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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.
About the author
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).Michela Spagnuolo is the Research Director at CNR-IMATI-GE, where she has been working since July 2001. Her research interests include geometric and semantic modeling of 3D objects, approaches based on computational topology for the analysis of shapes, and methods for the evaluation of similarity at the structural and semantic level. She authored more than 130 reviewed papers in scientific journals and international conferences, and is an associate editor of international journals in computer graphics (currently, The Visual Computer and Computers & Graphics). She actively works as chair of conferences and workshops, and she is a member of the steering committee of Shape Modeling International and of the EG Workshops on 3D Object Retrieval. In 2014, she was nominated as Fellow of the Eurographics Association.
