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- PHILIPP GROHS
- 2008

Geometric wavelet-like transforms for univariate and multivari-ate manifold-valued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of in-terpolatory wavelet transforms, which applies to Riemannian… (More)

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)

Nonlinear subdivision schemes that operate on manifolds are of use whenever manifold valued data have to be processed in a multiscale fashion. This paper considers the case where the manifold is a Lie group and the nonlinear subdivision schemes are derived from linear interpolatory ones by the so-called log-exp analogy. The main result of the paper is that… (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)

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 Céa lemma to nonlinear function spaces. In a second step we prove optimal interpolation error estimates for pointwise interpolation by… (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)

- Philipp Grohs
- 2009

In recent years directional multiscale transformations like the curvelet-or shearlet transformation have gained considerable attention. The reason for this is that these transforms are-unlike more traditional transforms like wavelets-able to efficiently handle data with features along edges. The main result confirming this property for shearlets is… (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 necessity to process data which live in nonlinear ge-ometries (e.g. motion capture data, unit vectors, subspaces, positive definite matrices) has led to some recent developments in nonlinear multiscale representation and subdivision algorithms. The present paper analyzes convergence and C 1 and C 2 smoothness of subdivision schemes which operate in… (More)