Image Processing Reference
In-Depth Information
TABLE 6.10
Weights as a Function of i
i
1
2
3
4
5
6
7
8
9
10
W
i
15.97
29.16
345.02
1323.87
6038.28
9411.7
1323.87
121.95
43.04
15.97
The weights for different values of i are shown in Table 6.10.
X
X
N
10
W
i
y
i
x
i
¼
S¼
W
i
y
i
x
i
½
1
¼
½
83
:
8135 144
:
9269
i¼
1
i¼
1
2
4
3
5
¼
P
10
1
W
i
x
i
P
10
1
W
i
x
i
P ¼
X
N
W
i
x
i
x
i
¼
X
10
176
:
0015 324
:
6467
T
i¼
i¼
W
i
x
i
½
1
½
x
i
1
¼
P
10
1
W
i
x
i
P
10
324
:
6467 634
:
3268
i¼
1
i¼
1
1
W
i
i¼
i¼
A¼SP
1
¼
[0
:
9789
0
:
2725]
¼
0
5
1
:
y ¼Ax ¼
[0
:
9789
0
:
2725]
0
:
2169
6.3.3 R
ECURSIVE
L
EAST
-S
QUARE
I
MPLEMENTATION
OF
M
OVING
-M
ATRIX
A
LGORITHM
The moving-matrix interpolation algorithm given by Equation 6.25 can be imple-
mented using recursive least square (RLS). The transformation matrix A is given by
!
X
"
#
1
X
N
N
A
¼
SP
1
W
i
y
i
x
i
W
i
x
i
x
i
¼
(
:
)
6
31
i
¼
1
i
¼
1
As can be seen, matrix inversion is required to compute the matrix A. Since the
matrix P
¼
P
i
¼
1
W
i
x
i
x
i
may be ill conditioned, we would like to compute A without
using matrix inversion. By using the RLS algorithm (Section 7.4.1.2), it is possible
to compute the inverse of P without using any matrix inversion. This can also be
achieved by recursive computation of A using the following algorithm:
¼
a
2
I and k
¼
0, where a
2
Step 1:
Initialization. Let B(0)
0 and I is the 3
3
identity matrix. Choose
e
to be a small positive number.
Step 2:
Change k
!
k
þ
1.
Step 3:
Compute
w
(
k
) ¼k
x
x
k
kþe
(
6
:
32
)
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