Abstract

Shape analysis requires methods for measuring distances between shapes, to define summary statistics, for example, or Gaussian-like distributions. One way to construct such distances is to specify a Riemannian metric on an appropriate space of maps, and then define shape distance as geodesic distance in a quotient space. For shapes in two dimensions, the 'elastic metric' combines tractability with intuitive appeal, with special cases that dramatically simplify computations while still producing state of the art results. For shapes in three dimensions, the situation is less clear. It is unknown whether the full elastic metric admits simplifying representations, and while a reduced version of the metric does, the resulting transform is difficult to invert, and its usefulness has therefore been questionable. In this talk, I will motivate the elastic metric for shapes in three dimensions, elucidate its interesting structure and its relation to the two-dimensional case, and describe what is known about the representation used to construct it. I will then focus on the reduced metric. This admits a representation that greatly simplifies computations, but which is probably not invertible. I will describe recent work that constructs an approximate right inverse for this representation, and show how, despite the theoretical uncertainty, this leads in practice to excellent results in shape analysis problems. This is joint work with Anuj Srivastava, Sebastian Kurtek, Hamid Laga, and Qian Xie.