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2001 | Buch

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Ripples in Mathematics

The Discrete Wavelet Transform

verfasst von: Arne Jensen, Anders la Cour-Harbo

Verlag: Springer Berlin Heidelberg

Enthalten in: Professional Book Archive

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Über dieses Buch

Yet another book on wavelets. There are many books on wavelets available, written for readers with different backgrounds. But the topic is becoming ever more important in mainstream signal processing, since the new JPEG2000 standard is based on wavelet techniques. Wavelet techniques are also impor tant in the MPEG-4 standard. So we thought that there might be room for yet another book on wavelets. This one is limited in scope, since it only covers the discrete wavelet trans form, which is central in modern digital signal processing. The presentation is based on the lifting technique discovered by W. Sweldens in 1994. Due to a result by I. Daubechies and W. Sweldens from 1996 this approach covers the same class of discrete wavelet transforms as the one based on two channel filter banks with perfect reconstruction. The goal of this book is to enable readers, with modest backgrounds in mathematics, signal analysis, and programming, to understand wavelet based techniques in signal analysis, and perhaps to enable them to apply such methods to real world problems. The book started as a set of lecture notes, written in Danish, for a group of teachers of signal analysis at Danish Engineering Colleges. The material has also been presented to groups of engineers working in industry, and used in mathematics courses at Aalborg University.

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Inhaltsverzeichnis

Frontmatter

1.. Introduction

Abstract

This book gives an introduction to the discrete wavelet transform, and to some of its generalizations. The transforms are defined and interpreted. Some examples of applications are given, and the implementation on the computer is described in detail. The book is limited to the discrete wavelet transform, which means that the continuous version of the wavelet transform is not presented at all. One of the reasons for this choice is the intention that the book should be accessible to readers with rather modest mathematical prerequisites. Another reason is that for readers with good mathematical prerequisites there exists a large number of excellent books presenting the continuous (and often also the discrete) versions of the wavelet transform.

Arne Jensen, Anders la Cour-Harbo

2.. A First Example

Abstract

In this chapter we introduce the discrete wavelet transform, often referred to as DWT, through a simple example, which will reveal some of its essential features. This idea is due to C.Mulcahy [20], and we use his example, with a minor modification.

Arne Jensen, Anders la Cour-Harbo

3.. The Discrete Wavelet Transform via Lifting

Abstract

We now introduce the discrete wavelet transform via a procedure, which is known as ‘lifting’ in the literature. This procedure was introduced by W. Sweldens in a series of papers [25, 26].

Arne Jensen, Anders la Cour-Harbo

4.. Analysis of Synthetic Signals

Abstract

The discrete wavelet transform has been introduced in the previous two chapters. The general lifting scheme, as well as some examples of transforms, were presented, and we have seen one application to a signal with just 8 samples. In this chapter we will apply the transform to a number of synthetic signals, in order to gain some experience with the properties of the discrete wavelet transform. We will process some signals by transformation, followed by some alteration, followed by inverse transformation, as we did in Chap. 2 to the signal with 8 samples. Here we use significantly longer signals. As an example, we will show how this approach can be used to remove some of the noise in a signal. We will also give an example showing how to separate slow and fast variations in a signal.

Arne Jensen, Anders la Cour-Harbo

5.. Interpretation

Abstract

In this chapter we start with an interpretation of the discrete wavelet transform based on the Haar building block. Then we will give interpretations of wavelet transforms based on more general building blocks. The last part of this chapter can be omitted on a first reading.

Arne Jensen, Anders la Cour-Harbo

6.. Two Dimensional Transforms

Abstract

In this chapter we will briefly show how the discrete wavelet transform can be applied to two dimensional signals, such as images. The 2D wavelet transform comes in two forms. One which consists of two 1D transforms, and one which is a true 2D transform. The first type is called separable, and the second nonseparable. We present some results and examples in the separable case, since it is a straightforward generalization of the results in the one dimensional case. At the end of the chapter we give an example of a nonseparable 2D DWT based on an adaptation of the lifting technique to the 2D case.

Arne Jensen, Anders la Cour-Harbo

7.. Lifting and Filters I

Abstract

Our discussion of the discrete wavelet transform has to this point been based on the time domain alone. We have represented and treated signals as sequences of sample points x = {x[n]}_{n ∈ z}. The index n ∈ Z represents equidistant sampling times, given a choice of time scale. But we will get a better understanding of the transform, if we also look at it in the frequency domain. The frequency representation is obtained from the time domain representation using the Fourier transform. We will refer to standard texts [22, 23] for the necessary background, but we recall some of the definitions and results below.

