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

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 ${\mathcal{R}}_{\text{p}}f$ at a single angle $\theta$, where $\text{p}=(\cos \theta ,\sin \theta )$ 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 $V(\phi ,{\mathbb{Z}}^2)$ generated by $\phi$ 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 $f\in V(\phi ,{\mathbb{Z}}^2)$. The central issue for the problem is to identify the eligible $\text{p}$ and sampling set $X_{\text{p}}\subseteq \mathbb{R}$ such that $f$ can be determined by its single-angle Radon (w.r.t $\text{p}$) samples at $X_{\text{p}}$. For the general generator $\phi$, we address the eligible $\text{p}$ for the two cases: (1) $\phi$ being nonvanishing (${\int }_{{\mathbb{R}}^2}\phi (\text{x})d\text{x}\ne 0$) and (2) being vanishing (${\int }_{{\mathbb{R}}^2}\phi (\text{x})d\text{x}=0$). We prove that eligible $X_{\text{p}}$ exists for general $\phi$. In particular, $X_{\text{p}}$ can be explicitly constructed if $\phi \in C^1({\mathbb{R}}^2)$. The single-angle problem corresponding to the case that $\phi$ being positive definite is addressed such that $X_{\text{p}}$ can be constructed easily.