LE DEBRUITAGE DES IMAGES SONAR EN UTILISANT LA THEORIE DES ONDELETTES SORIN MOGA ET ALEXANDRU ISAR

LE DEBRUITAGE DES IMAGES SONAR EN UTILISANT LA

THEORIE DES ONDELETTES

SORIN MOGA ET ALEXANDRU ISAR

A Second Order Statistical Analysis of the Hyperanalytic

Wavelet Transform

ISETc 2010, Timisoara, November 11, 2010

1Corina Nafornita, 1,2Ioana Firoiu, 1Dorina Isar, 2Jean-Marc Boucher and 1Alexandru Isar

1 Politehnica University of Timisoara, Romania 2Telecom Bretagne, France

Ioana Adam, Corina Nafornita, Jean-Marc Boucher, Alexandru Isar, A New Implementation of the Hyperanalytic Wavelet

Transform , ISSCS 2007 1/14

Hyperanalytic Mother Wavelet

2 2 2 1, , , and 1i j k ij=ji=k jk=kj=i ki=ik=-j ijk=

*

;

;

2

.2

1 i j k

1 i j k

1-k+i

1+k+i

Z x y z u

Z x y z u

Z x u y z

x u y z

, , ,

, ,

1 i H

j H k H H

h x

y x y

x y x y x y

x y x y

LE DEBRUITAGE DES IMAGES SONAR EN UTILISANT LA

THEORIE DES ONDELETTES

SORIN MOGA ET ALEXANDRU ISAR

Ioana Adam, Corina Nafornita, Jean-Marc Boucher, Alexandru Isar, A New Implementation of the Hyperanalytic Wavelet

Transform , ISSCS 2007 2/14

Hyperanalytic Wavelet Transform

, , , , .hHWT f x y f x y x y

, ,

, ,

,

, , , , .

H H

H H

HWT f x y DWT f x y

iDWT f x y jDWT f x yx y

kDWT f x yy x

f x y x y DWT f x yh h

LE DEBRUITAGE DES IMAGES SONAR EN UTILISANT LA

THEORIE DES ONDELETTES

SORIN MOGA ET ALEXANDRU ISAR

Ioana Adam, Corina Nafornita, Jean-Marc Boucher, Alexandru Isar, A New Implementation of the Hyperanalytic Wavelet

Transform , ISSCS 2007 3/14

Enhancement of directional selectivity

D04

2D-DWT

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

Communications 2010, Bucharest, June 11 4/14

Second Order Statistical Analysis

0, 1,2,3

2 , 4.k

x mD mx

k

k

m-scale, k-subband

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

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

Communications 2010, Bucharest, June 11 5/14

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

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

, ,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

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

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

21 2 1 2 2 1 2 1, ,k

w mwDR n n p p n n p p

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

xDR n n p p S n n p p

Dependencies

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

Communications 2010, Bucharest, June 11 6/14

Second Order Statistical Analysis

inter-scale and inter-band

inter-scale and intra-band

intra-scale and inter-band

intra-scale and intra-band

m-scale, k-subband

03,2,1 zE

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

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

7/14

inter-scale inter-scale

and inter-bandand inter-band

1

1

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

lim , , , , , , ,

lim , , , , , , ,

0.

r i

r i

z zm

z zm

R m m k k n n p p

R m m k k n n p p

1 2 1 2 1 2 1 2, , , , , , , 0 a .e.w . z zi iR m m k k n n p p

010

2030

40

010

2030

40-0.5

0

0.5

1

autocorrelation of input image

010

2030

40

010

2030

40-0.1

-0.05

0

0.05

0.1

intercorrelation of z-r z-i,m1=6,m2=7,k1=1,k2=2

8/14

inter-scale inter-scale

and intra-bandand intra-band

29

1 1 1 2 1 2, , , 2 ', , 2 ',q qz zr iR m m q k n n p p

1 1

1

21 1 2 1

2 1 1 1

2 , , , 2 ' ,

2 ' , , ,

H

fH

m q m qf x

m qy

R m m q k n n

p p R m m q k

1 1

1 1

2 1 2 1 1

1 2 1 2 1

2 ' , 2 ' ,

, , 2 ' , 2 '

H H Hm q m q

y x x

m q m q

n n p p R m

m q k n n p p

1

1

1 1 2 1

2 1

, , , 2 ' ,

2 ' .

H H Hm q

y x y

m q

R m m q k n n

p p

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

1 2 1 2 1 2, , , , , , 0 a .e.w . z zi iR m k k n n p p

9/14

intra-scale and intra-scale and

intra-bandintra-band

1 2 1 2

2 1 2 1

, , , 0,0

2 0,0 0,0

, .

f

H H H H

z zr r

f y x f y x f

R k n n p p S

S S

n n p p

010

2030

40

010

2030

40-0.5

0

0.5

1

autocorrelation of input image

010

2030

40

010

2030

40-0.05

0

0.05

0.1

0.15

intercorrelation of z-r z+r,m=1,k=2

10/14

z+r, m=7 k=1 k=2 k=3

010

2030

40

010

2030

40-0.5

0

0.5

1

autocorrelation of input image

12/14

13/14

Conclusions

Inter-scale dependence: Coefficients in subbands with same type of orientation are asymptotically decorrelated.

Inter-band dependence: bigger number of subbands, complex coefficients.

Inter-band and inter-scale: coefficients asymptotically decorrelated.

Subbands with opposite type of orientation – Cross-correlations are zero a.e.w. even for finite number of scales.

Inter-scale and intra-band: correlations independent of the mother wavelets.

Intra-scale and intra-band: whitening system.

14/14