Image Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 739

LL2 HL2

LL HL HL1

LH2 HH2

LH HH LH1 HH1

(a) Level-one transformation (b) Level-two transformation

FIGURE 14.41
The two-dimensional DWT for level 1 and level 2.

( Figure 14.41 (b)). The process proceeds to higher levels as desired. With the obtained wavelet
transform, we can quantize coefficients to achieve the compression requirement. For example, for the
second-level coefficients, we can omit HL1, LH1, HH1 to simply achieve a 4:1 compression ratio.
Decompression reverses the process, that is, we inversely transform columns and then rows of the
wavelet coefficients. We can apply the IDWT to the recovered LL band with the column and row
inverse transform processes, and continue until the inverse transform at level 1 is completed. Let us
look at an illustrative example.

EXAMPLE 14.16
Consider the following 4 4 image:

114 135 122 109

102 116 119 124

105 148 138 122

141 102 140 132

a. Perform 2D-DWT using the 2-tap Haar wavelet.
b. Using the result in (a), perform 2D-IDWT using the 2-tap Haar wavelet.
Solution:
a. The MATLAB function dwt() is applied to each row. The result for the first row is displayed below:
>> dwt([1 1]/sqrt(2), [114 135 122 109],1)' % Row vector coefficients

ans ¼176.0696 163.3417 14.8492 9.1924

The completed row transform is listed below:

176.0696 163.3417 14.8492 9 9.1924

154.1493 171.8269 9.8995 3.5355

178.8980 183.8478 30.4056 11.3137

171.8269 192.3330 27.5772 5.6569