Elastic Shape Analysis of Planar Objects Using Tensor Field Representations

Shape analysis of objects in images is a critical area of research, and several approaches, including those that utilize elastic Riemannian metrics, have been proposed. While elastic techniques for shape analysis of curves are pretty advanced, the corresponding results for higher-dimensional objects (surfaces and disks) are less developed. This paper studies shapes of solid planar objects that are embeddings of a compact domain—a unit square or a unit disk—in R 2 . Specifically, it introduces a mathematical representation of objects using tensor fields and uses a re-parametrization-invariant Riemannian metric on these tensor fields to analyze object shapes elastically. The essential contribution here is developing an efficient numerical technique to map tensor fields back to the object space, allowing one to approximate geodesic paths in these objects’ shape spaces. Finally, the paper extends this framework to reach landmark-driven registration and improve geodesic computations. The paper illustrates this framework using several simulated and natural objects.