The spherical mean Radon transform with centers on cylindrical surfaces

TL;DR

This paper develops explicit inversion formulas and efficient algorithms for reconstructing functions from spherical Radon transforms with centers on cylindrical surfaces, relevant to various imaging modalities.

Contribution

It introduces a novel decomposition of the spherical Radon transform on cylindrical centers into lower-dimensional transforms and provides explicit inversion formulas and efficient algorithms.

Findings

Explicit inversion formulas for cylindrical centers

Efficient backprojection algorithms with O(N^{4/3}) complexity

Numerical results demonstrating algorithm effectiveness

Abstract

Recovering a function from its spherical Radon transform with centers of spheres of integration restricted to a hypersurface is at the heart of several modern imaging technologies, including SAR, ultrasound imaging, and photo- and thermoacoustic tomography. In this paper we study an inversion of the spherical Radon transform with centers of integration restricted to cylindrical surfaces of the form $\Gamma \times \mathbb{R}^m$, where $\Gamma$ is a hypersurface in $\mathbb{R}^n$. We show that this transform can be decomposed into two lower dimensional spherical Radon transforms, one with centers on $\Gamma$ and one with a planar center-set in $\mathbb{R}^{m+1}$. Together with explicit inversion formulas for the spherical Radon transform with a planar center-set and existing algorithms for inverting the spherical Radon transform with a center-set $\mathbb{R}$, this yields reconstruction procedures for general cylindrical domains. In the special case of spherical or elliptical cylinders we obtain novel explicit inversion formulas. For three spatial dimensions, these inversion formulas can be implemented efficiently by backprojection type algorithms only requiring $\mathcal O(N^{4/3})$ floating point operations, where $N$ is the total number of unknowns to be recovered. We present numerical results demonstrating the efficiency of the derived algorithms.