Exponential polynomials and identification of polygonal regions from Fourier samples

TL;DR

This paper demonstrates that a small set of Fourier samples uniquely determines certain exponential polynomials and polygonal regions, with the sample size depending only on parameters like degree, number of vertices, and edge slopes.

Contribution

It introduces a nearly minimal sampling set for exponential polynomials and applies this to uniquely identify polygonal regions from Fourier samples, with bounds depending on geometric complexity.

Findings

A set of size O(D^2 N log N) determines exponential polynomials with degree < D and N terms.

Polygonal regions with up to k slopes are uniquely identified from O(k^2 N log N) Fourier samples.

Special case: axis-aligned polygons are determined from O(N log N) samples.

Abstract

Consider the set $E(D, N)$ of all bivariate exponential polynomials $$ f(\xi, \eta) = \sum_{j=1}^n p_j(\xi, \eta) e^{2\pi i (x_j\xi+y_j\eta)}, $$ where the polynomials $p_j \in \mathbb{C}[\xi, \eta]$ have degree $<D$, $n\le N$ and where $x_j, y_j \in \mathbb{T} = \mathbb{R}/\mathbb{Z}$. We find a set $A \subseteq \mathbb{Z}^2$ that depends on $N$ and $D$ only and is of size $O(D^2 N \log N)$ such that the values of $f$ on $A$ determine $f$. Notice that the size of $A$ is only larger by a logarithmic quantity than the number of parameters needed to write down $f$. We use this in order to prove some uniqueness results about polygonal regions given a small set of samples of the Fourier Transform of their indicator function. If the number of different slopes of the edges of the polygonal region is $\le k$ then the region is determined from a predetermined set of Fourier samples that depends only on $k$ and the maximum number of vertices $N$ and is of size $O(k^2 N \log N)$. In the particular case where all edges are known to be parallel to the axes the polygonal region is determined from a set of $O(N \log N)$ Fourier samples that depends on $N$ only. Our methods are non-constructive.