Locally-Weighted Elastic Comparison of Planar Shapes

Abstract

Registration of curves is a necessary component of statistical shape analysis. The goal of registration is to align collections of shapes so that common features are appropriately matched for further comparison and subsequent analyses. Traditional methods for registration typically rely on optimizing an energy functional over a set of appropriate shape-preserving transformations (i.e., rotations and reparameterizations). These functionals typically rely on the standard L^2 metric. In certain applications, it may make sense to use a more flexible metric which can align shapes most preferably with respect to a local shape feature (i.e., a certain curve segment selected from the overall shape). In this work, we define a weighted shape metric which allows for emphasis on local shape features. Registration can be performed with respect to this metric. We demonstrate the registration procedure using simulated curves as well as real data, and show the dependence of the optimal rotation and re-parameterization on the specified weights, as well as the resulting deformation path from one shape to another.