le debruitage des images sonar en utilisant la theorie des ondelettes sorin moga et alexandru isar
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ISETc 2010, Timisoara, November 11, 2010. A Second Order Statistical Analysis of the Hyperanalytic Wavelet Transform. LE DEBRUITAGE DES IMAGES SONAR EN UTILISANT LA THEORIE DES ONDELETTES SORIN MOGA ET ALEXANDRU ISAR. - PowerPoint PPT PresentationTRANSCRIPT
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