A bilinear inverse problem with forward operator inaccuracy applied to neonatal atlas-based diffuse optical tomography

TL;DR

This paper addresses the challenge of inaccuracies in the forward operator in inverse problems, proposing a bilinear tensor approach with algorithms and applying it to neonatal diffuse optical tomography to improve image reconstruction quality.

Contribution

It introduces a bilinear inverse problem framework incorporating forward operator inaccuracies and demonstrates its effectiveness in neonatal brain imaging.

Findings

Improved spatial localization in reconstructions.

Enhanced contrast-to-noise ratio.

Effective modeling of forward operator variations.

Abstract

Linear inverse problems are highly common in practical real-world applications from industry to medical imaging. The forward operator is often built on some approximations of the studied system. Handling inaccuracies in the forward operator in the context of inverse problems is a relatively unstudied problem. In this work, we assume that we have a set of candidate forward operator matrices and suggest principal component analysis (PCA) for modeling their variation from the mean. We combine the original linear problem with the included forward operator inaccuracy into a bilinear tensor inverse problem and present two optimization algorithms and Gibbs sampling for approximately solving the problem. As a real-world test case, we apply the algorithms to account for the inaccuracy that is present in the sensitivity profiles or Jacobian matrices in diffuse optical tomography when an atlas-based model of the head anatomy is used instead of the subject's own anatomical model in neonates over a wide range of gestational ages (29--44 weeks). We report visual and numerical improvements in the spatial localization and contrast-to-noise-ratio in reconstructions of simulated hemodynamic activity.