scholarly journals New Regularization Models for Image Denoising with a Spatially Dependent Regularization Parameter

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Tian-Hui Ma ◽  
Ting-Zhu Huang ◽  
Xi-Le Zhao
Keyword(s):  
Image Denoising ◽  
Hybrid Model ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Regularization Parameter ◽  
Alternating Minimization ◽  
Convergence Properties ◽  
Strictly Convex ◽  
Nonconvex Problem ◽  
Minimization Scheme

We consider simultaneously estimating the restored image and the spatially dependent regularization parameter which mutually benefit from each other. Based on this idea, we refresh two well-known image denoising models: the LLT model proposed by Lysaker et al. (2003) and the hybrid model proposed by Li et al. (2007). The resulting models have the advantage of better preserving image regions containing textures and fine details while still sufficiently smoothing homogeneous features. To efficiently solve the proposed models, we consider an alternating minimization scheme to resolve the original nonconvex problem into two strictly convex ones. Preliminary convergence properties are also presented. Numerical experiments are reported to demonstrate the effectiveness of the proposed models and the efficiency of our numerical scheme.

scholarly journals Parameter learning and fractional differential operators: Applications in regularized image denoising and decomposition problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sören Bartels ◽  
Nico Weber
Keyword(s):  
Image Denoising ◽  
Numerical Experiments ◽  
Fractional Laplacian ◽  
Differential Operators ◽  
Riesz Potential ◽  
Regularization Parameter ◽  
Computing Time ◽  
Bilevel Optimization ◽  
Learning Approaches ◽  
Decomposition Models

<p style='text-indent:20px;'>In this paper, we focus on learning optimal parameters for PDE-based image denoising and decomposition models. First, we learn the regularization parameter and the differential operator for gray-scale image denoising using the fractional Laplacian in combination with a bilevel optimization problem. In our setting the fractional Laplacian allows the use of Fourier transform, which enables the optimization of the denoising operator. We prove stable and explainable results as an advantage in comparison to machine learning approaches. The numerical experiments correlate with our theoretical model settings and show a reduction of computing time in contrast to the Rudin-Osher-Fatemi model. Second, we introduce a new regularized image decomposition model with the fractional Laplacian and the Riesz potential. We provide an explicit formula for the unique solution and the numerical experiments illustrate the efficiency.</p>

scholarly journals A STABILIZING SUBGRID FOR CONVECTION–DIFFUSION PROBLEM

2006 ◽  
Vol 16 (02) ◽  
pp. 211-231 ◽  
Author(s):  
ALI I. NESLITURK
Keyword(s):  
Numerical Experiments ◽  
A Priori ◽  
Diffusion Problem ◽  
Triangular Element ◽  
Discrete Solution ◽  
Convergence Properties ◽  
Small Diffusion ◽  
Galerkin Finite Element ◽  
Convection Diffusion ◽  
Priori Error Estimates

A stabilizing subgrid which consists of a single additional node in each triangular element is analyzed by solving the convection–diffusion problem, especially in the case of small diffusion. The choice of the location of the subgrid node is based on minimizing the residual of a local problem inside each element. We study convergence properties of the method under consideration and its connection with previously suggested stabilizing subgrids. We prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfy the same a priori error estimates that are typically obtained with SUPG and RFB methods. Some numerical experiments that confirm the theoretical findings are also presented.

scholarly journals A Hybrid Model for Image Denoising Combining Modified Isotropic Diffusion Model and Modified Perona-Malik Model

IEEE Access ◽  
2018 ◽  
Vol 6 ◽  
pp. 33568-33582 ◽  
Author(s):  
Na Wang ◽  
Yu Shang ◽  
Yang Chen ◽  
Min Yang ◽  
Quan Zhang ◽  
...  
Keyword(s):  
Image Denoising ◽  
Diffusion Model ◽  
Hybrid Model ◽  
Isotropic Diffusion

scholarly journals A well-posed multiscale regularization scheme for digital image denoising

2011 ◽  
Vol 21 (4) ◽  
pp. 769-777 ◽  
Author(s):  
V. Prasath
Keyword(s):  
Image Denoising ◽  
Digital Image ◽  
Regularization Parameter ◽  
Input Image ◽  
Smooth Transition ◽  
Functions Of Bounded Variation ◽  
Edge Preserving ◽  
Regularization Scheme ◽  
Adaptive Parameter ◽  
Well Posed

