Malik Equation Research Articles

Existence result and approximation of an optimal control problem for the Perona–Malik equation

We discuss some optimal control problem for the evolutionary Perona–Malik equations with the Neumann boundary condition. The control variable v is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution ud∈L2(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_d\\in L^2(\\Omega )$$\\end{document} and the current system state. Since we cannot expect to have a solution of the original boundary value problem for each admissible control, we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems for linear parabolic equations and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero.

ADMM algorithm for some regularized Perona–Malik equation and applications to image denoising

ADMM algorithm for some regularized Perona–Malik equation and applications to image denoising

A novel fractional‐order reaction diffusion system for the multiplicative noise removal

AbstractIn this paper, a fractional‐order nonlinear reaction diffusion system is proposed to remove the multiplicative Gamma noise. The new reaction diffusion system consists of three equations: the regularized Perona and Malik (PM) equation , which is used for presmoothing the image that is contaminated by noise; the time‐delay regularization equation, which is used for incorporating the past information into the diffusion process and adjusting oversmoothing; and the fractional‐order diffusion equation, which is used for removing the multiplicative Gamma noise and maintaining texture. The new reaction diffusion system is coupled, leading to the difficulty in theoretical analysis. To this end, we use decoupled and Schauder's fixed‐point theorem to obtain the existence and uniqueness of weak solution of the system. The explicit finite difference scheme is employed to implement the fractional‐order nonlinear reaction diffusion system. In addition, we test both texture images and nontexture images. Experimental results show that the new model achieves a better trade‐off between denoising performance and texture preservation than the other three models.

Image denoising by nonlinear nonlocal diffusion equations

In this paper, we propose and study a nonlinear nonlocal diffusion equation of the Perona–Malik type for the removal of Gaussian noise in images. Based on the fixed-point theorem, we prove the unique solvability of the equation. We also show that solutions of the nonlocal equation converge to the solution of the local spatially regularized Perona–Malik equation if the kernel is rescaled appropriately. The new equation inherits the merit of the nonlocal method that restores details and textures of noisy images but avoids speckles and artifacts in homogeneous regions. Comparisons with other nonlocal methods for image denoising are presented.

An image inpainting model based on the mixture of Perona–Malik equation and Cahn–Hilliard equation

In this paper, a fourth-order PDE model is proposed for image inpainting. This model is based on the mixture of Perona–Malik equation and Cahn–Hilliard equation. Using the idea of energy splitting, a numerical scheme is introduced for solving the proposed PDE model. Numerical experiments for testing the proposed model are provided at the end of the paper. The numerical results indicate that the proposed model can provide better inpainting performance with less computational time comparing to the classical PDE based inpainting models.

Selective diffusion involving reaction for binarization of bleed-through document images

Binarization has always been a challenging problem in document image processing because of various types of degradation. In this paper, we present a nonlinear reaction–diffusion model for binarization of bleed-through document images, which is the Perona–Malik equation involving diffusion coefficient based on structure tensor along with a nonlinear reaction term. The Perona–Malik diffusion is utilized to selectively smooth document images with bleed-through removal. Meanwhile, the nonlinear reaction term takes the responsibility for the desired binarization. In order to solve our model numerically, we develop a parallel–series splitting algorithm by combining finite differencing with two kinds of splitting methods in the literature. Our algorithm is tested on seven publicly available datasets (DIBCO 2009 to 2014 and 2016). The experimental results show that our method averagely outperforms six relevant models for the nineteen document images with bleed-through in the DIBCO series datasets.

Sinogram restoration for low-dose X-ray computed tomography using regularized Perona–Malik equation with intuitionistic fuzzy entropy

Edges and flat areas in the sinogram of low-dose X-ray computed tomography are difficult to distinguish. This lack of distinction leads to excessive smoothing in the sinogram restoration. To address this problem, we propose a sinogram restoration algorithm using the regularized Perona–Malik (P–M) equation with intuitionistic fuzzy entropy. Firstly, considering the sinogram fuzziness, a novel edge indicator function is constructed using both the gradient magnitude and intuitionistic fuzzy entropy. Secondly, using the constructed edge indicator function as the diffusion coefficient, a novel regularized P–M equation smoothing model is presented. The proposed model overcomes the shortage of traditional P–M equations, which are ill-conditioned. Moreover, it performs diffusion with different directions and intensities in different regions of the sinogram. The optimal solution of the proposed algorithm is obtained by using the additional operator splitting method. Finally, the reconstructed image is achieved by filtered back projection from the smoothed sinogram. Experimental results show that the presented method can retain important edges while smoothing noise and perform better than others.

