Goals and prerequisites

The lesson deals with basics of technique alternative to DCT presented in previous lesson. The technique is used in some modern image processing applications (JPEG-2000 is probably the best known one) and is called wavelet or subband coding. The two names come from two quite different mathematical concepts used in analog signals analysis. If these concepts are adopted for the processing of digital images of natural scenes - the differences gets much smaller or even vanish at all. Since both methods use filtering operation, Lesson 3 should be read first.

Wavelet / Subband Coding

In practice, the wavelet and subband coding techniques consist of there main parts. First, the input image is represented as a series of band-pass filtered and subsampled images. If filter banks used in this step are based on classical frequency analysis (DFT or Z transform) it is said that subband analysis is performed. On the other hand, if used filter banks are based on wavelet theory, it is said that wavelet transform is performed. It is worth to notice, that some filter banks may be obtained using any of the two theoretical backgrounds. Filter bank described on the next pages of this lesson is a good example - in the literature it is called "Haar filter bank" as well as "Haar wavelet transform".

Subsampled images, called from now on "subbands", are upsampled, filtered and added in the last step of technique presented in this lesson. This process - resulting in full-sized output image - is called subband synthesis (merging) or wavelet inverse transform. If filter banks used in those two described steps are properly designed; the output image will be the exact copy of the input image.

Between analysis and synthesis usually one performs additional coding of subbands leading to lossless or lossy compression. This processing is more critical for the whole system efficiency than filter bank type and, moreover, usually does not depend on theory behind filter banks.