Numerical inversion of a broken ray transform arising in single scattering optical tomography

TL;DR

This paper introduces a fast, robust numerical algorithm for inverting the broken ray transform in single scattering optical tomography, enabling improved interior imaging using light scattering data.

Contribution

It presents a novel numerical inversion method for the broken ray transform in SSOT, utilizing Fourier coefficient relations and truncated SVD for stability.

Findings

Accurate reconstruction demonstrated with noise-free data.

Robustness shown with noisy data.

Algorithm achieves fast and stable inversion.

Abstract

The article presents an efficient image reconstruction algorithm for single scattering optical tomography (SSOT) in circular geometry of data acquisition. This novel medical imaging modality uses photons of light that scatter once in the body to recover its interior features. The mathematical model of SSOT is based on the broken ray (or V-line Radon) transform (BRT), which puts into correspondence to an image function its integrals along V-shaped piecewise linear trajectories. The process of image reconstruction in SSOT requires inversion of that transform. We implement numerical inversion of a broken ray transform in a disc with partial radial data. Our method is based on a relation between the Fourier coefficients of the image function and those of its BRT recently discovered by Ambartsoumian and Moon. The numerical algorithm requires solution of ill-conditioned matrix problems, which is accomplished using a half-rank truncated singular value decomposition method. Several numerical computations validating the inversion formula are presented, which demonstrate the accuracy, speed and robustness of our method in the case of both noise-free and noisy data.