The Square Root Normal Field Distance and Unbalanced Optimal Transport

Bauer, Martin; Hartman, Emmanuel; Klassen, Eric

doi:10.1007/s00245-022-09867-y

Cited by 4 publications

(1 citation statement)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Consequently, there arises the phenomenon that distinct shapes are indistinguishable by the SRNF shape distance. This behavior was originally studied in [35] and was further discussed in [8], where it was shown that for each closed surface there exists a convex surface which is indistinguishable by the SRNF distance. Moreover, the image of I via φ is not convex, which implies that the SRNF distance is indeed only a first-order approximation of a geodesic distance function rather than a true geodesic distance on I, i.e., the SRNF distance does not come from geodesics (optimal deformations) in I.…”

Section: Related Work In Riemannian Shape Analysismentioning

confidence: 96%

Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework

Hartman¹,

Sukurdeep²,

Klassen³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Related Work In Riemannian Shape Analysismentioning

confidence: 96%

Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework

Hartman¹,

Sukurdeep²,

Klassen³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework

et al. 2023

View full text Add to dashboard Cite

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.

show abstract

Simple Unbalanced Optimal Transport

Khesin,

Modin,

Volk

2024

International Mathematics Research Notices

View full text Add to dashboard Cite

We introduce and study a simple model capturing the main features of unbalanced optimal transport. It is based on equipping the conical extension of the group of all diffeomorphisms with a natural metric, which allows a Riemannian submersion to the space of volume forms of arbitrary total mass. We describe its finite-dimensional version and present a concise comparison study of the geometry, Hamiltonian features, and geodesics for this and other extensions. One of the corollaries of this approach is that along any geodesic the total mass evolves with constant acceleration, as an object’s height in a constant buoyancy field.

show abstract

The Square Root Normal Field Distance and Unbalanced Optimal Transport

Cited by 4 publications

References 48 publications

Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework

Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework

Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework

Simple Unbalanced Optimal Transport

Contact Info

Product

Resources

About