Image Processing Reference
In-Depth Information
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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