Bayesian registration of functions and curves

TL;DR

This paper develops a Bayesian framework for the registration and comparison of functions and curves, utilizing the square root velocity function for warping invariance, with applications in shape analysis and proteomics.

Contribution

It introduces a Bayesian approach with Gaussian processes and MCMC algorithms for curve registration, addressing warping invariance and comparing ambient and quotient space estimators.

Findings

Effective Bayesian inference for curves using MCMC.

Comparable mean shape estimates in ambient and quotient spaces.

Successful application to shape classification and mass spectrometry alignment.

Abstract

Bayesian analysis of functions and curves is considered, where warping and other geometrical transformations are often required for meaningful comparisons. We focus on two applications involving the classification of mouse vertebrae shape outlines and the alignment of mass spectrometry data in proteomics. The functions and curves of interest are represented using the recently introduced square root velocity function, which enables a warping invariant elastic distance to be calculated in a straightforward manner. We distinguish between various spaces of interest: the original space, the ambient space after standardizing, and the quotient space after removing a group of transformations. Using Gaussian process models in the ambient space and Dirichlet priors for the warping functions, we explore Bayesian inference for curves and functions. Markov chain Monte Carlo algorithms are introduced for simulating from the posterior, including simulated tempering for multimodal posteriors. We also compare ambient and quotient space estimators for mean shape, and explain their frequent similarity in many practical problems using a Laplace approximation. A simulation study is carried out, as well as shape classification of the mouse vertebra outlines and practical alignment of the mass spectrometry functions.