Mathematical Foundations - Control of Color Imaging Systems: Analysis and Design - page 151

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

FIGURE 3.18 LENA image scatter data with the two principal components.

The two eigenvectors and the corresponding eigenvalues of covariance matrix
R are

,

0 : 7074

q 1 ¼ l ¼ 0 : 5039

1

0 : 7078

and

,

0 : 7068

l 2 ¼

q 2 ¼ 0 : 0014

0 : 7074

The two principal components are also shown in Figure 3.18.

Example 3.42

In this example, the gray-scale cameraman image is used to obtain the principal
components (eigenimages). These eigenimages are used as basis functions for com-
pressing the image. The image is partitioned into 8 64 sample

covariance matrix is formed. There are 64 basis functions. Each basis function is an
8

8 block and the 64

8 image. To compress the image, we use 16 of these principal component images
corresponding to the 16 largest singular values. The original image, the basis func-
tions, the compressed image, and the error image are shown in Figures 3.19 through
3.22, respectively. The peak signal to noise ratio (PSNR) of the compressed image is
65 dB. Note that the basis functions are not quantized. If we quantize the basis