A Wavelet Regularization for Nonlinear Diffusion Equations
نویسندگان
چکیده
We are concerned with a wavelet–based treatment of nonlinear diffusion equations in the context of image processing. In particular, we focus on the Perona– Malik model as a suitable instrument for smoothing images while preserving edges. We are not exploring a complete new method of solving the Perona– Malik equation but, inspired by Weickert et.al., we develop a new variant, based on wavelet technology, of regularizing this specific equation. By carefully choosing the generators, we are able to derive all inner products and integrals of the weak formulation with high efficiency. We prove that the proposed scheme overcomes the ill-posedness of the nonlinear Perona–Malik diffusion equation and illustrate the obtained results by some numerical experiments.
منابع مشابه
Applying Legendre Wavelet Method with Regularization for a Class of Singular Boundary Value Problems
In this paper Legendre wavelet bases have been used for finding approximate solutions to singular boundary value problems arising in physiology. When the number of basis functions are increased the algebraic system of equations would be ill-conditioned (because of the singularity), to overcome this for large $M$, we use some kind of Tikhonov regularization. Examples from applied sciences are pr...
متن کاملConvergence of Legendre wavelet collocation method for solving nonlinear Stratonovich Volterra integral equations
In this paper, we apply Legendre wavelet collocation method to obtain the approximate solution of nonlinear Stratonovich Volterra integral equations. The main advantage of this method is that Legendre wavelet has orthogonality property and therefore coefficients of expansion are easily calculated. By using this method, the solution of nonlinear Stratonovich Volterra integral equation reduces to...
متن کاملNumerical Solution of Caputo-Fabrizio Time Fractional Distributed Order Reaction-diffusion Equation via Quasi Wavelet based Numerical Method
In this paper, we derive a novel numerical method to find out the numerical solution of fractional partial differential equations (PDEs) involving Caputo-Fabrizio (C-F) fractional derivatives. We first find out the approximation formula of C-F derivative of function tk. We approximate the C-F derivative in time with the help of the Legendre spectral method and approximation formula o...
متن کاملA Regularization of Nonlinear Diffusion Equations in a Multiresolution Framework
We are developing a regularization technique for Perona–Malik diffusion equations that relies on multiresolution techniques. The main result of this paper is to show that the chosen discretization overcomes the ill-posedness of the nonlinear Perona–Malik model. The resulting algorithm is tested and the results are compared with pixel–based methods. keywords, phrases: Nonlinear diffusion, regula...
متن کاملThe Sine-Cosine Wavelet and Its Application in the Optimal Control of Nonlinear Systems with Constraint
In this paper, an optimal control of quadratic performance index with nonlinear constrained is presented. The sine-cosine wavelet operational matrix of integration and product matrix are introduced and applied to reduce nonlinear differential equations to the nonlinear algebraic equations. Then, the Newton-Raphson method is used for solving these sets of algebraic equations. To present ability ...
متن کامل