Novel Linear Image Denoising Approach Based on a Modified Gaussian Filter Kernel

A Gaussian noise reduction technique for grayscale images is proposed in this article. It uses a modified Gaussian filter kernel based on a hyperbolic second-order equation. The introduced mathematical model differs from the classic Gaussian model provided by the heat equations, by a localization property. Our filtering approach reduces the amount of Gaussian noise and also enhances the image contrast. Some image denoising experiments that prove the effectiveness of the proposed method are also described in this article.     

IMAGE DENOISING VIA TIME-DELAY REGULARIZATION COUPLED NONLINEAR DIFFUSION EQUATIONS

A novel nonlinear anisotropic diffusion model is proposed for image denoising which can be viewed as a novel regularized model that preserves the cherished features of Perona-Malik to some extent. It is characterized by a local dependence in the diffusivity which manifests itself through the presence of $p(x)$-Laplacian and time-delay regularization. The proposed model offers a new nonlinear anisotropic diffusion which makes it possible to effectively enhance the denoising capability and preserve the details while avoiding artifacts. Accordingly, the restored image is very clear and becomes more distinguishable. By Galerkin's method, we establish the well-posedness in the weak setting. Numerical experiments illustrate that the proposed model appears to be overwhelmingly competitive in restoring the images corrupted by Gaussian noise.     

广义全变差正则化图像去噪模型研究

本文研究了全变差正则化模型在图像去噪过程中易产生阶梯效应的问题,依据图像的局部结构特利用联合高斯滤波器和边缘检测算子的方法,构建了广义全变差正则化图像去噪模型,获得了在消除噪声的同时能够保留图像边缘细节和纹理信息的结果.实验结果表明,广义全变差正则化模型在平滑噪声的同时能够保留图像的边缘轮廓等细节信息,得到的复原图像在峰值信噪比、平均结构相似度和主观视觉效果方面均有所提高.     

A Fast Augmented Lagrangian Method for Euler's Elastica Models

In this paper, a fast algorithm for Euler's elastica functional is proposed, in which the Euler's elastica functional is reformulated as a constrained minimization problem. Combining the augmented Lagrangian method and operator splitting techniques, the resulting saddle-point problem is solved by a serial of subproblems. To tackle the nonlinear constraints arising in the model, a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution. We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic, real-world and medical images for image denoising, image inpainting and image zooming problems.     

A fast algorithm for the total variation model of image denoising

The total variation model of Rudin, Osher, and Fatemi for image denoising is considered to be one of the best denoising models. In the past, its solutions were based on nonlinear partial differential equations and the resulting algorithms were very complicated. In this paper, we propose a fast algorithm for the solution of the total variation model. Our algorithm is very simple and does not involve partial differential equations. We also provide a rigorous proof for the convergence of our algorithm.     

MRI denoising via sparse tensors with reweighted regularization

In recent years, image denoising based on sparse tensors has been one promising technique for denoising magnetic resonance images or video processing. This paper aims at developing a new sparse tensor model based on reweighted regularization of factor matrices for magnetic resonance images denoising. An improved Split-Bregman scheme is proposed which is simple in implementation and efficient in computation. Additionally, the convergence of proposed scheme is proved. Experiments show that the proposed algorithm is efficient, and the denoising results are better than the state-of-the-art image denoising methods. The average computational time of our method is slightly longer than the others under the same iteration, except LPGPCA and model in Ruru and Zhixun (2018) [22].     

A fractional partial differential equation based multiscale denoising model for texture image

Traditional integer‐order partial differential equation based image denoising approach can easily lead edge and complex texture detail blur, thus its denoising effect for texture image is always not well. To solve the problem, we propose to implement a fractional partial differential equation (FPDE) based denoising model for texture image by applying a novel mathematical method—fractional calculus to image processing from the view of system evolution. Previous studies show that fractional calculus has some unique properties that it can nonlinearly enhance complex texture detail in digital image processing, which is obvious different with integer‐order differential calculus. The goal of the modeling is to overcome the problems of the existed denoising approaches by utilizing the aforementioned properties of fractional differential calculus. Using classic definition and property of fractional differential calculus, we extend integer‐order steepest descent approach to fractional field to implement fractional steepest descent approach. Then, based on the earlier fractional formulas, a FPDE based multiscale denoising model for texture image is proposed and further analyze optimal parameters value for FPDE based denoising model. The experimental results prove that the ability for preserving high‐frequency edge and complex texture information of the proposed fractional denoising model are obviously superior to traditional integral based algorithms, as for texture detail rich images. Copyright © 2013 John Wiley & Sons, Ltd.     

