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An Introduction to Laplacian Spectral Distances and Kernels [electronic resource] : Theory, Computation, and Applications / by Giuseppe Patanè.

By: Patanè, Giuseppe [author.].
Contributor(s): SpringerLink (Online service).
Material type: materialTypeLabelBookSeries: Synthesis Lectures on Visual Computing: Computer Graphics, Animation, Computational Photography and Imaging: Publisher: Cham : Springer International Publishing : Imprint: Springer, 2017Edition: 1st ed. 2017.Description: XX, 120 p. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783031025938.Subject(s): Mathematics | Image processing -- Digital techniques | Computer vision | Mathematics | Computer Imaging, Vision, Pattern Recognition and GraphicsAdditional physical formats: Printed edition:: No title; Printed edition:: No titleDDC classification: 510 Online resources: Click here to access online
Contents:
List of Figures -- List of Tables -- Preface -- Acknowledgments -- Laplace Beltrami Operator -- Heat and Wave Equations -- Laplacian Spectral Distances -- Discrete Spectral Distances -- Applications -- Conclusions -- Bibliography -- Author's Biography.
In: Springer Nature eBookSummary: 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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List of Figures -- List of Tables -- Preface -- Acknowledgments -- Laplace Beltrami Operator -- Heat and Wave Equations -- Laplacian Spectral Distances -- Discrete Spectral Distances -- Applications -- Conclusions -- Bibliography -- Author's Biography.

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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