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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 17 (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.
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 12 (6 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.
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 ..."
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Cited by 9 (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.
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 7 (1 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
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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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.
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 5 (3 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.
BOUNDS ON SUPREMUM NORMS FOR HECKE EIGENFUNCTIONS OF QUANTIZED CAT MAPS
, 2006
"... Abstract. We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is ..."
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Cited by 4 (1 self)
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Abstract. We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are ≪ǫ N ǫ for all ǫ> 0 in the case of N square free. 1.
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 4 (1 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.
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 3 (2 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.
UNIFORM ESTIMATES FOR THE SOLUTIONS OF THE SCHRÖDINGER EQUATION ON THE TORUS AND REGULARITY OF SEMICLASSICAL MEASURES
"... Abstract. We establish uniform bounds for the solutions e it ∆ u of the Schrödinger equation on arithmetic flat tori, generalising earlier results by J. Bourgain. We also study the regularity properties of weak ∗ limits of sequences of densities of the form e it ∆ un  2 corresponding to highly os ..."
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Abstract. We establish uniform bounds for the solutions e it ∆ u of the Schrödinger equation on arithmetic flat tori, generalising earlier results by J. Bourgain. We also study the regularity properties of weak ∗ limits of sequences of densities of the form e it ∆ un  2 corresponding to highly oscillating sequences of initial data (un). We obtain improved regularity properties of those limits using previous results by N. Anantharaman and F. Macià on the structure of semiclassical measures for solutions to the Schrödinger equation on the torus. 1.