偏微分方程在图像去噪中的应用

本文介绍用于图像去噪的偏微分模型、方法的发展历程．从理论上分析了线性模型、简单非线性模型、复杂非线性模型、多步处理模型出现的背景和优缺点，并从空域和频域上对偏微分方程模型的去噪原理进行了分析．最后，指出了偏微分方程去噪与小波去噪结合的途径，据此对偏微分方程未来的发展方向进行了展望．     

A novel PDE based image restoration: Convection-diffusion equation for image denoising

In this paper we present a convection-diffusion equation for processing image denoising, edge preservation and compression. We compare it with a popular nonlinear diffusion model which has been widely implemented in image denoising for Gaussian white noise. Here we show that this convection-diffusion model effectively removes noise, especially for the mixture of Gaussian and salt-and-pepper noises. We propose the modified streamline diffusion method [Y. Shih, H.C. Elman, Modified streamline diffusion schemes for convection-diffusion problems, Comput. Methods Appl. Mech. Eng, 1998.] for the discretization of this convection-diffusion model to prevent internal layers because of the discontinuities while using the coarsening algorithm for the image compression. Numerical experiments have shown that our convection-diffusion model for removing both Gaussian and salt-and-pepper noises, efficiently and reliably preserves edges quite satisfactorily.     

On some new variational problems for image denoising

This paper is devoted to image denoising problems using multiresolution schemes related to variational problems. We start with the linear approach of Donoho and Johnstone, that is related to a well known diffusion‐type variational problem. In order to improve the behavior of this approach, we propose some new nonlinear variational problems more adapted to the problem of denoising. Moreover, the discretization is performed using nonlinear multiresolution schemes. In particular, we obtain some fast and well adapted schemes for the considered problem of denoising.     

Nonlinear multigrid method for solving the anisotropic image denoising models

In this paper, we study a nonlinear multigrid method for solving a general image denoising model with two L ¹-regularization terms. Different from the previous studies, we give a simpler derivation of the dual formulation of the general model by augmented Lagrangian method. In order to improve the convergence rate of the proposed multigrid method, an improved dual iteration is proposed as its smoother. Furthermore, we apply the proposed method to the anisotropic ROF model and the anisotropic LLT model. We also give the local Fourier analysis (LFAs) of the Chambolle’s dual iterations and a modified smoother for solving these two models, respectively. Numerical results illustrate the efficiency of the proposed method and indicate that such a multigrid method is more suitable to deal with large-sized images.     

Nonlocal patches based Gaussian mixture model for image inpainting

We consider the inpainting problem for noisy images. It is very challenge to suppress noise when image inpainting is processed. An image patches based nonlocal variational method is proposed to simultaneously inpainting and denoising in this paper. Our approach is developed on an assumption that the small image patches should be obeyed a distribution which can be described by a high dimension Gaussian Mixture Model. By a maximum a posteriori (MAP) estimation, we formulate a new regularization term according to the log-likelihood function of the mixture model. To optimize this regularization term efficiently, we adopt the idea of the Expectation Maximization (EM) algorithm. In which, the expectation step can give an adaptive weighting function which can be regarded as a nonlocal connections among pixels. Using this fact, we built a framework for non-local image inpainting under noise. Moreover, we mathematically prove the existence of minimizer for the proposed inpainting model. By using a splitting algorithm, the proposed model are able to realize image inpainting and denoising simultaneously. Numerical results show that the proposed method can produce impressive reconstructed results when the inpainting region is rather large.     

一个用于图像去噪、量化的反应扩散模型

提出一个用于图像去噪、量化的反应扩散模型,并通过计算机使用该模型对图像实例实行去噪和量化,验证了该模型在图像处理速度和效果上都优于已有模型.     

