A New Variational Model for Shape Graph Registration with Partial Matching Constraints

This paper introduces a new extension of Riemannian elastic curve matching to general class geometric structures, which we call (weighted) shape graphs, that allows for registration with partial constraints and topological inconsistencies. Weighted graphs are the union an arbitrary number component curves in Euclidean space potential connectivity between some their boundary points, together weight function defined on each curve. The framework higher-order invariant Sobolev metrics is particularly well suited constructing notions distances geodesics unparametrized curves. main difficulty adapting this setting absence consistency, typically results inadequate search exact two graphs. We overcome hurdle by defining inexact variational formulation problem any underlying topology, relying convenient measure representation given varifolds relax constraint. then prove existence minimizers when choose sufficient regularity total variation (TV) regularization function. propose numerical optimization approach adapts smoothed fast iterative shrinkage-thresholding algorithm (SFISTA) deal $TV$ norm minimization us reduce solving sequence smooth unconstrained problems. finally illustrate capabilities our model through several examples showcasing its ability tackle partially observed topologically varying data.