Information Theoretic Similarity Measures for Shape Matching

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Information Theoretic Similarity Measures for Shape Matching
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Hasanbelliu, Erion
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Principe, Jose C
Committee Members:
Rangarajan, Anand
Harris, John G
Rao, Murali

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alignment -- correntropy -- divergence -- information -- registration -- shape -- similarity
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Abstract:
This dissertation develops several information theoretic similarity measures to solve the shape matching problem. The similarity measures compare features extracted from the shape of the object, primarily point sets, and closed-form solutions for each method are provided. These solutions are computationally efficient, accurate, and robust against noise and outliers. Our methods have been applied to both affine and non-rigid transformations in computer vision and pattern recognition. The first algorithm uses a localized and non-linear similarity measure known as correntropy which is based on information theoretic learning and kernel methods. The method evaluates how similar two random variables are in a neighborhood of the joint space controlled by the kernel bandwidth. The nonlinearity introduced by the kernel provides a means to assess second and higher order moments of the joint PDF, which yields solutions that are more accurate than MSE for data that have non-Gaussian distributions. However, correntropy requires prior knowledge of the correspondence between the two point sets. To provide a good estimate of the initial correspondence, we use another measure which provides context based information on the point sets. The measure relies on the information content of the individual samples and provides an additional feature for each point set which quantifies the new information the sample brings to shape, which we call the surprise factor. We show that the measure is robust against any transformation and several types of noise. However, it does not perform well in cases of data with large amount of outliers. The next algorithm mitigates the correspondence problem by representing the point sets as probability density functions (PDFs). The problem of registering point sets is then treated as aligning distributions. Using PDFs instead of point sets not only avoids the need to establish a correspondence between the two sets, but also provides a more robust way of dealing with outliers and noise. The algorithm operates on the distance between the PDFs to recover the spatial transformation function needed to register them, but the price paid is higher computational complexity. The distance measure is based on the Cauchy-Schwarz inequality, and it is called the Cauchy-Schwarz divergence. These two methods can only be applied to a pair of point sets and cannot be easily extended to simultaneously align multiple shapes. Therefore, two more algorithms for groupwise registration are developed. The first divergence measure is based on the H\"{o}lder's inequality to compare multiple PDFs concurrently. This method is an extension of the Cauchy-Schwarz method, but its closed form solution is very computationally expensive. An approximation of the solution is provided which is fast to compute. The second algorithm for groupwise registration is based on R\'{e}nyi's quadratic entropy. The method compares the entropy of the union of the point sets against the sum of the entropies of the individual sets. The method provides closed form solution which is much faster than the H\"{o}lder's divergence. In addition, we extend the measure to provide a scale invariant solution by first dividing each of the entropies by the standard deviation of their respective distributions. In all our methods, we perform the non-rigid transformation using either thin-plate splines or Gaussian radial basis functions. We show that in certain cases Gaussian RBF outperforms TPS since TPS has a global effect on non-rigidity whereas Gaussian RBF's effect is localized, however this localization comes at a cost of having a free parameter. In addition, our algorithms have two free parameters: the kernel bandwidth used in the Gaussian kernel function to compare the random variables in correntropy or to estimate the PDFs in the other three methods, and the regularization parameter that controls the degree of smoothness of the transformation function. We determined that a slow decrease of the values of these parameters provides the best results. Slowly annealing these parameters avoids getting stuck in local extrema, provides a faster convergence rate, and stabilizes the algorithm. Finally, we tested the algorithms on various common synthetic and real datasets used in the field. We compared their performance against some of the traditional and also some of the state-of-the-art methods and the results are comparable or better than state-of-the-art algorithms, and have smaller computational complexity. In addition, we applied these algorithms to two real-world applications that fulfill specifications.
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In the series University of Florida Digital Collections.
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Includes vita.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Erion Hasanbelliu.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Principe, Jose C.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-05-31

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INFORMATIONTHEORETICSIMILARITYMEASURESFORSHAPEMATCH ING By ERIONHASANBELLIU ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c r 2012ErionHasanbelliu 2

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Tomyparents,mylittlesister,andmylove 3

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ACKNOWLEDGMENTS Iwouldliketotakethisopportunitytothankthepeoplethat helpedmethroughout thePhDdegree.First,Iwouldliketothankmyadviser,Dr.Jo seC.Principe,forhis guidance,continuoussupport,andpatience.Hisknowledge andadvicewereessential incompletingmyresearchwork,andhisconstantnudgeswere necessarytogetmeout oftheBrownianmotionandnallygraduate. Iwouldalsoliketothankmycommitteemembers:Dr.JohnHarr isforalwaysbeing practicalandhelpingmebringmytheoreticalworkdowntoea rth,Dr.AnandRangarajan forhisvastknowledgeintheeldandhisveryimportantsugg estions,andDr.Murali Raoforhisconstanthelpwithmymathematicalproblems,ava ilability,andadviseon copingwiththestressanddifcultiesofthePhDlife. IwouldliketoespeciallythankmycolleagueandfriendLuis SanchezGiraldofor ourcollaborations,willingnesstolistentomyideas,ande agernesstohelp.Ialsowould liketothanktheONRgroup:ToryCobbandKittipat`Bot'Kamp afortheirhelpand supportwiththeproject,AmirRubinforourcollaborations andhisinterestingideas,and ShannonChillingworthforshieldingmefromtheuniversity bureaucracy.Ishouldalso takethisopportunitytothanktheOfceofNavalResearchan dagainDr.Principefor fundingmyresearch. Iwouldliketothanksomespecialpeoplethatmadethelifein andoutofthelab alittlebetter:ShalomDarmanjian,AlexanderSinghAlvara do,SohanSeth,Goktug Cinar,andStefanCraciun.Iamalsoverythankfultomyfrien dsandcolleagues:Manu Rastogi,RaviShekhar,AustinBrockmeier,BilalFadlallah ,RakeshChalasani,Evan Kriminger,JihyeBae,RoshaPokharel,LinLi,MiguelTeixei ra,SonglinZhao,Pingping Zhu,JuliusD'souza,JieXu,JeremyAnderson,SavyasachiSi ngh,andSheng-FengYen. IalsowanttothankIl`Memming'Park,VaibhavGarg,BadongC hen,AysegulGunduz, SudhirRao,PuskalPokharel,andWeifengLiufortheirhelp. 4

