Image Processing Reference
In-Depth Information
6.6.2 M
ULTIDIMENSIONAL
S
MOOTHING
A
LGORITHM
The multidimensional smoothing algorithm is covered in this section. We
rst
discuss the 1-D smoothing algorithm and then extend the approach to multidimen-
sional (2-D through 4-D) case.
6.6.2.1 One-Dimensional Smoothing Algorithm
Assume that we have a function of one variable, f(x). The value of this function at n
different points is given in a vector f :
f
¼
T
½
fx
ðÞ
fx
ð Þ
fx
n
1
ð
Þ
(
6
:
76
)
nd a vector f that is a smooth approximation of f . Let the cost
function J
1
be a measure of closeness of f to f , that is,
The objective is to
X
n
1
2
f
i
f
i
¼k
f
f
k
¼ (
f
f
)
T
(
f
f
)
2
J
1
¼
(
6
:
77
)
i
¼
0
The other cost function, J
2
, is used to measure the smoothness of the
t
f. It is given
as the distance between the second derivative of function f and zero:
2
x
n
1
ð
2
d
f
(
x
)
d
x
2
J
2
¼
d
x
(
6
:
78
)
x
0
The cost function J
2
is approximated using
X
n
3
2
¼
f
T
TQf
J
2
¼
ð
f
i
2f
i
þ
1
þ
f
i
þ
2
Þ
(
6
:
79
)
i
¼
0
where
Q
¼
C
n
C
n
(
6
:
80
)
and
n
z
}|
{
2
4
3
5
000
0000
1
2100
01
.
210
.
.
.
00
.
.
.
.
.
.
C
n
¼
(
6
:
81
)
.
.
.
.
000
.
.
.
.
1
210
01
21
000
0000
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