Exponential polynomials and identification of polygonal regions from Fourier samples

TL;DR

The paper shows that bivariate exponential polynomials of the form $f(\xi,\eta)= \sum_{j=1}^n p_j(\xi,\eta) e^{2\pi i (x_j \xi + y_j \eta)}$, with $n \le N$ and $\deg p_j < D$, are uniquely determined by samples on a universal set of size $O(D^2 N \log N)$. It then transfers this identifiability result to polygonal regions by exploiting the Brion-Barvinok formula, proving that a polygon with at most $N$ vertices and a known slope set of size $k$ can be recovered from $\widehat{\mathbb{1}_P}$ sampled on a set $A(k,N)$ of size $O(k^2 N \log N)$; in the axis-aligned case ($k=2$), the required samples drop to $O(N \log N)$. The approach is non-constructive and distinguishes between known and unknown slopes, with the latter requiring a larger sampling set $A(2k,2N)$. Overall, the work establishes near-optimal, non-adaptive sampling bounds for identifying both exponential-polynomial signals and polygonal regions from Fourier data, highlighting a close link between sparse spectral models and geometric reconstruction.

Abstract

Consider the set $E(D, N)$ of all bivariate exponential polynomials $$ f(ξ, η) = \sum_{j=1}^n p_j(ξ, η) e^{2πi (x_jξ+y_jη)}, $$ where the polynomials $p_j \in \mathbb{C}[ξ, η]$ 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.

Figures (7)

• Figure 1: A set $E \subseteq {\mathbb T}$ consisting of $n$ arcs.
• Figure 2: The sampling set for Lemma \ref{['NlogN']}
• Figure 3: The partition of the set $X$ (projections of the points to the $x$-axis), to the sets $X_1, X_2, \cdots$.
• Figure 4: A polygonal region in the plane.
• Figure 5: A polygonal region in the plane with sides parallel to the axes.

Theorems & Definitions (28)

• Remark 1.1
• Remark 2.1: Determination of an exponential polynomial by its values on the integers
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof