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

TL;DR

The paper develops a graph Laplacian assisted regularization method for linear and nonlinear ill posed inverse problems that does not require knowledge of the noise level. It introduces E-IRMGL+Ψ, an iteratively updated graph regularization framework that uses averaged measurements hat{v}^{(m)} from repeated observations and origins in a gradient descent formulation of a two-term objective J(u). Two stopping rules, a data-driven heuristic and a statistical discrepancy principle, are analyzed under mild forward-model assumptions, proving stability and convergence for exact and noisy data, and finite termination under stochastic settings. The approach is validated on X-ray CT and phase retrieval CT, showing robust reconstructions and favorable performance across different measurement counts and initializers. The work contributes a theoretically grounded, noise-free regularization paradigm with practical usefulness for medical imaging and coherent imaging tasks, and points toward future extensions to infinite-dimensional settings.

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.

Figures (12)

• Figure 1: A schematic representation of the E-IRMGL+$\Psi$ approach with two noise level free stopping rules. Independent and identically distributed samples $v_1, v_2, \ldots, v_m$ (black points near $\hat{v}^{(m)}$) are averaged to obtain $\hat{v}^{(m)}$ (yellow). The primary reconstructor $\Psi$ is not usually a regularization technique, which is reflected in the piecewise linear trajectory of $\Psi(\hat{v}^{(m)})$ as $m \to \infty$. Starting from $\Psi(\hat{v}^{(m)})$, the iterative procedure generates approximations $u_k^{(m)}$. The approximate reconstructions obtained via the heuristic and statistical stopping rules are denoted by $u_{k_m^*}^{(m)}$ and $u_{k_m}^{(m)}$, respectively. As $m \to \infty$, both sequences converge to the ground-truth solution $u^\dagger$. The convergence of $u_{k_m^*}^{(m)}$ is indicated by the blue coiled arrow, whereas the convergence of $u_{k_m}^{(m)}$ is represented by the red wavy arrow.
• Figure 2: Exact solution for X-ray CT with 5 measured noisy sinograms and their empirical mean
• Figure 3: Reconstructed solutions of X-ray CT by using initializers with $m=10$
• Figure 4: Reconstructed solutions of X-ray CT with $m=5$ and $m=10$
• Figure 5: Reconstructed solutions of X-ray CT with $m=50$ and $m=100$

Theorems & Definitions (26)

• Definition 2.1
• Definition 2.2
• Remark 2.4
• Remark 2.7
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof