Image Processing Reference
In-Depth Information
Since m ¼
2 and r ¼
2, we have U ¼ U 1 . Therefore, we have
2
4
3
5
T
60 p 00
0
0
:
3651
0
0
:
9309
¼
0
:
707
0
:
707
A¼USV H
5 p 0
0
:
1826
0
:
9806
0
:
0716
0
:
707
0
:
707
0
:
9129 0
:
1961
0
:
3581
Example 3.39
Consider the 256
256 cameraman image. If we decompose the image into its
SVD components, each component
is an image of size 256
256. The ith
2
i u i v i and the whole image is
component of the image is
s
X
256
A ¼ USV T
1 s i u i v i
¼
Plot of the 256 singular values of the image normalized with respect to the largest
singular value is shown in Figure 3.13. As can be seen, most of the singular values
are small. The original image and the images reconstructed using the
rst 20 and
50 singular values are shown in Figures 3.14 through 3.16, respectively.
3.10.1 M ATRIX N ORM
Matrix norm like other vector-space norms must satisfy the properties of vector
norm. Let A be an m n matrix mapping vector space R n to vector space R m .By
de
nition, the p matrix norm is de
ned by
k A k p ¼ sup
x 2 R n
k x k p ¼
1 k Ax k p
(
3
:
142
)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
300
k
FIGURE 3.13
Singular values normalized with respect to the largest singular value.
 
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