$C^1$ Analysis of Hermite Subdivision Schemes on Manifolds

Moosmüller, Caroline

doi:10.1137/15m1033459

Cited by 17 publications

(20 citation statements)

References 28 publications

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“…To conclude C 1 convergence of U from convergence of S A , it is required that condition (15) is fulfilled whenever In the following we prove that the proximity condition (15) holds between a linear operator S A and the T M-valued operator U constructed from S A (13), where M is a surface or matrix group.…”

Section: The Proximity Condition For Hermite Schemesmentioning

confidence: 99%

“…While ⊕ is always smooth and often globally defined (this is the case in both matrix groups and complete surfaces [11,16]), is in general only smooth in some neighborhood of p. Our results in Section 5 are based on [15], which only considers "dense enough" input data. We therefore assume that is always smooth.…”

Section: Unified Notationmentioning

confidence: 99%

“…We also define r lin and w lin , which are the linear versions of r and w. This means that ⊕ and are replaced by + and − respectively and P m i p j (v j ) is replaced by v j . Therefore, in order to prove (15), we have to show the inequalities:…”

Section: The Proximity Condition For Hermite Schemesmentioning

confidence: 99%

“…In a recent paper [15] we propose an analogue of linear Hermite schemes in manifolds which are equipped with an exponential map. This construction works via conversion of vector data to point data, and makes use of the well-established methods of non-Hermite subdivision in manifold, see [9] for an overview.…”

mentioning

confidence: 99%

“…The C 1 convergence analysis of the nonlinear schemes we obtain by the parallel transport approach is provided from their linear counterparts by means of a proximity condition for Hermite schemes introduced by [15]. This condition allows to conclude C 1 convergence of the manifold-valued scheme if it is "close enough" to a C 1 convergent linear one.…”

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confidence: 99%

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Hermite subdivision on manifolds via parallel transport

Moosmüller

2017

Adv Comput Math

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We propose a new adaption of linear Hermite subdivision schemes to the manifold setting. Our construction is intrinsic, as it is based solely on geodesics and on the parallel transport operator of the manifold. The resulting nonlinear Hermite subdivision schemes are analyzed with respect to convergence and C 1 smoothness. Similar to previous work on manifold-valued subdivision, this analysis is carried out by proving that a so-called proximity condition is fulfilled. This condition allows to conclude convergence and smoothness properties of the manifold-valued scheme from its linear counterpart, provided that the input data are dense enough. Therefore the main part of this paper is concerned with showing that our nonlinear Hermite scheme is "close enough", i.e., in proximity, to the linear scheme it is derived from.

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Section: The Proximity Condition For Hermite Schemesmentioning

confidence: 99%

Section: Unified Notationmentioning

confidence: 99%

Section: The Proximity Condition For Hermite Schemesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Hermite subdivision on manifolds via parallel transport

Moosmüller

2017

Adv Comput Math

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Cubic Hermite interpolators on the space of probability measures

Adouani,

Samir,

Tran

2024

Math Methods in App Sciences

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In this paper, we introduce a novel two‐step modeling method to generalize cubic Hermite interpolators on the space of probability measures . First, we develop new approaches to capture the Riemannian geometric structure of when equipped with Fisher–Rao metric. Furthermore, we develop and detail all numerical tools on , namely, Levi–Civita connection, minimal geodesics, parallel transport, exponential map, and logarithm map. Then, we demonstrate that preliminary analysis results yield significant benefits in constructing an optimal cubic Hermite spline on as a nonlinear Riemannian manifold, precisely where conventional numerical methods fail.

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Geometric Subdivision and Multiscale Transforms

Wallner

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations like averages. This chapter discusses different kinds of geometric structures like metric spaces, Riemannian manifolds, and groups, and in what way we can make elementary operations geometrically meaningful. A nice example of this is the Riemannian metric naturally associated with the space of positive definite matrices and the intrinsic operations on positive definite matrices derived from it. We disucss averages first and then proceed to refinement operations (subdivision) and multiscale transforms. In particular, we report on the current knowledge as regards convergence and smoothness.

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$C^1$ Analysis of Hermite Subdivision Schemes on Manifolds

Cited by 17 publications

References 28 publications

Hermite subdivision on manifolds via parallel transport

Hermite subdivision on manifolds via parallel transport

Cubic Hermite interpolators on the space of probability measures

Geometric Subdivision and Multiscale Transforms

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