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In recent work nonlinear subdivision schemes which operate on manifold-valued data have been successfully analyzed with the aid of so-called proximity conditions bounding the difference between a linear scheme and the nonlinear one. The main difficulty with this method is the verification of these conditions. In the present paper we obtain a very clear… (More)

Subdivision is a powerful way of approximating a continuous object f (x, y) by a sequence ((S l p i,j) i,j∈Z) l∈N of discrete data on finer and finer grids. The rule S, that maps an approximation on a coarse grid, S l p, to the approximation on the next finer grid, S l+1 p, is called subdivision scheme. If for a given scheme S every continuous object f (x,… (More)

Linear stationary subdivision rules take a sequence of input data and produce ever denser sequences of subdivided data from it. They are employed in multiresolution modeling and have intimate connections with wavelet and more general pyramid transforms. Data which naturally do not live in a vector space, but in a nonlinear geometry like a surface, symmetric… (More)

We study the following modification of a linear subdivision scheme S: Let M be a surface embedded in Euclidean space, and P a smooth projection mapping onto M. Then the P-projection analogue of S is defined as T := P • S. As it turns out, the smoothness of the scheme T is always at least as high as the smoothness of the underlying scheme S or the smoothness… (More)

We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this we first generalize the well-known Cea lemma to nonlinear function spaces. In a second step we prove optimal interpolation error estimates for pointwise interpolation by… (More)

Anisotropic decompositions using representation systems based on parabolic scaling such as curve-lets or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains… (More)

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy (H 2 − K)/KdA. They are a Laguerre geometric counterpart of Will-more surfaces, the minimizers of (H 2 − K)dA, which are known to be an entity of M ¨ obius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of… (More)

This paper proves approximation order properties of various nonlin-ear subdivision schemes. Building on some recent results on the stability of nonlinear multiscale transformations, we are able to give very short and concise proofs. In particular we point out an interesting connection between stability properties and approximation order of nonlinear… (More)

Linear interpolatory subdivision schemes of C r smoothness have approximation order at least r + 1. The present paper extends this result to nonlinear univariate schemes which are in proximity with linear schemes in a certain specific sense. The results apply to nonlinear subdivision schemes in Lie groups and in surfaces which are obtained from linear… (More)

The present paper is concerned with the study of manifold-valued multiscale transforms with a focus on the Stiefel manifold. For this specific geometry we derive several formulas and algorithms for the computation of geometric means which will later enable us to construct multiscale transforms of wavelet type. As an application we study compression of… (More)