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In many applications, smooth processes generate data that is recorded under avariety of observation regimes, such as dense, sparse or fragmentedobservations that are often contaminated with error. The statistical goal ofregistering and estimating the individual underlying functions from discreteobservations has thus far been mainly approached sequentially without formaluncertainty propagation, or in an application-specific manner. We propose aunified Bayesian framework for simultaneous registration and estimation, whichis flexible enough to accommodate inference on individual functions undergeneral observation regimes. Our ability to do this relies on the specificationof strongly informative prior models over the amplitude component of functionvariability. We provide two strategies for this critical choice: a data-drivenapproach that defines an empirical basis for the amplitude subspace based ontraining data, and a shape-restricted approach when the relative location andnumber of local extrema is well-understood. The proposed methods build onelastic functional data analysis, which separately models amplitude and phasevariability inherent in functional data. We emphasize the importance ofuncertainty quantification and visualization of these two components as theyprovide complementary information about the estimated functions. We validatethe framework using simulations and real applications to medical imaging andbiometrics.

Bayesian Framework for Simultaneous Registration and Estimation of Noisy, Sparse and Fragmented Functional Data