Shape Space, Recognition, Minimal Distortion, Vision Groups and Applications

Steven Damelin, American Mathematical Society, University of Michigan Ann Arbor

Visual objects are often known up to some ambiguity, depending on the methods used to acquire them. The first-order approximation to any transformation is, by definition, affine, and the affine approximation to changes between images has been used often in computer vision. Thus it is beneficial to deal with objects known only up to an affine transformation. For example, feature points on a planar transform projectively between different views, and the projective transformation can in many cases be approximated by an affine transformation.

More generally, given two visual objects in a containing Euclidean space R^k,  one may study vision group actions between these two objects often with an underlying signature which are equivalent  under some symmetry or minimal distortion action with respect to a suitable metric inherited by this action. For example, Euclidean groups, similarity, Equi-Affine, projections, camera rotations and video groups.

The study of the space of ordered configurations of n distinct points in R^k up to similarity transformations was pioneered by Kendall who coined the name shape space. For different groups of transformations (rigid, similarity, linear, affine, projective for example) one obtains different shape spaces. Moreover, while these formulations allow often global optimal optimization, e.g. using convex objectives, many of the problems above require efficient approximation methods which work locally.

This framework has applications to biological structural molecule reconstruction problems, to recognition tasks and to matching features across images with minimal distortion. This talk will discuss various work with collaborators around this circle of ideas.

Steven Damelin holds a full time appointment with the American Mathematical Society and a courtesy appointment at the  University of Michigan. He did his Ph.D at Wiwatersrand under Doron Lubinsky, now at The Georgia Institute of Technology. He has worked  in Harmonic Analysis/Computational Applied Harmonic Analysis, One and Multi Dimensional Approximation theory, Singular operators,  Scatttering, Computer Vision and Optimization, Codes/Combinatorial Designs/Minimal Energy/Equidistribution, Orthogonal Polynomials/Potential Theory and Machine Learning/ Probability.