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Finally,Iwouldliketothankmyparentsandmysisterforthe irconstantloveand support.And,Iwouldliketothankmyloveforherdedication andalwaysbelievingin me.Youarethebestfamilyinthewholewideworld.Iamverylu ckyandverygratefulto haveyouinmylife. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTION ................................... 15 2OVERVIEW ...................................... 21 2.1Shape-BasedFeatures ............................ 21 2.1.1ImageSegmentation .......................... 22 2.1.2ShapeFeatures ............................. 24 2.2SimilarityMeasures .............................. 26 2.3InformationTheoreticSimilarityMeasures .................. 30 2.3.1Shannon'sEntropy ........................... 30 2.3.2R enyi'sEntropy ............................. 31 2.3.3Entropy-BasedSimilarityMeasures .................. 32 3POINTMATCHING-CORRENTROPY ...................... 34 3.1PointMatching ................................. 36 3.2Correntropy ................................... 38 3.3Non-RigidTransformationandRegularization ................ 42 3.3.1Thin-PlateSpline ............................ 42 3.3.2GaussianRadialBasisFunction .................... 43 3.4PointMatchingUsingCorrentropy ...................... 44 3.5Analysis:DealingWithNoise/Outliers .................... 47 3.6TPSvsGaussianRBF ............................. 50 3.7PointCorrespondence ............................. 52 3.7.1SurpriseEffectonAfneandNon-RigidTransformatio ns ...... 54 3.7.2SurpriseEffectonNoise ........................ 55 3.7.3PointCorrespondenceUsingSurprise ................ 56 3.8BinaryvsSoftAssignment ........................... 58 3.9Algorithm .................................... 62 4PDFMATCHING-CAUCHY-SCHWARZDIVERGENCE ............. 63 4.1Background ................................... 63 4.2Cauchy-SchwarzDivergence ......................... 65 4.3MatchingAlgorithm ............................... 66 6

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4.4ExperimentalResults ............................. 69 4.4.1Analysison2DFishData ....................... 69 4.4.2QuantitativeEvaluation ......................... 71 4.4.2.1Deformation ......................... 72 4.4.2.2Noise ............................. 74 4.4.2.3Outliers ............................ 77 4.4.2.4Rotation ............................ 79 4.4.2.5Occlusion ........................... 81 4.4.3Analysison3DBunnyData ...................... 83 4.4.4EvaluationonMPEG-7ShapeDatabase ............... 84 5GROUPWISESHAPEMATCHING ......................... 88 5.1Background ................................... 88 5.2H ¨ older'sDivergence .............................. 89 5.3MatchingAlgorithm ............................... 91 5.4ApproximationofH ¨ older'sDivergence .................... 93 5.4.1IntegratingOverOneoftheTerms .................. 93 5.4.2ReducingtheHypercubetoJustOneFace .............. 94 5.5GroupwiseAlignmentUsingR enyi'sSecondOrderEntropy ........ 95 5.5.1ProblemFormulation .......................... 97 5.5.2ScaleInvariance ............................ 100 5.6ExperimentalResults ............................. 101 5.6.1GroupwiseRegistrationofAtlasConstruction ............ 101 5.6.2GroupwiseRegistrationofBiasedDatasets ............. 105 6APPLICATIONS ................................... 108 6.1Side-ScanSonarImageryClassication ................... 108 6.1.1DataDescription ............................ 109 6.1.2TemplateGeneration .......................... 110 6.1.3ShapeMatchingAdaptationtotheApplication ............ 110 6.1.4Results ................................. 114 6.2RemoteContactlessStereoscopicMassEstimationSyste m ........ 118 6.2.1DataDescription ............................ 118 6.2.2Algorithm ................................ 120 6.2.3Results ................................. 121 7CONCLUSIONSANDFUTUREDIRECTIONS .................. 127 7.1Summary .................................... 127 7.2FutureDirections ................................ 129 REFERENCES ....................................... 130 BIOGRAPHICALSKETCH ................................ 137 7

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LISTOFTABLES Table page 5-1KSstatistic ...................................... 106 5-2Runtime ....................................... 107 6-1Confusionmatrix ................................... 117 6-2AsummaryoftheDTresults ............................ 118 6-3Estimatedweights .................................. 124 8

