The Continuous Wavelet Transform, from 1 to 3 Dimensions

Abstract

Wavelet analysis is a particular time- or space-scale representation of signals which has found a wide range of applications in physics and mathematics in the last few years. In order to understand its success, let us consider first the case of one-dimensional signals. As a matter of fact, most real life signals are nonstationary. They often contain transient components, sometimes physically significant, 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, especially in the attack, and are often very short.