A Fully Conservative Parallel Numerical Algorithm with Adaptive Spatial Grid for Solving Nonlinear Diffusion Equations in Image Processing

In this paper we present simple yet efficient parallel program implementation of grid-difference method for solving nonlinear parabolic equations, which satisfies both fully conservative property and second order of approximation on non-uniform spatial grid according to geometrical sanity of a task. The proposed algorithm was tested on Perona–Malik method for image noise ltering task based on differential equations. Also in this work we propose generalization of the Perona–Malik equation, which is a one of diffusion in complex-valued region type. This corresponds to the conversion to such types of nonlinear equations like Leontovich–Fock equation with a dependent on the gradient field according to the nonlinear law coefficient of diffraction. This is a special case of generalization of the Perona–Malik equation to the multicomponent case. This approach makes noise removal process more flexible by increasing its capabilities, which allows achieving better results for the task of image denoising.

Mixed Finite Element Approximation for Bivariate Perona–Malik Model Arising in 2D and 3D Image Denoising

In this paper, we present a new model of nonlinear diffusion arising in image processing field based on the Perona–Malik equation which we call the bivariate Perona–Malik model. The aim of this model is to remove image noise while preserving edges, boundaries, and textures. To solve this model we use a new algorithm based on mixed finite element method. In this context we prove mathematically the existence and uniqueness of the solution of the proposed model in a well chosen space. At last, we present the experimental results in 2D and 3D images filtering, which demonstrate the efficiency and effectiveness of our algorithm and finally, we compare it with other well known methods such as the finite difference method presented by Perona et al. (IEEE Trans Pattern Anal Mach Intell 12:629–639, 1990), the finite element method and the finite volume method studied by Handlovicova et al. (J Vis Commun Image Represent 13:217–237, 2002).

Wellposedness of a nonlocal nonlinear diffusion equation of image processing

Existence and uniqueness of solutions to non-smooth initial data are established for a slight modification of the degenerate regularization of the well-known Perona–Malik equation first proposed in Guidotti and Lambers (2009). The results heavily rely on the choice of an appropriate functional setting inspired by a recent approach to degenerate parabolic equations via so-called singular Riemannian manifolds (Amann, 2013, 2016).

Lie symmetry properties of nonlinear reaction-diffusion equations with gradient-dependent diffusivity

Complete descriptions of the Lie symmetries of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity in one and two space dimensions are obtained. A surprisingly rich set of Lie symmetry algebras depending on the form of diffusivity and source (sink) in the equations is derived. It is established that there exists a subclass in 1-D space admitting an infinite-dimensional Lie algebra of invariance so that it is linearisable. A special power-law diffusivity with a fixed exponent, which leads to wider Lie invariance of the equations in question in 2-D space, is also derived. However, it is shown that the diffusion equation without a source term (which often arises in applications and is sometimes called the Perona–Malik equation) possesses no rich variety of Lie symmetries depending on the form of gradient-dependent diffusivity. The results of the Lie symmetry classification for the reduction to lower dimensionality, and a search for exact solutions of the nonlinear 2-D equation with power-law diffusivity, also are included.

Iterative solvers for image denoising with diffusion models: A comparative study

In this paper we propose and compare the use of two iterative solvers using the Crank–Nicolson finite difference method, to address the task of image denoising via partial differential equations (PDEs) models such as Regularized Perona–Malik equation or C-model and Bazan model (Bilateral-filter-based model). The solvers which are considered in this paper are the Successive-over-Relaxation (SOR) and an advanced solver known as Hybrid Bi-Conjugate Gradient Stabilized (Hybrid BiCGStab) method. From numerical experiments, it is found that the Crank–Nicolson method with hybrid BiCGStab iterative solver produces better results and is more efficient than SOR and already existing, in terms of MSSIM and PSNR.

Convex Integration and Infinitely Many Weak Solutions to the Perona--Malik Equation in All Dimensions

We prove that for all smooth nonconstant initial data the initial-Neumann boundary value problem for the Perona--Malik equation in image processing possesses infinitely many Lipschitz weak solutions on smooth bounded convex domains in all dimensions. Such existence results have not been known except for the one-dimensional problems. Our approach is motivated by reformulating the Perona--Malik equation as a nonhomogeneous partial differential inclusion with linear constraint but some uncontrollable components of gradient. We establish a general existence result by a suitable Baire category method under a pivotal density hypothesis. We finally fulfill this density hypothesis by convex integration based on certain approximations from an explicit formula of lamination convex hull of some matrix set involved.

