Super-resolution via Prony-Type Polynomials

Anna Veselovska, Jürgen Prestin

Session: General

Keywords: Prony's method, exponential sums, Prony-type polynomials, super-resolution, common zeros

TL;DR: This paper presents a method for recovering hidden frequencies in three dimensions using Prony-type polynomials, offering improved stability in noisy data with moderate sample size.

Abstract: The problem of hidden periodicity in three dimensions is to recover frequency vectors $\omega_1, \dots, \omega_N \in [0,2\pi)^3$ using finitely many samples of the exponential sum $ f(n)=\sum_{j=1}^{N}a_j\exp{(-\mathrm{i}\langle \omega_j, n \rangle)},$ where $a_1,\ldots ,a_N \in \mathbb{C}\backslash\{0\}$ and $ n \in \mathbb{Z}^3$. Inspired by the previous approaches, we consider specifically constructed polynomials, which are called Prony-type polynomials, and show that the frequency vectors $ \omega_1, \dots, \omega_N $ can be recovered via a set of common zeros of such polynomials. By employing Cantor tuple functions, we position the method of Prony-type polynomials within the spectrum of sampling requirements between the methods proposed by Kunis et al. (2016) and by Cuyt and Wen-Shin (2016). While the Prony-type polynomial method demands more samples than the approach of Cuyt and Wen-Shin, numerical experiments indicate that it exhibits greater stability in the presence of noisy data.

Submission Number: 88