Parallel transport of local strains. Issue 5 (2nd November 2019) - Milicchio, F.; Varano, V.; Gabriele, S.; Teresi, L.; Puddu, P. E.; Piras, P.
Abstract: 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.