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LISTOFFIGURES Figure page 3-1CostFunctionsinthejointspace(A)MSE(B)Correntropy ............ 40 3-2Performancesurfacefordifferentkernelsizes. .................. 41 3-3ComparisonofcorrentropyagainstMSEonwhiteGaussian noise. ....... 48 3-4ComparisonofcorrentropyagainstMSEonimpulsivenois e. .......... 49 3-5ComparisonofcorrentropyagainstMSEonoutliers/occl usions. ........ 49 3-6ComparisonofMSEagainstcorrentropyofdifferentkern elbandwidths. .... 50 3-7ComparisonofTPSandGaussianRBFonvariousnon-rigidt ransformations. 51 3-8Non-rigidtransformationusingGaussianRBFofvarious sizes. ......... 52 3-9Effectofvariousafneandnonrigidtransformationson surprisefactor. ..... 55 3-10Changesinsurprisevalueduetonoiseaddedtothedatap oints. ........ 56 3-11Example1ofutilizingsurprisetodetermineshapecorr espondence. ...... 57 3-12Example2ofutilizingsurprisetodetermineshapecorr espondence. ...... 57 3-13Example3ofutilizingsurprisetodetermineshapecorr espondence. ...... 58 3-14Example4ofutilizingsurprisetodetermineshapecorr espondence. ...... 58 3-15Correspondenceusingsurprise ........................... 59 3-16AnexampleofIPalignmentonarotateddatasetcorrupte dbybothGaussian andimpulsivenoise. ................................. 61 4-1Theeffectofthekernelbandwidthontheconvergencerat e. .......... 70 4-2Exampleofnon-rigidregistrationfortheCS-Divalgori thm. ............ 70 4-3Amoredifcultexample,wherethereferencepointsetis corruptedtomake theregistrationmorechallenging. .......................... 71 4-4Exampleofafneandnon-rigidtransformation. .................. 72 4-5Anexampleofthedegreeofdeformationthatthetwoshape shaveundergone, andshapematchingperformanceofCorrentropyandCS-Div. .......... 73 4-6Matchingperformancecomparisonundervariousdegrees ofnon-rigidwarping. 74 4-7MedianRMSEresultsforvariousdegreesofnon-rigidwar ping. ......... 75 9

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4-8Anexampleofthelevelofnoisecorruptionthatthetwosh apeshaveundergone, andshapematchingperformanceofCorrentropyandCS-Div. .......... 76 4-9Matchingperformancecomparisonundervariousdegrees ofnoise. ...... 76 4-10MedianRMSEresultsforvariousdegreesofnoise. ................ 77 4-11Anexampleofthenumberofoutlierspointsthathavebee ninsertedinthe twoshapes,andshapematchingperformanceofCorrentropya ndCS-Div. ... 78 4-12Matchingperformancecomparisonundervariouslevels ofoutliers. ....... 79 4-13MedianRMSEresultsforvariouslevelsofoutliers. ................ 79 4-14ComparisonofCorrentropy,CS-Div,andPLNSundervari ouslevelsofoutliers fortheshshapes. .................................. 80 4-15Anexampleofapplyingtwodifferentrotationanglesto thetwoshapesand theshapematchingperformanceofCorrentropyandCS-Div. .......... 81 4-16Matchingperformancecomparisonundervariousrotati onangles. ....... 82 4-17MedianRMSEresultsforvariousrotationangles. ................. 82 4-18Anexampleofocclusionwhere30%ofeachpointsetisoccluded,andshape matchingperformanceofCorrentropyandCS-Div. ................ 84 4-19Matchingperformancecomparisonundervariouslevels ofocclusion. ...... 85 4-20MedianRMSEresultsforvariouslevelsofocclusion. ............... 85 4-213Dregistration:initialalignmentisoffby45 ................... 86 4-22ErrorrateofCauchy-Schwarzdivergenceonthe3Dbunny rotatedatvarious degrees ........................................ 86 4-23ExamplesofdifcultiesontheMPEG-7dataset. ................. 87 5-1CubeillustrationtoexplainthetermsofH ¨ older'sdivergenceupdate. ...... 92 5-2ExampleofmultipleshapesalignedusingtheEntropycos tfunction. ...... 102 5-3Exampleofunbiasedgroupwisenon-rigidregistrationo nrealCCdatasets. .. 103 5-4Exampleofunbiasedgroupwiseno-rigidregistrationon outliernoise. ...... 104 5-5Registrationresultsatdifferent annealingrates. ................. 105 5-6Exampleofbiasedgroupwiseno-rigidregistration. ................ 106 6-1Examplesofthesameobjectviewedfromfourdifferentan glesandranges. .. 109 10

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6-2Overlayofa3Dtemplatemodelanditsshadowprojectiono ntheobjecthighlight andshadowcontoursextractedfromasnippet. .................. 111 6-3Anexampleofasnippetwheretheobjecthighlightandsha dowcontoursare extracted,andthe3Dtemplatematching. ..................... 115 6-4Finalresultsof3Dtemplatematching ....................... 117 6-5Exampleofthetwoimagesprovidedbythestereoscopicca mera. ....... 119 6-6The3Dpointcloudextractedfromthestereoscopicimage ........... 119 6-7Thecroppedcowfromthe3Dpointcloud. ..................... 120 6-8Exampleofalignmentofthe3Dpointcloudagainstatempl ate. ......... 121 6-9Examplesofdifferentpointcloudsofthesameanimal. .............. 123 6-10Resultsoftheseconddataset. ........................... 124 6-11Exampleofunbiasedgroupwisenon-rigidalignmentoft he3Dpointclouds. .. 125 6-12Exampleofunbiasedgroupwisenon-rigidalignmentoft he3Dpointclouds viewedfromthetop. ................................. 126 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy INFORMATIONTHEORETICSIMILARITYMEASURESFORSHAPEMATCH ING By ErionHasanbelliu May2012 Chair:JoseC.PrincipeMajor:ElectricalandComputerEngineering Thisdissertationdevelopsseveralinformationtheoretic similaritymeasuresto solvetheshapematchingproblem.Thesimilaritymeasuresc omparefeaturesextracted fromtheshapeoftheobject,primarilypointsets,andclose d-formsolutionsforeach methodareprovided.Thesesolutionsarecomputationallye fcient,accurate,androbust againstnoiseandoutliers.Ourmethodshavebeenappliedto bothafneandnon-rigid transformationsincomputervisionandpatternrecognitio n. Therstalgorithmusesalocalizedandnon-linearsimilari tymeasureknownas correntropywhichisbasedoninformationtheoreticlearni ngandkernelmethods.The methodevaluateshowsimilartworandomvariablesareinane ighborhoodofthejoint spacecontrolledbythekernelbandwidth.Thenonlinearity introducedbythekernel providesameanstoassesssecondandhigherordermomentsof thejointPDF,which yieldssolutionsthataremoreaccuratethanMSEfordatatha thavenon-Gaussian distributions. However,correntropyrequirespriorknowledgeofthecorre spondencebetweenthe twopointsets.Toprovideagoodestimateoftheinitialcorr espondence,weuseanother measurewhichprovidescontextbasedinformationonthepoi ntsets.Themeasure reliesontheinformationcontentoftheindividualsamples andprovidesanadditional featureforeachpointsetwhichquantiesthenewinformati onthesamplebringsto shape,whichwecallthesurprisefactor.Weshowthatthemea sureisrobustagainst 12

