where u 2 L2(R2), both A and B are invertible matrix. The
value of determinant B is equal to 1. The elements of this
system are called synthetic wavelets. if w
AB
(u) is sat isfied
the following Parseval frame,
X
j;k;l
f ; u
j;l;k
2
¼ f
kk
2
8f 2 L
2
ðR
2
Þð2Þ
The shearlets are a special class of composite wavelets
where A is an anisotropic dilation matrix and B is a shear
matrix. When applying the Fourier transform to the ele-
ments u
j,l,k
in formula (1), we obtain
A ¼
40
02
; B ¼
11
01
ð3Þ
^
u
j;l;k
ðnÞ¼ det A
jj
j
2
uðnA
j
B
l
Þe
2pinA
j
B
l
k
ð4Þ
for any n ¼ðn
1
; n
2
Þ2R
2
, n
1
= 0, let u be given by
^
uðnÞ¼
^
uðn
1
; n
2
Þ¼
^
u
1
ðn
1
Þ
^
u
2
ð
n
2
n
1
Þð5Þ
where
^
u
1
;
^
u
2
2 C
1
ð
^
RÞ, supp
^
u
1
1
2
;
1
16
[
1
16
;
1
2
and supp
^
u
2
½1; 1. We assume that
X
j 0
^
u
1
ð2
2j
xÞ
2
¼ 1 for x
jj
1
8
ð6Þ
and
^
u
2
ðx 1Þ
jj
2
þ
^
u
2
ðxÞ
jj
þ
^
u
2
ðx þ1Þ
jj
2
¼1 for x
jj
1
ð7Þ
It follow from the last formula that, for any j C 0,
X
j 0
X
2
j
l¼2
j
^
uðnA
j
0
B
j
0
2
¼
X
j 0
^
u
1
ð2
2j
n
1
Þ
2
X
2
j
l¼2
j
^
u
2
ð2
j
n
2
n
1
þ lÞ
2
¼ 1
ð8Þ
For any (n
1
, n
2
) 2 D
C
, where D
C
¼ðn
1
; n
2
Þ2
^
R
2
:
n
1
jj
1
8
;
n
2
n
1
1g, the functions
^
u
ð0Þ
ðnA
j
0
B
l
0
Þ
form a
tiling of D
C
. Figure 2a show the tiling of the spatial fre-
quency plane induced by the shearlets. This property
described above implies that the collection
^
u
j;l;k
ðxÞ¼2
3j
2
u
ð0Þ
ðB
l
0
A
j
0
x kÞ : j 0; 2
j
l 2
j
; k 2 Z
2
ð9Þ
is a Parseval framer for L
2
ðD
C
Þ
V
¼ f 2 L
2
ðRÞ
2
: supp
^
f
n
D
C
g. And from the conditions on the support of
^
u
1
and
^
u
2
one can easily observe that the function u
j,l,k
have
frequency support,
supp
^
u
ð0Þ
j;k;l
ðn
1
; n
2
Þ : n
1
22
2j1
; 2
2j4
[
n
2
2j4
; 2
2j1
;
n
2
n
1
þ l2
j
2
j
ð10Þ
That is, each element
^
u
j;l;k
is support on a pair of
trapezoids, of approximate size 2
2j
9 2
j
, oriented along
lines of slope l2
-j
. Figure 2b show the frequency support
of shearlets.
Image registration is an important step in image fusion.
Registration accuracy effect the quality of the image
fusion. In order to reduce the effects of image registration,
this paper uses the translation invariant shearlet transform
of image decomposition. Decompose image requires
translation invariance, so it is need to build a pyramid
transform and sampling under shear wave filter. Therefore,
the translation invariant shearlet transform is divided into
two parts. The first part is multi-scale decomposition.
Images are decomposed by nonsubsampled pyramid. The
second part is the localization of direction. The localiza tion
of direction is completed by using shearlet filter. Standard
shearlet filter is a mapping from the pseudo polarization
grid system to Cartesian coordinate system. This process
can be done using two-dimensional convolution. Specific
process is also known as Meyer, the build of the window.
The Theory of Mean Shift
This article uses the mean shift clustering (Comaniciu and
Meer 1999, 2002) to elim inate the interference of random
and other noises. The mean shift algori thm is a nonpara-
metric clustering technology. It does not require a priori
knowledge and does not constrain shape of clusters. Let n
data points x
i
, i = 1, 2,…, n is in the d-dimensional space
R
d
, the multivariate kernel density estimation is composed
of kernel function K(x) and window radius h, it is expressed
as,
^
f ðxÞ¼
1
nh
d
X
n
i¼1
K
x x
i
h
ð11Þ
The radial symmetric kernels is defined as satisfying
KðxÞ¼c
k;d
kð x
kk
2
Þð12Þ
where c
k,d
is the normalized constant to ensure that the sum
of the kernel K(x) function is one, k(x) is profile function of
K(x). The formula (11) can be rewritten as
^
f
h;K
ðxÞ¼
c
k;d
nh
d
X
n
i¼1
k ð
x x
i
h
2
Þð13Þ
J Indian Soc Remote Sens
123