Stochastic Systems Group  

Towards A Statistical Theory Of Shapes of Curves
Anuj Srivastava
Florida State University
We seek a comprehensive theory that allows us to treat continuous curves as random quantities and to model/analyze their shapes with all the desired invariances. The basic framework is to form a Hilbert manifold of curves (open or closed, unit length or not, etc) endowed with a Riemannian metric. The invariances are introduced using the actions of similarity groups  rigid motion, scale, and reparameterization  in such a way that their actions form isometries and the Riemannian metric descends to the quotient space. The last invariance motivates the use of a specific squareroot function for representing shapes of curves. In this Riemannian framework, we have developed techniques for: (i) comparing individual shapes, (ii) computing sample statistics (means and covariance) of given shapes, (iii) deriving wrapped normal densities for modeling variability in shape classes, (iv) testing hypotheses about classifying test shapes into individual classes, and (v) estimating shapes in a Bayesian framework. I will demonstrate these ideas using examples from computer vision, human biometric, and medical image analysis.
Some extensions of this framework include: (1) quantifying asymmetry in shapes of objects, (2) looking for shapes in cluttered 2D point clouds, (3) predicting shapes of silhouettes of objects from novel viewing angles by transporting probability distributions on shape spaces, and (4) studying shapes of facial surfaces by representing them as indexed collection of curves. Time permitting, I will summarize research on these items.
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