Overview

We will briefly discuss the numerous applications of adapted wavelet analysis, some of which are just being explored. These include image compression, fast numerical methods for principal factor analysis and matrix application, acoustic signal processing and compression, and de-noising.

Many of these applications were first described in survey articles [27, 25, 29, 26, 32, 31, 115, 81, 112], while others depend on novel numerical algorithms which were individually analyzed [114, 111, 118, 9, 8, 11]. Still others were introduced as methods used for numerical experiments [34, 46, 51, 57, 110, 113] with specific signal processing problems. In a few cases we will go down a somewhat different path than the one described in the literature. For example, we will examine a fast approximate version of matching pursuit [72], and we will describe best basis transform coding in addition to various quantization methods to be applied after a fixed transform [77, 60].

There are plenty of other applications which are beyond the scope of this book. For example, we will not discuss applications such as [63, 73, 53], which depend on the continuous wavelet transform, or ones which depend on spline rather than QF or local trigonometric constructions of adapted wavelets [14, 41].

11.1 Picture Compression

We shall first consider the problem of storing, transmitting, and manipulating digital electronic images. Because of the file sizes involved, transmitting images will always consume large amounts of bandwidth, and storing images will always require hefty resources. Because of the large number N