Analysis of non-rigid shapes
I attended a talk given by Michael Bronstein yesterday, on the analysis of non-rigid shapes using metric geometry. That may sound a bit abstract, but in fact, it delved into just enough theory to whet the appetite, while emphasizing the fascinating and almost miraculous practial fruits of their techniques.
The Bronsteins (twin brothers) and their advisor Kimmel have been doing some fascinating work in — I guess their research falls under this rubric — computer vision and pattern recognition. Classical techniques for shape analysis: comparing and registering them, for example, assume that they are rigid; there are a few– snakes and some stochastic methods come to mind– which don’t make this assumption, but these tend to not be very robust. The novel approach that this group has come up with identifies an object with its intrinsic geometry rather than its extrinsic– i.e. the geodesic distances between points on a surface is considered more important than the distance in 
. The idea is that inelastic deformations will not change these geodesic distances, so they make a more precise and robust fingerprint for object identification.
The problem is how to compare geodesic distances on two candidate objects: if you did this naively, you’d have to look at each possible correspondence of the objects, find one which minimizes the overall difference in geodesic distances, and then use the ’stress’ of this mapping as a measure of the similarity between the objects– this operation is exponential in the number of points considered. Instead, they propose using multidimensional scaling techniques to find a minimally metric distorting embedding of the object into a Euclidean space; the image of the object under this embedding is called the canonical form. To compare two objects, one then compares their canonical forms using classical techniques for rigid shape analysis. Clever trick, and judging from some of their results, incredibly fruitful.
The math involved is a beautiful combination of optimization (for finding the minimally metric distorting embedding) and (metric) geometry. I’ll definitely be reading his book when it comes out.