Supervised deep learning of elastic SRV distances on the shape space of curves

Hartman, Emmanuel; Sukurdeep, Yashil; Charon, Nicolas; Klassen, Eric; Bauer, Martin

doi:10.48550/arxiv.2101.04929

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…Alternatively, we plan to investigate supervised deep learning approaches as a way to replace our optimization procedure by a simple forward pass through an appropriately trained neural network. Although still in their infancy within the field of elastic shape analysis, such methods have recently shown some success in simpler settings [38,26]. We can then derive the following upper bounds for each term:…”

Section: Discussionmentioning

confidence: 99%

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

Sukurdeep¹,

Bauer²,

Charon³

2021

Preprint

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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.

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Section: Discussionmentioning

confidence: 99%

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

Sukurdeep¹,

Bauer²,

Charon³

2021

Preprint

Self Cite

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“…where the infimum is taken over all orientation preserving smooth diffeomorphisms of the unit interval I. Various approaches for the efficient numerical solution of (discretisations of) this optimisation problem have been suggested, ranging from gradient based optimisation methods [12] over dynamical programming [7,15] and the reformulation as a Hamilton-Jacobi-Bellman equation [22] to machine learning methods [11,16]. There exists also an analytic algorithm for the case where the curves c 1 and c 2 are piecewise linear [13].…”

Section: Introductionmentioning

confidence: 99%

A square root velocity framework for curves of bounded variation

Grasmair¹

2022

Preprint

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The square root velocity transform is a powerful tool for the efficient computation of distances between curves. Also, after factoring out reparametrisations, it defines a distance between shapes that only depends on their intrinsic geometry but not the concrete parametrisation. Though originally formulated for smooth curves, the square root velocity transform and the resulting shape distance have been thoroughly analysed for the setting of absolutely continuous curves using a relaxed notion of reparametrisations. In this paper, we will generalise the square root velocity distance even further to a class of discontinuous curves. We will provide an explicit formula for the natural extension of this distance to curves of bounded variation and analyse the resulting quotient distance on the space of unparametrised curves. In particular, we will discuss the existence of optimal reparametrisations for which the minimal distance on the quotient space is realised.

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Supervised deep learning of elastic SRV distances on the shape space of curves

Cited by 2 publications

References 15 publications

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

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

A square root velocity framework for curves of bounded variation

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