An Introduction to Laplacian Spectral Distances and Kernels: Theory, Computation, and Applications

141 Pages·2017·4.311 MB·English

by  Giuseppe Patanè, Brian A. Barsky

Series ISSN: 2469-4215 PA
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Synthesis Lectures on Visual Computing A
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Computer Graphics, Animation, Computational Photography and Imaging
Series Editor: Brian R. Barsky, University of California, Berkeley A
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An Introduction to Laplacian Spectral Distance and Kernels N An Introduction to
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Theory, Computation, and Applications O
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Giuseppe Patanè, CNR-IMATI T
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In geometry processing and shape analysis, several applications have been addressed through the
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properties of the Laplacian spectral kernels and distances, such as commute-time, biharmonic, O
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diffusion, and wave distances. A
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Within this context, this book is intended to provide a common background on the L
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definition and computation of the Laplacian spectral kernels and distances for geometry C
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processing and shape analysis. To this end, we define a unified representation of the isotropic A
N
Theory, Computation, and Applications
and anisotropic discrete Laplacian operator on surfaces and volumes; then, we introduce the S
P
associated differential equations, i.e., the harmonic equation, the Laplacian eigenproblem, E
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and the heat equation. Filtering the Laplacian spectrum, we introduce the Laplacian spectral T
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distances, which generalize the commute-time, biharmonic, diffusion, and wave distances, and A
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their discretization in terms of the Laplacian spectrum. As main applications, we discuss the D
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design of smooth functions and the Laplacian smoothing of noisy scalar functions. S
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All the reviewed numerical schemes are discussed and compared in terms of robustness, N
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approximation accuracy, and computational cost, thus supporting the reader in the selection of
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the most appropriate with respect to shape representation, computational resources, and target A
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AThbiso vuoltum SeY isN a TprHintEedS IvSersion of a work that appears in the Synthesis Digital Library of Engineering M
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An Introduction to
Laplacian Spectral Distances
and Kernels
Theory, Computation, and Applications

iii
Synthesis Lectures on
Visual Computing
Computer Graphics, Animation, Computational
Photography, and Imaging
Editor
BrianA.Barsky,UniversityofCalifornia,Berkeley
Thisseriespresentslecturesonresearchanddevelopmentinvisualcomputingforanaudienceof
professionaldevelopers,researchers,andadvancedstudents.Topicsofinterestinclude
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Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedin
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AnIntroductiontoLaplacianSpectralDistancesandKernels:Theory,Computation,andApplications
GiuseppePatanè
www.morganclaypool.com
ISBN:9781681731391 paperback
ISBN:9781681731407 ebook
DOI10.2200/S00781ED1V01Y201705VCP029
APublicationintheMorgan&ClaypoolPublishersseries
SynthesisLecturesonVisualComputing:ComputerGraphics,Animation,
ComputationalPhotography,andImaging
Lecture#29
SeriesEditor:BrianA.Barsky,UniversityofCalifornia,Berkeley
SeriesISSN
Print2469-4215 Electronic2469-4223
An Introduction to
Laplacian Spectral Distances
and Kernels
Theory, Computation, and Applications
Giuseppe Patanè
CNR-IMATI
SYNTHESISLECTURESONVISUALCOMPUTING:COMPUTER
GRAPHICS,ANIMATION,COMPUTATIONALPHOTOGRAPHY,AND
IMAGING#29
M
&C Morgan&cLaypool publishers
ABSTRACT
Ingeometryprocessingandshapeanalysis,severalapplicationshavebeenaddressedthroughthe
propertiesoftheLaplacianspectralkernelsanddistances,suchascommute-time,biharmonic,
diffusion,andwavedistances.
Withinthiscontext,thisbookisintendedtoprovideacommonbackgroundonthedefi-
nitionandcomputationoftheLaplacianspectralkernelsanddistancesforgeometryprocessing
andshapeanalysis.Tothisend,wedefineaunifiedrepresentationoftheisotropicandanisotropic
discreteLaplacianoperatoronsurfacesandvolumes;then,weintroducetheassociateddifferen-
tialequations,i.e.,theharmonicequation,theLaplacianeigenproblem,andtheheatequation.
Filtering the Laplacian spectrum, we introduce the Laplacian spectral distances, which gener-
alize the commute-time, biharmonic,diffusion, and wavedistances, and their discretization in
termsoftheLaplacianspectrum.Asmainapplications,wediscussthedesignofsmoothfunc-
tionsandtheLaplaciansmoothingofnoisyscalarfunctions.
All the reviewed numerical schemes are discussed and compared in terms of robustness,
approximationaccuracy,andcomputationalcost,thussupportingthereaderintheselectionof
themostappropriatewithrespecttoshaperepresentation,computationalresources,andtarget
application.
KEYWORDS
Laplace-Beltrami operator, Laplacian spectrum, harmonic equation, Laplacian
eigenproblem, heat equation, diffusion geometry, Laplacian spectral distance and
kernels,spectralgeometryprocessing,shapeanalysis,numericalanalysis