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anytransformationandseveraltypesofnoise.However,itd oesnotperformwellin casesofdatawithlargeamountofoutliers. Thenextalgorithmmitigatesthecorrespondenceproblemby representingthepoint setsasprobabilitydensityfunctions(PDFs).Theproblemo fregisteringpointsetsis thentreatedasaligningdistributions.UsingPDFsinstead ofpointsetsnotonlyavoids theneedtoestablishacorrespondencebetweenthetwosets, butalsoprovidesamore robustwayofdealingwithoutliersandnoise.Thealgorithm operatesonthedistance betweenthePDFstorecoverthespatialtransformationfunc tionneededtoregister them,butthepricepaidishighercomputationalcomplexity .Thedistancemeasure isbasedontheCauchy-Schwarzinequality,anditiscalledt heCauchy-Schwarz divergence. Thesetwomethodscanonlybeappliedtoapairofpointsetsan dcannotbeeasily extendedtosimultaneouslyalignmultipleshapes.Therefo re,twomorealgorithmsfor groupwiseregistrationaredeveloped.Therstdivergence measureisbasedonthe H ¨ older'sinequalitytocomparemultiplePDFsconcurrently. Thismethodisanextension oftheCauchy-Schwarzmethod,butitsclosedformsolutioni sverycomputationally expensive.Anapproximationofthesolutionisprovidedwhi chisfasttocompute. Thesecondalgorithmforgroupwiseregistrationisbasedon R enyi'squadratic entropy.Themethodcomparestheentropyoftheunionofthep ointsetsagainst thesumoftheentropiesoftheindividualsets.Themethodpr ovidesclosedform solutionwhichismuchfasterthantheH ¨ older'sdivergence.Inaddition,weextendthe measuretoprovideascaleinvariantsolutionbyrstdividi ngeachoftheentropiesby thestandarddeviationoftheirrespectivedistributions. Inallourmethods,weperformthenon-rigidtransformation usingeitherthin-plate splinesorGaussianradialbasisfunctions.Weshowthatinc ertaincasesGaussianRBF outperformsTPSsinceTPShasaglobaleffectonnon-rigidit ywhereasGaussianRBF's effectislocalized,howeverthislocalizationcomesataco stofhavingafreeparameter. 13

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Inaddition,ouralgorithmshavetwofreeparameters:theke rnelbandwidthusedinthe Gaussiankernelfunctiontocomparetherandomvariablesin correntropyortoestimate thePDFsintheotherthreemethods,andtheregularizationp arameterthatcontrols thedegreeofsmoothnessofthetransformationfunction.We determinedthataslow decreaseofthevaluesoftheseparametersprovidesthebest results.Slowlyannealing theseparametersavoidsgettingstuckinlocalextrema,pro videsafasterconvergence rate,andstabilizesthealgorithm. Finally,wetestedthealgorithmsonvariouscommonsynthet icandrealdatasets usedintheeld.Wecomparedtheirperformanceagainstsome ofthetraditionaland alsosomeofthestate-of-the-artmethodsandtheresultsar ecomparableorbetterthan state-of-the-artalgorithms,andhavesmallercomputatio nalcomplexity.Inaddition,we appliedthesealgorithmstotworeal-worldapplicationsth atfulllspecications. 14

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CHAPTER1 INTRODUCTION Shapematchingisacommonproblemincomputervision,patte rnrecognition, medicalimaging,robotics,andmanyotherelds.Taskssuch ascontent-basedimage retrieval,facerecognition,objecttracking,andimagere gistrationallrequirematching offeatures/pointsets.Featuressuchaspoints,lines,and contoursareextractedfrom theimageofinterestandmatchedagainstfeaturesfromanot herimage(e.g.image registration)oragainstatemplate(e.g.objecttracking) Theabilitytocompareobjectsusingshapeinformationisan obviousrequirement sinceshapeisawell-denedconceptanditisconsideredthe primaryfeaturefornatural objectrecognition[ 1 ].However,extractinggoodfeaturestorepresenttheshape of theobjectindependentofitsorientationorsizeisverydif cultandanongoingledof research.Ingeneral,afeaturewithaverywideclassofinva riancelosesthepower todiscriminateamongessentialdifferences.Mostofthecu rrentfeaturesthatmeet thesecriteriaareverygeneralandnotgooddescriptorsoft heobject,i.e.aspectratio orcircularity.Thesefeaturesmayalsosufferinthepresen ceofnoiseinthedata,or occlusions.InChapter 2 ,webrieyreviewsomeofthemostcommonshape-based features. Oncetheshapefeaturesareextracted,asimilaritymeasure isusedtocompare them.Thechoiceofsimilaritymeasuredependsontheshapef eaturesandtheproblem. Agoodsimilaritymeasureshouldbecomputationallyefcie ntandinvarianttogeometric transformations.InthesecondpartofChapter 2 ,wereviewsomeofthesimilarity measuresthatareusedinshapematchingwork.Weprimarilyf ocusonmeasuresthat areapplicabletopointsets.Weconcludethechapterwithso mekeymeasuresbasedon informationtheorythatalsorelatetoourwork. Themosteffectiveshapematchingtechniquesdealwithtran sformingapattern andmeasuringitsresemblancetootherpatternsfromthedat abaseusingsome 15

