An introduction to Laplacian spectral distances and kernels : theory, computation, and applications

An introduction to Laplacian spectral distances and kernels : theory, computation, and applications /

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 th...

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Bibliographic Details
Main Author: Patanè, Giuseppe, 1974- (Author)
Format: Electronic eBook
Language:English
Published: Cham, Switzerland : Springer, [2017]
Series:Synthesis digital library of engineering and computer science.
Synthesis lectures on visual computing ; #29.
Subjects:
Geometry > Data processing.
Shapes > Computer simulation > Mathematics.
Computer graphics > Mathematics.
Laplacian operator.
Harmonic functions.
Geometry.
Shapes.
Computer simulation.
Mathematics.
Computer graphics.
Online Access:Connect to this title online
Table of Contents:
  • 1. Laplace-Beltrami operator
  • 1.1 Discrete Laplacians and spectral properties
  • 1.1.1 Laplacian on graphs, meshes, and volumes
  • 1.1.2 Laplacian matrix of point sets
  • 1.2 Harmonic equation
  • 1.3 Laplacian eigenproblem
  • 1.3.1 Discrete Laplacian eigenfunctions
  • 1.3.2 Stability of the Laplacian spectrum.
  • 2. Heat and wave equations
  • 2.1 Heat equation
  • 2.1.1 Heat equation on surfaces and volumes
  • 2.1.2 Optimal time value of the heat kernel
  • 2.1.3 Comparison of the heat kernel at different scales
  • 2.2 Wave equation
  • 2.3 Discrete heat equation and kernel
  • 2.3.1 Properties of the heat kernel
  • 2.3.2 Linear independence of the heat kernel at different points and scales
  • 2.4 Computation of the discrete heat kernel
  • 2.4.1 Linear approximation
  • 2.4.2 Polynomial approximation
  • 2.4.3 Rational approximation
  • 2.4.4 Special case: heat equation on volumes
  • 2.5 Discussion.
  • 3. Laplacian spectral distances
  • 3.1 Green kernel and linear operator
  • 3.2 Laplacian spectral operator and kernel
  • 3.2.1 Laplacian spectral kernel
  • 3.2.2 Spectrum of the spectral operator
  • 3.3 Laplacian spectral distances
  • 3.3.1 Well-posedness of the spectral kernels and distances
  • 3.3.2 Scale invariance and shape signatures
  • 3.4 Main examples of spectral distances
  • 3.4.1 Selection of the filter map
  • 3.4.2 Diffusion distances
  • 3.4.3 Commute-time and biharmonic distances
  • 3.4.4 Geodesic and transportation distances via heat kernel
  • 3.5 Spectrum-free approximation
  • 3.5.1 Polynomial filter
  • 3.5.2 Arbitrary filter: polynomial approximation
  • 3.5.3 Arbitrary filter: rational approximation
  • 3.5.4 Arbitrary filter: factorization of the rational approximation
  • 3.5.5 Convergence and accuracy.
  • 4. Discrete spectral distances
  • 4.1 Discrete spectral kernels and distances
  • 4.2 Native spectral spaces
  • 4.3 Computation of the spectral distances
  • 4.3.1 Truncated approximation
  • 4.3.2 Spectrum-free approximation
  • 4.3.3 A unified spectrum-free computation
  • 4.4 Discussion.
  • 5. Applications
  • 5.1 Design of scalar functions with constrained critical points
  • 5.2 Laplacian smoothing of scalar functions
  • 5.2.1 Related work on smoothing
  • 5.2.2 Unconstrained and constrained Laplacian smoothing of scalar functions.
  • 6. Conclusions
  • Bibliography
  • Author's biography.

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