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Algorithms for Comparing and Analyzing 3D Geometry ADISSERTATION SUBMITTEDTOTHEDEPARTMENTOFELECTRICALENGINEERING OFSTANFORDUNIVERSITY INPARTIALFULFILLMENTOFTHEREQUIREMENTS FORTHEDEGREEOF DOCTOROFPHILOSOPHY NiloyJ.Mitra September2006 © Copyright by Niloy J. Mitra  All Rights Reserved ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. LeonidasJ.Guibas PrincipalAdviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. MarcLevoy AssociateAdviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. MarkPauly ApprovedfortheUniversityCommitteeonGraduateStudies. iii iv Abstract Theworldaroundusconsistsofobjectsofvastlyvaryingshapes,sizes,andgeometriccom- plexity. A natural question is how to capture such geometric variations? We propose algo- rithmsforcapturing,comparing,andanalyzingsuchDgeometry. In the first part of this thesis, we propose algorithms to automate the various stages of a standard shape acquisition pipeline. Typically, a D scanner captures object geometry frommultipledirections. Fromeachviewpointwegetapartialgeometricmodelorascan. Scanregistrationinvolvesstitchingthesescanstogethertoformoneconsolidatedmodel.We presentanalgorithmfortheautomaticglobalrigidalignmentoftwoDshapes,withoutany assumptionsabouttheirinitialposes.Weimplicitlyevaluateallthepossiblecorrespondence assignmentsbetweenasetofselectedfeaturepointswithouthavingtoexplicitlyenumerate eachofthecorrespondences.Theroughinitialalignmentisfurtherrefinedusinglocalalign- ment whichisposedasaminimizationofthesquareddistancebetweentheunderlyingsur- faces. Welocallyapproximatethesquareddistancefieldusingquadraticfunctionsandthen developalinearsystemwhosesolutiongivesthelocalaligningrigidtransform. Aftermodel registration, we often get an incomplete representation of the scanned object with areas of missingdata. WepresentanovelapproachforplausibleDmodelcompletionusinggeomet- ricpriors. Ourmethodretrievessuitablecontextmodelsfromamodeldatabase,warpsthe retrievedmodelstoconformwiththeinputdata,andconsistentlyblendsthewarpedmodels toobtainthefinalconsolidatedDshape. Inthesecondpart, weintroducetwoshapeanalysistools. Wepresentanewalgorithm thatprocessesgeometricmodelsandefficientlydiscoversandextractsacompactrepresenta- tionoftheirEuclideansymmetries. Thesesymmetriescanbepartial,approximate,orboth. The extracted symmetry graph representation captures important high-level information v aboutthestructureofageometricmodel. Wealsoproposeacompactshapesignaturebased on probabilistic fingerprints. Our method is robust to noise, invariant to rigid transforms, handles articulated deformations, and effectively detects partial matches. These compact fingerprints are used to efficiently estimate similarity across multiple D shapes where di- rectlyevaluatingsimilarityisexpensiveandimpractical. Wedemonstratetheutilityofallouralgorithmsforawidevarietyofgeometryprocess- ing applications on a range of scanned geometric models of varying sizes, complexity, and details. vi Acknowledgements ThisthesisistheresultofmyworkatStanfordwherebyIhavebeenaccompaniedandsup- portedbymanypeople. Hereismyopportunitytoexpressmygratitudeforallofthem. ThefirstpersonIwouldliketothankismyadvisorLeonidasGuibas. Iconsidermyself extremelyluckyforbeingoneofLeo’sstudents. Hehasalwaysbeenasourceofinspiration and support over these years, been extremely patient with me, taught me how to conduct research, present research ideas, and ask the right questions. I cannot possibly express my gratitudeforhiminwords. IwouldliketothankMarkPaulywithwhomIhaveworkedextensively. Workingwith Mark is a pleasure. The long hours we spend discussing algorithms, tuning minor details, fixing codes, and running experiments resulted in a significant part of the work presented inthisthesis. I would also like to thank the other faculty members with whom I had the privilege of working during the course of my PhD life: Gunnar Carlson, Pat Hanrahan, Jean-Claude Latombe, Marc Levoy, Rajeev Motwani from Stanford, Joachim Giesen from Max-Planck- Institut für Informatik, Markus Gross from ETH Zurich, and Helmut Pottmann from Vi- enna University of Technology. Also I acknowledge the help received from the following: Pierre Alliez, Mario Botsch, Michael Hoffer, Richard Keiser, Doo Young Kwon, Bob Sum- ner,andMartinWicke. IalsothankManuelaCavegn,JohnGerth,AdaGlucksman,Heather