Fourier Analysis and q-Gaussian Functions: Analytical and Numerical Results
This work addresses theoretical issues for researchers in signal processing, but it is incremental as it builds on existing Fourier analysis without introducing new methods or applications.
The paper tackles the challenge of using the q-Gaussian kernel, a generalization of the Gaussian, in signal processing by analyzing its Fourier transform and properties in one and two dimensions, but does not report concrete numerical results or performance gains.
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.