Wavelets: An Elementary Treatment of Theory and Applications

The following sections are included:

• Approximations and Decompositions

• Preface

• Contents

This chapter gives a short elementary introduction to wavelets. We give a few properties of continuous wavelets, a few remarks on multiresolution analysis, and the construction for the first few spline wavelets as solutions of dilation equations. We also describe an example of the compactly supported orthonormal Daubechies wavelets. All this will be discussed in more depth in the following chapters.

This paper contains some mathematical preliminaries needed for a good understanding of wavelets. The first section discusses Hilbert spaces. The second section deals with Fourier transformation. Finally, the third section gives a short introduction to Riesz bases and frames.

The continuous affine wavelet transform is introduced and its main properties are discussed. The relationship with coherent states and with a square integrable representation of the ax+b group is given. Next wavelet type transforms are associated with more general groups. This is finally specialized to the case of the Heisenberg group, which leads to the windowed Fourier transform.

In this paper we present an elementary discussion of the discrete wavelet transform. A major problem is formed by the construction of an orthonormal wavelet basis of the Hilbert space of square integrable functions. It is shown that the concept of a multiresolution analysis is very helpful to make such a construction. With every multiresolution analysis one can associate a father wavelet, the translates of which define an orthonormal system, and a mother wavelet which forms the basis for the discrete wavelet transform. This transform can be associated with two filtering operations and their adjoints; in practice these are used to get a pyramid-like decomposition of a signal. The abstract theory is illustrated by means of two concrete examples, the sinc-wavelet and the Meyer-wavelet.

A wavelet analysis of a signal can separate information present at different scales. Moreover, wavelet bases are orthonormal. Therefore, the wavelet transformation can be used to remove redundancy from a signal. In this paper, it is demonstrated how the wavelet transform can be used for image compression.

In the first two sections of this paper we describe a construction of the compactly supported Daubechies' wavelets. In the third section we show how a Discrete Wavelet Transform of a function can be obtained. We end this paper with some interesting numerical illustrations.

In this chapter we describe a general strategy for the construction of wavelet bases when the loss of invariance under translation prevents the use of Fourier techniques.

Certain aspects of wavelet multiresolution analysis are closely related to concepts in the theory of multirate subband coding, a topic from the field of digital signal processing (DSP). In this tutorial we picture this relationship from the DSP point of view.

In many applications concerning wavelets, inner products of arbitrary functions f(x) with wavelets and scaling functions have to be calculated. This paper involves the calculation of these inner products from function evaluations of f(x). Both one point and multiple point quadrature formulae are considered, and a well-conditioned construction is presented.

An account is given of part of the theory in the paper “Fast wavelet transforms and numerical algorithms I” by Beylkin, Coifrnan, and Rokhlin. It will be shown that wavelet techniques can be used in order to compute, with rescribed accuracy, the action of a Calderón-Zygmund operator on a function on [0,1] in a number of operations which increases slightly faster than the inverse of the scale at which function and operator are considered. No results of numerical experiments are discussed.

This contribution consists of two parts. In the first part (Sections 1-5) we introduce a wavelet transform for finite-length, real sequences. By means of this finite wavelet transform (or FWT) a signal is expanded in terms of wavelet-like vectors, which are defined by discretisation and truncation of “continuous” wavelets. Such expansions are discussed from the point of view of the theory of frames. Some pictures of wavelet-transformed signals clearly exhibit the fact that the transform yields a time-frequency representation of the signals. In the part (Sections 6-8) we discuss an application in the area of seismic processing in which the advantage of the wavelet transform over the Fourier transform is made clear.

In this paper I discuss and describe the ideas of analysing fractal structures by means of wavelets as they are presented in [1-6, 8, 29, 30, 35] and elsewhere. The wavelet transform can serve as a kind of mathematical microscope to see the fractal nature of various objects at a point and to measure the local scaling symmetry parameters at points.

The following sections are included:

• Subject Index