Distribution on Warp Maps for Alignment of Open and Closed Curves

TL;DR

This paper introduces a new, easy-to-sample distribution on warp maps for curve alignment, accommodating landmark constraints and applicable to open and closed curves in various dimensions, enhancing Bayesian and optimization methods.

Contribution

It provides a constructive, point-process based distribution on warp maps that simplifies sampling and handles landmark constraints for curve alignment.

Findings

Distribution related to Dirichlet process for [0,1] warp maps.

Effective as a prior in Bayesian alignment models.

Useful as a proposal distribution in stochastic optimization.

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 $\mathbb{S}^1$ 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.