Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids
TL;DR: This paper introduces a method to extract 'Shape-DNA', a numerical fingerprint or signature, of any 2d or 3d manifold by taking the eigenvalues (i.e. the spectrum) of its Laplace-Beltrami operator and succeeds in computing eigen values for smoothly bounded objects without discretization errors caused by approximation of the boundary.
Abstract: This paper introduces a method to extract 'Shape-DNA', a numerical fingerprint or signature, of any 2d or 3d manifold (surface or solid) by taking the eigenvalues (i.e. the spectrum) of its Laplace-Beltrami operator. Employing the Laplace-Beltrami spectra (not the spectra of the mesh Laplacian) as fingerprints of surfaces and solids is a novel approach. Since the spectrum is an isometry invariant, it is independent of the object's representation including parametrization and spatial position. Additionally, the eigenvalues can be normalized so that uniform scaling factors for the geometric objects can be obtained easily. Therefore, checking if two objects are isometric needs no prior alignment (registration/localization) of the objects but only a comparison of their spectra. In this paper, we describe the computation of the spectra and their comparison for objects represented by NURBS or other parametrized surfaces (possibly glued to each other), polygonal meshes as well as solid polyhedra. Exploiting the isometry invariance of the Laplace-Beltrami operator we succeed in computing eigenvalues for smoothly bounded objects without discretization errors caused by approximation of the boundary. Furthermore, we present two non-isometric but isospectral solids that cannot be distinguished by the spectra of their bodies and present evidence that the spectra of their boundary shells can tell them apart. Moreover, we show the rapid convergence of the heat trace series and demonstrate that it is computationally feasible to extract geometrical data such as the volume, the boundary length and even the Euler characteristic from the numerically calculated eigenvalues. This fact not only confirms the accuracy of our computed eigenvalues, but also underlines the geometrical importance of the spectrum. With the help of this Shape-DNA, it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids. A patent application based on ideas presented in this paper is pending.
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15 Jul 2009
TL;DR: The Heat Kernel Signature, called the HKS, is obtained by restricting the well‐known heat kernel to the temporal domain and shows that under certain mild assumptions, HKS captures all of the information contained in the heat kernel, and characterizes the shape up to isometry.
Abstract: We propose a novel point signature based on the properties of the heat diffusion process on a shape. Our signature, called the Heat Kernel Signature (or HKS), is obtained by restricting the well-known heat kernel to the temporal domain. Remarkably we show that under certain mild assumptions, HKS captures all of the information contained in the heat kernel, and characterizes the shape up to isometry. This means that the restriction to the temporal domain, on the one hand, makes HKS much more concise and easily commensurable, while on the other hand, it preserves all of the information about the intrinsic geometry of the shape. In addition, HKS inherits many useful properties from the heat kernel, which means, in particular, that it is stable under perturbations of the shape. Our signature also provides a natural and efficiently computable multi-scale way to capture information about neighborhoods of a given point, which can be extremely useful in many applications. To demonstrate the practical relevance of our signature, we present several methods for non-rigid multi-scale matching based on the HKS and use it to detect repeated structure within the same shape and across a collection of shapes.
1,546 citations
TL;DR: This article uses multiscale diffusion heat kernels as “geometric words” to construct compact and informative shape descriptors by means of the “bag of features” approach, and shows that shapes can be efficiently represented as binary codes.
Abstract: The computer vision and pattern recognition communities have recently witnessed a surge of feature-based methods in object recognition and image retrieval applications. These methods allow representing images as collections of “visual words” and treat them using text search approaches following the “bag of features” paradigm. In this article, we explore analogous approaches in the 3D world applied to the problem of nonrigid shape retrieval in large databases. Using multiscale diffusion heat kernels as “geometric words,” we construct compact and informative shape descriptors by means of the “bag of features” approach. We also show that considering pairs of “geometric words” (“geometric expressions”) allows creating spatially sensitive bags of features with better discriminative power. Finally, adopting metric learning approaches, we show that shapes can be efficiently represented as binary codes. Our approach achieves state-of-the-art results on the SHREC 2010 large-scale shape retrieval benchmark.
