Philipp Grohs

Parabolic Molecules

Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significant 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 various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities.In this paper, with the introduction of the notion of parabolic molecules, we aim to provide a comprehensive framework which includes customarily employed representation systems based on parabolic scaling such as curvelets and shearlets. It is shown that pairs of parabolic molecules have the fundamental property to be almost orthogonal in a particular sense. This result is then applied to analyze parabolic molecules with respect to their ability to sparsely approximate data governed by anisotropic features. For this, the concept of sparsity equivalence is introduced which is shown to allow the identification of a large class of parabolic molecules providing the same sparse approximation results as curvelets and shearlets. Finally, as another application, smoothness spaces associated with parabolic molecules are introduced providing a general theoretical approach which even leads to novel results for, for instance, compactly supported shearlets.

Laguerre minimal surfaces, isotropic geometry and linear elasticity

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy

A General Proximity Analysis of Nonlinear Subdivision Schemes

\int (H^2-K)/K d\!A

Smoothness Properties of Lie Group Subdivision Schemes

. They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of

Smoothness Analysis of Subdivision Schemes on Regular Grids by Proximity

\int (H^2-K)d\!A

ε-subgradient algorithms for locally lipschitz functions on Riemannian manifolds

, which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.

Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

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 understanding of which properties a nonlinear scheme has to satisfy in order to fulfill proximity conditions. To this end we introduce a novel polynomial generation property for linear subdivision schemes and obtain a characterization of this property via simple multiplicativity properties of the moments of the mask coefficients. As a main application of our results we prove that the Riemannian analogue of a linear subdivision scheme which is defined by replacing linear averages by the Riemannian center of mass satisfies proximity conditions of arbitrary order. As a corollary we conclude that the Riemannian analogue always produces limit curves which are at...

Approximation order from stability for nonlinear subdivision schemes

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 space, or a Lie group (e.g., motion capture data), require different handling. One way to deal with Lie group valued data has been proposed by Donoho [talk at the IMI Approximation and Computation Meeting, Charleston, SC, 2001]: It is to employ a logexponential analogue of a linear subdivision rule. While a comprehensive discussion of applications is given by Ur Rahman et al. [Multiscale Model. Simul., 4 (2005), pp. 1201–1232], this paper analyzes convergence and smoothness of such subdivision processes and shows that the nonlinear schemes essentially have the same properties regarding

Energy Propagation in Deep Convolutional Neural Networks

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FFRT - A Fast Finite Ridgelet Transform for Radiative Transport

and