Parallel transport of local strains

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 ${\mathcal{B}}_X$ and ${\mathcal{B}}_Y$ both undergoing their own deformation, and describing these two deformations with the diffeomorphisms ${\mathrm{\Phi }}_X$ and ${\mathrm{\Phi }}_Y$, then when is it possible to say that they are experiencing the same deformation? Given a correspondence between the points of ${\mathcal{B}}_X$ and ${\mathcal{B}}_Y$ (i.e. a bijective map), then a naïve possible answer could be that the displacement vector $\mathbf{u}$, 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 ${\mathbf{C}}_X:={\mathrm{\nabla }}^{\top }{\mathrm{\Phi }}_X\mathrm{\nabla }{\mathrm{\Phi }}_X={\mathrm{\nabla }}^{\top }{\mathrm{\Phi }}_Y\mathrm{\nabla }{\mathrm{\Phi }}_Y=:{\mathbf{C}}_Y$ holds or, in other words, the local metric (non linear strain) induced by the two diffeomorphisms is the same for all the corresponding points.

Keywords:

• Transporting deformations
• strain integration
• finite element methods
• shape analysis

Notes

No potential conflict of interest was reported by the authors.