Wavelet analysis of signals and images, a grand tour,

We review the general properties of the wavelet transform, both in its continuous and its discrete versions, in one or two dimensions, and we describe some of its applications in signal and image processing. We also consider its extension to higher dimensions and to the space-time context, for the analysis of moving objects. 1. MOTIVATION: WHAT IS WAVELET ANALYSIS? Wavelet analysis is a particular timeor space-scale representation of signals which 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, let us consider first the case of one-dimensional signals. It is a fact that most real life signals are nonstationary. They often contain transient components, sometimes very significant physically, and mostly cover a wide range of frequencies. In addition, there is frequently 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 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 long, whereas consonants contain a wide spectrum, up to very high frequencies, especially in the attack, but they are very short. Clearly standard Fourier analysis is inadequate for treating such signals, since it looses all information about the time localization of a given frequency component. In addition, it is very uneconomical. If a segment of the signal is almost flat, i.e., uninteresting, one still has to sum an infinite series for reproducing it. Worse yet, Fourier analysis is highly unstable with respect to perturbation, because of its global character. For instance, if one adds an extra term, with a very small amplitude, to a linear superposition of sine waves, the signal will barely be modified, but the Fourier spectrum will be completely perturbed. This does not happen if the signal is represented in terms of localized components. Therefore, signal analysts turn to time-frequency (TF) representations. The idea is that one needs two parameters. One, called a, characterizes the frequency, the other one, b, indicates the position in the signal. This concept of a TF representation is in fact quite old and familiar. The most obvious example is simply a musical score! If one requires, in addition, the transform to be linear, a general TF transform will take the form: s(x) 7→ S(b, a) = ∫ ∞ −∞ ψba(x) s(x) dx, (1.1) where s is the signal and ψba the analyzing function (we denote the time variable by x, in view of the extension to higher dimensions). Within this class, two TF transforms stand out as particularly simple and efficient, the windowed or short time Fourier transform (STFT) and the wavelet transform (WT). For both of them, the analyzing function ψba is obtained by a group action on a basic (or mother) function ψ, only the group differs. The essential difference between the two is in the way the frequency parameter