Transporting deformations via integration of local strains

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 BXand BYboth undergoing its own deformation, and describing these two deformations with the diffeomorphisms ΦXand ΦY, then when it is possible to say that they are experiencing the same deformation? Given a correspondence between the points of BXand BY(i.e. a bijective map), then a naive 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 CX:= ∇TΦX∇ΦX= ∇TΦY∇ΦY=: CYholds or, in other words, the local metric (non linear strain) induced by the two diffeomorphisms is the same for all the corresponding points.