Image Restoration with Fractional-Order Total Variation Regularization and Group Sparsity

Abstract

In this paper, we present a novel image denoising algorithm, specifically designed to effectively restore both the edges and texture of images. This is achieved through the use of an innovative model known as the overlapping group sparse fractional-order total variation regularization model (OGS-FOTVR). The OGS-FOTVR model ingeniously combines the benefits of the fractional-order (FO) variation domain with an overlapping group sparsity measure, which acts as its regularization component. This is further enhanced by the inclusion of the well-established L2-norm, which serves as the fidelity term. To simplify the model, we employ the alternating direction method of multipliers (ADMM), which breaks down the model into a series of more manageable sub-problems. Each of these sub-problems can then be addressed individually. However, the sub-problem involving the overlapping group sparse FO regularization presents a high level of complexity. To address this, we construct an alternative function for this sub-problem, utilizing the mean inequality principle. Subsequently, we employ the majorize-minimization (MM) algorithm to solve it. Empirical results strongly support the effectiveness of the OGS-FOTVR model, demonstrating its ability to accurately recover texture and edge information in images. Notably, the model performs better than several advanced variational alternatives, as indicated by superior performance metrics across three image datasets, PSNR, and SSIM.

Adaptive Numerical Regularization for Variational Denoising Model with Complementary Approach

“…The inverse and ill-posed problems depend on the smoothing parameters [3,4,6,18,24]. Most of the regularization methods are based on the uniform selection of the smoothing parameters [5,10,11,14,19,21] where they use the finite difference methods as simulation methods for the solution of the arising partial differential equations from the variational models. This work is dedicated towards the application of the adaptive finite element method along with the design of an adaptive algorithm for the optimization of the computed solution in the spirit of [7,11,14,22].…”

“…It has been observed in the literature that the ill-posed inverse problems arising in the Science and engineering generally solved by the regularization methods These strategies play an effective role in the solution of many ill-posed inverse problems but due to their non-linear nature many complexities may arise in the solution strategies, therefore practitioners look forward to the linear approaches with modern and novel smoothing techniques [9]. Such strategies have been successfully applied in the image sciences where the famous applications are image denoising [2,3,3,6,11,18,[21][22][23], optical flow problems [9,10,13] and stereo vision problems [21] . The Partial Differential Equation (PDEs) based approaches have been extensively used in image processing due to their modeling flexibility and numerical implementations [3,6,9,13,18,22,25].…”

“…In this study for reconstruc-tion of original image u from the observed image f, where an adaptive regularization approach has been proposed. The proposed adaptive regularization is based on a priori error estimate that is applied on the smoothness parameters to cope with the damaged areas of image domain where the mean square error (MSE) is large and the obtained solution is less regular [5,11].…”

Boundedness in the Bloch Space of Symmetric Domain for a Class of Multi-Valent Meromorphic Functions Given by a Fractional Integral

“…In signal processing, image processing, and statistics, this feature is frequently exploited. (2) The convolution of Bessel functions: In both science and engineering, Bessel functions, a family response to Bessel's differential equation, are used in a variety of different contexts. Two Bessel functions are combined to create a third Bessel function [1].…”

“…The output of these multiplications is then added together to create the kernel's central value, which is subsequently superimposed over the present location in the image. For each position in the image that is viable, this process is completed [2].…”