Abstract
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 C1 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.
Highlights
Hermite subdivision is an iterative method for constructing a curve together with its derivatives from discrete point-vector data
The present paper investigates manifold analogues of Hermite subdivision rules which work directly with vectors and employ the parallel transport operators available in Riemannian manifolds and in Lie groups
In the following we prove that the proximity condition (15) holds between a linear operator SA and the takes arguments in (T M)-valued operator U constructed from SA (13), where M is a surface or matrix group
Summary
Hermite subdivision is an iterative method for constructing a curve together with its derivatives from discrete point-vector data. It has mainly been studied in the linear. 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. The C1 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]. Throughout this paper we use as an instructive example a certain non-interpolatory Hermite scheme which is the de Rham transform [5] of a scheme proposed by [13]
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