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similaritymeasure.Giventwopatternsandasimilaritymea sure,thematchingproblem isapproachedinvariousforms[ 2 ]: computationproblem -computethedissimilaritybetweenthetwopatterns, decisionproblem -foragiventhreshold,decidewhetherthedissimilaritybe tween thetwopatternsissmallerthanthethreshold, optimizationproblem -ndthetransformationthatminimizesthedissimilarity betweenthetransformedpatternandtheotherpattern. Ourapproachprimarilyfollowstheoptimizationproblemwh ereaprototypetemplate orinputimagerepresentingtheobjectofinterestismanipu latedthroughasetof possibletransformationstoaccommodateforthevariabili tyduetoobjectposition,size, orientation,etc.Nevertheless,themethodsthatwedevelo pinthisworkcanalsobe appliedtotheothertwoapproaches. Thepointmatchingprocessiscomposedoftwosubtasks:1)id entifyingthe correspondencebetweenthepoints/featuresinthetwoshap es/images,and2)nding thetransformation/mappingthatalignsthetwoobjects.Th esimplestapproachto solvingthesesubtasksisusinganiterativealgorithmsuch asEMwhichimprovesboth thecorrespondenceandthetransformationfunctioninanre cursivemanner.Mostof thesemethodsusemeansquarederrorasthecostfunctionwhi chprovidessuboptimal solutionsforvariableswithnon-Gaussiandistributions, suchasthosecorruptedby impulsivenoiseorwithoccludedregions. InChapter 3 ,wediscussanalgorithmwhichusesalocalizedandnon-line ar similaritymeasurebasedoninformationtheoreticlearnin gandkernelmethods.The similaritymeasureisknownascorrentropyandhasbeenshow ntobedirectlyrelated totheprobabilityofhowsimilartworandomvariablesarein aneighborhoodofthejoint spacecontrolledbythekernelbandwidth,thuscontrolling theobservationwindowin whichthesimilarityisassessed[ 3 ].Thislocalizationensuresthatcorrentropyisrobust 16

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againstimpulsivenoise,allowingthealgorithmtobehaveb etterinthepresenceof outliersornoisedifferentfromGaussiandistribution,an dasaresultoutperformMSE. Thismethodisprimarilyusedincaseswherethecorresponde nceproblemis easilysolvedorwhenpriorknowledgeisprovided.Tosolvet hecorrespondence problemweutilizetwoapproaches:a)contextbased,usingt hesurprisefactor,and b)soft-assignmentallowingforaglobal-to-localsearcha ndasmoothtransitioninthe transformationspace.Therstapproachusesameasurewhic hquantiestheamount ofinterestinginformationthatadatasamplecarrieswithr especttoamodel,orinour casetheshape.Themeasurecomputestheinformationgained bythenewsample andisknownasthesurprisefactor.Thismeasureprovidesan additionalfeatureforour shapeandallowsustodeterminethecorrespondencebetween shapes.Thesurprise factorisrobusttoanytypeofafneornon-rigidtransforma tion.Inaddition,itperforms wellagainstvarioustypesofnoiseexceptforheavyoutlier noise.Thesecondapproach reliesonamany-to-manycorrespondencebetweenthepoints inthetwosetswhichis slowlyreducedtoone-to-one. Thecorrentropycostfunctionevaluatesthesimilarityoft hetworandomvariables byrstmappingthemintoafeaturespace.Thetransformatio nandcomparisonin thefeaturespacearerepresentedthroughakernelfunction .Inourwork,wechose Gaussianasthekernelfunctionbecauseitissymmetric,shi ftinvariant,andapproaches zeroasthedistancebetweenpointsincreases.Thisdecayin gfactoriscontrolledbythe kernelbandwidth,whichisafreeparameter.Weuseannealin gtoslowlydecreasethe valueofthekernelbandwidth.Thisensuresthatthealgorit hmdoesnotgetstuckata localmaximaandalsoprovidesafasterconvergencerate. Toperformthenon-rigidtransformation,weprovidetwoalt ernatives:thin-plate splines(TPS)orGaussianradialbasisfunctions(RBF).Bot happroachesworkwell, howeversinceTPShasaglobaleffect,wherethenon-rigidtr ansformationatapoint isaffectedbyallthecontrolpoints,incertaincasesitper formsworsethanGaussian 17

