  A novel approach to perform unsupervised sequential learning for functional
data is proposed. Our goal is to extract reference shapes (referred to as
templates) from noisy, deformed and censored realizations of curves and images.
Our model generalizes the Bayesian dense deformable template model
(Allassonni\`ere et al., 2007), 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 algorithm, extending
the work from Capp\'e and Moulines (2009). Our 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 our findings.
