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Transporting deformations from a template to a different one is a typical task of the shape analysis. In particular, it is necessary to perform such a kind of transport when performing group-wise statistical analyses in Shape or Size and Shape Spaces. A typical example is when one is interested in separating the difference in function from the difference in shape. The key point is: given two different templates B X and B Y both undergoing their own deformation, and describing these two deformations with the diffeomorphisms Φ X and Φ Y , then when is it possible to say that they are experiencing the same deformation? Given a correspondence between the points of B X and B Y (i.e. a bijective map), then a naïve possible answer could be that the displacement vector u , associated to each corresponding point couple, is the same. In this manuscript, we assume a different viewpoint: two templates undergo the same deformation if for each corresponding point couple of the two templates the condition C X : = ∇ ⊤ Φ X ∇ Φ X = ∇ ⊤ Φ Y ∇ Φ Y = : C Y holds or, in other words, the local metric (non linear strain) induced by the two diffeomorphisms is the same for all the corresponding points. [ABSTRACT FROM AUTHOR]

SUMMARY Ontogeny is usually studied by analyzing a deformation series spanning over juvenile to adult shapes. In geometric morphometrics, this approach implies applying generalized Procrustes analysis coupled with principal component analysis on multiple individuals or multiple species datasets. The trouble with such a procedure is that it mixes intra- and inter-group variation. While MANCOVA models are relevant statistical/mathematical tools to draw inferences about the similarities of trajectories, if one wants to observe and interpret the morphological deformation alone by filtering inter-group variability, a particular tool, namely parallel transport, is necessary. In the context of ontogenetic trajectories, one should firstly perform separate multivariate regressions between shape and size, using regression predictions to estimate within-group deformations relative to the smallest individuals. These deformations are then applied to a common reference (the mean of per-group smallest individuals). The estimation of deformations can be performed on the Riemannian manifold by using sophisticated connection metrics. Nevertheless, parallel transport can be effectively achieved by estimating deformations in the Euclidean space via ordinary Procrustes analysis. This approach proved very useful in comparing ontogenetic trajectories of species presenting large morphological differences at early developmental stages. [ABSTRACT FROM AUTHOR]

The aim of this paper is to emphasise the role of the primary strain-line patterns in a human left ventricle (LV) within the complex system that is the heart. In particular, a protocol is proposed for the measurement of the principal strain lines (PSL) in the walls of the LV; this protocol is tested by means of a computational model which resembles a human LV. When the analysis is focused on the epicardial surface, PSL can be used to derive information on the directions of muscle fibres during the entire cardiac cycle, not only the systolic phase. [ABSTRACT FROM AUTHOR]