Gentner,andHoaNguyenfortheirhelpwithadministrativetasks. Iacknowledgethesup- port from the funding agencies: Darpa, ITR, NIH, and NSF. I express my sincere thanks totheJosephW.andHonMaiGoodmanStanfordGraduateFellowshipthatenabledmeto workontopicsIfeltexcitedabout. IwouldliketothankNatashaGelfandandAnNguyenforworkingwithmeonvarious vii topicsduringmyPhDyears. IwouldliketospeciallythankQingFang,DanielRussel,and AfraZomorodianforthemanyhourswespendtogetherdiscussingandarguingoverthings not related to my work. I express my many thanks to all the present and past members of the Leo Guibas group, and also my friends in the Stanford Graphics lab for the good and badtimeswespendtogetherleavingmewithlotsoffondmemoriestocherish. Itakethisopportunitytothankafewofmyfriends:AnupamDatta,AmalEkbal,Gaurav Garg,MaheshHardikar,InamRehman,DebasisSahoo,andPadmaSundaram. Alsothanks tothemanyothersfromtheStanfordCricketClub,theStanfordAlpineClub,theStanford ClimbingGym,andtheStanfordOutingClubforhelpingmehavefun,enjoylife,andremain saneduringmystayatStanford. Idedicatethisthesistomyfamily: myparents,mybrother,andmylongtimegirlfriend and now fiancé, Devasree. This is a good place to express my sincere thanks to my parents foralwaysbeingbymyside,forgivingmeagoodeducation,andencouragingmetopursue mydreams. IoweittothemforanygoodvaluesthatIhaveinmyself.Mybrotherhasalways beenmygoodfriendwhotaughtmetoquestionallthingsaroundandthinklogically.Kudos toDevasreeforherpatienceduringmylongPhDyears,helpingmekeepfaithinmyself,and forallherloveandtenderness. Thankyouall,withoutyourhelpIwouldnothavebeenheretoday. viii Contents Abstract v Acknowledgements vii 1 Introduction 1  . Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  I Shape Registration  2 GlobalRegistration 13  . RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . IntegralVolumeDescriptor . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. DefinitionofDescriptor . . . . . . . . . . . . . . . . . . . . . . . . .   . FeaturePointSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. BasicAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . DistanceMetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . CorrespondenceSearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. ComputingPotentialCorrespondences . . . . . . . . . . . . . . . . .   .. MatchingAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. GreedyBound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. PartialMatching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. ObjectRegistration . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  .. SymmetryDetection . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. ArticulatedMatching . . . . . . . . . . . . . . . . . . . . . . . . . . .   . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3 LocalRegistration 35  . RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . RegistrationofPointCloudData . . . . . . . . . . . . . . . . . . . . . . . . .   . Registrationusingthesquareddistancefunction . . . . . . . . . . . . . . . .   .. RegistrationinD . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. RegistrationinD . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . SquaredDistanceFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. On-DemandComputation . . . . . . . . . . . . . . . . . . . . . . . .   .. QuadraticApproximantsusingdTree . . . . . . . . . . . . . . . . .   .. ICPasspecialcasesofquadraticapproximant . . . . . . . . . . . . .   . ConvergenceIssues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4 ShapeCompletionusingGeometricPriors 57  . RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . DataClassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . DatabaseRetrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . Non-rigidAlignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. DistortionMeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. GeometricError . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   .. Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . Blending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . ResultsandDiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x

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rithms for capturing, comparing, and analyzing such D geometry. I dedicate this thesis to my family: my parents, my brother, and my longtime girlfriend .. However, there are some basic properties shared by all the popular