# Optimal recovery of a radiating source with multiple frequencies along one line

## TL;DR

This paper investigates the unique recovery of a radiating source from line measurements across multiple frequencies, accounting for known attenuation, and introduces a generalized Laplace transform approach with numerical demonstrations.

## Contribution

It establishes conditions for unique source recovery using multi-frequency data and develops a generalized Laplace transform framework for this inverse problem.

## Key findings

- Unique determination of source density up to levelset averaging
- Development of a generalized Laplace transform method
- Numerical examples demonstrating the approach

## Abstract

We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation. This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.