Graph-Regularized Estimation for Inverse Problems in Computational Optical Imaging

Abstract

Computational optical imaging reconstructs signals from limited and noisy measurements, leading to ill-posed inverse problems that require regularization. Graph regularization (GR) has shown promise for modeling complex spatial correlations, but its relationship to classical Total Variation (TV) regularization remains unclear. Using ghost imaging as a representative inverse imaging application, this paper establishes a unified theoretical framework for graph-regularized estimation. We derive explicit expressions for the bias and variance, obtain an upper bound on the mean squared error (MSE), and develop an information-theoretic minimax lower bound for TV regularization. The analysis identifies regimes in which GR can be advantageous relative to TV, highlighting the different inductive biases of graph-based smoothness and TV-based local sparsity. In particular, GR is especially effective under low-frequency priors, high noise, strong graph connectivity, and severe undersampling. We also implement a graph-regularized reconstruction framework based on spatial and gradient similarities. Simulations and experiments show that the proposed method achieves competitive and often improved reconstruction quality relative to TV and sparsity-based baselines, especially under noisy and undersampled conditions. These results clarify the role of GR in computational optical imaging.