A note on the variation of geometric functionals

TL;DR

This paper clarifies the proper derivation of Euler-Lagrange and gradient descent equations in geometric functionals, ensuring the resulting equations are geometrically meaningful and consistent in image processing applications.

Contribution

It provides a correct methodology for deriving geometric and sensible Euler-Lagrange and gradient descent equations in the calculus of variations for geometric functionals.

Findings

Correct derivation of Euler-Lagrange equations for geometric functionals

Ensures gradient descent equations are geometrically meaningful

Addresses issues with non-sensical equations in existing methods

Abstract

Calculus of Variation combined with Differential Geometry as tools of modelling and solving problems in image processing and computer vision were introduced in the late 80's and the 90s of the 20th century. The beginning of an extensive work in these directions was marked by works such as Geodesic Active Contours (GAC), the Beltrami framework, level set method of Osher and Sethian the works of Charpiat et al. and the works by Chan and Vese to name just a few. In many cases the optimization of these functional are done by the gradient descent method via the calculation of the Euler-Lagrange equations. Straightforward use of the resulted EL equations in the gradient descent scheme leads to non-geometric and in some cases non sensical equations. It is costumary to modify these EL equations or even the functional itself in order to obtain geometric and/or sensical equations. The aim of this note is to point to the correct way to derive the EL and the gradient descent equations such that the resulted gradient descent equation is geometric and makes sense.