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Part of: Synthesis digital library of engineering and computer science.

Includes bibliographical references (pages 93-112).

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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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.

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Title from PDF title page (viewed on July 21, 2017).