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RBF,whichhasalocaleffectcontrolledbythekernelbandwi dthoftheGaussian.This, ontheotherhand,introducesanotherfreeparameterthatwe havetodetermineprior toevaluatingthesimilaritymeasure.Inaddition,weneedt oconstraintheexibilityof thenon-rigidtransformationtoensurethatourshapeshave asmoothcontour.We controlthedegreeofsmoothnessusingaregularizationpar ameter.Similartothekernel bandwidth,weslowlydecreaseitsvaluesoastoallowthealg orithmtoinitiallyfocus onglobal(afne)transformationstoalignthetwoshapes,t henslowlyintroducelocal (non-rigid)transformations.Thisalsoensuresthattheal gorithmisstable. Inourwork,wedealwithverynoisydatawhereimagesegmenta tionorpointcloud extractionisverydifcultandtheresultingobjectcontou rsdonotrepresentwellthe objectofinterest.Inaddition,thereisahighdegreeofvar iabilityinthetransformations presentinthedatasetandacorrespondencebetweenpointse tsisverydifcultto obtain.Sincecorrentropyreliesonthecorrespondencebet weenthetwoshapes,and inthesedifcultcasesthecorrespondenceisnotachievabl eeventhroughthesurprise factor,welookedatadifferentapproachtopointsetregist ration.Inthisapproach,the shapesarerepresentedasprobabilitydensityfunctions(P DFs),andtoachievethe shaperegistration,wealignthetwoPDFsinstead.Thisappr oachisveryimportant becauseiteliminatestheneedforexplicitcorrespondence betweenthesets.Byusing PDFsinsteadofthepoints,themethodalsobecomesrobustag ainstnoiseorother perturbationsindata. InChapter 4 ,werepresentpointsetsasPDFsandusetheCauchy-Schwarz divergencemeasuretocompareandalignthePDFs.Thegoalis tondthePDF transformationrequiredtominimizethedivergencebetwee nthePDFsofthereference setandthetransformedset,andindoingso,itprovidesamet hodtocomputethe mappingfunctionusedinthesimilaritymeasurewithoutnee dingaone-to-one correspondencebetweenthetwopointsets.Thismethodisro bustagainstnoise andocclusions. 18

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Incertainapplications,suchasmedicalimaging,thereisa lsoaneedtosimultaneouslycompareandalignmultipleshapes.Inaddition,ther eisnotargetshapewhere alltheothersaregoingtobealignedagainst.So,themethod needstodetermine an`average'shapewherealltheshapeswillbealignedto.Si nceneitherofthetwo algorithmscoveredinChapters 3 and 4 canbeeasilyadaptedtogroupwisecomparison, inChapter 5 ,weintroducetwonewmethodsforgroupwiseregistration.T herstis basedonH ¨ older'sinequalityandisderivedasanextensiontotheCauc hy-Schwarz divergencemeasure,wherethecross-productofallthePDFs actsasthesimilarity measure.Weprovideaclosed-formsolutionforthemethod,h oweveritisvery computationallyexpensive.Toreducethecomplexityweals oprovideanapproximation whichfairlycompareswiththeoriginalmethod. Inaddition,weprovideanothersimilaritymeasurebasedon R enyi'squadratic entropy.ItfollowsasimilarapproachtotheJensen-Shanno ndivergencewhereit comparestheentropyofthe`overall'shape,theunionofall theshapes,againstthesum oftheentropiesoftheindividualshapes.Sincethismethod usesentropyandentropy dependsonthestandarddeviationofthedistribution,topr ovideascale-invariant solution,werstdividetheentropiesbytheirrespectives tandarddeviation.Both methodsarecomparedagainstthebestmethodavailabletoda te,andtheyperform betterandfaster. InChapter 6 ,thesesimilaritymeasuresarethentestedontworeal-worl dapplications: sonarcontactfusion,andremotecontactlessmassestimati on.Inthesonarcontact fusionproject,weareprovidedwithside-scansonarimages ofvariousminetypes.The overallgoaloftheprojectistoextractinformationfromth eseimagessoastogroup thesnippetscomingfromthesamemine.Weshowhowshapematc hingcanbeused asatooltoextractinformationontheshapeoftheminespres entintheimages.We provideresultsofshapematchingasaclassicationtool,a ndalsoasafeaturetoa morecomplexsystem. 19

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Intheremotecontactlessmassestimationapplication,wea rerequiredtoestimate theweightofananimal,i.e.cow,byusinginformationtaken fromastereoscopicimage. Weusethestereoimagetoextracta3Dpointsetofthetargeta nimal,andthenapply oursimilaritymeasurestoaligna3Dtemplateoftheanimala gainstthepointcloud.The dimensionsofthealigned3Dtemplatewerethenusedtoestim atetheweightofthe animal. Weconcludewithasummaryofourworkandsomefuturedirecti ons. 20

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CHAPTER2 OVERVIEW Shapematchingcomprises:identifyingthefeaturestorepr esenttheshape,and determiningadistance(similarity)measuretocomparethe m.Agoodshapedescriptor shouldbesimpletocompute,providesufcientinformation contenttodistinguishthe shape,berobustagainstnoiseandocclusions,andbeinvari anttotransformations(i.e. rotation,scaleandtranslation).Asimilaritymeasuresho uldalsobesimpletocompute, indicatethedegreeofresemblanceoftheshapes,preferabl ysatisfyallthemetric properties,andbeinvarianttogeometricaltransformatio ns.Inthischapter,wereview theshapematchingconstituents.Westartwiththemostcomm onshapefeaturesand similaritymeasures,andattheend,focusoninformationth eoreticsimilaritymeasures relatedtoourwork. 2.1Shape-BasedFeatures Shapeisafundamentalfeatureindescribinganobject.Itis adescriptorofthe geometricalinformationofanobjectanddoesnotchangeeve nwhentheposition, sizeandorientationoftheobjectarechanged[ 4 ].Therefore,shapefeaturesshould beinvarianttotranslation,rotation,andscaletobesucce ssfullyusedinmatching. Shape-basedfeaturestypicallyrequiresomepre-processi ng,suchasregionidentication orsegmentation,whichdiffersfromcolor-basedortexture -basedfeatureswhichare mainlyglobalattributesofanimage.Beforeapplyingshape descriptors,animage needstobedividedintopartsextractingtheobject(s)ofin terestfromthebackgroundor segmentingtheregion(s)ofinterest.Thesegmentationpro cessusuallyisautomated, buttherearecaseswherehumaninteractionisrequiredduet oitsdifculty.Infact, segmentationofnontrivialshapesisoneofthemostdifcul ttasksinimageprocessing [ 5 ],anditisanopenproblemincomputervision[ 6 ].Sincesegmentationaccuracy determinestheeventualsuccessorfailureofshapematchin gtechniques,considerable 21

