Symmetry Analysis for a Fourth-order Noise-reduction Partial Differential Equation

TL;DR

This paper uses Lie symmetry theory to analyze a fourth-order PDE for image noise reduction, deriving similarity solutions and simplifying static solutions to second-order ODEs, including nonstatic closed-form solutions.

Contribution

It applies Lie symmetry analysis to a specific fourth-order PDE in image processing, deriving similarity solutions and simplifying static solutions.

Findings

Static solutions reduce to maximally symmetric second-order ODEs

Nonstatic closed-form solutions are obtained

Lie symmetries help simplify complex PDEs in image processing

Abstract

We apply the theory of Lie symmetries in order to study a fourth-order $1+2$ evolutionary partial differential equation which has been proposed for the image processing noise reduction. In particular we determine the Lie point symmetries for the specific 1+2 partial differential equations and we apply the invariant functions to determine similarity solutions. For the static solutions we observe that the reduced fourth-order ordinary differential equations are reduced to second-order ordinary differential equations which are maximally symmetric. Finally, nonstatic closed-form solutions are also determined.