The updates and changes from the earlier version of the book have to a large extent been driven by the comments of readers and reviewers. The second edition has been extended with Chapters 2, 24, and 25 that cover dimensional analysis, the asymptotic evaluation of integrals, and variational calculus, respectively. In a few places, new material has been inserted, such as Section 19.5 that covers the reciprocity of wave propagation. A number of teachers and students remarked that the level of difficulty of the problems in the first edition was highly variable. The problems in the second edition contain more hints and advice to make these problems more tractable.

What is wavelet analysis?

Wavelet analysis is a particular time- or space-scale representation of signals that has found a wide range of applications in physics, signal processing and applied mathematics in the last few years. In order to get a feeling for it and to understand its success, we consider first the case of one-dimensional signals. Actually the discussion in this introductory chapter is mostly qualitative. All the mathematically relevant properties will be described precisely and proved systematically in the next chapter for the two-dimensional case, which is the proper subject of this book.

It is a fact that most real life signals are nonstationary (that is, their statistical properties change with time) and they usually cover a wide range of frequencies. Many signals contain transient components, whose appearance and disappearance are physically very significant. Also, characteristic frequencies may drift in time (e.g., in geophysical time series – one calls them pseudo-frequencies). In addition, there is often a direct correlation between the characteristic frequency of a given segment of the signal and the time duration of that segment. Low frequency pieces tend to last for a long interval, whereas high frequencies occur in general for a short moment only. Human speech signals are typical in this respect: vowels have a relatively low mean frequency and last quite a long time, whereas consonants contain a wide spectrum, up to very high frequencies, especially in the attack, but they are very short.

In the previous chapter, we have discussed a number of applications of the 2-D CWT that belong essentially to the realm of image processing. Besides these, however, there are plenty of applications to genuine physical problems, in such diverse fields as astrophysics, geophysics, fluid dynamics or fractal analysis. Here the CWT appears as a new analysis tool, that often proves more efficient than traditional methods, which in fact rarely go beyond standard Fourier analysis. We will review some of these applications in the present chapter, without pretention of exhaustivity, of course. Our treatment will often be sketchy, but we have tried to provide always full references to the original papers.

Astronomy and astrophysics

Wavelets and astronomical images

Astronomical imaging has distinct characteristics. First, the Universe has a marked hierarchical structure, almost fractal. Nearby stars, galaxies, quasars, galaxy clusters and superclusters have very different sizes and live at very different distances, which means that the scale variable is essential and a multiscale analysis is in order, instead of the usual Fourier methods. This suggests wavelet analysis. Now, the main problem is that of detecting particular features, relations, groupings, etc., in images, which leads us to prefer the continuous WT over the discrete WT. Finally, there is in general no privileged direction, nor particular oriented features, in astrophysical images.

Wavelets are everywhere nowadays. Be it in signal or image processing, in astronomy, in fluid dynamics (turbulence), in condensed matter physics, wavelets have found applications in almost every corner of physics. In addition, wavelet methods have become standard in applied mathematics, numerical analysis, approximation theory, etc. It is hardly possible to attend a conference on any of these fields without encountering several contributions dealing with them. Correspondingly, hundreds of papers appear every year and new books on the topic get published at a sustained pace, with publishers strongly competing with each other. So, why bother to publish an additional one?

The answer lies in the finer distinction between various types of wavelet transforms. There is, indeed, a crucial difference between two approaches, namely, the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT). Furthermore, one has to distinguish between problems in one dimension (signal analysis) and problems in two dimensions (image processing), since the status of the literature is very different in the two cases.

Take first the one-dimensional case. Beginning with the classic textbook of Ingrid Daubechies [Dau92], several books, such as those of M. Holschneider [Hol95], B. Torrésani [Tor95] or A. Arnéodo et al. [Arn95], cover the continuous wavelet transform, in a more or less mathematically oriented approach.

The topic of this book is the application of mathematics to physical problems. Mathematics and physics are often taught separately. Despite the fact that education in physics relies on mathematics, it turns out that students consider mathematics to be disjoint from physics. Although this point of view may strictly be correct, it reflects an erroneous opinion when it concerns an education in the sciences. The reason for this is that mathematics is the only language at our disposal for quantifying physical processes. One cannot learn a language by just studying a textbook. In order to truly learn how to use a language one has to go abroad and start using that language. By the same token one cannot learn how to use mathematics in the physical sciences by just studying textbooks or attending lectures; the only way to achieve this is to venture into the unknown and apply mathematics to physical problems.