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attentionneedstobeprovided.Here,webrieyoverviewima gesegmentationinrelation toshapematchingandshapefeatureextraction.2.1.1ImageSegmentation Imagesegmentationinvolvespartitioninganimageintoase tofnon-overlapping, homogeneousandmeaningfulregions,suchthatpixelsineac hpartitionedregion possessanidenticalsetofpropertiesorattributes[ 5 ].Segmentationoftheregions dependsonthepropertiesandconnectivityamongthepixels inaregion.Segmentation generallyisbasedontwobasicpropertiesofintensityvalu es: discontinuity and similarity amongpixels.Discontinuityallowstopartitiontheimageb asedonabruptchanges inintensity,suchasedgesorlinesinanimage.Similaritya llowstopartitiontheimage intoregionsthataresimilaraccordingtoasetofpredened criteriasuchashaving homogeneousintensity.Techniquessuchasthresholding,d iscontinuitydetection, region-basedsegmentation(regiongrowing,splitting,an dmerging),andsegmentation bymorphologicalwatershedsallfallinthiscategory. Intensitybasedsegmentation quiteoftenproducespuriousedgesorgapsthat donotnecessarilycorrespondtoboundaryobjects.Thelimi tationofthesemethodsis duetotheircompleterelianceoninformationcontainedint helocalneighborhoodofthe image.However,inanimage,thereusuallyisinterferenceb yothersignalsorartifacts thatriseduringsampling.Theseinterferencescauseprobl emswhenusingcommon techniquesforsegmentationsincetheyrelyonlocalinform ationandignorebothmodel basedinformationandhighorderorganizationoftheimage[ 4 ]. Theapplicationofpriorknowledgesuchas geometricalinformation improves thesegmentationprocess.Theactivecontourmethodsprovi deaneffectivewayfor segmentation,inwhichtheboundariesoftheobjectsaredet ectedbyevolvingcurves. Kassetal.introduced snakes asoneoftherstapproachestoutilizevisualanalysisof shape[ 7 ].Avariationofactivecontourmodels,curveevolution,is usedin[ 8 ]forimage segmentationusingadaptiveows.Othertechniquesinvolv eusingenergydiffusion 22

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[ 9 ],andgraphpartitioning[ 10 ].Severalofthesemethods,requireuserinput,usually toprovideastartingpointforsegmentation.Thissomewhat defeatsthepurposeof automatingthesegmentationprocess.Anattempttoaddress thisproblemisprovided in[ 11 ]whereanattraction-basedgroupingmethodisintroducedi nspatial-color-texture space. Anotheralternativeis probabilisticimagesegmentation whereanimage isrepresentedusingaGaussianmixturemodel(GMM).In[ 12 ],animageisrst modeledasasetofcoherentregionswhereeachregionisrepr esentedbyaGaussian distribution.ThesetofallregionsisrepresentedbyaGMM, andtheexpectation/ maximization(EM)algorithmisusedtodeterminetheparame tersofthemixture. Oncethemodelislearned,eachpixeloftheimageisafliate dwiththemostprobable Gaussianresultinginaprobabilisticimagesegmentation. Othermethodswhichprovide regionscoherentincolorandtextureinclude:JSEG[ 13 ]whichusescolorquantization andspatialsegmentationtodivideanimageintocolor-text ureregions,andBlobworld regionsegmentation[ 14 ]whichsegmentsanimageintoregionsbymodelingthejoint distributionofcolor,texture,andpositionfeatureswith amixtureofGaussiansestimated viatheEMalgorithm. Mostoftheshape-basedfeaturesusedinshapematchingrequ ireaccurate segmentationofimagestodetecttheobjectortheregionbou ndaries,whichisin itselfanill-posedproblem[ 2 ].Theinherentsensitivityinmostsegmentationalgorithm s preventstheuseofautomaticsegmentationinbroaddomains orforsensoryconditions whereclutterandocclusionaretobeexpected.Segmentatio ninunconstrainedcontext isverydifcultandsometimesmeaningless.Duetothesedif culties,mostofthework focusedonusingsimilaritymeasuresbasedonshapefeature sisrestrictedtonarrow domaindatasets. 23

