Constructing Laplace Operator from Point Clouds in ℝ<sup><i>d</i></sup>

The Laplace-Beltrami operator, a fundamental object associated with Riemannian manifolds, encodes all intrinsic geometry of manifolds and has many desirable properties. Recently, we proposed the point integral method (PIM), a novel numerical method for discretizing the Laplace-Beltrami operator on point clouds (Li et al. in Commun Comput Phys 22(1):228-258, 2017). In this paper, we analyze the convergence of PIM for Poisson equation with Neumann boundary condition on submanifolds that are isometrically embedded in Euclidean spaces. Keywords: Laplace-Beltrami operator, Neumann boundary, Point cloud, Point integral method, Convergence analysisMathematics Subject Classification: 65N12, 65N25, 65N75 BackgroundThe partial differential equations on manifolds arise in a wide variety of applications. In many problems, including material science [10,20], fluid flow [22,25], biology and biophysics [2,3,21,37], people need to study the physical process, for instance diffusion and convection, in curved surfaces which introduce different kinds of PDEs in surfaces. It has been several decades to develop numerical methods for solving PDEs in surfaces. Many methods have been developed, such as surface finite element method [19], level set method [9,48], grid-based particle method [31,32] and closest point method [35,43].Recently, manifold model attracts more and more attentions in data analysis and image processing [4,11,13,23,26,29,30,36,[40][41][42]47]. In the manifold model, data or images are represented as a point cloud, which is defined as a collection of points that are embedded in a high-dimensional Euclidean space. One fundamental assumption in the manifold model is that the point cloud samples a smooth manifold. Thus, the information of the manifold is very useful to understand the data or images. PDEs on the manifold, particularly the Laplace-Beltrami equation, encode several intrinsic information of the manifold, thus helping reveal the underlying structures in the data or images. To get the information encoded in PDEs, we need to solve them in the unstructured point cloud. Given that the point cloud is embedded in a high-dimensional space, the traditional methods for PDEs on 2D surfaces do not work.

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“…Lemma 5 (e.g., [8]) Assume M is a submanifold isometrically embedded in R d with reach σ > 0. For any two x, y on M with |x − y| ≤ σ /2,…”

Section: Proof Of Propositionmentioning

confidence: 99%

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Shi

Sun

2017

Res Math Sci

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“…Gaussian kernel weights have been advocated by Belkin et al [1] as providing good properties such as convergence to the continuous Laplace-Beltrami operator. Interestingly, the above weights correspond to a spectral decomposition where the original manifold M is considered in a higher dimensional space R 3+l where l is the dimension of the co-domain of F , e.g.…”

Section: Computational Strategymentioning

confidence: 99%

Scale-space representation of scalar functions on 2D manifolds

Zaharescu¹,

Boyer

Horaud

2011

2011 IEEE International Conference on Computer Vision Workshops (ICCV Workshops)

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In this paper we develop a novel approach for the scalespace representations of scalar functions defined over Riemannian manifolds. One of the main interest in such representations stems from the task of 3D modelling where 2D surfaces, endowed with various physical properties, are recovered from images. Multi-scale analysis allows to structure the information with respect to its intrinsic scale, hence enabling a wide range of low-level computations, similar to what is usually used for representing images. In contrast to the Euclidean image domain, where scale spaces can be easily obtained through convolutions with Gaussian kernels, surfaces require a more general approach that must handle non-Euclidean spaces. Such a generalized scalespace framework is the main contribution of this paper, which builds on the spectral decomposition available with the heat-diffusion framework to derive a computational approach for representing scalar functions on 2D Riemannian manifolds using an intrinsic scale parameter. In addition, we propose a feature detector and a region descriptor, based on these representations, extending the widely used DOG detector and HOG descriptor to manifolds. Experiments on real datasets with various physical properties, i.e., scalar functions, demonstrate the validity and the interest of this approach.

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“…Several discrete Laplace operators have been developed to compute a Laplace-like operator either from a mesh approximating the hidden manifold [5,10,16], or more generally, simply from a set of points sampled from a hidden manifold [1,6]. For high dimensional data, the most practical and also most popular discrete Laplace operator is perhaps the weighted graph Laplace operator -the reason being its simplicity and practicality.…”

Section: Introductionmentioning

confidence: 99%

“…For high dimensional data, the most practical and also most popular discrete Laplace operator is perhaps the weighted graph Laplace operator -the reason being its simplicity and practicality. To compute it, one only needs to build the proximity graph from the input points, while other operators require a mesh structure either locally [6] or globally [5,10,16], which is expensive for high dimensional data. Furthermore, Belkin and Niyogi showed that, for points uniformly randomly sampled from the hidden manifold, the Gaussian-weighted graph Laplacian converges to the manifold Laplacian both pointwise [4] and in spectrum [3].…”

Section: Introductionmentioning

confidence: 99%

Weighted Graph Laplace Operator under Topological Noise

Dey¹,

Ranjan²,

Wang³

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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Recently, various applications have motivated the study of spectral structures (eigenvalues and eigenfunctions) of the so-called Laplace-Beltrami operator of a manifold and their discrete versions. A popular choice for the discrete version is the so-called Gaussian weighted graph Laplacian which can be applied to point cloud data that samples a manifold. Naturally, the question of stability of the spectrum of this discrete Laplacian under the perturbation of the sampled manifold becomes important for its practical usage. Previous results showed that the spectra of both the manifold Laplacian and discrete Laplacian are stable when the perturbation is "nice" in the sense that it is restricted to a diffeomorphism with minor area distortion. However, this forbids, for example, small topological changes.We study the stability of the spectrum of the weighted graph Laplacian under more general perturbations. In particular, we allow arbitrary, including topological, changes to the hidden manifold as long as they are localized in the ambient space and the area distortion is small. Manifold Laplacians may change dramatically in this case. Nevertheless, we show that the weighted graph Laplacians computed from two sets of points, uniformly randomly sampled from a manifold and a perturbed version of it, have similar spectra. The distance between the two spectra can be bounded in terms of the size of the perturbation and some intrinsic properties of the original manifold.

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Constructing Laplace Operator from Point Clouds in ℝ^d

Cited by 117 publications

References 21 publications

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Scale-space representation of scalar functions on 2D manifolds

Weighted Graph Laplace Operator under Topological Noise

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Constructing Laplace Operator from Point Clouds in ℝd

Cited by 117 publications

References 21 publications

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Scale-space representation of scalar functions on 2D manifolds

Weighted Graph Laplace Operator under Topological Noise

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Product

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Constructing Laplace Operator from Point Clouds in ℝ^d