A Scalable Sequential Framework for Dynamic Inverse Problems via Model Parameter Estimation
Aryeh Keating, Mirjeta Pasha
TL;DR
This work tackles dynamic inverse problems that are ill-posed and memory-intensive by proposing a memory-efficient, sequential framework based on a prior-informed, dimension-reduced Kalman filter with smoothing (RKFS). It augments RKFS with Expectation-Maximization to automatically estimate time-varying motion models and noise covariances, using three motion-model variants (including optical-flow and dynamic mode decomposition-based approaches) to capture dynamics. The method EMIRKFS-M demonstrates superior reconstruction quality, memory efficiency, and computational cost compared with all-at-once and baseline RKFS across limited-angle CT, MNIST, and emoji datasets. These advances enable scalable, online reconstruction of dynamic image sequences under undersampling with automatic parameter tuning, broadening applicability to real-time imaging scenarios.
Abstract
Large-scale dynamic inverse problems are often ill-posed due to model complexity and the high dimensionality of the unknown parameters. Regularization is commonly employed to mitigate ill-posedness by incorporating prior information and structural constraints. However, classical regularization formulations are frequently infeasible in this setting due to prohibitive memory requirements, necessitating sequential methods that process data and state information online, eliminating the need to form the full space-time problem. In this work, we propose a memory-efficient framework for reconstructing dynamic sequences of undersampled images from computerized tomography data that requires minimal hyperparameter tuning. The approach is based on a prior-informed, dimension-reduced Kalman filter with smoothing. While well suited for dynamic image reconstruction, practical deployment is challenging when the state transition model and covariance parameters must be initialized without prior knowledge and estimated in a single pass. To address these limitations, we integrate regularized motion models with expectation-maximization strategies for the estimation of state transition dynamics and error covariances within the Kalman filtering framework. We demonstrate the effectiveness of the proposed method through numerical experiments on limited-angle and single-shot computerized tomography problems, highlighting improvements in reconstruction accuracy, memory efficiency, and computational cost.