# Spectral Geometry of Shapes

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## Spectral Geometry of Shapes

Author | : Jing Hua,Zichun Zhong |

Publsiher | : Academic Press |

Total Pages | : 195 |

Release | : 2020-01-15 |

ISBN 10 | : 0128138424 |

ISBN 13 | : 9780128138427 |

Language | : EN, FR, DE, ES & NL |

**Spectral Geometry of Shapes Book Review:**

Spectral Geometry of Shapes presents unique shape analysis approaches based on shape spectrum in differential geometry. It provides insights on how to develop geometry-based methods for 3D shape analysis. The book is an ideal learning resource for graduate students and researchers in computer science, computer engineering and applied mathematics who have an interest in 3D shape analysis, shape motion analysis, image analysis, medical image analysis, computer vision and computer graphics. Due to the rapid advancement of 3D acquisition technologies there has been a big increase in 3D shape data that requires a variety of shape analysis methods, hence the need for this comprehensive resource. Presents the latest advances in spectral geometric processing for 3D shape analysis applications, such as shape classification, shape matching, medical imaging, etc. Provides intuitive links between fundamental geometric theories and real-world applications, thus bridging the gap between theory and practice Describes new theoretical breakthroughs in applying spectral methods for non-isometric motion analysis Gives insights for developing spectral geometry-based approaches for 3D shape analysis and deep learning of shape geometry

## Shape Analysis Using Spectral Geometry

Author | : Jiaxi Hu |

Publsiher | : Unknown |

Total Pages | : 95 |

Release | : 2015 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : OCLC:916479798 |

Language | : EN, FR, DE, ES & NL |

**Shape Analysis Using Spectral Geometry Book Review:**

Finally we prove the shape spectrum is a continuous function to a scale function on the conformal factor of the manifold. The derivatives of the eigenvalues are analytically expressed with those of the scale function. The property applies to both continuous domain and discrete triangle meshes. On the triangle meshes, a spectrum alignment algorithm is developed. Given two closed triangle meshes, the eigenvalues can be aligned from one to the other and the eigenfunction distributions are aligned as well. This extends the shape spectra across non-isometric deformations, supporting a registration-free analysis of general motion data.

## Shape Optimization and Spectral Theory

Author | : Antoine Henrot |

Publsiher | : De Gruyter Open |

Total Pages | : 474 |

Release | : 2017-05-08 |

ISBN 10 | : 9783110550856 |

ISBN 13 | : 3110550857 |

Language | : EN, FR, DE, ES & NL |

**Shape Optimization and Spectral Theory Book Review:**

"Shape optimization and spectral theory" is a survey book aiming to give an overview of recent results in spectral geometry and its links with shape optimization. It covers most of the issues which are important for people working in PDE and differential geometry interested in sharp inequalities and qualitative behaviour for eigenvalues of the Laplacian with different kind of boundary conditions (Dirichlet, Robin and Steklov). This includes: existence of optimal shapes, their regularity, the case of special domains like triangles, isospectrality, quantitative form of the isoperimetric inequalities, optimal partitions, universal inequalities and numerical results. Much progress has been made in these extremum problems during the last ten years and this edited volume presents a valuable update to a wide community interested in these topics. List of contributors Antunes Pedro R.S., Ashbaugh Mark, Bonnaillie-Noel Virginie, Brasco Lorenzo, Bucur Dorin, Buttazzo Giuseppe, De Philippis Guido, Freitas Pedro, Girouard Alexandre, Helffer Bernard, Kennedy James, Lamboley Jimmy, Laugesen Richard S., Oudet Edouard, Pierre Michel, Polterovich Iosif, Siudeja Bartlomiej A., Velichkov Bozhidar

## The Application of Spectral Geometry to 3D Molecular Shape Comparison

Author | : Matthew Seddon |

Publsiher | : Unknown |

Total Pages | : 135 |

Release | : 2017 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : OCLC:1063493392 |