It is the goal of this book to do exactly that; problems are presented in order to apply mathematical techniques and knowledge to physical concepts. These examples are not presented as well-developed theory. Instead, they are presented as a number of problems that elucidate the issues that are at stake. In this sense this book offers a guided tour: material for learning is presented but true learning will only take place by active exploration. In this process, the interplay of mathematics and physics is essential; mathematics is the natural language for physics while physical insight allows for a better understanding of the mathematics that is presented.

Introduction

We live in a world where objects (cars, animals, men, birds, aeroplanes, the Sun, etc.) that surround us are constantly in relative motion. One would like to extract the motion information from the observation of the scene and use it for various purposes, such as detection, tracking and identification. In particular, tracking of multiple objects is of great importance in many real world scenarios. The examples include traffic monitoring, autonomous vehicle navigation, and tracking of ballistic missile warheads. Tracking is a complex problem, often requiring to estimate motion parameters – such as position, velocity – under very challenging situations. Algorithms of this type typically have difficulty in the presence of noise, when the object is obscured, in situations including crossing trajectories, and when highly maneuvering objects are present.

Most motion estimation (ME) techniques such as the ones based on block matching, optical flow, and phase difference [Jah97,280,281] assume that the object is constant from frame to frame. That is, the signature of the object does not change with time. Consequently, these techniques tend to have difficulty handling complex motion, particularly when noise is present.

The time-dependent continuous wavelet transform (CWT) is attractive as a tool for analysis, in that important motion parameters can be compactly and clearly represented.

In the previous chapters, we have thoroughly discussed the 2-D CWT and some of its applications. Then we have made the connection with the group theoretical origins of the method, thus establishing a general framework, based on the coherent state formalism. In the present chapter, we will apply the same technique to a number of different situations involving higher dimensions: wavelets in 3-D space ℝ3, wavelets in ℝn (n > 3), and wavelets on the 2-sphere S2. Then, in the next chapter, we will treat time-dependent wavelets, that is, wavelets on space–time, designed for motion analysis.

In all cases, the technique is the same. First one identifies the manifold on which the signals are defined and the appropriate group of transformations acting on the latter. Next one chooses a square integrable representation of that group, possibly modulo some subgroup. Then one constructs wavelets as admissible vectors and derives the corresponding wavelet transform.

Three-dimensional wavelets

Some physical phenomena are intrinsically multiscale and three-dimensional. Typical examples may be found in fluid dynamics, for instance the appearance of coherent structures in turbulent flows, or the disentangling of a wave train in (mostly underwater) acoustics, as discussed above. In such cases, a 3-D wavelet analysis is clearly more adequate and likely to yield a deeper understanding [56].

Group theory and matrix geometry of wavelets

In Chapters 1 and 2, we have studied systematically the continuous wavelet transform in one and two dimensions, respectively. As already emphasized there, the properties of the transforms in the two cases are remarkably similar. In 2-D we have formalized them in the three propositions 2.2.1, 2.2.2 and 2.2.3, and essentially the same statements may be made in 1-D. A moment's reflection shows that one could write out, without difficulty, an entirely parallel mathematical description in any dimension n ≥ 1. Clearly there must be some unifying principle underlying the picture. The question is, of course, what is this principle? As so often in such situations, the answer is to be found in group representation theory, i.e., by looking at the underlying geometry of the space of signals. The various transformations (translation, rotation, zoom, etc.) that a signal may undergo, determine a set of mathematical symmetries, which, interestingly enough, can be expressed in simple matrix terms and, as will be made clear in the following, the signal space itself – as a mathematical object – emerges as a consequence of this geometry.

But we have been using group theory all along! Indeed, to draw on a literary analogy, like Molière's Monsieur Jourdain speaking in prose without knowing so, we have been using group-theoretical language throughout our analysis! It is the aim of the present chapter to demonstrate this fact.

The material of this chapter is usually not covered in a book on mathematics. The field of mathematics deals with numbers and numerical relationships. It does not matter what these numbers are; they may account for physical properties of a system, but they may equally well be numbers that are not related to anything physical. Consider the expression g = df/dt. From a mathematical point of view these functions can be anything, as long as g is the derivative of f. The situation is different in physics. When f(t) is the position of a particle, and t denotes time, then g(t) is a velocity. This relation fixes the physical dimension of g(t). In mathematical physics, the physical dimension of variables imposes constraints on the relation between these variables. In this chapter we explore these constraints. In Section 2.2 we show that this provides a powerful technique for spotting errors in equations. In the remainder of this chapter we show how the physical dimensions of the variables that govern a problem can be used to find physical laws. Surprisingly, while most engineers learn about dimensional analysis, this topic is not covered explicitly in many science curricula.

Two rules for physical dimensions

In physics every physical parameter is associated with a physical dimension. The value of each parameter is measured with a certain physical unit. For example, when I measure how long a table is, the result of this measurement has dimension “length”.