Characterization of signals by the ridges of their wavelet transforms

International audienceWe present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of one-dimensional signals. These detection procedures are shown to be robust to additive white noise. We also derive and test a new reconstruction procedure. The latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge. This provides with a very efficient way to code the information contained in the signal

On the Analytic Wavelet Transform

An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of non-negligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias

On the extraction of instantaneous frequencies from ridges in time-frequency representations of signals

The extraction of oscillatory components and their properties from different time-frequency representations, such as windowed Fourier transform and wavelet transform, is an important topic in signal processing. The first step in this procedure is to find an appropriate ridge curve: a sequence of amplitude peak positions (ridge points), corresponding to the component of interest. This is not a trivial issue, and the optimal method for extraction is still not settled or agreed. We discuss and develop procedures that can be used for this task and compare their performance on both simulated and real data. In particular, we propose a method which, in contrast to many other approaches, is highly adaptive so that it does not need any parameter adjustment for the signal to be analysed. Being based on dynamic path optimization and fixed point iteration, the method is very fast, and its superior accuracy is also demonstrated. In addition, we investigate the advantages and drawbacks that synchrosqueezing offers in relation to curve extraction. The codes used in this work are freely available for download.Comment: 13 pages, 7 figures, plus 4 supplementary figure

Multi-Ridge Detection and Time-Frequency Reconstruction

International audienceThe ridges of the wavelet transform, the Gabor transform or any time-frequency representation of a signal contain crucial information on the characteristics of the signal. Indeed they mark the regions of the time-frequency plane where the signal concentrates most of its energy. We introduce a new algorithm to detect and identify these ridges.The procedure is based on an original form of Markov Chain Monte Carlo algorithm specially adapted to the present situation. We show that this detection algorithm is especially useful for noisy signals with multi-ridge transforms. It is a common practice among practitioners to reconstruct a signal from the skeleton of a transform of the signal (i.e. the restriction of the transform to the ridges). After reviewing several known procedures we introduce a new reconstruction algorithm and we illustrate its efficiency on speech signals

Blind Multiridge Detection and Reconstruction Using Ultrasonic Signals

Time-frequency signal analysis has been widely applied in the modern radar, acoustic, sonar and ultrasonic signal processing techniques. Recently, the nondestructive testing (NDT) techniques via the ultrasonic instrumentation have shown the striking capability of the quality control for the material fabrication industry. In this thesis, we first provide a general mathematical model for the ultrasonic signals collected by pulse-echo sensors and then design a totally blind, novel, signal processing NDT technique relying on neither a priori signal information nor any manual effort. The signature signal can be blindly extracted by using the automatic optimal frame size selection for further modeling and characterization of the ultrasonic signal using Gabor analysis. This modeled signature signal is used for multiridge detection and for reconstruction of the signal. The detected ridge information can be used to estimate the transmission and attenuation coefficients, shear modulus, and Young’s modulus associated with any arbitrary material sample for fabrication quality control. Thus, our algorithm can be applied for ultrasonic signal characterization and ridge detection in non-destructive testing for new material fabrication. Experimental results show that the ridge detection performance by our proposed method is superior to that of the existing techniques