Language | : EN, FR, DE, ES & NL |

**The Application of Spectral Geometry to 3D Molecular Shape Comparison Book Review:**

## Old and New Aspects in Spectral Geometry

Author | : M.-E. Craioveanu,Mircea Puta,Themistocles RASSIAS |

Publsiher | : Springer Science & Business Media |

Total Pages | : 446 |

Release | : 2013-03-14 |

ISBN 10 | : 940172475X |

ISBN 13 | : 9789401724753 |

Language | : EN, FR, DE, ES & NL |

**Old and New Aspects in Spectral Geometry Book Review:**

It is known that to any Riemannian manifold (M, g ) , with or without boundary, one can associate certain fundamental objects. Among them are the Laplace-Beltrami opera tor and the Hodge-de Rham operators, which are natural [that is, they commute with the isometries of (M,g)], elliptic, self-adjoint second order differential operators acting on the space of real valued smooth functions on M and the spaces of smooth differential forms on M, respectively. If M is closed, the spectrum of each such operator is an infinite divergent sequence of real numbers, each eigenvalue being repeated according to its finite multiplicity. Spectral Geometry is concerned with the spectra of these operators, also the extent to which these spectra determine the geometry of (M, g) and the topology of M. This problem has been translated by several authors (most notably M. Kac). into the col loquial question "Can one hear the shape of a manifold?" because of its analogy with the wave equation. This terminology was inspired from earlier results of H. Weyl. It is known that the above spectra cannot completely determine either the geometry of (M , g) or the topology of M. For instance, there are examples of pairs of closed Riemannian manifolds with the same spectra corresponding to the Laplace-Beltrami operators, but which differ substantially in their geometry and which are even not homotopically equiva lent.

## Dirac Operators and Spectral Geometry

Author | : Giampiero Esposito |

Publsiher | : Cambridge University Press |

Total Pages | : 209 |

Release | : 1998-08-20 |

ISBN 10 | : 0521648629 |

ISBN 13 | : 9780521648622 |

Language | : EN, FR, DE, ES & NL |

**Dirac Operators and Spectral Geometry Book Review:**

A clear, concise and up-to-date introduction to the theory of the Dirac operator and its wide range of applications in theoretical physics for graduate students and researchers.

## Spectrum Geometry

Author | : Spectrum |

Publsiher | : Carson-Dellosa Publishing |

Total Pages | : 128 |

Release | : 2015-02-15 |

ISBN 10 | : 1483816621 |

ISBN 13 | : 9781483816623 |

Language | : EN, FR, DE, ES & NL |

**Spectrum Geometry Book Review:**

With the help of Spectrum Geometry(R) for grades 6 to 8, children develop problem-solving math skills they can build on. This standards-based workbook focuses on middle school geometry concepts like points, lines, rays, angles, triangles, polygons, circles, perimeter, area, and more. --Middle school is known for its challengesÑlet Spectrum(R) ease some stress. Developed by education experts, the Spectrum Middle School Math series strengthens the important home-to-school connection and prepares children for math success. Filled with easy instructions and rigorous practice, Spectrum Geometry helps children soar in a standards-based classroom!

## Spectral Geometric Methods for Deformable 3D Shape Retrieval

Author | : Chunyuan Li |

Publsiher | : Unknown |

Total Pages | : 135 |

Release | : 2013 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : OCLC:1032921560 |

Language | : EN, FR, DE, ES & NL |

**Spectral Geometric Methods for Deformable 3D Shape Retrieval Book Review:**

## The Authority of Material Vs the Spirit

Author | : Douglas D Hunter |

Publsiher | : Trafford Publishing |

Total Pages | : 135 |

Release | : 2006-12-22 |

ISBN 10 | : 1412240433 |

ISBN 13 | : 9781412240437 |

Language | : EN, FR, DE, ES & NL |

**The Authority of Material Vs the Spirit Book Review:**

A new mathematically-based structure for language allows for a new context with which one can make verifiable predictions about: material, life, mind, and the spiritual intent of (creative) existence.

## The Changing Shape of Geometry

Author | : Mathematical Association,Mathematical Association of America |

Publsiher | : Cambridge University Press |

Total Pages | : 541 |

Release | : 2003-01-09 |

ISBN 10 | : 9780521531627 |

ISBN 13 | : 0521531624 |

Language | : EN, FR, DE, ES & NL |

**The Changing Shape of Geometry Book Review:**

Collection of popular articles on geometry from distinguished mathematicians and educationalists.

