Discrete Spectra of Convolutions on Disks using Sturm-Liouville Theory
Arash Ghaani Farashahi, Gregory S. Chirikjian
TL;DR
The paper develops a discrete spectral framework for convolving functions supported on disks by exploiting polar Fourier analysis and Sturm-Liouville theory. It constructs orthonormal disk bases $\\Psi_{nm}^a(r,\\theta)$ under two boundary conditions, zero-value and derivative, yielding discrete eigenvalues $\\lambda_n=\\rho_{nm}^2$ from zeros of $J_m$ or $J_m'$. It then provides explicit closed-form coefficients $C_{n,m}^a[f]$ in terms of lattice Fourier coefficients $\\widehat{f}$, and derives comprehensive convolution formulas for both zero-padding and basis functions, together with Plancherel-type energy identities for each boundary condition. These results preserve rotational structure, enable rotation-invariant descriptors on disk-supported data, and avoid torus periodization, with boundary-condition choice tailored to the problem's nature.
Abstract
This paper presents a systematic study for analytic aspects of discrete spectra methods for convolution of functions supported on disks, according to the Sturm-Liouville theory. We then investigate different aspects of the presented theory in the cases of zero-value boundary condition and derivative boundary condition.