Fast Algorithm for Best Anisotropic Walsh Bases and Relatives

Abstract

We introduce a new family of N log N best basis search algorithms for functions of more than one variable. These algorithms search the collection of anisotropic wavelet packet and cosine packet bases and output a minimum entropy basis for a given function. These algorithms are constructed after treating the model problem of computing best Walsh packet bases. Several intermediate algorithms for conducting mixed isotropic/anisotropic best basis searches in the function's various coordinate directions are also presented.

Optimal tilings and best basis search in large dictionaries

“…In other words, we can impose that during our recursive binary splitting process, an n 1 × n 2 rectangle may only be split either into two n 1 /2 × n 2 rectangles, or into two n 1 × n 2 /2 rectangles. This scenario is similar to the anisotropic wavelet packets [1], [5]. 2 It can be shown that in this case, the total number of possible rectangular tiles is O (N 1 N 2 ), and therefore the computation of the costs has time and space complexity O (N 1 N 2 ).…”

“…A more flexible framework called anisotropic 2-D wavelet packets [1], [5], allows for any sequence of rowwise and columnwise applications of the decomposition operation (7), and is therefore a generalization of the classical wavelet packets.…”

“…The union of the atomic bases at the leaves of such a tree is called an anisotropic wavelet packet basis. A fast algorithm for finding the best anisotropic wavelet packet basis for an additive cost was provided in [1], [5]. It is easy to see that a specialization of our algorithm of Fig.…”

“…Search methods for various dictionaries that correspond to different sets of possible timefrequency tilings have been proposed, such as the double-tree algorithm [7], time-frequency trees [19], [21], space-frequency trees [8], adaptive Haar-Walsh tilings [13], anisotropic wavelet packets [1], [5], anisotropic cosine packets [1], and mixed isotropic/anisotropic packets [1].…”

“…We argue in Section III that the algorithms of Sections II are special cases of a general dynamic programming algorithm for finding the best object in a multitree dictionary. We moreover show in Section IV that another special case is a recursive version of the algorithm from [1], [5] to find the best 2-D anisotropic wavelet packet basis. Yet another special case of our algorithm is the best multitree local cosine basis [9].…”