There is a very large literature on the theoretical underpinnings of the wavelet transform. However, theory must be complemented with a significant amount of practical work. Selection of method, implementation, validation of results, comparison with alternatives – these are all centrally important for the applied scientist or engineer. Turning theory into practice is the theme of this book. Various applications have benefited from the wavelet and other multiscale transforms. In this book, we describe many such applications, and in this way illustrate the theory and practice of such transforms. We describe an ‘embedded systems’ approach to wavelets and multiscale transforms in this book, in that we introduce and appraise appropriate multiscale methods for use in a wide range of application areas.

Astronomy provides an illustrative background for many of the examples used in this book. Chapters 5 and 6 cover problems in remote sensing. Chapter 3, dealing with noise in images, includes discussion on problems of wide relevance. At the time of writing, the authors are applying these methods to other fields: medical image analysis (radiology, for mammography; echocardiology), plasma physics response signals, and others.

Chapter 1 provides an extensive review of the theory and practice of the wavelet transform. This chapter then considers other multiscale transforms, offering possible advantages in regard to robustness. The reader wishing early ‘action’ may wish to read parts of Chapter 1 at first, and dip into it again later, for discussion of particular methods.

Image registration of remotely sensed data is a procedure that determines the best spatial fit between two or more images that overlap the same scene, acquired at the same time or at different times, by identical or different sensors. This is an important step, as it is frequently necessary to compare data taken at different times on a point-to-point basis, for many applications such as the study of temporal changes for example. Therefore we need to obtain a new dataset in such a way that its image under an appropriate transform is registered, geometrically, with previous datasets.

The inventory of natural resources and the management of the environment requires appropriate and complex perception of the objects to be studied. Often a multiresolution approach is essential for the identification of the phenomena studied, as well as for the understanding of the dynamical processes underlying them. In this case, the processing of data taken with different ground resolutions by different or identical sensors is necessary.

Another important situation where the need for different images acquired with a different ground resolution sensor arises is when the generalization to larger surface areas of an identification or an interpretation model, based on small areas, is required (Achard and Blasco, 1990). This is the case for studies on a continental scale. Examples of this application can be found in Justice and Hiernaux (1986), Hiernaux and Justice (1986) and Prince, Tucker and Justice (1986). Therefore, the data must be geometrically registered with the best possible accuracy.

Definitions

Differences in images of real world scenes may be induced by the relative motion of the camera and the scene, by the relative displacement of two cameras or by the motion of objects in the scene. These differences are important because they contain enough information to allow a partial reconstruction of the three-dimensional structure of the scene from its two-dimensional projections. When such differences occur between two images, we say that there is a disparity between them, which may be represented by a vector field mapping one image onto the other (Barnard and Thompson, 1980). The evaluation of the disparity field has been called the correspondence problem (Duda and Hart, 1973). Time-varying images of the real world can provide kinematical, dynamical and structural information (Weng, Huang and Ahuja, 1989). The disparity field can be interpreted into meaningful statements about the scene, such as depth, velocity and shape.

Disparity analysis, in the sense of stereovision, may be broadly defined as the evaluation of the existing geometrical differences, in a given reference frame, between two or more images of the same or similar scenes. The differences in remote sensing are mainly the result of different imaging directions. The goal of the analysis is to assign disparities, which are represented as two-dimensional vectors in the image plane, to a collection of well-defined points in one of the images. Disparity analysis is useful for image understanding in several ways.

A software package has been implemented by CEA (Saclay, France) and Nice Observatory. This software includes almost all applications presented in this book. Its goal is not to replace existing image processing packages, but to complement them, offering to the user a complete set of multiresolution tools. These tools are executable programs, which work on different platforms, independent of any image processing system, and allow the user to carry out various operations using multiresolution on his or her images such as a wavelet transform, filtering, deconvolution, etc.

The programs, written in C++, are built on three classes: the ‘image’ class, the ‘multiresolution’ class, and the ‘noise-modeling class’. Figure B.1 illustrates this architecture. A multiresolution transform is applied to the input data, and noise modeling is performed. Hence the multiresolution support data structure can be derived, and the programs can use it in order to know at which scales, and at which positions, significant signal has been detected. Several multiresolution transforms are available (see Figure B.2), allowing much flexibility. Fig. 2.7 in Chapter 2 summarizes how the multiresolution support is derived from the data and our noise-modeling.

A set of IDL (Interactive Data Language) routines is included in the package which interfaces IDL and the C++ executables. A comprehensive package, MR/1, is available. Further details on it can be found at http://www.multiresolution.com or from email addresses [email protected] or [email protected].

Waves are everywhere

It is useful to be completely at home with the properties of waves because they occur in so many different fields of physics. Thus quantum mechanics, optics, electromagnetism, stretched strings and membranes, seismology and sound are just some of the topics in which waves are very common. Even from the limited viewpoint of doing well in examinations, it pays to learn about waves because they are likely to occur in many different types of physics problems, as well as in mathematics papers.

For concreteness, most of the language we will use will be that for transverse waves on an elastic string; with suitable modification it can be applied to other types of examples. The string is taken as lying along the x axis when no wave is present, and in order to avoid end-effects will usually be assumed to extend to infinity in both directions. The transverse displacement is in the y direction. Thus we are interested in how y varies as we look at the wave; it will be a function of both the position x along the string and the instant t that we look at it.

Waves can usually be of almost any shape. For example, we could arrange that the initial displacement on the string (i.e. what would be seen in a photograph of the string) looks like a single square pulse, an infinite repetition of square pulses, five cycles of a sine wave, some arbitrary complicated shape, or an infinitely repeating sinusoidal wave (see fig. 14.1).

The ideas that motivated me to write this text, and my philosophy about what it should contain and how it should be used, were expounded at length in the Preface to Volume 1. Here I would like only to reemphasise that, in order to derive optimum benefit from this book, it is essential for the reader to work through the problems at the end of each chapter. This is the only way of making sure that the material covered has been absorbed. The problems are relatively few in number, so that it should be reasonable to attempt them all.

I wish to express my thanks to the various people who have offered me advice concerning the topics covered in this volume. They include David Acheson, Ian Aitchison, David Andrews, Peter Clifford, Gideon Engler, Raymond Hide, Moshe Kugler, Elaine Lyons, John Roe, Lee Segal, Robert Thorne and Andrew Tolley, as well as the Jesus College first-year physics students of 1996 who read and commented on the text.

I am most grateful to Brenda Willoughby for her sterling work in typing this document, and to Irmgard Smith for producing the beautiful diagrams.

Introduction

What is a partial differential equation?

It is tea-time, and you have decided to make some smoked salmon sandwiches. A slight inconvenience is that you have only just removed the loaf of bread from the freezer and need to let it defrost. How long is this going to take?

The way in which the temperature rises for any small region in the interior of the loaf depends on the rate at which heat is conducted into that region. This in turn depends on the temperature gradients within the loaf. Thus there is a relationship between the spatial and the time derivatives of the temperature T of the bread. This relationship is a partial differential equation in that it involves partial derivatives of T (with respect to x and with respect to t).

This is typical of partial differential equations. In contrast to ordinary differential equations, which have only one independent variable (see Chapter 5), here we consider differential equations which involve at least two independent variables. Because the dependent variable of our partial differential equation (T in the above example) is a function of these independent variables, the derivatives are necessarily partial ones. The solution of the equation then involves finding a specific functional dependence for, say, T in terms of position and time, which satisfies the particular requirements of the given problem.

Specific examples

Here we describe and derive some of the more common examples of partial differential equations.