Geometric analysis of planar shapes using geodesic paths

We propose a differential geometric representation of planar shapes using "direction" functions of their boundaries. Each shape becomes an element of a constrained function space, an infinite-dimensional manifold, and pairwise differences between are quantified using the lengths of geodesics connecting them on this space. A gradient-based shooting method is used for finding geodesics between any two shapes. Some applications of this shape metric are illustrated including clustering of objects based on their shapes and computation of intrinsic mean shapes.