Single-angle Radon samples based reconstruction of functions in refinable shift-invariant space

TL;DR

This paper explores the possibility of reconstructing functions in refinable shift-invariant spaces using Radon transform samples at a single angle, with theoretical guarantees and numerical validation.

Contribution

It establishes that functions in certain SISs can be reconstructed from single-angle Radon samples, extending CT and optical imaging capabilities.

Findings

Functions in general SISs are determined by single-angle Radon samples.

Reconstruction in spline-generated SISs is possible at almost every angle due to positive definiteness.

Numerical simulations confirm the theoretical results.

Abstract

The traditional approaches to computerized tomography (CT) depend on the samples of Radon transform at multiple angles. In optics, the real time imaging requires the reconstruction of an object by the samples of Radon transform at a single angle (SA). Driven by this and motivated by the connection between Bin Han's construction of wavelet frames (e.g [13]) and Radon transform, in refinable shift-invariant spaces (SISs) we investigate the SA-Radon sample based reconstruction problem. We have two main theorems. The fist main theorem states that, any compactly supported function in a SIS generated by a general refinable function can be determined by its Radon samples at an appropriate angle. Motivated by the extensive application of positive definite (PD) functions to interpolation of scattered data, we also investigate the SA reconstruction problem in a class of (refinable) box-spline generated SISs. Thanks to the PD property of the Radon transform of such spline, our second main theorem states that, the reconstruction of compactly supported functions in these spline generated SISs can be achieved by the samples of Radon transform at almost every angle. Numerical simulation is conducted to check the result.