On a fractional reaction-diffusion system applied to image decomposition and restoration

Abstract In this paper, we propose a new model using a fractional reaction–diffusion system for image restoration and image decomposition which is motivated by work of S. Osher, A. Sole and L. Vese’s. Our model decomposes the degraded image into cartoon component belonging to a fractional Sobolev space and textured component belonging to a negative Hilbert space. We are interested in the existence and uniqueness of weak solutions of the system obtained from the model. To overcome the difficulties of the singularity in the system, we use a regularization method. We also give a numerical algorithm for solving the system, and present numerical experimental results to show the efficiency of our model.

• Publication year

  2019

• Venue

  Computers and Mathematics with Applications

• Publication date

  2019-09-01

• Fields of study

  Mathematics, Physics, Computer Science, Engineering

• Identifiers
  DOI 10.1016/J.CAMWA.2019.05.030
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Fractional p-Laplacian evolution equations

  2016cited by this paper
• On fractional p-Laplacian parabolic problem with general data

  2016cited by this paper
• A fractional-order derivative based variational framework for image denoising

  2016cited by this paper
• Fractional calculus in image processing: a review

  2016cited by this paper
• On the Evolutionary Fractional p-Laplacian

  2015cited by this paper
• A Total Fractional-Order Variation Model for Image Restoration with Nonhomogeneous Boundary Conditions and Its Numerical Solution

  2015cited by this paper
• The Dirichlet Problem for the fractional p-Laplacian evolution equation

  2015cited by this paper
• A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations

  2015cited by this paper
• The fractional Cheeger problem

  2013cited by this paper
• Fractional-order TV-L2 model for image denoising

  2013cited by this paper
• On a reaction-diffusion system applied to image decomposition and restoration

  2011cited by this paper
• Hitchhiker's guide to the fractional Sobolev spaces

  2011cited by this paper
• Numerical Approximations for a Nonlocal Evolution Equation

  2011cited by this paper
• Fast Cartoon + Texture Image Filters

  2010cited by this paper
• Image Denoising Methods. A New Nonlocal Principle

  2010cited by this paper
• A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

  2008cited by this paper
• Nonlocal Operators with Applications to Image Processing

  2008cited by this paper
• Analysis of the TV regularization and $H^{-1}$ fidelity model for decomposing animage into cartoon plus texture

  2007cited by this paper
• Nonlocal Linear Image Regularization and Supervised Segmentation

  2007cited by this paper
• Fractional-Order Anisotropic Diffusion for Image Denoising

  2007cited by this paper
• A Review of Image Denoising Algorithms, with a New One

  2005cited by this paper
• Image quality assessment: from error visibility to structural similarity

  2004cited by this paper
• Image Decomposition and Restoration Using Total Variation Minimization and the H1

  2003cited by this paper
• Nonlinear total variation based noise removal algorithms

  1992cited by this paper

• A Novel Image Denoising Method Using the Caputo Time-Fractional Convection-Diffusion Model

  2025cites this paper
• On a class of forward-backward reaction-diffusion systems with local and nonlocal coupling for image restoration

  2025cites this paper
• Image Inpainting with Fractional Laplacian Regularization: An Lp Norm Approach

  2025cites this paper
• Full fractional total variation method for eliminating salt and pepper noise

  2025cites this paper
• Integral solution to a parabolic equation involving the fractional p-Laplacian operator

  2025cites this paper
• Nonlocal Cahn-Hilliard type model for image inpainting

  2024cites this paper
• A novel image denoising technique with Caputo type space–time fractional operators

  2024cites this paper
• On the long-time behavior of the continuous and discrete solutions of a nonlocal Cahn-Hilliard type inpainting model

  2024cites this paper
• Variable-order fractional 1-Laplacian diffusion equations for multiplicative noise removal

  2023cites this paper
• The fractional Laplacian based image inpainting

  2023cites this paper
• A class of nonlinear parabolic systems having standard growth and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e685" altimg="si3.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> data

  2022cites this paper
• Nonlocal Adaptive Biharmonic Regularizer for Image Restoration

  2022cites this paper
• Asymptotic stabilisation of coupled delayed time fractional reaction diffusion systems with boundary input disturbances via backstepping sliding-mode control

  2022cites this paper
• An improved spatially controlled reaction–diffusion equation with a non-linear second order operator for image super-resolution

  2021cites this paper
• A non-convex ternary variational decomposition and its application for image denoising

  2021cites this paper
• Parameter learning and fractional differential operators: Applications in regularized image denoising and decomposition problems

  2021cites this paper
• A novel fractional‐order reaction diffusion system for the multiplicative noise removal

  2021cites this paper
• An Efficient Parallel Approximate Algorithm for Solving Time Fractional Reaction-Diffusion Equations

  2020cites this paper
• Parameter learning and fractional differential operators: application in image regularization and decomposition

  2020cites this paper