Overview

In the continuous wavelet transform, a function ? ("psi"), which in practice looks like a little wave, is used to create a family of wavelets ?( at + b) where a and b are real numbers, a dilating (compressing or stretching) the function ? and b translating (displacing) it. The word continuous refers to the transform, not the wavelets, although people sometimes speak of "continuous wavelets."

The continuous wavelet transform turns a signal f( t) into a function with two variables (scale and time), which one can call c( a, b):

This transformation is in theory infinitely redundant, but it can be useful in recognizing certain characteristics of a signal. In addition, the extreme redundancy is less of a problem than one might imagine; a number of researchers have found ways of rapidly extracting the essential information from these redundant transforms.

One such method reduces a redundant transform to its skeleton. When certain signals are represented by a continuous wavelet transform, all the significant information of the signal is contained in curves, or "ridges," says Bruno Torr sani of the French Centre National de Recherche Scientifique, who works at the University of Aix-Marseille II. These are essentially the points in the time-frequency plane "where the natural frequency of the translated and dilated wavelet coincides with the local frequency, or one of the local frequencies, of the signal." These ridges form the skeleton of the transform.

Torr sani,...