A well-posed multiscale regularization scheme for digital image denoisingWe propose an edge adaptive digital image denoising and restoration scheme based on space dependent regularization. Traditional gradient based schemes use an edge map computed from gradients alone to drive the regularization. This may lead to the oversmoothing of the input image, and noise along edges can be amplified. To avoid these drawbacks, we make use of a multiscale descriptor given by a contextual edge detector obtained from local variances. Using a smooth transition from the computed edges, the proposed scheme removes noise in flat regions and preserves edges without oscillations. By incorporating a space dependent adaptive regularization parameter, image smoothing is driven along probable edges and not across them. The well-posedness of the corresponding minimization problem is proved in the space of functions of bounded variation. The corresponding gradient descent scheme is implemented and further numerical results illustrate the advantages of using the adaptive parameter in the regularization scheme. Compared with similar edge preserving regularization schemes, the proposed adaptive weight based scheme provides a better multiscale edge map, which in turn produces better restoration.

scholarly journals Performance of Restarted Homotopy Perturbation Method for TV-Based Image Denoising Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-18 ◽  
Author(s):  
Yu Du Han ◽  
Jae Heon Yun
Keyword(s):  
Perturbation Method ◽  
Image Denoising ◽  
Numerical Experiments ◽  
Homotopy Perturbation Method ◽  
Nonlinear Pde ◽  
Finite Difference Schemes ◽  
Numerical Implementation ◽  
Homotopy Perturbation ◽  
Efficiency And Reliability ◽  
Nonlinear Denoising

We first propose a restarted homotopy perturbation method (RHPM) for solving a nonlinear PDE problem which repeats HPM process by computing only the first few terms instead of computing infinite terms, and then we present an application of RHPM to TV- (Total Variation-) based image denoising problem. The main difficulty in applying RHPM to the nonlinear denoising problem is settled by using binomial series techniques. We also provide finite difference schemes for numerical implementation of RHPM. Lastly, numerical experiments for several test images are carried out to demonstrate the feasibility, efficiency, and reliability of RHPM by comparing the performance of RHPM with that of existing TM and recently proposed RHAM methods.

Image Denoising with Contour-Based Directionlet Transform

2014 ◽  
Vol 513-517 ◽  
pp. 3607-3611
Author(s):  
Huan An Xu ◽  
Guo Hua Peng ◽  
Zhe Liu
Keyword(s):  
Image Processing ◽  
Image Denoising ◽  
Numerical Experiments ◽  
Adaptive Threshold ◽  
Perceptual Quality ◽  
Laplacian Pyramid ◽  
Low Frequencies ◽  
Threshold Algorithm ◽  
Image Transform

A novel mutiscale and directionally adaptive image transform called contour based directionlet tansform is presented. Directionlet transform (DT) has shown its charming performance in image processing, but it has scrambled frequencies. Laplacian Pyramid is employed here to separate the low frequencies before applying DT for avoiding the drawback. And an adaptive threshold algorithm is proposed for denoising. Numerical experiments are performed to assess the applicability of the proposed method. The obtained results show that the proposed scheme outperforms Wavelet and Directionlet transforms in terms of numerical and perceptual quality.

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

scholarly journals A PROJECTION APPROACH TO THE NUMERICAL ANALYSIS OF LIMIT LOAD PROBLEMS

2011 ◽  
Vol 21 (06) ◽  
pp. 1291-1316 ◽  
Author(s):  
GUILLAUME CARLIER ◽  
MYRIAM COMTE ◽  
IOAN IONESCU ◽  
GABRIEL PEYRÉ
Keyword(s):  
Numerical Analysis ◽  
Gradient Descent ◽  
Numerical Scheme ◽  
Limit Load ◽  
Deformation Field ◽  
Dual Formulation ◽  
Strictly Convex ◽  
Convex Perturbation ◽  
Projection Approach

This paper proposes a numerical scheme to approximate the solution of (vectorial) limit load problems. The method makes use of a strictly convex perturbation of the problem, which corresponds to a projection of the deformation field under bounded deformation and incompressibility constraints. The discretized formulation of this perturbation converges to the solution of the original landslide problem when the amplitude of the perturbation tends to zero. The projection is computed numerically with a multi-step gradient descent on the dual formulation of the problem.

scholarly journals FINITE-DIFFERENCE ANALYSIS FOR THE LINEAR THERMOPOROELASTICITY PROBLEM AND ITS NUMERICAL RESOLUTION BY MULTIGRID METHODS

2012 ◽  
Vol 17 (2) ◽  
pp. 227-244 ◽  
Author(s):  
Natalia Boal ◽  
Francisco Jos´e Gaspar ◽  
Francisco Lisbona ◽  
Petr Vabishchevich
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Stability And Convergence ◽  
System Of Equations ◽  
Convergence Properties ◽  
Discrete Energy ◽  
Numerical Resolution ◽  
Multigrid Convergence

This paper deals with the numerical solution of a two-dimensional thermoporoelasticity problem using a finite-difference scheme. Two issues are discussed: stability and convergence in discrete energy norms of the finite-difference scheme are proved, and secondly, a distributive smoother is examined in order to find a robust and efficient multigrid solver for the corresponding system of equations. Numerical experiments confirm the convergence properties of the proposed scheme, as well as fast multigrid convergence.

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