Metrics in the Space of Shapes for Optimization and Dynamics

In this talk, we discuss three problems in computer vision and medical
image analysis: (1) object detection/extraction from images, (2)
visual object tracking from time-varying imagery, and (3) object
comparison/statistical operations on objects. Objects are partially
characterized by their shape, and therefore all three problems require
mathematical tools to deal with shapes. An important concept in the
study of shape is that of a metric structure on the space of shapes
(i.e., a similarity score between shapes). Although shape metrics are
vital for object comparison and defining statistical operations on
objects, they have not been considered of use for object extraction
and visual object tracking in the computer vision community. In this
talk, I will show that the study of shape metrics, specifically
Riemannian metrics, are central to problems (1) and (2), although not
known before in the computer vision community, and that they play a
fundamental role in applications. I will illustrate the idea on the so
called Sobolev-type metric – a metric which is natural for
applications in computer vision because of its smoothness
properties. In the first part of the talk, I will display the
applications to object extraction from images based on active
contours. In the second part of the talk, I will present recent work
on defining a dynamical model-based approach for tracking the shape
and deformation of deforming objects from time-varying imagery.