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
W i y i x i
½
1
¼
½
83
:
8135 144
:
9269
1
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
W i x i
½
1
½
x i
1
¼
P
10
1 W i x i P
10
324
:
6467 634
:
3268
1
1
1 W i
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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