Image Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 731

EXAMPLE 14.15
Given the following DCT coefficients from a 2 2 image, determine the pixel pð0; 0Þ:

! " 120 80

30 30

#

F u; v ¼

Solution:
Applying Equation (14.19) of the inverse 2D-DCT with N ¼ M ¼ 2,

i ¼ 0, and j ¼ 0, it follows that

cðuÞcðvÞF ðu; v cos u p
4

cos v p
4

p 0; 0 ¼ 1
u ¼0

1

v ¼0

! ! 2 ! 2 !

p
2

p
2

p p

F 0; 0 þ F 0; 1

¼

2 2

! 2 !

þ F 0; 1

! !

p
2

p

p
2

p
2

F 1; 0

þ

2

30 ¼ 100

¼ 1

2 120 þ 1 2 80 þ 1 2 30 þ 1 2

We apply the MATLAB function idct2() to verify the inverse DCT and get the following pixel values:
>> p ¼ idct2([120 80; 30 30])
p ¼ 100.0000

50.0000

100.0000 10.0000

14.7.2 Two-Dimensional JPEG Grayscale Image Compression Example
To understand JPEG image compression, we examine an 8 8 grayscale subblock. Table 14.5 shows
a subblock of the grayscale image in Figure 14.37 that is to be compressed. Applying 2D-DCT leads to
Table 14.6 .
These DCT coefficients have a big DC component of 1198 but small AC component values. These
coefficients are further normalized (quantized) with a quality factor Q, defined in Table 14.7 .

Table 14.5 8 8 Subblock

150 148 140 132 150 155 155 151

155 152 143 136 152 155 155 153

154 149 141 135 150 150 150 150

156 150 143 139 154 152 152 155

156 151 145 140 154 152 152 155

154 152 146 139 151 149 150 151

156 156 151 142 154 154 154 154

151 154 149 139 151 153 154 153