A relaxed approach for curve matching with elastic metrics
This work provides a unified and computationally efficient framework for curve matching under elastic metrics, benefiting shape analysis and registration tasks.
The paper develops a method for computing geodesics between unparametrized curves under elastic metrics, integrating H^2-metrics and scale-invariant variants. By using a varifold-based relaxed variational formulation, it avoids optimizing over reparametrizations and can quotient out similarity groups, with numerical examples showing competitive performance.
In this paper we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of $H^2$-metrics with constant coefficients and scale-invariant $H^2$-metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration.