An Introduction to Laplacian Spectral Distances and Kernels

Theory, Computation, and Applications
Erscheinungsjahr 2017
ISBN: 978-3-031-01465-9
Verlag: Springer International Publishing

Buch, Englisch, 120 Seiten, Format (B × H): 191 mm x 235 mm, Gewicht: 277 g

Reihe: Synthesis Lectures on Visual Computing: Computer Graphics, Animation, Computational Photography and Imaging

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.

Zielgruppe

Professional/practitioner

Autoren/Hrsg.

Patanè, Giuseppe

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein
• Mathematik | Informatik EDV | Informatik Informatik Bildsignalverarbeitung

Inhaltsverzeichnis

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.

Über Autor(innen)

Giuseppe Patane is a researcher at CNR-IMATI (2006-today) Institute for Applied Mathematics and Information Technologies-Italian National Research Council. Since 2001, his research activities have been focused on the definition of paradigms and algorithms for modeling and analyzing digital shapes and multidimensional data. He received a Ph.D. in Mathematics and Applications from the University of Genova (2005) and a Post-Lauream Degree Master in Applications of Mathematics to Industry from the F. Severi National Institute for Advanced Mathematics, Department of Mathematics and Applications-University of Milan (2000).