Geometric and Topological Mesh Feature Extraction for 3D Shape Analysis

190 Pages·2020·11.041 MB·English

by  Jean-Luc Mari, Franck Hétroy-Wheeler, Gérard Subsol

Geometric and Topological Mesh Feature
Extraction for 3D Shape Analysis
Geometric Modeling and Applications Set
coordinated by
Marc Daniel
Volume 3
Geometric and Topological
Mesh Feature Extraction for
3D Shape Analysis

Jean-Luc Mari
Franck Hétroy-Wheeler
Gérard Subsol
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
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© ISTE Ltd 2019
The rights of Jean-Luc Mari, Franck Hétroy-Wheeler and Gérard Subsol to be identified as the authors of
this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2019946683

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-041-6
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1. Geometric Features based on Curvatures . . . . . . 1
1.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.Somemathematicalremindersofthedifferentialgeometry
ofsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1.Fundamentalformsandnormalcurvature . . . . . . . . . . . 2
1.2.2.Principalcurvaturesandshapeindex . . . . . . . . . . . . . 5
1.2.3.Principaldirectionsandlinesofcurvature . . . . . . . . . . 6
1.2.4.Weingartenequationsandshapeoperator . . . . . . . . . . . 9
1.2.5.Practicalcomputationofdifferentialparameters . . . . . . . 12
1.2.6.Euler’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.7.Meusnier’stheorem . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.8.Localapproximationofthesurface . . . . . . . . . . . . . . 16
1.2.9.Focalsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.Computationofdifferentialparametersonadiscrete
3Dmesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.2.Somenotations. . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.3.Computingnormalvectors . . . . . . . . . . . . . . . . . . . 20
1.3.4.Locallyfittingaparametricsurface . . . . . . . . . . . . . . 22
1.3.5.Discretedifferentialgeometryoperators . . . . . . . . . . . 22
1.3.6.Integrating2Dcurvatures . . . . . . . . . . . . . . . . . . . . 28
vi GeometricandTopologicalMeshFeatureExtractionfor3DShapeAnalysis
1.3.7.Tensorofcurvature: Taubin’sformula . . . . . . . . . . . . . 28
1.3.8.Tensorofcurvaturebasedonthenormalcycle
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.9.Integralestimators . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3.10.Processingunstructured3Dpointclouds . . . . . . . . . . 38
1.3.11.Discussionofthemethods. . . . . . . . . . . . . . . . . . . 38
1.4.Featurelineextraction . . . . . . . . . . . . . . . . . . . . . . . . 46
1.4.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.4.2.Linesofcurvature . . . . . . . . . . . . . . . . . . . . . . . . 47
1.4.3.Crest/ridgelines . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.4.4.Featurelinesbasedonhomotopicthinning . . . . . . . . . . 79
1.5.Region-basedapproaches . . . . . . . . . . . . . . . . . . . . . . 84
1.5.1.Meshsegmentation . . . . . . . . . . . . . . . . . . . . . . . 84
1.5.2.Shapedescriptionbasedongraphs. . . . . . . . . . . . . . . 87
1.6.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Chapter 2. Topological Features . . . . . . . . . . . . . . . . . . . 99
2.1.Mathematicalbackground . . . . . . . . . . . . . . . . . . . . . 99
2.1.1.Atopologicalviewonsurfaces . . . . . . . . . . . . . . . . . 100
2.1.2.Algebraictopology . . . . . . . . . . . . . . . . . . . . . . . 103
2.2.Computationofglobaltopologicalfeatures . . . . . . . . . . . . 106
2.2.1.Connectedcomponentsandgenus . . . . . . . . . . . . . . . 106
2.2.2.Homologygroups . . . . . . . . . . . . . . . . . . . . . . . . 107
2.3.Combininggeometricandtopologicalfeatures . . . . . . . . . . 111
2.3.1.Persistenthomology . . . . . . . . . . . . . . . . . . . . . . . 112
2.3.2.ReebgraphandMorse–Smalecomplex . . . . . . . . . . . . 115
2.3.3.Homologygenerators . . . . . . . . . . . . . . . . . . . . . . 118
2.3.4.Measuringholes . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.4.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Chapter 3. Applications . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.2.Medicine: linesofcurvatureforpolypdetectioninvirtual
colonoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.3.Paleo-anthropology: crest/ridgelinesforshapeanalysisof
