Analysis of Planar Shapes Using Geodesic Lengths on a Shape Manifold

For analyzing shapes of planar, closed curves, we propose a mathematical representation of closed curves using “direction” functions (integrals of the signed curvature functions). Shapes are represented as elements of an infinite-dimensional manifold and their pairwise differences are quantified using the lengths of geodesics connecting them on this manifold. Exploiting the periodic nature of these representations, we use a Fourier basis to discretize them and use a gradient-based shooting method for finding geodesics between any two shapes. Lengths of geodesics provide a metric for comparing shapes. Some applications of this shape metric are illustrated including: (i) clustering of objects based on their shapes, and (ii) statistical analysis of shapes including computation of intrinsic means and covariances.