Lie group of transformations to study anisotropic diffusion of scalar images
type: Item
instance of: scholarly article
title: Lie group of transformations to study anisotropic diffusion of scalar images
zbMATH Open document ID: 1260.22013
DOI: 10.1134/S1995080212030080
author: J. Jena
published in: Lobachevskii Journal of Mathematics
publication date: 2013-01-23T00:00:00Z
review text: A Lie group of transformations associated with an anisotropic diffusion equation is derived using invariance properties. The equation is of great importance in the study of the geometrical diffusion of scalar images. The author employs the method of Lie group of transformations after transforming the equation to a system of first-order partial differential equations. Constant conformal invariance of the considered system of equations is used to obtain a system of equations for the evaluation of the infinitesimals. Particular cases are considered by asigning particular values to the arbitrary function constants associated with the infinitesimals. It was observed that the edge stopping functions used by Peron and Malik can be obtained in particular cases. In view of the assumptions, the number of independent variables in the evolution equation are reduced, which is useful for the numerical implementation in the study of the anisotropic diffusion of scaler images. Infinitesimals of the Lie group of transformations with respect to the independent and dependent variables along with invariant surface conditions are used to obtain a general form of edge stopping function. The edge stopping functions used by Perona and Malik and their successors are obtained as particular cases. The equation in three independent variables is reduced in terms of fewer independent variables.
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 22E20
Mathematics Subject Classification ID: 22E05
Mathematics Subject Classification ID: 22E65
zbMATH DE Number: 6128343
zbMATH Keywords: Lie group of transformations
zbMATH Keywords: partial differential equations
zbMATH Keywords: self-similar solutions
zbMATH Keywords: image processing
zbMATH Keywords: edge stopping function
Wikidata QID: Q115247396
reviewed by: Jerzy August Gawinecki
MaRDI profile type: Publication
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full work available at URL: s1995080212030080
OpenAlex ID: W1987233141
description: scientific article; zbMATH DE number 6128343