## Spectral Methods for Isometric Shape Matching and Symmetry Detection

Author | : Anonim |

Publsiher | : Stanford University |

Total Pages | : 135 |

Release | : 2011 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : STANFORD:xb493fy4331 |

Language | : EN, FR, DE, ES & NL |

**Spectral Methods for Isometric Shape Matching and Symmetry Detection Book Review:**

Shape matching and symmetry detection are among the most basic operations in digital geometry processing with applications ranging from medical imaging to industrial design and inspection. While the majority of prior work has concentrated on rigid or extrinsic matching and symmetry detection, many real objects are non-rigid and can exhibit a variety of poses and deformations. In this thesis, we present several methods for analyzing and matching such deformable shapes. In particular, we restrict our attention to shapes undergoing changes that can be well approximated by intrinsic isometries, i.e. deformations that preserve geodesic distances between all pairs of points. This class of deformations is much richer than rigid motions (extrinsic isometries) and can approximate, for example, articulated motions of humans. At the same time, as we show in this thesis, there exists a rich set of spectral quantities based on the Laplace-Beltrami operator that are invariant to intrinsic isometries, and can be used for both shape matching and symmetry detection. One of the principal observations of this thesis is that in many cases spectral invariants are \emph{complete}, and characterize a given shape up to isometry. This allows us to devise efficient methods for intrinsic symmetry detection, multiscale point similarity and isometric shape matching. Our methods are robust and all come with strong and often surprising theoretical guarantees.

## Old and New Aspects in Spectral Geometry

Author | : M. -E. Craioveanu,Mircea Puta,Themistocles Rassias |

Publsiher | : Unknown |

Total Pages | : 460 |

Release | : 2014-01-15 |

ISBN 10 | : 9789401724760 |

ISBN 13 | : 9401724768 |

Language | : EN, FR, DE, ES & NL |

**Old and New Aspects in Spectral Geometry Book Review:**

## On Perturbative Methods in Spectral Geometry

Author | : Mikhail Panine |

Publsiher | : Unknown |

Total Pages | : 122 |

Release | : 2017 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : OCLC:1036282641 |

Language | : EN, FR, DE, ES & NL |

**On Perturbative Methods in Spectral Geometry Book Review:**

The goal of spectral geometry is to establish how much information about the geometry of compact Riemannian manifolds is contained in the spectra of natural differential operators, especially Laplacians, defined on them. Ideally, one would like to be able to recover the Riemannian manifold, up to isometry, from the spectra of one or several such operators. This would be a very powerful result, as it would introduce an invariant way to describe the shape of Riemannian manifolds. The consequences of such a result would range from practical applications such as shape recognition to theoretical insights into quantum gravity. However, the most general form of such statements is known to be false. There are a number of known counterexamples, that is isospectral but not isometric manifolds. Indeed, there are even techniques to construct such counterexamples. Nonetheless, it is believed that almost all Riemannian manifolds can be identified by their spectra. In other words, the counterexamples are expected to be exceedingly rare special cases. This has been shown to be the case in some restricted classes of manifolds. The proof in the general case has remained elusive. The main goal of this thesis is to move towards such a proof by studying the structure of isospectral sets of metrics. The main tool we use for this purpose is perturbation theory, a method ubiquitous in physics, but strangely underused in spectral geometry. Consequently, a secondary goal of this work is to demonstrate the usefulness of perturbation theory to the study of spectral geometry. We begin by a numerical exploration of spectral geometry in a perturbative regime. Then, we show that sets of isospectral conformally equivalent metrics on boundaryless manifolds of dimension two contain no convex subsets. This is an entirely new type of result in spectral geometry. We argue that it could lead to a proof of the rarity of counterexamples in the program of identifying shapes by their spectra. The thesis also includes reviews of the fundamentals of the spectral theory of Laplace-type operators, of major results in spectral geometry and of perturbation theory.

