  In the study of shapes of human organs using computational anatomy,
variations are found to arise from inter-subject anatomical differences,
disease-specific effects, and measurement noise. This paper introduces a
stochastic model for incorporating random variations into the Large Deformation
Diffeomorphic Metric Mapping (LDDMM) framework. By accounting for randomness in
a particular setup which is crafted to fit the geometrical properties of LDDMM,
we formulate the template estimation problem for landmarks with noise and give
two methods for efficiently estimating the parameters of the noise fields from
a prescribed data set. One method directly approximates the time evolution of
the variance of each landmark by a finite set of differential equations, and
the other is based on an Expectation-Maximisation algorithm. In the second
method, the evaluation of the data likelihood is achieved without registering
the landmarks, by applying bridge sampling using a stochastically perturbed
version of the large deformation gradient flow algorithm. The method and the
estimation algorithms are experimentally validated on synthetic examples and
shape data of human corpora callosa.
