We may now conclude our overview of the “world according to (2-D) wavelets” [Bur98]. We have thoroughly analyzed the 2-D continuous wavelet transform, given some ideas about the discrete or discretized versions, discussed a large number of applications and generalizations (3-D, sphere, space–time). Where do we go now?

Why wavelets in the first place? When should one use them instead of other methods? Suppose we are facing a new signal or image. The very first question to ask is, what do we want to know or to measure from it? Depending on the answer, wavelets will or will not be useful. If we think they might be, we must next (i) choose a wavelet technique, discrete or continuous; (ii) then select a wavelet well adapted to the signal/image at hand, and (iii) determine the relevant parameter ranges. We emphasize that this approach is totally different from the standard one, based on Fourier methods. There is indeed no parameter to adjust here, the Fourier transform is universal. Wavelets on the other hand are extremely flexible, and the tool must be adapted each time to the situation at hand.

As for the first choice, discrete versus continuous WT, it is a fact that the vast majority of authors use the former, in particular if some data compression is required.

The 2-D CWT has been used by a number of authors, in a wide variety of problems [Com89,Mey91,Mey93]. In all cases, its main use is for the analysis of images, since image synthesis or compression problems are rather treated with the DWT. In particular, the CWT can be used for the detection or determination of specific features, such as a hierarchical structure, edges, filaments, contours, boundaries between areas of different luminosity, etc. Of course, the type of wavelet chosen depends on the precise aim. An isotropic wavelet (e.g. a Mexican hat) often suffices for pointwise analysis, but a directional wavelet (e.g. a Morlet or a conical wavelet) is necessary for the detection of oriented features in the signal. Somewhat surprisingly, a directional wavelet is often more efficient in the presence of noise.

In the next two chapters, we will review a number of such applications, including some nonlinear extensions of the CWT. First, in the present chapter, we consider various aspects of image processing. Then, in Chapter 5, we will turn to several fields of physics where the CWT has made an impact. Some of the applications are rather technical and use specific jargon. We apologize for that and refer the reader to the original papers for additional information.

Up to now, we have developed the 2-D CWT and a number of generalizations, relying in each case on the group-theoretical formalism. Given a class of finite energy signals and a group of transformations, including dilations, acting on them, one derives the corresponding continuous WT as soon as one can identify a square integrable representation of that group.

On the other hand, we have also briefly sketched the discrete WT and several transforms intermediate between the two. One conclusion of the study is that the pure DWT is too rigid, whereas redundancy is helpful, in that it increases both flexibility and robustness to noise of the transform. Indeed, the wavelet community has seen in the last few years a growing trend towards more redundancy and the development of tools more efficient than wavelets, such as ridgelets, curvelets, warplets, etc. The key word here is geometry: the new transforms and approximation methods take much better into account the geometrical features of the signals. To give a simple example, a smooth curve is in fact a 1-D object and it is a terrible waste (of times or bits) to represent it by a 2-D transform designed for genuine 2-D images.

It is therefore fitting to conclude the book by a chapter that covers these new developments.

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.

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 last chapter has already familiarized us with the use of group theoretical methods for the construction and analysis of wavelets and gaborettes. We aim in this chapter to first indicate the general applicability of these techniques and then to look at the case of the two-dimensional continuous transform, using the SIM(2) group. Later, we look at general matrix groups of the type that can be used for constructing other types of wavelet transforms in two dimensions. We shall be led, in this manner, to studying a class of semidirect product type groups, certain coadjoint orbits of which are isomorphic to the group itself. In all these cases, the common features of such a matrix-group analysis will be: (a) the group will refer to a set of possible symmetry transformations which the signal may undergo; (b) the space over which the signals are defined (as L2-functions) is intrinsic to the group; (c) the parameters in terms of which the wavelet transform is expressed are the parameters of the group itself, i.e., symmetry parameters of the signal, and (d) these parameter spaces, which arise as coadjoint orbits of the group, are also identifiable with phase spaces of signals.

Referring back to the 2-D wavelet transform introduced in Chapter 2, we shall see that this transform is again related to a square integrable representation of a matrix group.