Irina Kogan, NCSU
2 February 2015
An essential problem for image recognition is the object-image correspondence problem. Given an object in 3-space Z, and a 2D set X, we want to know if X is an image of Z. We’re want to remain agnostic about everything else: where the camera was that took the picture, where it was pointed, and even the internal structure of the camera. Perhaps the image X could arise as a picture of Z in several different ways.
This is a hard problem. There are 12 degrees of freedom, so solving the problem by brute force is probably too expensive.
Dr. Kogan reminds us of Cartan’s theory of signatures. Given a group acting on a manifold, this theory provides a set of invariants (the signature) so that to tell whether two smooth submanifolds are equivalent under the action, we just compute their signatures. If the signatures agree, then the two submanifolds are locally related by the group action.
To get from ‘local’ to ‘global’, we could assume that the sets in question are rigid–say, algebraic sets. But algebraic sets aren’t necessarily smooth submanifolds, so some additional work is needed to develop a theory of signatures that’s up to the task.
Dr. Kogan and her collaborators’ work lies in developing just such a theory, that reduces the problem from 12 degrees of freedom down to only 3.
