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