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.

Potential fields play an important role in physics and geophysics because they describe the behavior of gravitational and electric fields as well as a number of other fields. Conversely, measurements of potential fields provide important information about the internal structure of bodies. For example, measurements of the electric potential at the Earth's surface when a current is sent into the Earth give information about the electrical conductivity while measurements of the Earth's gravity field or geoid provide information about the mass distribution within the Earth.

An example of this can be seen in Figure 21.1 in which the gravity anomaly over the northern part of the Yucatan peninsula in Mexico is shown [49]. The coast is visible as a thin white line. Note the ring structure that is visible in the gravity signal. These rings have led to the discovery of the Chicxulub crater which was caused by the massive impact of a meteorite. Note that the diameter of the impact crater is about 150 km! This crater is presently hidden by thick layers of sediments: at the surface the only apparent imprint of this crater is the presence of underground water-filled caves called “cenotes” at the outer edge of the crater. It was the measurement of the gravity field that made it possible to find this massive impact crater.

The equation that the gravitational or electric potential satisfies depends critically on the Laplacian of the potential.

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.

We all continue to feel a frustration because of our inability to foresee the soul's ultimate fate. Although we do not speak about it, we all know that the objectives of our science are, from a general point of view, much more modest than the objectives of, say, the Greek sciences were; that our science is more successful in giving us power than in giving us knowledge of truly human interest. [E. P. Wigner, 1972, The place of consciousness in modern physics, in Consciousness and Reality, eds. C. Muses and A. M. Young, New York, Outerbridge and Lizard, pp. 132–141].

In this book we have explored many methods of mathematics as used in the physical sciences. Mathematics plays a central role in the physical sciences because it is the only language we have for expressing quantitative relations in the world around us. In fact, mathematics not only allows us to express phenomena in a quantitative way, it also has a remarkable predictive power in the sense that it allows us to deduce the consequences of natural laws in terms of measurable quantities. In fact, we do not quite understand why mathematics gives such an accurate description of the world around us [120].

It is truly stunning how accurate some of the predictions in (mathematical) physics have been. The orbits of the planetary bodies can now be computed with an extreme accuracy. Morrison and Stephenson [72] compared the path of a solar eclipse at 181 BC with historic descriptions made in a city in eastern China that was located in the path of the solar eclipse.

Many problems in mathematical physics exhibit a spherical or cylindrical symmetry. For example, the gravity field of the Earth is to first order spherically symmetric. Waves excited by a stone thrown into water are usually cylindrically symmetric. Although there is no reason why problems with such a symmetry cannot be analyzed using Cartesian coordinates (i.e. (x, y, z)-coordinates), it is usually not very convenient to use such a coordinate system. The reason for this is that the theory is usually much simpler when one selects a coordinate system with symmetry properties that are the same as the symmetry properties of the physical system that one wants to study. It is for this reason that spherical coordinates and cylindrical coordinates are introduced in this section. It takes a certain effort to become acquainted with these coordinate systems, but this effort is well spent because it makes solving a large class of problems much easier.

Introducing spherical coordinates

In Figure 4.1 a Cartesian coordinate system with its x-, y-, and z-axes is shown as well as the location of a point r. This point can be described either by its x-, y-, and z-components or by the radius r and the angles θ and ϕ shown in Figure 4.1. In the latter case one uses spherical coordinates. Comparing the angles θ and ϕ with the geographical coordinates that define a point on the globe one sees that ϕ can be compared with longitude and θ can be compared with colatitude, which is defined as (latitude – 90 degrees).

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.

In most situations, the equations that we would like to solve in mathematical physics are too complicated to solve analytically. One of the reasons for this is often that an equation contains many different terms which make the problem simply too complex to be manageable. However, many of these terms may in practice be very small. Ignoring these small terms can simplify the problem to such an extent that it can be solved in closed form. Moreover, by deleting terms that are small one is able to focus on the terms that are significant and that contain the relevant physics. In this sense, ignoring small terms can actually give a better physical insight into the processes that really do matter.

Scale analysis is a technique in which one estimates the different terms in an equation by considering the scale over which the relevant parameters vary. This is an extremely powerful tool for simplifying problems. A comprehensive overview of this technique with many applications is given by Kline and in Chapter 6 of Lin et al. Interesting examples of the application of scaling arguments to biology are given by Vogel.

With the application of scale analysis one caveat must be made. One of the major surprises of classical physics of the twentieth century was the discovery of chaos in dynamical systems. In a chaotic system small changes in the initial conditions lead to a change in the time evolution of the system that grows exponentially with time.