Simultaneous Patch-Group Sparse Coding with Dual-Weighted ℓp Minimization for Image Restoration

Sparse coding (SC) models have been proven as powerful tools applied in image restoration tasks, such as patch sparse coding (PSC) and group sparse coding (GSC). However, these two kinds of SC models have their respective drawbacks. PSC tends to generate visually annoying blocking artifacts, while GSC models usually produce over-smooth effects. Moreover, conventional ℓ1 minimization-based convex regularization was usually employed as a standard scheme for estimating sparse signals, but it cannot achieve an accurate sparse solution under many realistic situations. In this paper, we propose a novel approach for image restoration via simultaneous patch-group sparse coding (SPG-SC) with dual-weighted ℓp minimization. Specifically, in contrast to existing SC-based methods, the proposed SPG-SC conducts the local sparsity and nonlocal sparse representation simultaneously. A dual-weighted ℓp minimization-based non-convex regularization is proposed to improve the sparse representation capability of the proposed SPG-SC. To make the optimization tractable, a non-convex generalized iteration shrinkage algorithm based on the alternating direction method of multipliers (ADMM) framework is developed to solve the proposed SPG-SC model. Extensive experimental results on two image restoration tasks, including image inpainting and image deblurring, demonstrate that the proposed SPG-SC outperforms many state-of-the-art algorithms in terms of both objective and perceptual quality.

Generalized Iteration Method via Green's Function for the Solution of the Fourth-order Boundary Value Problems

The aim of this paper is to extend and generalize Picard-Green’s fixed point iteration method for the solution of fourth-order Boundary Value Problems. Several numerical applications to linear and nonlinear fourth-order Boundary Value Problems are discussed to illustrate the main results.

Iterative Methods for Solving Systems of Multi-Valued Logical Equations in the Simulation of Object Control Digital Systems

The article is devoted to the analysis of methods for solving systems of multivalued logical equations by iteration methods. Iterative methods for solving such systems of equations are a mathematical description of the main process of functional-logical simulation, which is used at the stage of designing digital systems for objects control to verify the correctness of the design. Consideration of multi-valued values of logical signals at the outputs of blocks and elements of digital systems is explained by the fact that in some cases, to analyze the correctness of time relationships when simulating the hardware of digital systems, a several valued representation of logical signals is used, as well as that recently, logical elements are being developed that implement four or more valued logic. Based on the analysis of the structure of the system of logical equations used in digital hardware simulation, using graph and logical models, an analysis is made of the existence of solutions and their number. Iterative methods of a simple and generalized iteration are analyzed, a relationship is shown between the number of solutions of the system of equations and its graph representation, which reflects a given circuit of connecting elements of the hardware of a digital control system. For the generalized iteration method, options with a different structure of the iteration trace are considered, in particular, it is shown that, with a certain structure of the iteration trace, the generalized iteration turns into a simple iteration or Seidel iteration. It is shown that the generalized iteration most adequately describes the process of simulating the switching of logical signals in a simulated circuit of digital control systems hardware. The correspondence between various options of functional-logical simulation of digital systems and the used methods of iterative solution of systems of logical equations is shown.