humanfossils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.4.Geology: extractionoffracturelinesonvirtualoutcrops . . . . 137
Contents vii
3.5.Planetaryscience: detectionoffeaturelinesfortheextraction
ofimpactcratersonasteroidsandrockyplanets . . . . . . . . . . . . 140
3.6.Botany: persistenthomologytorecoverthebranching
structureofplants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Preface
Three-dimensional surface meshes, composed of collections of planar
polygons, are the most common discrete representation of the surface of a
virtual shape. These 3D surface meshes need to be inspected in order to
understand or evaluate their overall structure or some details. This can be
done by extracting relevant geometric or topological features. Such shape
characteristics can simplify the way the object is looked at, can help
recognitionandcandescribeandcategorizeitaccordingtospecificcriteria.
Shape characteristics can be defined in many ways. This book takes the
point of view of discrete mathematics, which aims to propose discrete
counterparts to concepts mathematically defined in continuous terms. More
specifically, in this book, we review how standard geometric and topological
notions of surfaces can be defined and computed for a 3D surface mesh, as
well as their use for shape analysis. In particular, recent methods are
described to extract feature lines having a meaning related to either geometry
or topology. Differential estimators such as discrete principal curvatures are
detailedastheyplayacriticalroleinthecomputationofsalientstructures.An
emphasis is then placed on topology since the global structure and the
connectivity of features play an important role in the understanding of a
shape. Several applications are finally developed, showing that each of them
needs specific adjustments to generic approaches. These applications are
relatedtomedicine,geology,botanyandothersciences.
Focusing on shape features, the topic of this book is narrower but more
detailed than other shape analysis books, which do not, or only briefly, refer
to feature definition and computation. It is intended not only for students,
x GeometricandTopologicalMeshFeatureExtractionfor3DShapeAnalysis
researchers and engineers in computer science and shape analysis, but also
numerical geologists, anthropologists, biologists and other scientists looking
for practical solutions to their shape analysis, understanding or recognition
problems. Wehope that our book will be a useful review of existingwork for
allofthem.
Finally, we would like to thank Marc Daniel for giving us the opportunity
towritethisbookandAldoGonzalez-Lorenzoforhisreadingofchapter2and
forhisconstructiveremarks.
Jean-LucMARI
FranckHÉTROY-WHEELER
GérardSUBSOL
August2019
Introduction
I.1. Context: 3D shape analysis
Shapes, whether from the natural world or manufactured, are more and
more often digitized for visualization or measurement purposes, among
others. This process generally results in 3D surface meshes, which are
composed of collections of planar polygons. Such meshes nowadays are the
most common discrete representation of the surface of a virtual shape. These
meshes are automatically, or sometimes interactively, examined, in order to
understand, evaluate or match their overall structure or some details. This
process is called 3D shape analysis and can be done by extracting relevant
geometricortopological features.Suchshapecharacteristicscansimplifythe
way the object is looked at, can help recognition and can describe and
categorize it according to specific criteria. In this book, we will review
various mathematical definitions of mesh features and some algorithms to
compute them. We will then give a few application examples where these
featuresareusedtogloballyorlocallyanalyzea3Dshape.
It is important to note that this book is not about 3D shape analysis, but
only about feature definition and computation. 3D shape analysis has a wider
spectrum than feature detection. Among concepts that will not be tackled in
this book are global shape descriptor/signature definition and spectral shape
analysis.Theinterestedreadercanreferto[BIA14]or[LÉV10]tolearnmore
aboutthesenotions.
Before going into details about surface features from a mathematical and
computationalpointofview,anddetailingsomeapplicationsinChapters1–3,