Parallel transport 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 (Formula presented.) and (Formula presented.) both undergoing their own deformation, and describing these two deformations with the diffeomorphisms (Formula presented.) and (Formula presented.), then when is it possible to say that they are experiencing the same deformation? Given a correspondence between the points of (Formula presented.) and (Formula presented.) (i.e. a bijective map), then a naïve possible answer could be that the displacement vector (Formula presented.), 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 (Formula presented.) holds or, in other words, the local metric (non linear strain) induced by the two diffeomorphisms is the same for all the corresponding points.