Images taken and stored digitally are often degraded by noise, so that the perceived image quality is significantly decreased in the presence of noise and human gaze behavior is affected by this noise as well. In recent years, image denoising has been modeled as partial differential equations (PDEs), which can be said to eliminate noise as well as preserve image detail is one of their main advantages. Numerical simulation of these PDEs, which are often nonlinear, high-order and high-dimensional, is one of the most challenging topics in this field. In this paper, one of the most powerful meshless numerical approaches, the operator splitting radial basis function (RBF) collocation method, is introduced to solve two modified versions of the Perona-Malik (PM) equation. The method splits each problem into several equations each of which is solved in one direction so as to make the major problem solvable in a smaller size. Furthermore, being in one direction makes it possible to solve the equations without any need for domain decomposition and its complexities. An unconditionally stable method (Crank-Nicolson) is used for discretizing time in this paper. Using meshless approaches in the image processing problems makes sense since the structure of digital images consist of pixels that corresponds to the distributed nodes of the meshless method so the complex time consuming domain construction of the finite element methods is avoided to get higher speeds in obtaining the desired denoised results. Some images with different noise levels are denoised using the presented method in the section of numerical results to show the desirable performance of the method. Also, the quality of the denoised images are concluded in tables which can be used to compare the obtained results. 

Efficient image denoising technique using the meshless method: Investigation of operator splitting RBF collocation method for two anisotropic diffusion-based PDEs

We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.

Recovering the initial condition in the One-Phase Stefan problem.

Down-sampling in the process of saving and transferring images often leads to the appearance of jaggies or step like edges in the objects of images where they should be smooth to seem natural. The application of blurring filters for the purpose of anti-aliasing images seems inappropriate since it leaves a blurring effect that destroys the edges in a way that they become indistinguishable. In 2012 Ziou and Horé proposed a novel partial differential equation (PDE) in which anti-aliased images efficiently. However, solving this equation as it is severely nonlinear and defined in a large domain due to the size of images is a challenging problem. In this paper, we propose a new numerical method that is based on a combination of iterative operator splitting (IOS) technique with Gaussian radial basis functions (RBFs) method for solving this equation in an efficient way. The proposed method converts the problem to several one-dimensional linear problems that are solved in smaller sizes to overcome the large domain of the problem. The Crank-Nicolson method is applied for the discretization of the time dimension. The proposed method is tested on different gray-scale, color, and outline images to illustrate the efficiency of the proposed method visually and numerically in the section of numerical results. 

Anti-aliasing of gray-scale/color/outline images: Looking through the lens of numerical approaches for PDE-based models

&lt;p style='text-indent:20px;'&gt;We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.&lt;/p&gt; 

Recovering the initial condition in the one-phase Stefan problem

For the first time in mathematical finance field, we propose the local weak form meshless methods for option pricing; especially in this paper we select and analysis two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpolation (LRPI) based on Wu's compactly supported radial basis functions (WCS-RBFs). LBIE and LRPI schemes are the truly meshless methods, because, a traditional non-overlapping, continuous mesh is not required, either for the construction of the shape functions, or for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the $\theta$-weighted scheme is employed for the time derivative. Stability analysis of the methods is analyzed and performed by the matrix method. In fact, based on an analysis carried out in the present paper, the methods are unconditionally stable for implicit Euler (\theta = 0) and Crank-Nicolson (\theta = 0.5) schemes. It should be noted that LBIE and LRPI schemes lead to banded and sparse system matrices. Therefore, we use a powerful iterative algorithm named the Bi-conjugate gradient stabilized method (BCGSTAB) to get rid of this system. Numerical experiments are presented showing that the LBIE and LRPI approaches are extremely accurate and fast.

Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options

In this paper, the dynamical modeling and behavior analysis of the inverse boundary Stefan problem which promising the understanding of modeling brain tumor treatment, are studied. To umerical simulate these models and to overcome their difficulties such as, non-linearity, free boundary property and having a non-rectangular domain, we propose the use of a strongly meshless technique based on radial basis functions in conjunction with a (quasi)linearization algorithm. Numerical examples are given to show the good accuracy and stability of the presented method.

Recovering a moving boundary from Cauchy data in an inverse problem which arises in modeling brain tumor treatment: the (quasi)linearization idea combined with radial basis functions (RBFs) approximation

A. Hajiollow

Papers

Recovering a moving boundary from Cauchy data in an inverse problem which arises in modeling brain tumor treatment: the (quasi)linearization idea combined with radial basis functions (RBFs) approximation