Image Processing Reference
In-Depth Information
According to the property of cross-correlation and sine function, we have
N
1
1
R
()
mR
( )
m
g
() (
n x
n
m
)
xg
g
x
ij
( )
ij
N
ij
i
(1)
j
i
(1)
j
ij
n
0
N
1
1
g
n

n
m
()sin(
)
i j
( )
ij
ij
ij
N
n
N
0
1
(3.15)
1
g
(
n
)[sin(
 
n
) cos(
m
)
cos(
 
n
)sin(
m
)]
i j
( )
ij
ij
ij
ij
ij
ij
N
n
0
N
1
N
1
1
1
cos(
mg n
)
(
)sin(
 
n
)
sin(
m
)
g
(
n
)cos(
 
n
)
ij
i j
( )
ij
ij
ij
i j
( )
ij
ij
N
N
n
0
n
0
Similarly, for other cross-correlation, we have
N
1
1
R
()
m
l
() (
n x
n
m
)
xl
ij
( )
ij
N
ij i
(1)
j
n
0
(3.16)
N
1
N
1
1
1
cos(
ml
)
(
n
)sin(
 
n
)
sin(
ml
)
(
n
) cos(
 
n
)
ij
i j
( )
ij
ij
ij
i j
( )
ij
ij
N
N
n
0
n
0
N
1
1
R
()
m
g
()
n x
(
n
m
)
gx
ij
ij
( )
N
ij
i
(1)
j
n
0
(3.17)
N
1
N
1
1
1
cos(
m
)
g
(
n
)sin(
n
)
sin(
m
)
g
(
n
) cos(
n
)
ij
ij
i j
( )
ij
ij
ij
i j
( )
ij
N
N
n
0
n
0
N
1
1
R
()
m
l
()
n x
(
n
m
)
lx
ij
ij
( )
N
ij
i
(1)
j
n
0
(3.18)
N
1
N
1
1
1
cos(
ml
)
(
n
)sin(
n
)
sin(
ml
)
(
n
) cos(
n
)
ij
ij
i j
( )
ij
ij
ij
i j
( )
ij
N
N
n
0
n
0
Then, the B can be obtained as follows:
N
1
N
1
1
B
cos(
m
)[
g
(
n
)sin(
 
n
)
l
(
n
)sin(
 
n
)
ij
i j
( )
ij
ij
i j
( )
ij
ij
N
n
0
n
0
f
1
f
1
s
s
gn
()sin(
n
)
ln
()sin(
n
)]
ij
i j
( )
ij
ij
i j
( )
ij
n
0
n
0
(3.19)
N
1
N
1
1
sin(
mg
)[
(
n
) cos(
n
)
g
(
n
) cos(
n
)
ij
ij
i j
( )
ij
i j
( )
i
j
ij
N
n
0
n
0
N
1
N
1
ln
()cos(
n
)
l
()cos(
n
 
n
)]
ij
i j
( )
ij
i j
( )
ij
ij
n
0
n
0
The sum of formula (3.19) is equal to zero in the range [0,N-1].
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