Abstract
A novel approach to perform unsupervised sequential learning for functional data is proposed. The goal is to extract reference shapes (referred to as templates) from noisy, deformed and censored realizations of curves and images. The proposed model generalizes the Bayesian dense deformable template model, a hierarchical model in which the template is the function to be estimated and the deformation is a nuisance, assumed to be random with a known prior distribution. The templates are estimated using a Monte Carlo version of the online Expectation–Maximization (EM) algorithm. The designed sequential inference framework is significantly more computationally efficient than equivalent batch learning algorithms, especially when the missing data is high-dimensional. Some numerical illustrations on curve registration problem and templates extraction from images are provided to support the methodology.
Highlights
Functional data analysis is concerned with the analysis of curves and shapes, which often display common patterns and variations
We propose the Monte Carlo online EM (MCoEM), an online algorithm in which the curves/shapes are processed one at a time and only once, allowing to estimate the unknown parameters of the mixture of deformable templates model
We have proposed a statistical framework to perform sequential and unsupervised inference of a deformable template model, with application to curve synchronization and shape extraction
Summary
Functional data analysis is concerned with the analysis of curves and shapes, which often display common patterns and variations (in amplitude, orientations, time–space warping, etc. . . ). This alternative has been considered, among others by Liu and Yang (2009) and Frey and Jojic (2003), in which the transformed mixture of Gaussian models was used Another way to handle the E-step, suggested in Gaffney and Smyth (2004) and Bernhardt et al (2015), consists in performing an approximate Bayesian integration, which amounts to replace the posterior distribution of the hidden data conditionally to the observation by a Gaussian distribution, obtained from a Laplace approximation. We propose the Monte Carlo online EM (MCoEM), an online algorithm in which the curves/shapes are processed one at a time and only once, allowing to estimate the unknown parameters of the mixture of deformable templates model.
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