String Methods for Stochastic Image and Shape Matching

Abstract: Matching of images and analysis of shape differences is traditionally pursued by energy minimization of paths of deformations acting to match the shape objects. In the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, iterative gradient descents on the matching functional lead to matching algorithms informally known as Beg algorithms. When stochasticity is introduced to model stochastic variability of shapes and to provide more realistic models of observed shape data, the corresponding matching problem can be solved with a stochastic Beg algorithm, similar to the finite temperature string method used in rare event sampling. In this paper, we apply a stochastic model compatible with the geometry of the LDDMM framework to obtain a stochastic model of images and we derive the stochastic version of the Beg algorithm which we compare with the string method and an expectation-maximization optimization of posterior likelihoods. The algorithm and its use for statistical inference is tested on stochastic LDDMM landmarks and images.