  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.
