Title: 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
$\mathbb{S}^1$ 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.