894 citations
14 Jun 2006
TL;DR: This paper studies a specific type of hierarchical function bases, defined by the eigenfunctions of the Laplace-Beltrami operator, and explains in practice how to compute an approximation of the eigens of a differential operator and shows possible applications in geometry processing.
Abstract: 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.
446 citations
Cites background or methods from "Laplace-Beltrami spectra as 'Shape-..."
...polynomials), and use the divergence formula to transform the equation into a generalized eigenvalue problem, as done in [24]....
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...However, the spectrum contains much information, which led to the idea of using it as a signature for shape matching and classification, as explained in the “shape DNA” approach [24]....
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TL;DR: Mindboggle’s algorithms are evaluated using the largest set of manually labeled, publicly available brain images in the world and compare them against state-of-the-art algorithms where they exist and results are publicly available.
Abstract: Mindboggle (http://mindboggle.info) is an open source brain morphometry platform that takes in preprocessed T1-weighted MRI data and outputs volume, surface, and tabular data containing label, feature, and shape information for further analysis. In this article, we document the software and demonstrate its use in studies of shape variation in healthy and diseased humans. The number of different shape measures and the size of the populations make this the largest and most detailed shape analysis of human brains ever conducted. Brain image morphometry shows great potential for providing much-needed biological markers for diagnosing, tracking, and predicting progression of mental health disorders. Very few software algorithms provide more than measures of volume and cortical thickness, while more subtle shape measures may provide more sensitive and specific biomarkers. Mindboggle computes a variety of (primarily surface-based) shapes: area, volume, thickness, curvature, depth, Laplace-Beltrami spectra, Zernike moments, etc. We evaluate Mindboggle’s algorithms using the largest set of manually labeled, publicly available brain images in the world and compare them against state-of-the-art algorithms where they exist. All data, code, and results of these evaluations are publicly available.
403 citations
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16,450 citations
TL;DR: A near-real-time computer system that can locate and track a subject's head, and then recognize the person by comparing characteristics of the face to those of known individuals, and that is easy to implement using a neural network architecture.
Abstract: We have developed a near-real-time computer system that can locate and track a subject's head, and then recognize the person by comparing characteristics of the face to those of known individuals. The computational approach taken in this system is motivated by both physiology and information theory, as well as by the practical requirements of near-real-time performance and accuracy. Our approach treats the face recognition problem as an intrinsically two-dimensional (2-D) recognition problem rather than requiring recovery of three-dimensional geometry, taking advantage of the fact that faces are normally upright and thus may be described by a small set of 2-D characteristic views. The system functions by projecting face images onto a feature space that spans the significant variations among known face images. The significant features are known as "eigenfaces," because they are the eigenvectors (principal components) of the set of faces; they do not necessarily correspond to features such as eyes, ears, and noses. The projection operation characterizes an individual face by a weighted sum of the eigenface features, and so to recognize a particular face it is necessary only to compare these weights to those of known individuals. Some particular advantages of our approach are that it provides for the ability to learn and later recognize new faces in an unsupervised manner, and that it is easy to implement using a neural network architecture.
14,562 citations
"Laplace-Beltrami spectra as 'Shape-..." refers methods in this paper
...Similar to the approach of Turk and Pentland [61], who use eigenvectors of PCA (called eigenfaces) for face recognition, it can then be decided quickly if a new geometric object has a match in the database represented by the point set of Shape-DNA by measuring the distance of its Shape-DNA to shape space and by checking if its projection into shape space lies in or close to a cluster....
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Book•
01 Jan 1947
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.
7,426 citations
TL;DR: In this article, the authors proposed a geometrically motivated algorithm for representing high-dimensional data, based on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold and the connections to the heat equation.
Abstract: One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.
7,210 citations