Geometric Analysis of Continuous, Planar Shapes

We propose two differential geometric representations of planar shapes using: (i) direction functions and (ii) curvature functions, of their boundaries. Under either representation, planar shapes are treated as elements of infinite-dimensional shape spaces. Pairwise differences between the shapes are quantified using the lengths of geodesics connecting them on the shape spaces. We specify the geometry of the two shape spaces and utilize numerical methods for finding geodesics on them. Some applications of this shape analysis are illustrated including: (i) interpolation between shapes, (ii) clustering of objects according to their shapes, and (iii) computation of intrinsic mean shapes.