates.

2. The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales. That is

3. The only function that is common to all is .That is

4. Any function can be represented with arbitrary precision. That is,

1 0 1 2V V V V V V

jV ( ) 0f x

0V

2V L R

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• Wavelet function

/ 2, ( ) 2 (2 )j j

j k x x k

spans the difference between any two adjacent scaling subspaces and

jV 1jV

for all that spans the space k Z jW

where , ( )j j kk

W span x

The wavelet function can be expressed as a weighted sum of shifted, double-resolution scaling functions. That is,

where the are called the wavelet function coefficients.

( ) ( ) 2 (2 )n

x h n x n ( )h n

It can be shown that ( ) ( 1) (1 )nh n h n

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2 1 1 0 0 1V V W V W W

0V0W1W

1 0 0V V W

Figure 2 The relationship between scaling and wavelet function spaces.

The scaling and wavelet function subspaces in Fig. 2 are related by

We can express the space of all measurable, square-integrable function as

or

1j j jV V W

20 0 1 2L V W W W R

22 1 0 1 2L W W W W W R

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3 Discrete Wavelet Transform

• Wavelet series expansion

0 0

0

, ,( ) ( ) ( ) ( ) ( )j j k j j kk j j k

f x c k x d k x

where j0 is an arbitrary starting scale

0 0 0, ,( ) ( ), ( ) ( ) ( )j j k j kc k f x x f x x dx

, ,( ) ( ), ( ) ( ) ( )j j k j kd k f x x f x x dx

called the approximation or scaling coefficients

called the detail or wavelet coefficients

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3 Discrete Wavelet Transform

• Discrete Wavelet Transform

where j0 is an arbitrary starting scale

called the approximation or scaling coefficients

called the detail or wavelet coefficients

0

0

0 , ,

1 1( ) ( , ) ( ) ( , ) ( )j k j k

k j j k

f x W j k x W j k xM M

the function f(x) is a sequence of numbers

0

1

0 ,0

1( , ) ( ) ( )

M

j kx

W j k f x xM

1

,0

1( , ) ( ) ( )

M

j kx

W j k f x xM

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4 The Fast Wavelet Transform

• Fast Wavelet Transform (FWT) – computationally efficient implementation of the DWT– the relationship between the coefficients of the DWT at adjacent

scales – also called Mallat's herringbone algorithm – resembles the twoband subband coding scheme

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( ) ( ) 2 (2 )n

x h n x n Scaling x by 2j, translating it by k, and letting m = 2k + n

1(2 ) ( ) 2 2(2 ) ( 2 ) 2 2j j j

n m

x k h n x k n h m k x m

1(2 ) ( 2 ) 2 (2 )j j

m

x k h m k x m Similarity,

Consider the DWT. Assume and ( ) ( )x x ( ) ( )x x

( ) ( ) 2 (2 )n

x h n x n 0

0

1

0 ,0

1( , ) ( ) ( )

1( 2 ) 2 (2 )

M

j kx

j

m

W j k f x xM

h m k x mM

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4 The Fast Wavelet Transform / 2

, ( ) 2 (2 )j jj k x x k

1

,0

/ 2

/ 2 1

( 1) / 2 1

1( , ) ( ) ( )

1( )2 (2 )

1( )2 ( 2 ) 2 (2 )

1( 2 ) ( )2 (2 )

( 2 ) ( 1, )

M

j kx

j j

x

j j

x m

j j

m x

m

W j k f x xM

f x x kM

f x h m k x mM

h m k f x x mM

h m k W j m

Similarity, ( , ) ( 2 ) ( 1, )

m

W j k h m k W j m

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2 , 0

2 , 0

( , ) ( ) ( 1, )

( , ) ( ) ( 1, )

n k k

n k k

W j k h n W j n

W j k h n W j n

Figure 3 An FWT analysis filter bank.


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Figure 4 An FWT-1 synthesis filter bank.

By subband coding theorem, perfect reconstrucion for two-band orthonormal filters requires for i = {0, 1}. That is, the synthesis and analysis filters must be time-reversed versions of one another. Since the FWT analysis filter are and , the required FWT -1 synthesis filtersare and .

( ) ( )i ig n h n

0 ( ) ( )h n h n 1( ) ( )h n h n

0 0( ) ( ) ( )g n h n h n 1 1( ) ( ) ( )g n h n h n

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Wavelet Transform vs. Fourier Transform

• Fourier transform– Basis function cover the entire signal range,

varying in frequency only

• Wavelet transform– Basis functions vary in frequency (called “scale”)

as well as spatial extend• High frequency basis covers a smaller area• Low frequency basis covers a larger area

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Wavelet Transform vs. Fourier Transform

Time-frequency distribution for (a) sampled data, (b) FFT, and (c) FWT basis

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5 Wavelet Transforms in Two Dimension

Figure 5 The two-dimensional FWT the analysis filter.

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( 1, , )W j m n

( , , )W j m n ( , , )HW j m n

( , , )VW j m n ( , , )DW j m n

Figure 6 Two-scale of two-dimensional decomposition

( 1, , )W j m n

( , , )W j m n ( , , )HW j m n

( , , )VW j m n ( , , )DW j m n

two-dimensional decomposition

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Figure 7 The two-dimensional FWT the synthesis filter bank.

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6 Image Compression

Wavelet coding

QuantizationEntropy coding

image bitstream

• Quantization– uniform scalar quantization – separate quantization step-sizes for each subband

• Entropy coding– Huffman coding– Arithmetic coding

( , )( , ) sign( ( , ))

j

j jj

W m nq m n y m n

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7 Simulation Result

I1: Original image with width W and height H

C: Encoded jpeg stream from I1 I2: Decoded image from C

CR (Compression Ratio) = sizeof(I1) / sizeof(C)

RMS (Root mean square error) =

I1 C I2encoder decoder

2

1 21 1

( , ) ( , ) /( )H W

y x

I x y I x y H W

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