## A Riemannian Framework for Shape Analysis of Curves and Surfaces

I will present a comprehensive framework for analyzing shapes of 2D and 3D objects, by focusing on their boundaries as curves and surfaces. An important distinction is to treat boundaries not as point sets or level sets, as is commonly done, but as parameterized objects. However, parameterization adds an extra variability in the representation, as different re-parameterizations of an object do not change its shape. This variability are handled by defining quotient spaces of object representations, modulo re-parameterization and rotation groups, and inheriting a Riemannian metric on the quotient space from the larger space. For curves in Euclidean spaces, we use an elastic Riemannian metric that can be viewed as an extension of the classical Fisher-Rao metric, used in information geometry, to higher dimensions. Furthermore, we define a specific square-root representation that reduces this complicated metric to the standard L2 metric and, thus, greatly simplifying computations such as geodesic paths, sample means, tangent PCA, and stochastic modeling of observed shapes. For surfaces, we have proposed a similar square-root representation and an elastic Riemannian metric, that allows parameterization-invariant shape analysis

of 3D objects.

I will demonstrate these ideas using applications from computer vision, biometrics and activity recognition, protein structure analysis, and anatomical shape analysis.

Anuj Srivastava's Website: http://stat.fsu.edu/~anuj/