Fourier Analysis and q-Gaussian Functions: Analytical and Numerical Results

TL;DR

This paper explores the properties of q-Gaussian functions, a nonextensive generalization of Gaussians, analyzing their Fourier transforms and implications for signal processing.

Contribution

It provides a comprehensive theoretical and numerical analysis of q-Gaussian functions in one and two dimensions, focusing on their Fourier transforms and related properties.

Findings

Derived analytical expressions for the Fourier transform of 1D and 2D q-Gaussians.

Identified key issues in computing the Fourier transform of 2D q-Gaussians.

Analyzed the q-Gaussian kernel using space window, cut-off frequency, and Heisenberg inequality.

Abstract

It is a consensus in signal processing that the Gaussian kernel and its partial derivatives enable the development of robust algorithms for feature detection. Fourier analysis and convolution theory have central role in such development. In this paper we collect theoretical elements to follow this avenue but using the q-Gaussian kernel that is a nonextensive generalization of the Gaussian one. Firstly, we review some theoretical elements behind the one-dimensional q-Gaussian and its Fourier transform. Then, we consider the two-dimensional q-Gaussian and we highlight the issues behind its analytical Fourier transform computation. We analyze the q-Gaussian kernel in the space and Fourier domains using the concepts of space window, cut-off frequency, and the Heisenberg inequality.