Nonlinear tomographic reconstruction via nonsmooth optimization

TL;DR

This paper introduces a nonsmooth, nonconvex optimization approach for nonlinear tomographic reconstruction in CT, demonstrating faster convergence and better conditioning than traditional linear methods, especially for high dynamic range images.

Contribution

It proposes a novel nonsmooth, nonconvex loss function and a subgradient method that converges geometrically, improving reconstruction speed and sample efficiency in nonlinear CT imaging.

Findings

Converges at a geometric rate under a statistical model.

Achieves faster reconstruction with fewer samples.

Offers improved conditioning over previous methods.

Abstract

We study iterative signal reconstruction in computed tomography (CT), wherein measurements are produced by a linear transformation of the unknown signal followed by an exponential nonlinear map. Approaches based on pre-processing the data with a log transform and then solving the resulting linear inverse problem are tempting since they are amenable to convex optimization methods; however, such methods perform poorly when the underlying image has high dynamic range, as in X-ray imaging of tissue with embedded metal. We show that a suitably initialized subgradient method applied to a natural nonsmooth, nonconvex loss function produces iterates that converge to the unknown signal of interest at a geometric rate under the statistical model proposed by Fridovich-Keil et al. (arXiv:2310.03956). Our recovery program enjoys improved conditioning compared to the formulation proposed by the latter work, enabling faster iterative reconstruction from substantially fewer samples.