# Image reconstruction from radially incomplete spherical Radon data

## TL;DR

This paper presents a new method for uniquely reconstructing images from incomplete spherical Radon data, with practical algorithms demonstrated through numerical examples in medical and radar imaging.

## Contribution

It introduces a novel inversion formula for radially incomplete spherical Radon data and provides a robust computational algorithm for practical image reconstruction.

## Key findings

- Unique reconstruction in spherical shells with incomplete data
- Reconstruction formulas applicable inside, outside, and across the data sphere
- Algorithm demonstrates high accuracy and efficiency in numerical tests

## Abstract

We study inversion of the spherical Radon transform with centers on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Our result is valid when the support of the image function is inside the data acquisition sphere, outside that sphere, as well as on both sides of the sphere. Furthermore, in addition to the uniqueness result our method of proof provides reconstruction formulas for all those cases. We present a robust computational algorithm based on our inversion formula and demonstrate its accuracy and efficiency on several numerical examples.