Date of Award
6-26-2026
Date Published
August 2026
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
Advisor(s)
Lixin Shen
Keywords
Image restoration;Inverse Problem;Moreau envelope;Nonconvex model;Proximity operator;Sparsity-promoting function
Subject Categories
Mathematics | Physical Sciences and Mathematics
Abstract
Image restoration aims to recover an unknown clean image from observations degraded by blur, subsampling, missing pixels, or noise. This dissertation develops and analyzes variational image restoration models based on structured sparsity promoting functions (SSPFs) and their Moreau-enveloped regularization. The main objective is to design nonconvex regularization models that promote structured sparsity in transformed image representations while remaining analytically tractable and computationally effective. The first part of the dissertation studies an SSPF regularized image restoration model. Under the identifiability condition $\ker(A)\cap \ker(B)=\{\bm0\}$, where $A$ is a linear forward degradation operator and $B$ is a transform operator used to extract image features, we prove the existence of global minimizers. We derive explicit formulas for the associated blockwise proximal mappings and develop efficient numerical schemes for the resulting composite proximal subproblems. Several optimization algorithms are proposed for the model, including inertial proximal, difference-of-convex, double-proximal-gradient, and ADMM-type methods. Subsequential convergence to stationary points is established under appropriate assumptions. Numerical experiments in this part demonstrate the effectiveness of the SSPF regularized model and compare its performance with classical total variation and total generalized variation approaches. The second part of the dissertation introduces a Moreau-enveloped SSPF regularized model, in which the original nonsmooth SSPF penalty is replaced by its Moreau envelope. This construction yields a smoother variational model while retaining the structural information encoded by the SSPF regularizer. We prove the existence of global minimizers for the Moreau-enveloped model under the same identifiability condition, establish a quantitative relation between the original and smoothed models, and develop several numerical methods with subsequential convergence analysis. Numerical experiments show that the Moreau-enveloped model is competitive in the tested restoration settings and is particularly effective for severely blurred images with large piecewise smooth regions, where it better preserves gradual intensity transitions and reduces blocky artifacts. Overall, this dissertation provides a theoretical and computational framework for nonconvex image restoration based on structured sparsity, combining existence theory, proximal analysis, algorithm design, convergence analysis, and numerical validation.
Access
Open Access
Recommended Citation
Wei, Jianchen, "Image Restoration Regularized by Structured Sparsity Promoting Functions: Theory, Algorithms, and Applications" (2026). Dissertations - ALL. 2369.
https://surface.syr.edu/etd/2369
