Fundamentals of Digital Image Processing - Control of Color Imaging Systems: Analysis and Design - page 38

p ( x )

1

x

-1 1

FIGURE 2.16 PDF of Example 2.5.

Step 3: For the current set of decision levels, find the optimum reconstruction levels
by using Equation 2.51. If the changes in the reconstruction levels with respect to the
previous values are less than a threshold, stop. Otherwise, go to Step 2. Now the
process is illustrated using a simple example.

Example 2.5

The PDF of random variable X is shown in Figure 2.16. For this random variable,

a. Design the 2 bit (4 levels) uniform quantizer and calculate the MSE.
b. Design the 2 bit (4 levels) Lloyd Max quantizer and calculate the MSE.

-

S OLUTION
The step size for uniform quantizer is

x max x min

4

2
4

Dx ¼ ¼ ¼ 0 : 5. The reconstruction

and decision levels are

d 0 ¼ 1 d 1 ¼ 0 : 5 d 2 ¼ 0 d 3 ¼ 0 : 5 d 4 ¼ 1

r 1 ¼ 0 : 75 r 2 ¼ 0 : 25 r 3 ¼ 0 : 25 r 4 ¼ 0 : 75

The MSE is computed as

d i

ð ð 0:5 ð

0

X

L

(x r i ) 2 p(x)dx ¼ 75) 2 p(x)dxþ 25) 2 p(x)dx D ¼ (x þ 0 : (xþ 0 :

i¼ 1

d i1 1 0 : 5

ð ð

0 : 5 1

25) 2 p(x)dx þ 75) 2 p(x)dx þ (x 0 : (x 0 :

0 0 : 5

or

2 ð 2 ð

0 : 5 1

25) 2 (1 75) 2 (1 D ¼ (x 0 : x)dxþ (x 0 : x)dx ¼ 0 : 0208

0 0 : 5