Frequency Analysis Of Signals and Images Using G-Lets

Abstract

This paper presents a method of frequency analysis for discrete signals using G-lets. From a group of transformations and representation theory, a finite basis of the signal space is obtained. The projections of the signal onto this basis are called G-lets. G-lets, due to the nature of transformations used, contain oscillations in such a manner that the difference between consecutive G-let coefficients is proportional to the local frequency. The signal frequency, in turn, is proportional to the difference in amplitude of the signal at any point. A dilation operation is defined to capture the frequencies without use of a windowing function, by highlighting the highest frequency of the existing signal in a G-let. Considering features of a signal as a combination of frequencies, feature extraction of 1-D signals and images are examined. The beginning and end of each feature are identified by the spread of low frequencies in the neighborhood of a high frequency. Results are demonstrated using dihedral groups, for simple 1-D signals, an ECG signal, and ‘Lena’ image. A qualitative comparison is provided with wavelets and Fourier analysis.