## Spectral Analysis of Nonlinear Elastic Shapes

Author | : James F. Doyle |

Publsiher | : Springer Nature |

Total Pages | : 409 |

Release | : 2020-11-26 |

ISBN 10 | : 3030594947 |

ISBN 13 | : 9783030594947 |

Language | : EN, FR, DE, ES & NL |

**Spectral Analysis of Nonlinear Elastic Shapes Book Review:**

This book concerns the elastic stability of thin-walled structures — one of the most challenging problems facing structural engineers because of its high degree of nonlinearity — and introduces the innovative approach of using spectral analysis of the shapes and the stiffness to gain insights into the nonlinear deformations. The methodology greatly facilitates correlating the shape changes with the stiffness changes. Professor Doyle also develops specific computer procedures that complement finite element methods so that the ideas and methods are applicable to general structural problems. Basic validity of the procedures is established using key archetypal problems from buckling/post-buckling of columns, arches, curved plates, and cylindrical shells, all worked out in significant detail. The book is ideal for a wide variety of structural engineers, particularly those in aerospace and civil fields. Researchers in computational mechanics also find a rich source of new ideas for post-processing data from nonlinear analyses.

## Geometric Approaches for 3D Shape Denoising and Retrieval

Author | : Anis Kacem |

Publsiher | : Unknown |

Total Pages | : 135 |

Release | : 2013 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : OCLC:1032935423 |

Language | : EN, FR, DE, ES & NL |

**Geometric Approaches for 3D Shape Denoising and Retrieval Book Review:**

## An Introduction to Laplacian Spectral Distances and Kernels

Author | : Giuseppe Patanè |

Publsiher | : Morgan & Claypool Publishers |

Total Pages | : 139 |

Release | : 2017-07-05 |

ISBN 10 | : 1681731401 |

ISBN 13 | : 9781681731407 |

Language | : EN, FR, DE, ES & NL |

**An Introduction to Laplacian Spectral Distances and Kernels Book Review:**

In geometry processing and shape analysis, several applications have been addressed through the properties of the Laplacian spectral kernels and distances, such as commute time, biharmonic, diffusion, and wave distances. Within this context, this book is intended to provide a common background on the definition and computation of the Laplacian spectral kernels and distances for geometry processing and shape analysis. To this end, we define a unified representation of the isotropic and anisotropic discrete Laplacian operator on surfaces and volumes; then, we introduce the associated differential equations, i.e., the harmonic equation, the Laplacian eigenproblem, and the heat equation. Filtering the Laplacian spectrum, we introduce the Laplacian spectral distances, which generalize the commute-time, biharmonic, diffusion, and wave distances, and their discretization in terms of the Laplacian spectrum. As main applications, we discuss the design of smooth functions and the Laplacian smoothing of noisy scalar functions. All the reviewed numerical schemes are discussed and compared in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate with respect to shape representation, computational resources, and target application.

## ECAI 2020

Author | : G. De Giacomo,A. Catala,B. Dilkina |

Publsiher | : IOS Press |

Total Pages | : 3122 |

Release | : 2020-09-11 |

ISBN 10 | : 164368101X |

ISBN 13 | : 9781643681016 |

Language | : EN, FR, DE, ES & NL |

**ECAI 2020 Book Review:**

This book presents the proceedings of the 24th European Conference on Artificial Intelligence (ECAI 2020), held in Santiago de Compostela, Spain, from 29 August to 8 September 2020. The conference was postponed from June, and much of it conducted online due to the COVID-19 restrictions. The conference is one of the principal occasions for researchers and practitioners of AI to meet and discuss the latest trends and challenges in all fields of AI and to demonstrate innovative applications and uses of advanced AI technology. The book also includes the proceedings of the 10th Conference on Prestigious Applications of Artificial Intelligence (PAIS 2020) held at the same time. A record number of more than 1,700 submissions was received for ECAI 2020, of which 1,443 were reviewed. Of these, 361 full-papers and 36 highlight papers were accepted (an acceptance rate of 25% for full-papers and 45% for highlight papers). The book is divided into three sections: ECAI full papers; ECAI highlight papers; and PAIS papers. The topics of these papers cover all aspects of AI, including Agent-based and Multi-agent Systems; Computational Intelligence; Constraints and Satisfiability; Games and Virtual Environments; Heuristic Search; Human Aspects in AI; Information Retrieval and Filtering; Knowledge Representation and Reasoning; Machine Learning; Multidisciplinary Topics and Applications; Natural Language Processing; Planning and Scheduling; Robotics; Safe, Explainable, and Trustworthy AI; Semantic Technologies; Uncertainty in AI; and Vision. The book will be of interest to all those whose work involves the use of AI technology.

