A note on the variation of geometric functionals
This work provides a methodological correction for researchers in computer vision and image processing using variational methods, but it is incremental as it builds on established frameworks like Geodesic Active Contours and level set methods.
The paper addresses the issue that deriving Euler-Lagrange equations for geometric functionals in image processing often leads to non-geometric or nonsensical gradient descent equations, and it proposes a method to correctly derive these equations to ensure geometric validity and sensical results.
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.