Image Processing Reference
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f 1
1 R 1
2 S 1
3 T 1
Multiply both sides of Equation 6.100 by the operator
2 S 1
1 R 1
R 1
2 S 1
3 T 1
u R 1
3 T 1
w S 1
3 T 1
¼ f 1
þ f 2
1 R 1
c T 1
2 S 1
þ f 3
(
6
:
101
)
De
ne
F ¼ f 1 R 1
2 S 1
3 T 1
(
6
:
102
)
and
F ¼ f 1 R 1
2 S 1
3 T 1
(
6
:
103
)
Then,
F ¼ F 1 u þ F 2 w þ F 3 c
(
6
:
104
)
Therefore,
F ijk ¼ F ijk u ii þ F ijk w jj þ F ijk c kk
(
6
:
105
)
F ijk
u ii þ w jj þ c kk
F ijk ¼
(
6
:
106
)
Once F is determined, f can be calculated from
f ¼ F 1 R 2 S 3 T
(
6
:
107
)
We now extend the theory to the 4-D case. The algorithm for 4-D is as follows:
Step 1: Form and diagonalize the positive de
a C n C n ,
a C m C m ,
a C k C k ,
nite matrices
and
a C l C l þ I l , where I l
is the l l identity matrix:
a C n C n ¼ R u R 1
a C m C m ¼ S w S 1
(
6
:
108
)
a C k C k ¼ T c T 1
a C l C l þ I l ¼ V g V 1
Step 2: Compute
F ¼ f 1 R 1
2 S 1
3 T 1
4 V 1
(
:
)
6
109
Step 3: Compute
F ijkl
u ii þ w jj þ c kk þ g ll
F ijkl ¼
(
6
:
110
)
Step 4: Find f using Equation 6.107.
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