Image Processing Reference
In-Depth Information
xxx
(


xx
)
(23)
3
2
3,
nn
n i s
3,
1
3,1
3,0
1
xxx
(


xx
)
(24)
4
4,2
n
4,2
n
n s
1
4,1
4,0
2
21
With substituting the required multiplicative inverses and values of moduli, i.e. P 1 =2 n ,
P 2 =2 n +1, P 3 =2 2n +1 and P 4 =2 n -1 in the New CRT-I formulas (11)-(14), we achieve the
following conversion equation:
3
n
3
n
2
2
n
1
n
2
n
2(
xx
  
)(2
2
2 (2 1 (
 
xx
)
n
3
2
4
3
Xx

2
(25)
1
n
2
n
2
n
2 2 2 (
xx
)
1
4
4
n
21
This main conversion equation can be simplified based on the following two well-known
modulo (2 n -1) arithmetic properties.
Property 1 : The residue of a negative residue number (− v ) in modulo (2 n − 1) is the one's
complement of v , where 0≤ v < 2 n − 1 (Hariri et al. 2008).
Property 2 : The multiplication of a residue number v by 2 P in modulo (2 n − 1) is carried out
by P bit circular left shift, where P is a natural number (Hariri et al. 2008).
Now, (25) can be rewritten as follows
Xx

2 n
Z
(26)
1
Where
Zv v v v v v
   
(27)
1
2
31
32
41
42 21
4
n
Next, the binary vectors v i 's which have been simplified based on properties 1 and 2 are as
below
v xxx

xxx

xxx

xxx

xx

(28)
1
1,1
1,0
1,
n
1
1,1
1,0
1,
n
1
1,1
1,0
1,
n
1
1,1
1,0
1,
n
1
1,3
1,2
2
n
n
n
n
2
vx

xx
1 1

(29)

2
2,
n
1
2,1
2,0

3
n
n
vxx
1 1

x xx

1 1

x xx

  
(30)


31
3,1
3,0
3,
n
3,1
3,0
3,
n
3,3
3,2
n
1
n
1
2
n
1
n
1
v
0
x

x
x
0

00
x

x
x
0

00
(31)


32
3,
n
3,1
3,0
3,
n
3,1
3,0
 
n
1
n
2
n
1
n
1
vx xx

0 0

x


x x
(32)

41
4,
n
4,1
4,0
4,2
n
4,
n
2
4,
n
1

21
n
n
1
n
v
 
1 1
x

x
x
1 1

(33)


42
4,2
n
4,1
4,0

n
n
1
21
n
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