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In this paper we study a class of Riemannian metrics on the space ofunparametrized curves and develop a method to compute geodesics with givenboundary conditions. It extends previous works on this topic in severalimportant ways. The model and resulting matching algorithm integrate within onecommon setting both the family of $H^2$-metrics with constant coefficients andscale-invariant $H^2$-metrics on both open and closed immersed curves. Thesefamilies include as particular cases the class of first-order elastic metrics.An essential difference with prior approaches is the way that boundaryconstraints are dealt with. By leveraging varifold-based similarity metrics wepropose a relaxed variational formulation for the matching problem that avoidsthe necessity of optimizing over the reparametrization group. Furthermore, weshow that we can also quotient out finite-dimensional similarity groups such astranslation, rotation and scaling groups. The different properties andadvantages are illustrated through numerical examples in which we also providea comparison with related diffeomorphic methods used in shape registration.

A relaxed approach for curve matching with elastic metrics