Wigner distribution and trasform properties for space-time nonstationary signal representation

This presentation addresses a synthetic overview and comparison of the main and novel techniques related to the problem of early signal processing for machine vision and visual recognition applications, under two sources of variability: geometric and photometric, for bioengineering applications. In bioengineering applications, the concept of stationary time or space data series is a mathematical idealization, which is never realized by practical applications. In general, the spectral content of the time or space series changes with time, and the time-averaged or space-averaged amplitudes found by traditional methods are inadequate to effectively describe such phenomena. While the Fourier Transform of the entire time or space series does contain information about the spectral components in a time or space series, for a large class of practical applications this information is inadequate. It would be desirable to have a joint Time Frequency Representation (TFR) of physical data. A TFR with frequency-dependent resolution while at the same time maintaining the direct relationship, through time-averaging or space-averaging, with the Fourier spectrum could be even more effective. Over the years, several techniques of examining the time-varying nature of the spectrum have been proposed; among them are the Short Time Fourier Transform, the Continuous Wavelet Transform, the Gabor Transform, and the bilinear class of time frequency distributions known as Cohen's class of which the Wigner Distribution is a member. After a brief introduction on the main research topics for machine vision and human vision analytic approaches, this paper is divided into four main sections and it will take a brief look at thefollowing analysis and representation techniques: Part One - Basic Traditional Transform Relationships: Laplace Transform, Mellin Transform, Hilbert Transform And Analytical Signal. Part Two - The Main Limitations of Traditional Analysis And Representation Techniques: Fourier Analysis, Analytic Signal & Instantaneous Frequency. Part Three - Linear TFRs: Short Time Fourier Transform (Spectrograms), Continuous Wavelet Transform (Scalograms), Discrete Orthonormal Wavelet Transform, Phase Correcting the Wavelet Transform: S Transform (Bobograms), Discrete S Transform, Gabor's Signal Expansion And Gabor Transform. Part Four - Bilinear TFRs: Cohen's Class of Generalized Time-Frequency Representations, Wigner (Ville) Distribution Function and Transform.