Riemannian Metrics on the Space of Solid Shapes

We present a new framework for multidimensional shape analysis. The proposed framework represents solid objects as points on an infinite-dimensional Riemannian manifold and distances between objects as minimal length geodesic paths. Intershape distance forms the foundation for shape-based statistical analysis. The proposed method incorporates a metric that naturally prevents self-intersections of object boundaries and thus produces a well-defined interior and exterior along every geodesic path. This paper presents an implementation of the geodesic computations for 2D shapes and gives examples of geodesic paths that demonstrate the advantages of enforcing well-defined boundaries. This compares favorably with equivalent paths under a linear L metric, which do not prevent self-intersection of the boundary, and thus do not produce valid solid objects.