Accurate and Efficient Computation of Laplacian Spectral Distances and Kernels

Computer Graphics Forum

2016

Self Cite

In geometry processing and shape analysis, several applications have been addressed through the properties of the spectral kernels and distances, such as commute-time, biharmonic, diffusion, and wave distances. Our survey is intended to provide a background on the properties, discretization, computation, and main applications of the Laplace-Beltrami operator, the associated differential equations (e.g., harmonic equation, Laplacian eigenproblem, diffusion and wave equations), Laplacian spectral kernels and distances (e.g., commute-time, biharmonic, wave, diffusion distances). While previous work has been focused mainly on specific applications of the aforementioned topics on surface meshes, we propose a general approach that allows us to review Laplacian kernels and distances on surfaces and volumes, and for any choice of the Laplacian weights. All the reviewed numerical schemes for the computation of the Laplacian spectral kernels and distances are discussed in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate method with respect to shape representation, computational resources, and target application.

Section: Laplacian Spectral Kernels and Distancesmentioning

confidence: 99%

STAR ‐ Laplacian Spectral Kernels and Distances for Geometry Processing and Shape Analysis

Computer Graphics Forum

2016

Self Cite

“…With respect to previous work and our recent results on the definition and computation of discrete spectral distances [39,38], the main novelties of this paper are…”

Section: Novelties With Respect To Previous Workmentioning

confidence: 98%

“…2.1). Recalling the definition of the spectral operator and kernel introduced in [39,37,38], we review equivalent representations of the spectral distances in terms of the spectral norm of the δ-functions, the Laplacian spectrum, the spectral operator, and the spectral kernel (Sect. 2.2).…”

Section: Laplacian Spectral Distancesmentioning

confidence: 99%

“…3). From the computational point of view, they have two main drawbacks: a high computational cost and storage overhead, which prevent the evaluation of a large number of Laplacian eigenpairs, and numerical instabilities with respect to the surface discretisation [38,39].…”

Section: Laplace-beltrami Operatormentioning

confidence: 99%

“…According to [38,39], let ρ : R + → R be a positive filter map that is square integrable and admits the power series ρ(s) = ∑ +∞ n=0 α n s n . Considering the orthonormal Laplacian eigensystem (λ n , φ n ) +∞ n=0 , the spectral representation of the functions ∆ i f and…”

Section: Spectral Operator and Kernelmentioning

confidence: 99%

See 2 more Smart Citations

A unified definition and computation of Laplacian spectral distances

2019

Pattern Recognition

Self Cite

Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion, wave kernel distances) are easily defined through a filtering of the Laplacian eigenpairs. They play a central role in several applications, such as dimensionality reduction with spectral embeddings, diffusion geometry, image smoothing, geometric characterisations and embeddings of graphs. Extending the results recently derived in the discrete setting [38,39] to the continuous case, we propose a novel definition of the Laplacian spectral kernels and distances, whose approximation requires the solution of a set of inhomogeneous Laplace equations. Their discrete counterparts are equivalent to a set of sparse, symmetric, and well-conditioned linear systems, which are efficiently solved with iterative methods. Finally, we discuss the optimality of the Laplacian spectrum for the approximation of the spectral kernels, the relation between the spectral and Green kernels, and the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues.

Spectral Laplace Transform of Signals on Arbitrary Domains

2023

J Sci Comput

The Laplace transform is a central mathematical tool for analysing 1D/2D signals and for the solution to PDEs; however, its definition and computation on arbitrary data is still an open research problem. We introduce the Laplace transform on arbitrary domains and focus on the spectral Laplace transform, which is defined by applying a 1D filter to the Laplacian spectrum of the input domain. The spectral Laplace transform satisfies standard properties of the 1D and 2D transforms, such as dilation, translation, scaling, derivation, localisation, and relations with the Fourier transform. The spectral Laplace transform is enough general to be applied to signals defined on different discrete data, such as graphs, 3D surface meshes, and point sets. Working in the spectral domain and applying polynomial and rational polynomial approximations, we achieve a stable computation of the spectral Laplace Transform. As main applications, we discuss the computation of the spectral Laplace Transform of functions defined on arbitrary domains (e.g., 2D and 3D surfaces, graphs), the solution of the heat diffusion equation, and graph signal processing.