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Original Articles

Parallel transport of local strains

ORCID Icon, ORCID Icon, , , ORCID Icon &
Pages 520-528 | Received 16 Jan 2018, Accepted 17 May 2018, Published online: 07 Jun 2018
 

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 BX and BY both undergoing their own deformation, and describing these two deformations with the diffeomorphisms ΦX and ΦY, then when is it possible to say that they are experiencing the same deformation? Given a correspondence between the points of BX and BY (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 CX:=ΦXΦX=ΦYΦY=:CY holds or, in other words, the local metric (non linear strain) induced by the two diffeomorphisms is the same for all the corresponding points.

Notes

No potential conflict of interest was reported by the authors.

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