Horizontal and vertical ridges associated to continuous wavelet transforms

By studying the wavelet transform behavior of asymptotic signals, the so-called ridge and skeleton leading to a simplified representation of a given signal have been defined. This skeleton allows the estimation of the frequency and the amplitude modulation laws associated with each elementary contribution. In the particular case of signals composed of spectral lines (monochromatic signals modulated in amplitude), the ridges can be set to the horizontal by disentangling the components, which allows the estimation of the frequency and the amplitude modulation law of each spectral line. This technique permits a complete modeling of a sound signal, leading to various transformations such as transposition, cross synthesis, and time stretching. This mathematical formalism cannot be applied to signals containing fast transitions or isolated transients. The problem of the detection and characterization of such signals is addressed by means of vertical ridges. Locally homogeneous signals are emphasized in order to model transients corresponding to a discontinuity of a given derivative