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Alignment of curve data is an integral part of their statistical analysis,and can be achieved using model- or optimization-based approaches. Theparameter space is usually the set of monotone, continuous warp maps of adomain. Infinite-dimensional nature of the parameter space encourages samplingbased approaches, which require a distribution on the set of warp maps.Moreover, the distribution should also enable sampling in the presence ofimportant landmark information on the curves which constrain the warp maps. Foralignment of closed and open curves in $\mathbb{R}^d, d=1,2,3$, possibly withlandmark information, we provide a constructive, point-process based definitionof 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 thedesiderata for decomposition of the alignment problem with landmark constraintsinto multiple unconstrained ones. For warp maps on $[0,1]$, the distribution isrelated to the Dirichlet process. We demonstrate its utility by using it as aprior distribution on warp maps in a Bayesian model for alignment of twounivariate curves, and as a proposal distribution in a stochastic algorithmthat optimizes a suitable alignment functional for higher-dimensional curves.Several examples from simulated and real datasets are provided.

Distribution on Warp Maps for Alignment of Open and Closed Curves