Inverse transport with isotropic time-harmonic sources

TL;DR

This paper investigates how increasing the modulation frequency of isotropic, time-harmonic sources improves the stability and accuracy of reconstructing the scattering coefficient in a 2D transport equation, relevant to Optical Tomography.

Contribution

It demonstrates that higher modulation frequencies lead to more accurate and stable reconstructions of the scattering coefficient in a practical medical imaging setting.

Findings

Reconstruction accuracy improves with increasing frequency 

Stable reconstruction of low-frequency components of the scattering coefficient

Analysis based on single scattering singularities and stationary phase methods

Abstract

This paper concerns the reconstruction of the scattering coefficient in a two-dimensional transport equation from angularly averaged measurements when the probing source is isotropic and time-harmonic. This is a practical setting in the medical imaging modality called Optical Tomography. As the modulation frequency of the source increases, we show that the reconstruction of the scattering coefficient improves. More precisely, as the frequency $\omega$ increases, we show that all frequencies of the scattering coefficient lower than $b$ are reconstructed stably with an accuracy that improves as $\omega$ increases and $b$ decreases. The proofs are based on an analysis of the single scattering singularities of the transport equation and on careful analyses of oscillatory integrals by stationary phase arguments.