COMPARING SHAPE TRAJECTORIES OF BIOLOGICAL SOFT TISSUES IN THE SIZE-AND-SHAPE SPACE

Biological soft tissues can suffer for very large size and shape changes during their motion. We aim to study the peculiar deformation parameters of this motion, that can be well represented as trajectories in a particular Riemannian manifold named size-and-shape space. Given a sample of different individuals that experience different morphological trajectories it is possible to compare the shape of trajectories themselves. This comparison is not trivial because the Size-and-Shape Space is non-Euclidean. Dealing with soft tissues the differences between individuals can be very large, as a consequence the morphological trajectories can start from highly scattered points on the Size-and-Shape Space. In such a case ordination methods based on the tangent space (as the PCA) should be performed on a different tangent space for each individual to have a good approximation. In order to compare shape data belonging to different tangent spaces is necessary to perform a parallel transport of the data toward a common tangent space. In this paper we propose a data centering technique based on the Levi-Civita parallel transport in the Size-and-Shape Space useful to compare trajectories. Moreover we give a simple criterion to discriminate in which cases the proposed technique can be well approximated by a suitable hierarchical alignment followed by a linear shift. We illustrate this technique by performing analysis on real human hearts whose 2D left ventricular motion data set are considered