Nonlinear RMM-GKS for Large-Scale Dynamic and Streaming Inverse Problems with Uncertain Forward Operators

TL;DR

This paper introduces a nonlinear optimization framework combining majorization-minimization and Krylov subspace methods for large-scale inverse problems with uncertain forward models, enabling efficient and high-quality dynamic imaging reconstruction.

Contribution

It extends existing methods to nonlinear settings with uncertain operators, incorporating streaming data processing and temporal regularization for large-scale dynamic imaging.

Findings

Achieves high-quality reconstructions with bounded memory in CT and photoacoustic tomography.

Develops alternating minimization and variable projection formulations for joint image and parameter estimation.

Demonstrates effectiveness on large-scale dynamic imaging problems with uncertain forward models.

Abstract

Many practical imaging systems suffer from uncertainty in acquisition geometry -- such as projection angles in computed tomography or sensor positions in photoacoustic tomography -- leading to nonlinear inverse problems that require joint estimation of both the image and the forward model parameters. Standard approaches that assume a known linear forward operator fail to account for these uncertainties, resulting in significant reconstruction artifacts. We propose a nonlinear recycled majorization-minimization generalized Krylov subspace (NL-RMM-GKS) framework for large-scale inverse problems with uncertain forward operators. The method extends MM-GKS to nonlinear settings by combining majorization-minimization for nonsmooth regularization with Krylov subspace projection and subspace recycling, ensuring bounded memory usage. Two complementary formulations are developed: an alternating minimization approach that alternates between image updates and Gauss-Newton parameter estimation, and a variable projection approach that eliminates the image variable and optimizes directly over the parameters using inexact inner solves. We further introduce streaming variants that process data sequentially, enabling reconstruction from large or dynamically acquired datasets without storing the full operator. For dynamic problems, we incorporate two temporal regularization strategies -- optical flow and anisotropic total variation -- as plug-in choices within the framework. We carry out rigorous numerical experiments in fan-beam computed tomography and photoacoustic tomography to demonstrate that our proposed framework achieves high-quality reconstructions with bounded memory requirements, making it suitable for large-scale dynamic imaging problems.