Using the vector distance functions to evolve manifolds of arbitrary codimension

Abstract

We present a novel method for representing and evolving objects of arbitrary dimension. The method, called the Vector Distance Function (VDF) method, uses the vector that connects any point in space to its closest point on the object. It can deal with smooth manifolds with and without boundaries and with shapes of different dimensions. It can be used to evolve such objects according to a variety of motions, including mean curvature. If discontinuous velocity fields are allowed the dimension of the objects can change. The evolution method that we propose guarantees that we stay in the class of VDF’s and therefore that the intrinsic properties of the underlying shapes such as their dimension, curvatures can be read off easily from the VDF and its spatial derivatives at each time instant. The main disadvantage of the method is its redundancy: the size of the representation is always that of the ambient space even though the object we are representing may be of a much lower dimension. This disadvantage is also one of its strengths since it buys us flexibility.