Parallel transport of local strains. Issue 5 (2nd November 2019)

• Record Type: Journal Article
• Authors: Milicchio, F.; Varano, V.; Gabriele, S.; Teresi, L.; Puddu, P. E.; Piras, P.
• Is Part Of: Computer methods in biomechanics and biomedical engineering. Volume 7:Issue 5/6(2019)
• Page Start: 520
• Page End: 528
• Publication Date: 2019-11-02
• Subjects: Transporting deformations -- strain integration -- finite element methods -- shape analysis; Imaging systems in biology -- Periodicals; Imaging systems in medicine -- Periodicals; Biomechanics -- Data processing -- Periodicals; Biomedical engineering -- Periodicals; 616.0757
• DOI: 10.1080/21681163.2018.1479313
• Languages: English
• ISSNs: 2168-1163

Abstract

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.