Fundamentals of Digital Image Processing - Control of Color Imaging Systems: Analysis and Design - page 35

Q ( x )

r L

r 3

r 2

r 1

x

d 0 d 1 d 2 d 3 d L -1 d L

FIGURE 2.14 Quantization function Q ( x ) .

x max x mi L ¼ x max x min

2 B

D ¼ ( 2 : 41 )

The decision boundaries are related to the quantization step size D by

d i d i 1 ¼ D 1 i L ( 2 : 42 )

The reconstruction levels are chosen to be halfway between the decision boundaries
in order to minimize the mean-squared error (MSE).

d i þ d i 1
2

r i ¼ 1 i L ( 2 : 43 )

Example 2.4

Design a uniform 2 bit (4-level) quantizer for a signal with dynamic range between
0 and 1.

S OLUTION
The decision boundaries and reconstruction levels are

d 0 ¼ 0, d 1 ¼ 0 : 25, d 2 ¼ 0 : 5, d 3 ¼ 0 : 75, d 4 ¼ 1

r 1 ¼ 0 : 125, r 2 ¼ 0 : 375, r 3 ¼ 0 : 625, r 4 ¼ 0 : 875

2.4.2.2 Signal-to-Quantization Noise Ratio (SQNR)

Let s assume

that the signal distribution (histogram) over a given quantizer decision interval
can be approximated by a uniform probability density function (PDF) as shown in

D denote the quantization step size of a uniform quantizer and let '