Arne Jensen, Anders la Cour-Harbo

8.. Wavelet Packets

Abstract

In the first chapters we have introduced the lifting technique, which is a method for defining and implementing the discrete wavelet transform. The definitions were given in Chap. 3, and simple examples were given in Chap. 4. In particular, we saw applications of the wavelet analysis to noise reduction. The applications were based on the one scale building block, applied several times. In this chapter we want to extend the use of these building blocks to define many new transforms, all called wavelet packets. In many applications these new transforms are the basis for the successful use of wavelets. Some examples are given in Chap. 13.

Arne Jensen, Anders la Cour-Harbo

9.. The Time-Frequency Plane

Abstract

Time-frequency analysis is an important tool in modern signal analysis. By using information on the distribution of the energy in a signal with respect to both time and frequency, one hopes to gain additional insight into the nature of signals.

Arne Jensen, Anders la Cour-Harbo

10.. Finite Signals

Abstract

In the previous chapters we have only briefly, and in a casual way, considered the problems arising from having a finite signal. In the case of the Haar transform there are no problems, since it transforms a signal of even length to two parts, each of half the original length. In the case of infinite length signals there are obviously no problems either. But in other cases we may need for instance sample s[−1], and our given signal starts with sample s[0]. We will consider solutions to this problem, which we call the boundary problem. Theoretical aspects are considered in this chapter. It is important to understand that there is no universal solution to the boundary problem. The preferred solution depends on the kind of application one has in mind The implementations are discussed in Chap. 11. The reader mainly interested in implementations and applications can skip ahead to this chapter.

Arne Jensen, Anders la Cour-Harbo

11.. Implementation

Abstract

In order to implement the methods presented in the previous chapters some software package is obviously needed. We have chosen to use MATLAB, since it is well suited to perform signal processing. It is available for many platforms, and the programming language is independent of the platform. Although many things are explained in detail, we still assume some familiarity with MATLAB. If you have never used MATLAB, you should start by becoming familiar with the facilities, using the introduction shipped with MAT-LAB. We will also assume that you have worked through all the examples in Chap. 13, before you start on this chapter.

Arne Jensen, Anders la Cour-Harbo

12.. Lifting and Filters II

Abstract

There are basically three forms for representing the building block in a DWT: The transform can be represented by a pair of filters (usually low pass and high pass filters) satisfying the perfect reconstruction conditions from Chap. 7, or it can be given as lifting steps, which are either given in the time domain as a set of equations, or in the frequency domain as a factored matrix of Laurent polynomials. The Daubechies 4 transform has been presented in all three forms in previous chapters, but so far we have only made casual attempts to convert between the various representations. When trying to do so, it turns out that only one conversion requires real work, namely conversion from filter to matrix and equation forms. In Chap. 7 we presented the theorem, which shows that it is always possible to do this conversion, but we did not show how to do it. This chapter is therefore dedicated to discussing the three basic forms of representation of the wavelet transform, as well as the conversions between them. In particular, we give a detailed proof of the `from filter to matrix/equation’ theorem stated in Chap. 7. The proof is a detailed and exemplified version of the proof found in I. Daubechies and W. Sweldens [7].

Arne Jensen, Anders la Cour-Harbo

13.. Wavelets in Matlab

Abstract

In the previous chapters we have presented wavelets through the lifting technique. We have also shown that this approach is equivalent with the approach based on filters. This means that one can use any of the available software packages (all based on the filter approach) to experiment with wavelet analysis.

Arne Jensen, Anders la Cour-Harbo

14.. Applications and Outlook

Abstract

In this chapter we give some information on applications of wavelet based transforms. We only give brief descriptions and references, since each topic requires a different background of the reader. We also give some directions for further study of the vast wavelet literature.

Arne Jensen, Anders la Cour-Harbo

Backmatter

Titel: Ripples in Mathematics
verfasst von: Arne Jensen
Anders la Cour-Harbo
Copyright-Jahr: 2001
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-56702-5
Print ISBN: 978-3-540-41662-3
DOI: https://doi.org/10.1007/978-3-642-56702-5

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