Graph Laplacian assisted regularization method under noise level free heuristic and statistical stopping rule

TL;DR

This paper introduces a graph Laplacian regularization method for solving ill-posed inverse problems that operates without noise level knowledge, using iterative graph updates and two stopping rules, with proven convergence and demonstrated effectiveness in CT applications.

Contribution

It presents a novel graph-based regularization framework with dynamic Laplacian updates and noise-free stopping criteria, advancing inverse problem solutions.

Findings

Effective in X-ray CT and phase retrieval CT reconstructions.

Robustness demonstrated under different stopping rules.

Convergence and stability established theoretically.

Abstract

In this work, we address the solution of both linear and nonlinear ill-posed inverse problems by developing a novel graph-based regularization framework, where the regularization term is formulated through an iteratively updated graph Laplacian. The proposed approach operates without prior knowledge of the noise level and employs two distinct stopping criteria namely, the heuristic rule and the statistical discrepancy principle. To facilitate the latter, we utilize averaged measurements derived from multiple repeated observations. We provide a detailed convergence analysis of the method in statistical prospective, establishing its stability and regularization properties under both stopping strategies. The algorithm begins with the computation of an initial reconstruction using any suitable techniques like Tikhonov regularization (Tik), filtered back projection (FBP) or total variation (TV), which is used as the foundation for generating the initial graph Laplacian. The reconstruction is made better step by step using an iterative process, during which the graph Laplacian is dynamically re-calibrated to reflect how the solution's structure is changing. Finally, we present numerical experiments on X-ray Computed Tomography (CT) and phase retrieval CT, demonstrating the effectiveness and robustness of the proposed method and comparing its reconstruction performance under both stopping rules.