An inexact matching approach for the comparison of plane curves with general elastic metrics

Sukurdeep, Yashil; Bauer, Martin; Charon, Nicolas

doi:10.1109/ieeeconf44664.2019.9049031

Cited by 4 publications

(1 citation statement)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A second avenue for improvement would be to develop a faster numerical pipeline to estimate the distance and/or geodesics between shape graphs in view of applications of the method to the statistical analysis of large shape datasets. This could be achieved for instance by focusing on first-order Sobolev metrics (at the expense of solid theoretical results on the existence of solutions), for which the computation of the metric can be significantly simplified thanks to the availability of a general simplifying transformation known as the F a,b transform [37,45]. 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.…”

Section: Discussionmentioning

confidence: 99%

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

Sukurdeep¹,

Bauer²,

Charon³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%

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

Sukurdeep¹,

Bauer²,

Charon³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation

Bauer¹,

Charon²,

Klassen³

et al. 2021

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

View full text Add to dashboard Cite

Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation

Bauer

Charon

Klassen

et al. 2023

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

Self Cite

View full text Add to dashboard Cite

We introduce a family of Finsler metrics, called the L p -Fisher-Rao metrics Fp, for p ∈ (1, ∞), which generalizes the classical Fisher-Rao metric F2, both on the space of densities Dens+(M ) and probability densities Prob(M ). We then study their relations to the Amari-Cencov α-connections ∇ (α) from information geometry: on Dens+(M ), the geodesic equations of Fp and ∇ (α) coincide, for p = 2/(1 − α). Both are pullbacks of canonical constructions on L p (M ), in which geodesics are simply straight lines. In particular, this gives a new interpretation of α-geodesics as being energy minimizing curves. On Prob(M ), the Fp and ∇ (α) geodesics can still be thought as pullbacks of natural operations on the unit sphere in L p (M ), but in this case they no longer coincide unless p = 2. On this space we show that geodesics of the α-connections are pullbacks of projections of straight lines onto the unit sphere, and they cease to exists when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman-Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of Fp, and study their relation to ∇ (α) .

show abstract

An inexact matching approach for the comparison of plane curves with general elastic metrics

Cited by 4 publications

References 16 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

Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation

Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation

Contact Info

Product

Resources

About