Hessian Schatten‐norm and adaptive dictionary for image recovery

Abstract

From many fewer acquired measurements than that suggested by the Nyquist sampling theory, compressive sensing (CS) theory demonstrates that a signal can be reconstructed with high probability when it exhibits sparsity in a certain domain. Recent CS methods have employed analytical sparsifying transforms such as wavelets, curvelets, and finite differences. In this paper, we propose a novel algorithm for image recovery, which minimizes a linear combination of three terms corresponding to least square data fitting, adaptive dictionary, and Hessian Schatten-norm regularization. We split the problem into some subproblems which turn the minimization task into much simpler. Numerical experiments are conducted on several test images with a variety of sampling patterns and ratios in both noiseless and noise scenarios. The results demonstrate the superior performance of the proposed algorithm.

Plug‐and‐play algorithms for convex non‐convex regularization: Convergence analysis and applications

“…This adjustment utilizes the strong convexity of the data-fidelity term 𝑓 (x) to balance out the non-convexity of the regularizer 𝜙 b (x). In particular, Selesnick et al [16,18] showed that if 𝜙 b (x) is defined as (15),…”

“…The primary drawback lies in its inherent bias as an estimator, leading to an underestimation of the large values within sparse solutions. In contrast, the non-convex regularization can effectively reduce estimation deviation but may also lead to the objective function getting trapped in local optima [13][14][15]. To leverage the benefits of both non-convex regularization and convex optimization, the convex non-convex (CNC) strategy has been proposed, which allows for non-convex regularization while maintaining the convexity of the optimization problem [16][17][18][19][20].…”