Median Shapes

When I wrote the paper with Simon Morgan pointing out the $L^1\text{TV}$ functional was actually computing the flat norm for boundaries, we suggested this gave us a computational route to statistics in spaces of shapes. While earlier work certainly touched on this idea of using the flat norm for inference in shape spaces — see this paper on shape recognition, it was not until my student Yunfeng Hu collaborated with myself and Bala Krishnamoorthy (my collaborator, also a co-mentor of Yunfeng’s), that we started addressing the idea of statistics in shape spaces in the original paper with Simon Morgan.

The results can be found here: https://arxiv.org/abs/1802.04968 , in a paper with the title Median Shapes, with authors Yunfeng Hu, Matthew Hudelson, Bala Krishnamoorthy, Altansuren Tumurbaatar, Kevin R. Vixie. Tumurbaatar wrote the first complete version of the code used, and Matthew Hudelson contributed a pivotal new result on graphs inspired by a problem in the paper, while Bala led the computational end of things and I led, in collaboration with Yunfeng Hu (and Bala keeping us honest!), the theoretical parts of the paper. It was a fair bit of work.

We went over the more difficult results a few times, finding improvements and corrections. Of course, there may be a few things here and there to improve, but for now, it is done.

Yunfeng probably spent the most time writing up the piece proving that near regular points on the median, the collection of minimal surfaces meeting the median have a tangent structure we describe as a book.  While this is clear to experienced geometric analysts,  there are lots of little details and we wanted most of the paper to be more accessible to a wider audience. There are lots of other pieces here and there that took time to think about and write up (and rewrite). For example, when showing the set of medians need not contain any regular members,  the part where we show that we need only consider graphs when searching for a minimizer was not easy. And of course, as in most all of geometric analysis, there are problems you solve without too much effort at a high level, but find that writing down is tedious, though at times enlightening due to the fact that those little details turn out to be hard and illuminating.

Because the problem of computing the median reduces to a linear program, while the mean reduces to a quadratic program, we focused on the median problem. Some parts of the paper are a bit long winded, for the reason that we wanted it to have more details that would usually be in a paper communicating to others that understand geometric analysis.

Anyway, have a look. If you find yourself interested, there is already code you can use to compute medians, though we hope eventually to have faster code.