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Spectral Methods for Mesh Processing and Analysis
 EUROGRAPHICS 2007
, 2007
"... Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early works in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis ..."
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Cited by 31 (0 self)
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Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early works in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis of mesh geometry based on a combinatorial Laplacian aids our understanding of the lowpass filtering approach to mesh smoothing. Over the past ten years or so, the list of applications in the area of geometry processing which utilize the eigenstructures of a variety of mesh operators in different manners have been growing steadily. Many works presented so far draw parallels from developments in fields such as graph theory, computer vision, machine learning, graph drawing, numerical linear algebra, and highperformance computing. This stateoftheart report aims to provide a comprehensive survey on the spectral approach, focusing on its power and versatility in solving geometry processing problems and attempting to bridge the gap between relevant research in computer graphics and other fields. Necessary theoretical background will be provided and existing works will be classified according to different criteria — the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embeddings used — and described in adequate length. Finally, despite much empirical success, there still remain many open questions pertaining to the spectral approach, which we will discuss in the report as well.
RECENT DEVELOPMENTS IN MATHEMATICAL QUANTUM CHAOS
, 2009
"... This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigen ..."
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Cited by 23 (3 self)
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This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigenfunctions with anomalous concentration. We cover the lower bounds on entropies of quantum limit measures due to Anantharaman, Nonnenmacher, and Rivière on compact Riemannian manifolds with Anosov flow. These lower bounds give new constraints on the possible quantum limits. We also cover the nonQUE result of Hassell in the case of the Bunimovich stadium. We include some discussion of Hecke eigenfunctions and recent results of Soundararajan completing Lindenstrauss ’ QUE result, in the context of matrix elements for Fourier integral operators. Finally, in answer to the potential question ‘why study matrix elements’ it presents an application of the author to the geometry of nodal sets.
Matching shapes by eigendecomposition of the laplacebeltrami operator
, 2010
"... We present a method for detecting correspondences between nonrigid shapes, that utilizes surface descriptors based on the eigenfunctions of the LaplaceBeltrami operator. We use clusters of probable matched descriptors to resolve the sign ambiguity in matching the eigenfunctions. We then define a m ..."
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Cited by 22 (5 self)
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We present a method for detecting correspondences between nonrigid shapes, that utilizes surface descriptors based on the eigenfunctions of the LaplaceBeltrami operator. We use clusters of probable matched descriptors to resolve the sign ambiguity in matching the eigenfunctions. We then define a matching cost that measures both the descriptor similarity, and the similarity between corresponding geodesic distances measured on the two shapes. We seek for correspondence by minimizing the above cost. The resulting combinatorial problem is then reduced to the problem of matching a small number of feature points using quadratic integer programming. 1.
Counting nodal lines which touch the boundary of an analytic domain
, 2008
"... Abstract. We consider the zeros on the boundary ∂Ω of a Neumann eigenfunction ϕλ of a real analytic plane domain Ω. We prove that the number of its boundary zeros is O(λ) where ∆ϕλ = λ 2 ϕλ. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O ..."
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Cited by 20 (8 self)
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Abstract. We consider the zeros on the boundary ∂Ω of a Neumann eigenfunction ϕλ of a real analytic plane domain Ω. We prove that the number of its boundary zeros is O(λ) where ∆ϕλ = λ 2 ϕλ. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(λ). It follows that the number of nodal lines of ϕλ (components of the nodal set) which touch the boundary is of order λ. This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains. 1.
Sign and area in nodal geometry of Laplace eigenfunctions, arXiv math.AP/0402412
"... The paper deals with asymptotic nodal geometry for the LaplaceBeltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc center ..."
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Cited by 18 (2 self)
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The paper deals with asymptotic nodal geometry for the LaplaceBeltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line {f = 0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mild local asymmetry may appear. The proof combines methods due to DonnellyFefferman and Nadirashvili with a new result on harmonic functions in the unit disc. 1 Introduction and main results Consider a compact manifold S endowed with a C ∞ Riemannian metric
LOCAL AND GLOBAL ANALYSIS OF EIGENFUNCTIONS ON RIEMANNIAN MANIFOLDS
, 2009
"... This is a survey on eigenfunctions of the Laplacian on Riemannian manifolds (mainly compact and without boundary). We discuss both local results obtained by analyzing eigenfunctions on small balls, and global results obtained by wave equation methods. Among the main topics are nodal sets, quantum li ..."
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Cited by 16 (2 self)
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This is a survey on eigenfunctions of the Laplacian on Riemannian manifolds (mainly compact and without boundary). We discuss both local results obtained by analyzing eigenfunctions on small balls, and global results obtained by wave equation methods. Among the main topics are nodal sets, quantum limits, and L p norms of global eigenfunctions. The emphasis is on the connection between the behavior of eigenfunctions and the dynamics of the geodesic flow, reflecting the relation between quantum mechanics and the underlying classical mechanics. We also discuss the analytic continuation of eigenfunctions of real analytic Riemannian manifolds (M, g) to the complexification of M and its applications to nodal geometry. Besides eigenfunctions, we also consider quasimodes and random linear combinations of eigenfunctions with close eigenvalues. Many examples are discussed.
THE CALDERÓN PROBLEM IN TRANSVERSALLY ANISOTROPIC GEOMETRIES
, 2013
"... We consider the anisotropic Calderón problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work [8], it was shown that a metric in a fixed conformal class is uniquely determined by boundary measuremen ..."
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Cited by 12 (4 self)
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We consider the anisotropic Calderón problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work [8], it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Fourier transform / attenuated geodesic ray transform (or integral against a more general semiclassical limit measure) of an unknown coefficient. In particular, one obtains uniqueness results whenever the geodesic ray transform on the transversal manifold is injective. The second result shows that the boundary measurements in an infinite cylinder uniquely determine the transversal metric. The first result is proved by using complex geometrical optics solutions involving Gaussian beam quasimodes, and the second result follows from a connection between the Calderón problem and Gel’fand’s inverse problem for the wave equation.
Spectral Mesh Processing
"... Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early work in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis ..."
Abstract

Cited by 11 (1 self)
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Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early work in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis of mesh geometry based on a combinatorial Laplacian aids our understanding of the lowpass filtering approach to mesh smoothing. Over the past fifteen years, the list of applications in the area of geometry processing which utilize the eigenstructures of a variety of mesh operators in different manners have been growing steadily. Many works presented so far draw parallels from developments in fields such as graph theory, computer vision, machine learning, graph drawing, numerical linear algebra, and highperformance computing. This paper aims to provide a comprehensive survey on the spectral approach, focusing on its power and versatility in solving geometry processing problems and attempting to bridge the gap between relevant research in computer graphics and other fields. Necessary theoretical background is provided. Existing works covered are classified according to different criteria: the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embeddings used. Despite much empirical success, there still remain many open questions pertaining to the spectral approach. These are discussed as we conclude the survey and provide our perspective on possible future research.
Geometrical structure of Laplacian eigenfunctions
, 2013
"... We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and com ..."
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Cited by 10 (3 self)
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We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is placed onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.
Semiclassical measures for the Schrödinger equation on the torus
, 2011
"... Abstract. In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain ..."
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Cited by 8 (3 self)
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Abstract. In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the RadonNikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the L 2norm of a solution on any open subset of the torus controls the full L 2norm. hal00476829, version 2 13 Sep 2011 1.