## Progress in Inverse Spectral Geometry

Author | : Stig I. Andersson,Michel L. Lapidus |

Publsiher | : Birkhäuser |

Total Pages | : 197 |

Release | : 2012-12-06 |

ISBN 10 | : 3034889380 |

ISBN 13 | : 9783034889384 |

Language | : EN, FR, DE, ES & NL |

**Progress in Inverse Spectral Geometry Book Review:**

Most polynomial growth on every half-space Re (z) ::::: c. Moreover, Op(t) depends holomorphically on t for Re t> O. General references for much of the material on the derivation of spectral functions, asymptotic expansions and analytic properties of spectral functions are [A-P-S] and [Sh], especially Chapter 2. To study the spectral functions and their relation to the geometry and topology of X, one could, for example, take the natural associated parabolic problem as a starting point. That is, consider the 'heat equation': (%t + p) u(x, t) = 0 { u(x, O) = Uo(x), tP which is solved by means of the (heat) semi group V(t) = e- ; namely, u(·, t) = V(t)uoU· Assuming that V(t) is of trace class (which is guaranteed, for instance, if P has a positive principal symbol), it has a Schwartz kernel K E COO(X x X x Rt, E* ®E), locally given by 00 K(x, y; t) = L>-IAk(~k ® 'Pk)(X, y), k=O for a complete set of orthonormal eigensections 'Pk E COO(E). Taking the trace, we then obtain: 00 tA Op(t) = trace(V(t)) = 2::>- k. k=O Now, using, e. g., the Dunford calculus formula (where C is a suitable curve around a(P)) as a starting point and the standard for malism of pseudodifferential operators, one easily derives asymptotic expansions for the spectral functions, in this case for Op.

## Steklov Geometry Processing

Author | : Wang Yu (S.M.) |

Publsiher | : Unknown |

Total Pages | : 80 |

Release | : 2018 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : OCLC:1051458896 |

Language | : EN, FR, DE, ES & NL |

**Steklov Geometry Processing Book Review:**

We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.

## Diffusion Driven Wavelet Design for Shape Analysis

Author | : Tingbo Hou,Hong Qin |

Publsiher | : CRC Press |

Total Pages | : 208 |

Release | : 2014-10-22 |

ISBN 10 | : 1482220296 |

ISBN 13 | : 9781482220292 |

Language | : EN, FR, DE, ES & NL |

**Diffusion Driven Wavelet Design for Shape Analysis Book Review:**

From Design Methods and Generation Schemes to State-of-the-Art Applications Wavelets are powerful tools for functional analysis and geometry processing, enabling researchers to determine the structure of data and analyze 3D shapes. Suitable for researchers in computer graphics, computer vision, visualization, medical imaging, and geometric modeling as well as graduate and senior undergraduate students in computer science, Diffusion-Driven Wavelet Design for Shape Analysis presents recent research results in wavelet designs on 3D shapes and their applications in shape analysis. It explains how to apply the design methods to various types of 3D data, such as polygonal meshes, point clouds, manifolds, and volumetric images. Extensions of Wavelet Generation on Volumetric and Manifold Data The first part of the book introduces design methods of wavelets on manifold data, incorporating interdisciplinary knowledge from differential geometry, functional analysis, Fourier transform, spectral graph theory, and stochastic processes. The authors show how wavelets are purely determined by the shape geometry and how wavelet transforms are computed as inner products of wavelet kernels and input functions. Wavelets for Solving Computer Graphics Problems The second part presents applications in shape analysis/representation. The book looks at wavelets as spectral tools for geometry processing with filters in a joint space-frequency domain and examines wavelets as detail extractors for shape feature definition and detection. Going beyond these fundamental applications, the book also covers middle- and high-level applications, including shape matching, shape registration, and shape retrieval. Easy-to-Understand Implementations and Algorithms Unlike many other wavelet books, this one does not involve complicated mathematics. Instead, the book uses simplified formulations and illustrative examples to explain deep theories. Code and other materials are available on a supplementary website.