Radial weak solutions for the Perona–Malik equation as a differential inclusion

The Perona–Malik equation is an ill-posed forward–backward parabolic equation with some application in image processing. In this paper, we study the Perona–Malik type equation on a ball in an arbitrary dimension n and show that there exist infinitely many radial weak solutions to the homogeneous Neumann boundary problem for smooth nonconstant radially symmetric initial data. Our approach is to reformulate the n-dimensional equation into a one-dimensional equation, to convert the one-dimensional problem into an inhomogeneous partial differential inclusion problem, and to apply a Baire's category method to the differential inclusion to generate infinitely many solutions.

New estimation method of the contrast parameter for the Perona–Malik diffusion equation

The aim of this contribution is to make an efficient smoothing algorithm that preserves edges and provides valuable information for any segmentation process. The non-linear anisotropic diffusion (AD) model of Perona–Malik is considered to enhance the edges in the process of diffusion through a variable diffusion coefficient. However, the diffusion coefficient is very sensitive to the so-called contrast or gradient threshold parameter. This article proposes a novel methodology for the estimation of this contrast parameter based on a partition of the image using the K-means algorithm and a least-square fit to approximate the diffusion coefficient (KMLS). The experimental results show that the quality of the edge detection process improves when the proposed algorithm in the smoothing AD is used instead of the traditional techniques. The comparison is performed using two objective edge detection performance measures, the so-called Pratt's figure of merit and the symmetric average distance. Both measures show great improvements if we use the Perona–Malik equation with the KMLS estimator.

The Perona–Malik inequality and application to image denoising

This paper deals with the mathematical and numerical study of a variational model of the image denoising equation derived from the Perona–Malik equation. The diffusion resulting from the proposed model is a combination of the Perona–Malik and p-Laplacian operators. The existence and uniqueness of the asymptotic behavior as p→∞ of the proposed model are established. We also give some numerical implementation details and show experimental results on examples images which prove the efficiency and effectiveness of our model.

Anisotropic second and fourth order diffusion models based on Convolutional Virtual Electric Field for image denoising

This paper proposes novel second order and fourth order anisotropic diffusion models for image denoising. These two models are based on CONvolutional Virtual Electric Field (CONVEF). The proposed second order anisotropic diffusion model was introduced by introducing the CONVEF into the Perona–Malik (P–M) equation and the fourth order one is coined by combining CONVEF and the You–Kaveh (Y–K) equation. The employment of the CONVEF model allows to improve the estimation of high order derivatives and makes the proposed methods more robust to noise. In addition, the proposed CONVEF-based Y–K model is an anisotropic diffusion equation which can avoid the over-smoothness of edges and provide better edge protection capability except for better denoising. Some experiments are conducted to demonstrate the efficiency of the proposed methods.

PASSING TO THE LIMIT IN MAXIMAL SLOPE CURVES: FROM A REGULARIZED PERONA–MALIK EQUATION TO THE TOTAL VARIATION FLOW

We prove that solutions of a mildly regularized Perona–Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension. The proof is based on the general principle that "the limit of gradient-flows is the gradient-flow of the limit". To this end, we exploit a general result relating the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.

A backward–forward regularization of the Perona–Malik equation

It is shown that the Perona–Malik equation (PME) admits a natural regularization by forward–backward diffusions possessing better analytical properties than PME itself. Well-posedness of the regularizing problem along with a complete understanding of its long time behavior can be obtained by resorting to weak Young measure valued solutions in the spirit of Kinderlehrer and Pedregal (1992) [1] and Demoulini (1996) [2]. Solutions are unique (to an extent to be specified) but can exhibit “micro-oscillations” (in the sense of minimizing sequences and in the spirit of material science) between “preferred” gradient states. In the limit of vanishing regularization, the preferred gradients have size 0 or ∞ thus explaining the well-known phenomenon of staircasing. The theoretical results do completely confirm and/or predict numerical observations concerning the generic behavior of solutions.

Slow Time Behavior of the Semidiscrete Perona–Malik Scheme in One Dimension

We consider the long time behavior of the semidiscrete scheme for the Perona–Malik equation in one dimension. We prove that approximated solutions converge, in a slow time scale, to solutions of a limit problem. This limit problem evolves piecewise constant functions by moving their plateaus in the vertical direction according to a system of ordinary differential equations. Our convergence result is global-in-time, and this forces us to face the collision of plateaus when the system singularizes. The proof is based on energy estimates and gradient-flow techniques, according to the general idea that “the limit of the gradient flows is the gradient flow of the limit functional.” Our main innovations are a uniform Hölder estimate up to the first collision time included, a well preparation result with a careful analysis of what happens at the discrete level during collisions, and renormalizing the functionals after each collision in order to have a nontrivial Gamma-limit for all times.