Image Processing Reference
In-Depth Information
The
first term in Equation 6.86 is a measure of smoothness with respect to the
variable x and the second term with respect to the variable y. Equation 6.86 can be
written as
h
i
h
i
T
2
F
þk
C
m
f
T
k
2
F
¼ Tr
C
n
fC
n
f
Þ
T
þ Tr
C
m
f
T
C
m
f
T
J
2
¼k
C
n
f
k
ð
þ Tr
f
T
C
n
C
n
f
fC
m
C
m
f
T
¼ Tr
(
6
:
87
)
where
C
n
and C
m
are matrices de
ned by Equation 6.81
Tr stands for the trace of a matrix
Since we want a
fit that is both smooth and approximates the data well, we let the
cost function be a linear combination of J
1
and J
2
, that is, J
¼
J
1
þa
J
2
. We then try
to
a
can be used to control the
relative importance of the smoothness and the goodness of the
find the minimum of J. Similar to the 1-D case,
fit. To minimize J,we
set the gradient of J with respect to f equal to zero:
q
J
q
f
¼
q
J
1
q
f
þ a
q
J
2
q
f
¼
0
(
6
:
88
)
since
q
J
1
q
f
¼
(
f
f
)
2
(
6
:
89
)
and
q
J
2
q
f
¼
a
fC
n
C
n
þ
a
fC
m
C
m
2
2
(
6
:
90
)
Substituting Equations 6.90 and 6.89 into Equation 6.88 results in
f
þ
f
a
C
m
C
m
¼
f
a
C
n
C
n
þ
I
n
(
6
:
91
)
where I
n
is an n
n identity matrix. This equation is a special case of the Sylvester
equation (AX
þ
XB
¼
C), which can be solved by the MATLAB lyap function. Note
that if either of n or m is one, then this reduces to the same equation used in the 1-D case.
6.6.2.3 Three-Dimensional Smoothing Algorithm
The 4-D smoothing algorithm can easily be derived from the 3-D case. Therefore, we
first consider the 3-D case. In the 3-D case, we have a function of three variables,
x, y, and z, that we are trying to approximate. The values of this function at evenly
spaced points in the 3-D space are stored in an in
m
l tensor f . Let f be the
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