  In medical image analysis, constructing an atlas, i.e. a mean representative
of an ensemble of images, is a critical task for practitioners to estimate
variability of shapes inside a population, and to characterise and understand
how structural shape changes have an impact on health. This involves
identifying significant shape constituents of a set of images, a process called
segmentation, and mapping this group of images to an unknown mean image, a task
called registration, making a statistical analysis of the image population
possible. To achieve this goal, we propose treating these operations jointly to
leverage their positive mutual influence, in a hyperelasticity setting, by
viewing the shapes to be matched as Ogden materials.
  The approach is complemented by novel hard constraints on the $L^\infty$ norm
of both the Jacobian and its inverse, ensuring that the deformation is a
bi-Lipschitz homeomorphism. Segmentation is based on the Potts model, which
allows for a partition into more than two regions, i.e. more than one shape.
The connection to the registration problem is ensured by the dissimilarity
measure that aims to align the segmented shapes. A representation of the
deformation field in a linear space equipped with a scalar product is then
computed in order to perform a geometry-driven Principal Component Analysis
(PCA) and to extract the main modes of variations inside the image population.
Theoretical results emphasizing the mathematical soundness of the model are
provided, among which existence of minimisers, analysis of a numerical method
of resolution, asymptotic results and a PCA analysis, as well as numerical
simulations demonstrating the ability of the modeling to produce an atlas
exhibiting sharp edges, high contrast and a consistent shape.
