Distribution on Warp Maps for Alignment of Open and Closed Curves

Abstract

Alignment of curve data is an integral part of their statistical analysis, and can be achieved using model- or optimization-based approaches. The parameter space is usually the set of monotone, continuous warp maps of a domain. Infinite-dimensional nature of the parameter space encourages sampling based approaches, which require a distribution on the set of warp maps. Moreover, the distribution should also enable sampling in the presence of important landmark information on the curves which constrain the warp maps. For alignment of closed and open curves in ${\mathbb{R}}^d,d=1,2,3$, possibly with landmark information, we provide a constructive, point-process based definition of a distribution on the set of warp maps of [0, 1] and the unit circle $\mathbb{S}$, that is, (1) simple to sample from, and (2) possesses the desiderata for decomposition of the alignment problem with landmark constraints into multiple unconstrained ones. For warp maps on [0, 1], the distribution is related to the Dirichlet process. We demonstrate its utility by using it as a prior distribution on warp maps in a Bayesian model for alignment of two univariate curves, and as a proposal distribution in a stochastic algorithm that optimizes a suitable alignment functional for higher-dimensional curves. Several examples from simulated and real datasets are provided.

Keywords:

• Functional data
• Point processes
• Simulated annealing
• Stochastic curve registration

Supplementary Materials

Proofs of all technical results, comments on an alternative method to define a distribution on warp maps of ${\mathbb{S}}^1$, descriptions of datasets used, and additional numerical results are presented in the supplementary materials.

Acknowledgments

The authors thank Ian Dryden, Huiling Le, and Eric Klassen for helpful discussions. We also thank two anonymous reviewers for their suggestions.

Additional information

Funding

This research was partially supported by NSF DMS 1613054 and NIH R01 CA214955 (to KB and SK). SK was also partially supported by NSF CCF 1740761.