Discrete Wavelet Transformations, 2nd Edition

CHAPTER 10WAVELET PACKETS

Suppose we compute j = 1, …, i iterations of the wavelet transform of matrix A using a prescribed filter (or biorthogonal filter pair). Then counting the matrix itself, we have i + 1 representations of A. In previous chapters, we have seen that the number of iterations of the wavelet transform varies between applications. In image compression, we learn that increasing the number of iterations concentrates the energy of the transformed image to relatively few elements. In an application such as edge detection, using a high iteration transform might make it difficult to detect changes in the image and in the signal de‐noising, we learn that the differences portion of the first iteration is comprised essentially of noise.

Since a variety of representations of the data are desirable, it is natural to ask if we can create even more representations of the data via the wavelet transformation. This question was considered by Coifman, Meyer and Wickerhauser in [25]. They decided to perform a full decomposition of the data. That is, instead of iteratively applying the wavelet transformation to the preceding averages portion, they suggested applying it to all preceding portions.

Section 10.1 describes the wavelet packet transformation and the large number or redundant representations that result. In Section 10.2, we introduce the notion of a cost function that is used in conjunction with an algorithm for selecting the “best” representation. The algorithm is stated and ...