A deformation invariant representation of surfaces, the GPS embedding, is introduced using the eigenvalues and eigenfunctions of the Laplace-Beltrami differential operator. Notably, since the definition of the GPS embedding completely avoids the use of geodesic distances, and is based on objects of global character, the obtained representation is robust to local topology changes. The GPS embedding captures enough information to handle various shape processing tasks as shape classification, segmentation, and correspondence. To demonstrate the practical relevance of the GPS embedding, we introduce a deformation invariant shape descriptor called G2-distributions, and demonstrate their discriminative power, invariance under natural deformations, and robustness.

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Laplace-Beltrami eigenfunctions for deformation invariant shape representation

Preface * 1 Basic Concepts in Banach Spaces * 2 Hahn-Banach and Banach Open Mapping Theorems * 3 Weak Topologies * 4 Locally Convex Spaces * 5 Structure of Banach Spaces * 6 Schauder Bases * 7 Compact Operators on Banach Spaces * 8 Differentiability of Norms * 9 Uniform Convexity * 10 Smoothness and Structure * 11 Weakly Compactly Generated Spaces * 12 Topics in Weak Toplogy * References * Index

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Functional Analysis and Infinite-Dimensional Geometry

A hilbert space problem book

One of the challenges in geometry processing is to automatically reconstruct a higher-level representation from raw geometric data. For instance, computing a parameterization of an object helps attaching information to it and converting between various representations. More generally, this family of problems may be thought of in terms of constructing structured function bases attached to surfaces. In this paper, we study a specific type of hierarchical function bases, defined by the eigenfunctions of the Laplace-Beltrami operator. When applied to a sphere, this function basis corresponds to the classical spherical harmonics. On more general objects, this defines a function basis well adapted to the geometry and the topology of the object. Based on physical analogies (vibration modes), we first give an intuitive view before explaining the underlying theory. We then explain in practice how to compute an approximation of the eigenfunctions of a differential operator, and show possible applications in geometry processing.

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Laplace-Beltrami Eigenfunctions Towards an Algorithm That "Understands" Geometry

We give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x in R^n from the signs of O(s log^2(n/s)) random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our result is universal in the sense that with high probability, one measurement scheme will successfully recover all sparse vectors simultaneously. The argument is based on solving an equivalent geometric problem on random hyperplane tessellations.

One-bit compressed sensing by linear programming

Basic theory of M-ideals.- Geometric properties of M-ideals.- Banach spaces which are M-ideals in their biduals.- Banach spaces which are L-summands in their biduals.- M-ideals in Banach algebras.- M-ideals in spaces of bounded operators.

M-Ideals in Banach Spaces and Banach Algebras

A Banach space X is said to have the Daugavet property if every operator T: X -* X of rank 1 satisfies 11 Id +Tl= 1 + flTIl. We show that then every weakly compact operator satisfies this equation as well and that X contains a copy of t1. However, X need not contain a copy of L1. We also study pairs of spaces X C Y and operators T: X -* Y satisfying I I J + T I I -_ 1-4- 1f T I I, where J: X -* Y is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with f Id +Tll 1 + Il T l I is as small as possible and give characterisations in terms of a smoothness condition.

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Banach spaces with the daugavet property

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Recent Progress on the Daugavet Property

We study $M$-ideals of compact operators by means of the property~$(M)$ introduced in \\cite{Kal-M}. Our main result states for a separable Banach space $X$ that the space of compact operators on $X$ is an $M$-ideal in the space of bounded operators if (and only if) $X$ does not contain a copy of $\\ell_{1}$, has the metric compact approximation property, and has property~$(M)$. The investigation of special versions of property~$(M)$ leads to results on almost isometric structure of some classes of Banach spaces. For instance, we give a simple necessary and sufficient condition for a Banach space to embed almost isometrically into an $\\ell_{p}$-sum of finite-dimensional spaces resp.\\ into $c_{0}$, and for $2<p<\\iy$ we prove that a subspace of $L_{p}$ embeds almost isometrically into $\\ell_{p}$ if and only if it does not contain a subspace isomorphic to $\\ell_{2}$.

Property (M), M-ideals, and almost isometric structure of Banach spaces.

We present an example of a Banach space whose numerical index is strictly greater than the numerical index of its dual, giving a negative answer to a question which has been latent since the beginning of the seventies. We also show a particular case in which the numerical index of the space and the one of its dual coincide

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Dirk Werner

Papers

M-Ideals in Banach Spaces and Banach Algebras

Banach spaces with the daugavet property

Recent Progress on the Daugavet Property

Property (M), M-ideals, and almost isometric structure of Banach spaces.

Numerical index of Banach spaces and duality