A new variational model for shape graph registration with partial matching constraints

Abstract

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

Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

“…A second avenue for future research would be to replace the L 2 or Fisher-Rao penalties by different regularization metrics for the weight change function in either a static or metamorphosis setting, with the purpose of imposing spatially smoother weights. In the special case of rectifiable varifolds, one could for instance introduce higher-order Sobolev or total variation norms of the weight change function, by analogy with what has been considered in the context of functional shapes in [19] or elastic shape analysis in [26].…”

“…Through some of the presented simulations, we will show that it can provide robustness to different types of density imbalances in structured and unstructured geometric data. But we shall also illustrate its potential to deal with partially observed data with curves and surfaces, which has been a recurrent and challenging issue for many different shape analysis models [22][23][24][25][26][27].…”

“…Moreover, as we shall see, the generalization from measures to higher-dimensional varifolds and the corresponding change in the transport action induces further significant differences with these two models. Finally, related to the aforementioned challenge of partial data registration, we should mention the two recent works [25] and [26] which both examine alternative approaches that also rely, to some degree, on the varifold representation. The key difference consists in the fact that [25] rather modifies the fidelity term used in the registration problem into a pseudo-distance that allows for partial overlap of the matched shape and the target while our approach actively models and estimate local varifold weight change itself.…”

“…The key difference consists in the fact that [25] rather modifies the fidelity term used in the registration problem into a pseudo-distance that allows for partial overlap of the matched shape and the target while our approach actively models and estimate local varifold weight change itself. On the other hand, the method of [26], while also based on the estimation of weight changes, focuses on the space of curves equipped with intrinsic Sobolev metrics and does not fall in the setting of Riemannian metrics between varifolds that we follow in the present work.…”