Image Processing Reference
In-Depth Information
n
m
l tensor that is a smooth approximation of f . Similar to the other cases, we
de
ne J
1
as a measure of closeness of f to f to be
X
X
X
n
m
l
2
f
fijk
f
fijk
J
1
¼
(
6
:
92
)
i
¼
1
j
¼
1
k
¼
1
Let J
2
be a measure of the smoothness of
f. Using the discrete second derivative,
we have
X
X
X
X
X
X
n
2
m
l
n
m
2
l
2
2
J
2
¼
f
fijk
2f
(
i
þ
1
)
jk
þ
f
(
i
þ
2
)
jk
þ
f
fijk
2f
i
(
j
þ
1
)
k
þ
f
i
(
j
þ
2
)
k
i
¼
1
j
¼
1
k
¼
1
i
¼
1
j
¼
1
k
¼
1
X
n
X
m
X
l
2
2
þ
f
fijk
2f
ij
(
k
þ
1
)
þ
f
ij
(
k
þ
2
)
(
6
:
93
)
i
¼
1
j
¼
1
k
¼
1
The three terms in the above equation are measures of smoothness with respect to
the three variables x, y,andz, respectively. Note that it is an approximation of the integral
of the square of the second derivative of f along each of the three directions. Let
n
stand for the n-mode multiplication of a tensor with a matrix and
h
,
i
stand for tensor
multiplication. The de
nitions of these operations are given in Appendix C. Then,
J
2
¼
h
f
1
C
n
, f
1
C
n
i þ
h
f
2
C
m
, f
2
C
m
i þ
h
f
3
C
l
, f
3
C
l
i
(
6
:
94
)
Similar to the 2-D case, let the overall cost function be
J
¼
J
1
þ a
J
2
(
6
:
95
)
The minimum of J is obtained by setting the gradient of J with respect to f equal to
zero. This yields
q
J
q
f
¼
q
J
1
q
f
þ a
q
J
2
q
f
¼
0
(
6
:
96
)
þ a
f
2
C
m
C
m
þ a
f
3
C
l
C
l
¼
f
f
þ a
f
1
C
n
C
n
0
(
6
:
97
)
þ
f
2
þ
f
3
¼
f
a
C
n
C
n
a
C
m
C
m
a
C
l
C
l
þ
I
l
f
1
(
6
:
98
)
a
C
n
C
n
,
a
C
m
C
m
, and
a
C
l
C
l
þ
I
l
are all positive de
Since
nite diagonalizable matri-
ces, we can de
ne
a
C
n
C
n
¼
R
u
R
1
a
C
m
C
m
¼
S
w
S
1
a
C
l
C
l
þ
I
l
¼
T
c
T
1
(
6
:
99
)
where
u
,
w
, and
c
are diagonal matrices. Substituting Equation 6.99 into Equation
6.98 results in
þ
f
2
S
w
S
1
þ
f
3
T
c
T
1
f
¼
f
1
R
u
R
1
(
6
:
100
)
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