A new nonlocal model for the restoration of textured images

In this paper, we focus on the mathematical and numerical study of a new nonlocal reaction-diffusion system for image denoising. This model is motivated by involving the decomposition approach of $H^{-1}$ norm suggested by Meyer [25] which is more appropriate to represent the oscillatory patterns and small details in the textured image. Based on Schaeffer''s fixed point theorem, we prove the existence and uniqueness of solution of the proposed model. To illustrate the efficiency and effectiveness of our model, we test the denoising experimental results as well we compare with some existing models in the literature.     

A combination model for image denoising

In this paper, we propose an efficient combination model of the second-order ROF model and a simple fourth-order partial differential equation (PDE) for image denoising. The split Bregman method is used to convert the nonlinear combination model into a linear system in the outer iteration, and an algebraic multigrid method is applied to solve the linear system in the inner iteration. Furthermore, Krylov subspace acceleration is adopted to improve convergence in the outer iteration. At the same time, we prove that the model is strictly convex and exists a unique global minimizer. We have also conducted a variety of numerical experiments to analyze the parameter selection criteria and discuss the performance of the fourth-order PDE in the combination model. The results show that our model can reduce blocky effects and our algorithm is efficient and robust to solve the proposed model.     

Image denoising using the Gaussian curvature of the image surface

A number of high‐order variational models for image denoising have been proposed within the last few years. The main motivation behind these models is to fix problems such as the staircase effect and the loss of image contrast that the classical Rudin–Osher–Fatemi model [Leonid I. Rudin, Stanley Osher and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259–268] and others also based on the gradient of the image do have. In this work, we propose a new variational model for image denoising based on the Gaussian curvature of the image surface of a given image. We analytically study the proposed model to show why it preserves image contrast, recovers sharp edges, does not transform piecewise smooth functions into piecewise constant functions and is also able to preserve corners. In addition, we also provide two fast solvers for its numerical realization. Numerical experiments are shown to illustrate the good performance of the algorithms and test results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1066–1089, 2016     

基于广义交叉认证的多小波阈值的图像降噪

提出一种新的小波收缩阈值降噪方法,该方法是通过对噪声图像进行多小波变换,然后用广义交叉认证的方法来确定小波阈值参数．由于本文采用的是多小波变换,而多小波一般同时具有正交性和线性相位,另外广义交叉认证方法不需要对噪声的强度进行估计,所以这种方法有比较好的降噪效果．实验结果表明该方法与基于小波变换的广义交叉认证的图像降噪方法相比较,其降噪效果有一定的提高;同时也表明在一定的条件下,其降噪效果要明显好于传统的Wiener滤波方法．     

基于PSO_Trainlm BP模型的图像去噪研究

针对在使用BP模型进行图像去噪时,模型存在的对初始权阈值敏感、易陷入局部极小值和收敛速度慢的问题.为了提高模型去噪效率,提出采用改进粒子群神经网络模型进行图像去噪.首先运用改进粒子群算法对BP神经网络权阈值进行初始寻优,再用trainlm BP算法对优化的网络权阈值进一步精确优化,随后建立基于粒子群算法的BP神经网络去噪模型,并将其应用到图像去噪研究中.仿真结果表明,新模型结合了粒子群算法的全局寻优能力和BP算法的局部搜索能力,减小了模型对初始权阈值的敏感性,有效防止了模型陷入局部极小值的可能,提高了图像去噪模型的速度和质量.     

A projection method based on the splitting Bregman iteration for the image denoising

By analyzing the connection between the projection operator and the shrink operator, we propose a projection method based on the splitting Bregman iteration for image denoising problem in this paper. Compared with the splitting Bregman method, the proposed method has a more compact form so that it is more fast and efficient. Following from the operator theory, the convergence of the proposed method is proved. Some numerical comparisons between the proposed method and the splitting Bregman method are arranged for solving two basic image denoising models.     

A two-step model for image denoising using a duality strategy and surface fitting

In order to alleviate the staircase effect or the edge blurring in the course of the image denoising, we propose a two-step model based on the duality strategy. In fact, this strategy follows the observation that the dual variable of the restored image can be looked at as the normal vector. So we first obtain the dual variable and then reconstruct the image by fitting the dual variable. Following the augmented Lagrangian strategy, we propose a projection gradient method for solving this two-step model. We also give some convergence analyses of the proposed projection gradient method. Several numerical experiments are tested to compare our proposed model with the ROF model and the LLT model.