Communications-2010, Bucharest, June 11 A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform Corina Nafornita 1, Ioana Firoiu 1,2,

Communications-2010, Bucharest, June 11

A Second Order Statistical Analysis of the

2D Discrete Wavelet Transform

Corina Nafornita1, Ioana Firoiu1,2, Dorina Isar1, Jean-Marc Boucher2, Alexandru Isar1

1 Politehnica University of Timisoara, Romania2 Telecom Bretagne, France

C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

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Goal

• Computation of the correlation functions:– inter-scale and inter-band dependency,– inter-scale and intra-band dependency,– intra-scale and intra-band dependency.

• Computation of expected value and variance of the wavelet coefficients.

• Results useful for the design of different signal processing systems based on the wavelet theory.


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2D-DWT

2D DWT coefficients level m, subband k

where

1 11 1 1 2 1 2

k kx m m,n ,pD n , p x , , ,

1 2 1 2k k km,n,p m,n m,p,

1 4

2 3m,nk

m,nm,n

, k ,

, k ,

2 4

1 3

m,pkm,p

m,p

, k ,

, k ,

22 2 andm

mm,n n

22 2m

mm,n n


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D04


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Expectations

1 2 , , 1 2

*1 2 , , 1 2 1 2

, , ,

, ,

kx m

k kx m m n pD

km n p

E D E x

E x d d

* *1 2 , , 1 2 1 2 , , 1 2 1 2

, , 1 2

, , ,

, 0,0

kx m

k km n p x m n pD

kx m n p

E x d d d d

F

1 22, , 1 2 1 2, 2 2 ,2

mj n pk m k m mm n p e F F

0, 1,2,3

2 , 4.k

x mD mx

k

k

m-scale, k-subband


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Dependencies

intrintraa--sscale and intrcale and intraa-band-band

inter-scale and inter-bandinter-scale and inter-band

inter-scale and intra-bandinter-scale and intra-band

intra-scale and inter-bandintra-scale and inter-band


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Inter-scale and Inter-bandCorrelation

m2 = m1+q, k1 ≠ k2

The inter-scale and inter-band dependency of the wavelet coefficients depends on the:

• autocorrelation of the input signal,

• intercorrelation of the mother wavelets that generate the sub-bands

1 21 2

1 21 2

*

1 2 1 2,k kx xm m

k kx xm mD D

R n n p p E D D

1 21 2

1 1 12 1

1 2 1 2

22 1 2 1 2 1 2 1

2 ' , 2 '

2 2 ' , 2 ' ', '

k kx xm m

k k

q qD D

m q m q m qx

R n n p p

R n n p p R n n p p

xR2 1k kR


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Inter-scale and Inter-band White Gaussian Noise

• Input image: bi-dimensional i.i.d. white Gaussian noise with variance and zero mean:

• Generally the 2D DWT correlates the input signal.

2w

1 2 1 2, ,x w

11 2 2 11 2

2 21 2 1 2 2 1 2 12 ' , 2 ' 2 ', ' .k k k k

w wm m

m qq qwD D

R n n p p R n n p p


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Inter-scale and Intra-band Correlation

m2 = m1+q, k1 =k2=k.

Orthogonal wavelets:

1 2

1 1 1

1 2 1 2

22 1 2 1 2 1 2 1

2 ' , 2 '

2 2 ' , 2 ' ', '

k kx m x m

k

q qD D

m q m q m qx

R n n p p

R n n p p R n n p p

, ,kR n p n p

1 1 1

1 2

21 2 1 2 2 1 2 12 ' , 2 ' 2 2 ' , 2 ' .k k

x m x m

m q m q m qq qxD DR n n p p R n n p p

The intercorrelation of the wavelet coefficients depends solely of the autocorrelation of the input signal.


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Inter-scale and Intra-bandWhite Gaussian Noise

1 2 1 2, ,x w

Input image: bi-dimensional i.i.d. white Gaussian noise with variance and zero mean:

1 1 1

1 2

2 21 2 1 2 2 1 2 12 ' , 2 ' 2 2 ' , 2 ' .k k

x m x m

m q m q m qq qwD DR n n p p n n p p

The wavelet coefficients with different resolutions of a white Gaussian noise are not correlated inside a sub-band.


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Inter-scale and Intra-bandAsymptotic Regime

1 2 1 2 1 2 2 1 2 12 ' , 2 ' 2 0,0 ', 'k kx x

q q qxD DR n n p p S n n p p

The intra-band coefficients are asimptotically decorrelated for orthogonal wavelets.


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Intra-scale and Intra-band Correlation

m2 = m1= m, k2 = k1= k.

21 2 1 2 2 1 2 1' , ' 2 2 ' , 2 'k

x m

m m mxDR n n p p R n n p p

The autocorrelation of the wavelet coefficients depends solely on the autocorrelation of the input signal.


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Intra-scale and Intra-bandVariances

2

221 1

-11 2 1 2 1 22

, 0,0

12 ,2 2 ,2 0,0 .

4

k kx m x m

kx mD D

m m m mx xR

E D n p R

S d d S

F

4 4 4 4

22 4 2

1 1

-1 2 21 2

, 0,0

2 ,2 0,0 2 .

x m x m x m x mx mD D D D

m m mx x

E D n p R

S

F

For k=1 or 2 or 3 :

For k=4 :


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Intra-scale and Intra-band White Gaussian Noise

2w

1 2 1 2, ,x w

21 2 1 2 2 1 2 1, ,k

w mwDR n n p p n n p p

Input image: bi-dimensional i.i.d. white Gaussian noise with variance and zero mean:

In the same band and at the same scale, the 2D DWT does not correlate the i.i.d. white Gaussian noise.

2 20,0kx m

w wD S


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Intra-scale and Intra-band Asymptotic Regime

1 2 1 2 2 1 2 1, 0,0 ,kx

xDR n n p p S n n p p

Asymptotically the 2D DWT transforms every colored noise into a white one. Hence this transform can be regarded as a whitening system in an intra-band and intra-scale scenario.

For k=1 or 2 or 3 :

22 -1

1 1 1 2, lim 2 ,2 0,0 0,0kx

k m mx m x xD m

E D n p S S

F


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Conclusions• 2D DWT : sub-optimal bi-dimensional whitening system.

Contributions• formulas describing inter-scale and inter-band; inter-scale and intra-

band and intra-scale and intra-band dependencies of the coefficients of the 2D DWT,

• expected values and variances of the wavelet coefficients belonging to the same band and having the same scale.

Use• design of different image processing systems which apply 2D DWT

for compression, denoising, watermarking, segmentation, classification…

• develop a second order statistical analysis of some complex 2D WTs.

Documents

Communications-2010, Bucharest, June 11 A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform Corina Nafornita 1, Ioana Firoiu 1,2,