Hermite subdivision on manifolds via parallel transport

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 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.

Hermite multiwavelets for manifold-valued data

“…We use a linear Hermite subdivision operator S A , with mask A of the form (2), to define a manifold-valued analogue T A satisfying the properties of Definition 8. This is based on the parallel transport construction of [32].…”

“…Results for manifold-valued subdivision schemes on topics such as convergence, smoothness, and approximation order, are often derived from their linear counterparts via a proximity condition [14,16,31,32,42,43,44]. A comparison between a linear and a manifold-valued operator only makes sense in a chart or an embedding of M. In this paper we use charts and thus assume that T M ⊂ V 2 .…”

“…In [32,Corollary 1] it is shown that if the base point sequence is chosen as either m i = p i or as the geodesic midpoint between p i and p i+1 , and the input data is bounded, then the proximity condition between S A and T A is satisfied. Therefore, in this paper, we choose the base point sequence as either one of those sequences.…”

“…We use the mentioned tight connection between subdivision schemes and wavelets to obtain manifold-valued Hermite wavelet schemes, using the construction presented in [31,32]. This works in a similar fashion as the scalar constructions of [17,18].…”

A note on spectral properties of Hermite subdivision operators

“…Hermite subdivision schemes find applications in geometric modeling (if derivatives are of interest) [16,31,32], in approximation theory (linear and manifold-valued) [19,22,25,26], and they can be used for the construction of multiwavelets [7,8,21] and the analysis of biomedical images [6,30].…”