NONSEPARABLE WALSH - TYPE FUNCTIONS ON R d

We study wavelet packets in the setting of a multiresolution analysis of L2(Rd) generated by an arbitrary dilation matrix A satisfying |det A| = 2. In particular, we consider the wavelet packets associated with a multiresolution analysis with a scaling function given by the characteristic function of some set (called a tile) in Rd. The functions in this class of wavelet packets are called generalized Walsh functions, and it is proved that the new functions share two major convergence properties with the Walsh system defined on [0, 1). The functions constitute a Schauder basis for Lp(Rd), 1 < p < ∞, and the expansion of Lp-functions converge pointwise almost everywhere. Finally, we introduce a family of compactly supported wavelet packets in R2 of class Cr(R2), 1 ≤ r < ∞, modeled after the generalized Walsh function. It is proved that this class of smooth wavelet packets has the same convergence properties as the generalized Walsh functions. INTRODUCTION Wavelet analysis was originally introduced in order to improve seismic signal processing by switching from short-time Fourier analysis to new algorithms better suited to detect and analyze abrupt changes in signals. It corresponds to a decomposition of phase space in which the trade-off between time and frequency localization has been chosen to provide better and better time localization at high frequencies in return for poor frequency localization. This makes the analysis well adapted to the study of transient phenomena and has proven a very successful approach to many problems in signal processing, numerical analysis, and quantum mechanics. Nevertheless, for stationary signals wavelet analysis is outperformed by short-time Fourier analysis. Wavelet packets were introduced by Coifman et al. [5] to improve the poor frequency localization of wavelet bases at high frequencies and thereby provide a more efficient decomposition of signals containing both transient and stationary components. So far most work on wavelet packets has been done in one dimension or using separable wavelet packets in higher dimensions (i.e., tensor products of one dimensional wavelet packets). However, separable wavelet and wavelet packet bases both have several drawbacks for the application to fields like image analysis since they impose an unavoidable line structure on the plane. For example, the zero set of a separable wavelet packet at high frequencies will contain a large number (same order of magnitude as the frequency) of horizontal and vertical lines that may create artifacts in the reconstructed image. Another potential problem is in the Fourier domain where separable two-dimensional wavelet packets have four characteristic peaks making it hard to selectively localize a unique frequency. Coifman and Meyer introduced the socalled Brushlets in [11] to remove the “uncertainty” in frequency localization, however 1 NONSEPARABLE WALSH-TYPE FUNCTIONS ON Rd 2 the Brushlets are essentially Fourier transforms of smooth local trigonometric bases and are therefore no longer functions associated with a multiresolution structure. Another example of nonseparable orthonormal bases with good frequency resolution is Donoho’s Ridgelets [7]. The aim of the present paper is to construct nonseparable wavelet packet bases for L2(Rd) with nice convergence properties. In section 1 we introduce wavelet packets associated with the class of multiresolution analyses of L2(Rd) for which there are associated wavelet bases generated by only one wavelet. Section 1 is rather brief due to the fact that the construction is similar to the well known one dimensional theory of wavelet packets. The wavelet packets constructed provide the same large number of orthonormal bases as wavelet packets in one-dimension, and they provide a good platform for doing image analysis using the well known “best basis” algorithm of Coifman and Wickerhauser. The paper [3] contains several numerical experiments with the wavelet packets of Section 1. In Section 2 we study a special type of multiresolution analysis that generalizes the well known Haar multiresolution analysis from L2(R). Section 3 contains results on a special wavelet packets construction that can be considered the multidimensional generalization of the Walsh system on [0, 1). We prove that this multidimensional generalization share the two most important convergence properties of the classical Walsh system: the new system is a Schauder basis for Lp(Rd), 1 < p < ∞, and the expansion of every Lp-function in the system converges pointwise a.e. Section 4 contains the main result of the present paper. There we consider a class of smooth wavelet packets, called Walsh-type wavelet packets, which shares a number of properties with the Walsh functions from Section 3. In Theorem 4.10 (and Corollary 4.11) it is proved that the Walsh-type wavelet packet expansion of a function from Lp, 1 < p < ∞, converges pointwise a.e. More restricted results in the one dimensional setting were considered by the author in [15]. Periodic versions of the smooth wavelet packets of Section 4 are considered in Section 5, and finally Section 6 contains some explicit examples of filters that can be used to generate Ck(R2) wavelet packets for any k ≥ 1. 1. NONSTATIONARY WAVELET PACKETS We begin by recalling some facts about multiresolution analyses associated with a general dilation matrix that we will use later in this section to define the wavelet packets we have in mind. The reader can find a more extensive discussion of the topic in [21]. Let A be a d× d-matrix such that A : Zd → Zd. If the eigenvalues of A all have absolute value strictly greater than 1 then we call A a dilation matrix. Example 1.1. The 2 × 2 matrices