NDLI: Convergence Analysis of Finite Volume Scheme for Nonlinear Tensor Anisotropic Diffusion in Image Processing

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Mikula, Karol Drblkov, Olga
Copyright Year	2007
Abstract	In this article we design the semiimplicit finite volume scheme for coherence enhancing diffusion in image processing and prove its convergence to the weak solution of the problem. The finite volume methods are natural tools for image processing applications since they use piecewise constant representation of approximate solutions similarly to the structure of digital images. They have been successfully applied in image processing, e.g., for solving the PeronaMalik equation or curvature-driven level set equations, where the nonlinearities are represented by a scalar function dependent on a solution gradient. Design of suitable finite volume schemes for tensor diffusion is a nontrivial task here we present the first such scheme with a convergence proof for the practical nonlinear model used in coherence-enhancing image smoothing. We provide basic information about this type of nonlinear diffusion including a construction of its diffusion tensor, and we derive a semiimplicit finite volume scheme for this nonlinear model with the help of covolume mesh. This method is well known as the diamond-cell method owing to the choice of covolume as a diamond-shaped polygon. Further, we prove a convergence of a discrete solution given by our scheme to the weak solution of the problem. The proof is based on Kolmogorov's compactness theorem and a bounding of a gradient in the tangential direction by using a gradient in the normal direction. Finally computational results illustrated in figures are discussed.
Starting Page	37
Ending Page	60
Page Count	24
File Format	PDF
ISSN	00361429
DOI	10.1137/070685038
e-ISSN	10957170
Issue Number	1
Volume Number	46
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2007-12-12
Access Restriction	Subscribed
Subject Keyword	image processing numerical solution Stability and convergence of numerical methods nonlinear tensor diffusion Image processing convergence Nonlinear initial value problems for linear parabolic equations semiimplicit scheme diamond-cell finite volume method Finite volume methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Error Estimates for a Combined Finite Volume--Finite Element Method for Nonlinear Convection--Diffusion Problems

Entropy Formulation for Parabolic Degenerate Equations with General Dirichlet Boundary Conditions and Application to the Convergence of FV Methods

SemiImplicit Covolume Method in 3D Image Segmentation

Semi-discrete finite difference multiscale scheme for a concrete corrosion model: a priori estimates and convergence

Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations

An unconditionally stable finite element scheme for anisotropic curve shortening flow

Linearized FE Approximations to a Nonlinear Gradient Flow

Numerical Methods for Semilinear Parabolic Equations

Numerical Methods for Time-Periodic Solutions of Nonlinear Parabolic Boundary Value Problems

Convergence Analysis of Finite Volume Scheme for Nonlinear Tensor Anisotropic Diffusion in Image Processing

Similar Documents

Error Estimates for a Combined Finite Volume--Finite Element Method for Nonlinear Convection--Diffusion Problems

Entropy Formulation for Parabolic Degenerate Equations with General Dirichlet Boundary Conditions and Application to the Convergence of FV Methods

SemiImplicit Covolume Method in 3D Image Segmentation

Semi-discrete finite difference multiscale scheme for a concrete corrosion model: a priori estimates and convergence

Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations

An unconditionally stable finite element scheme for anisotropic curve shortening flow

Linearized FE Approximations to a Nonlinear Gradient Flow

Numerical Methods for Semilinear Parabolic Equations

Numerical Methods for Time-Periodic Solutions of Nonlinear Parabolic Boundary Value Problems

Convergence Analysis of Finite Volume Scheme for Nonlinear Tensor Anisotropic Diffusion in Image Processing