Determination of compactly supported functions in shift-invariant space by single-angle Radon samples

TL;DR

This paper investigates the unique reconstruction of compactly supported functions within shift-invariant spaces from Radon transform samples taken at a single angle, providing conditions and explicit constructions for eligible sampling sets.

Contribution

It characterizes when functions in shift-invariant spaces can be reconstructed from single-angle Radon samples and offers explicit methods to construct sampling sets for various generator functions.

Findings

Eligible sampling sets exist for general generators.

Explicit construction of sampling sets is possible if the generator is in C^1.

Single-angle Radon sampling can uniquely determine certain compactly supported functions.

Abstract

While traditionally the computerized tomography of a function $f\in L^{2}(\mathbb{R}^{2})$ depends on the samples of its Radon transform at multiple angles, the real-time imaging sometimes requires the reconstruction of $f$ by the samples of its Radon transform $\mathcal{R}_{\emph{\textbf{p}}}f$ at a single angle $\theta$, where $\emph{\textbf{p}}=(\cos\theta, \sin\theta)$ is the direction vector. This naturally leads to the question of identifying those functions that can be determined by their Radon samples at a single angle $\theta$. The shift-invariant space $V(\varphi, \mathbb{Z}^2)$ generated by $\varphi$ is a type of function space that has been widely considered in many fields including wavelet analysis and signal processing. In this paper we examine the single-angle reconstruction problem for compactly supported functions $f\in V(\varphi, \mathbb{Z}^2)$. The central issue for the problem is to identify the eligible $\emph{\textbf{p}}$ and sampling set $X_{\emph{\textbf{p}}}\subseteq \mathbb{R}$ such that $f$ can be determined by its single-angle Radon (w.r.t $\emph{\textbf{p}}$) samples at $X_{\emph{\textbf{p}}}$. For the general generator $\varphi$, we address the eligible $\emph{\textbf{p}}$ for the two cases: (1) $\varphi$ being nonvanishing ($\int_{\mathbb{R}^{2}}\varphi(\emph{\textbf{x}})d\emph{\textbf{x}}\neq0$) and (2) being vanishing ($\int_{\mathbb{R}^2}\varphi(\emph{\textbf{x}})d\emph{\textbf{x}}=0$). We prove that eligible $X_{\emph{\textbf{p}}}$ exists for general $\varphi$. In particular, $X_{\emph{\textbf{p}}}$ can be explicitly constructed if $\varphi\in C^{1}(\mathbb{R}^{2})$. The single-angle problem corresponding to the case that $\varphi$ being positive definite is addressed such that $X_{\emph{\textbf{p}}}$ can be constructed easily.