A relaxed approach for curve matching with elastic metrics

Metric registration of curves and surfaces using optimal control

“…The assumptions on γ are much milder: typical choices include for instance the linear kernel γ( t, t ) = t, t or the restricted Gaussian kernel on S d−1 γ( t, t ) = e 2 σ 2 (1− t, t ) . Although we do not obtain, with this framework, a true distance on S, we point out that the result of Theorem 1 may in fact still hold with the same assumptions for a larger class than embedded shapes: for instance, it was extended to immersed curves with transverse self-intersections in [7]. In all cases, and for the purpose of this work, d var gives in practice a simple and explicit notion of shape proximity to be used as a relaxation of the exact matching constraint.…”

“…As mentioned earlier, d Var defined by ( 12) and ( 13) is only a pseudo-distance on S unless the RKHS W associated with the kernel is such that the mapping [µ] is injective. Unfortunately, it can be seen that, in the case of oriented varifold representation (11), this is actually impossible for the full space of unparametrized immersed curves, no matter the choice of W (see for example [7] for a counterexample with curves). However, sufficient conditions exist for d var to be a distance on the subset of embedded submanifolds Emb(M, R d )/ Diff(M ).…”

“…More details on this approach can be found in [7,10]. The full multiplicity of the boundary knots in t simplify the boundary constraints, since we have…”

“…The code can be downloaded at github: https://www.github.com/h2metrics/h2metrics, see[7] for a detailed description of the implementation.…”