Functional Currents : a new mathematical tool to model and analyse functional shapes

TL;DR

This paper introduces functional currents, a mathematical framework that combines geometric and functional data to better model, analyze, and process complex shapes with associated signals, especially in medical imaging.

Contribution

The paper develops the concept of functional currents, extending mathematical currents to handle both shape geometry and associated signals simultaneously.

Findings

Functional currents effectively represent combined geometric and functional information.

The framework enables new algorithms for shape analysis and registration.

Applications demonstrated in shape redundancy reduction and shape registration.

Abstract

This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometrico-functional data, that is a data structure that involves a deformable shape (roughly a finite dimensional sub manifold) together with a function defined on this shape taking value in another manifold. Indeed, if mathematical currents have already proved to be very efficient theoretically and numerically to model and process shapes as curves or surfaces, they are limited to the manipulation of purely geometrical objects. We show that the introduction of the concept of functional currents offers a genuine solution to the simultaneous processing of the geometric and signal information of any functional shape. We explain how functional currents can be equipped with a Hilbertian norm mixing geometrical and functional content of functional shapes nicely behaving under geometrical and functional perturbations and paving the way to various processing algorithms. We illustrate this potential on two problems: the redundancy reduction of functional shapes representations through matching pursuit schemes on functional currents and the simultaneous geometric and functional registration of functional shapes under diffeomorphic transport.