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2.1.2ShapeFeatures Efcientshapefeaturesmustbeinvarianttoafnetransfor mations,robustagainst noise,statisticallyindependent,invarianttoocclusion s,andreliable[ 15 ].Theshape descriptorsshouldbeaclose-to-completerepresentation oftheshapecontent,andthey shouldallowforasimplemeansofmeasuringthedistancebet weenthem.Thereexist manydifferentclassicationmethodsforshape-basedfeat ures.Here,wewillbriey coverthemostcommonfeaturesandgroupthembasedonthepro cessingapproach used. Themostbasicshapefeaturesarederivedfromtheshapecont ourandareoften called shapesignature [ 16 ].Signaturesareusuallyone-dimensionalfunctionsthat oftenactaspreprocessingtootherfeatureextractionalgo rithms.Somecommonlyused signaturesare:contourcurvature,shapearea,centroiddi stance,complexcoordinates, andturningangles.Theseshapesneedtobenormalizedorrep resentedintermsof relativevaluestoaccountfortransformationinvariance. Ellipse-basedshapefeatures arecomputedusingapolygonalapproximationof theobjecttofocusoncapturingtheoverallshapeandignore minorvariationsalong thecontour[ 17 ].Theapproximationremovesanyredundantpointsandnoise inthe shape.Someofthefeaturesthatcanbederivedusingthepoly gonalapproximationare: shapecompactnesswhichmeasurestheroundnessoftheshape [ 18 ],eccentricitywhich representstheratioofthemajortominoraxisofthebesttt ingellipseoftheshape[ 19 ], andsoliditywhichmeasurestheconcavityoftheshape[ 20 21 ]. Spatialinterrelationfeatures describeboththeboundaryandtheregionsofthe shapeusingsimplegeometricfeaturessuchaslength,area, curvatureandrelative orientation,anddistanceandlocation.Someofthemostwel l-knownfeaturesthatfall underthisgroupare:adaptivegridresolutionwherethecel lresolutionvariesbasedon theshapecontentinthatarea[ 22 ],boundingboxwhichcomputeshomeomorphisms between2Dlatticesandshapes,chaincodeswhereashapeisr epresentedinterms 24

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oflinesegmentswithgivenorientation[ 23 ]orsmoothcurvedecomposition[ 24 ],shape matrixwhereashapeisdescribedasaMNmatrixeitherinasquaremodel[ 25 ]orin apolarmodel[ 26 ],andshapecontextwhichdescribesthewholeshapeinahist ogram ofpolarcoordinatesfromtheperspectiveofeachkeypointe itheronthecontouror interioroftheshape[ 27 ]. Scale-spacefeatures relyonthescalespacetheorywhereacurveisembedded inacontinuousfamilyofgraduallysimpliedversions.Thi sisimportantbecausedueto differentscalesitispossibletoseparatetherelevantcha racteristicsoftheshapefrom theirrelevantdetails.Comparisonscanbeperformedatdif ferentscalestodetermine differentlevelsofsimilarity.Themostcommonfeatureist hecurvaturescale-space whichwasselectedasacontourshapedescriptorforMPEG-7[ 28 29 ].Thisfeature providesamulti-scalerepresentationofthecurvatureoft heshapecontour.Itcaptures themainfeaturesofashapeatdifferentscales,isrobustag ainstnoise,invariantto slightafnetransformations,anditisalsofasttocompute Moment-basedfeatures evolvedfromtheconceptofmomentsinphysics.They areeasytocomputeandsupposedlyinvarianttothebasictra nsformationthatcanbe foundinashapesuchasrotation,translation,andscale.Th erearevarioustypesof momentsthathavebeenusedforshapeclassication:geomet ricmoments[ 30 ]based onthenormalizedcentralmoments,Hu'smoments[ 31 ]whichareinvarianttothethree commontransformationsbutcarryinformationredundancys incethebasisarenot orthogonalandarealsoverysensitivetonoise,Zernikemom ents[ 32 ]whichsolvethe orthogonalityproblemandarealsorobusttonoise,andradi alChebyshevmoments[ 30 ] whichusediscreteorthogonalmoments. Inadditiontospatialshapefeatures,therearemanyfeatur esthatareextractedby rsttransformingtheshapeinanotherdomain. Fouriershapefeatures arebasedon Fouriertransform.Theycapturetheglobalshapefeaturesb ytherstfewlowfrequency termsandthenershapefeaturesbythehigherfrequencyter ms[ 16 ].Theytendto 25

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capturethegeneralshapecharacteristicsbetterthanspat ialfeatures[ 33 ],butthey sufferfromrotationandscalingvariances.Wavelet-based shapefeaturessimilarto Fourierfeaturesdecomposetheshapedetailsintodifferen tcomponents. Wavelet features providemulti-resolutionrepresentation,butmaynotbein varianttoafne transformations.In[ 34 ],adescriptorisobtainedbyapplyingFouriertransformal ongthe polarangleaxisandwavelettransformalongtheradiusaxis .Thisfeatureisshowntobe invarianttotranslation,rotation,andscaling[ 15 ]. Thesefeaturesarebasedontheglobalshape,andtheyrelyon thequalityof segmentationofthecompleteobject.Asaresult,theyareno trobustagainstocclusion orallowforpartialmatching.Manyofthesefeaturesareals osensitivetomajor transformationssuchasscaling,rotationortranslation. Othersimplerfeaturesexist thatworkwellincasesofpartialocclusion,outliers,ordi fferentorientationsandscale. Pointsarethesimplestshapefeaturewhereanobjectisrepr esentedasapointsetofits contourand/oritsinterior.Inourwork,werepresentshape susingpointsetsbecause theyaresimpletoextractandworkwith.Nevertheless,oura lgorithmscouldalsobe appliedtoothershape-basedfeatures.Inaddition,wealso useprobabilitydensity functionsestimatedbythepointsetstorepresentshapes. Moreinformationontheseandotherfeaturescanbefoundont hesesurveys: [ 2 15 23 35 – 40 ].Mingqianget.alin[ 15 ]provideatablelistingmostofthefeatures discussedhereandmanyothersalongwiththeirkeyproperti essuchasinvariance toafnetransformations,robustnessagainstnoiseoroccl usion,andcomputational complexity. 2.2SimilarityMeasures Oncethefeaturesareextractedfromtheobjects,thenextst epistocompare them.Asimilaritymeasureshouldbechosentocomparethefe aturesofbothobjects inthecaseofpaircomparison,orallobjectsinthecaseofgr oupwisecomparisons. Thesimilaritymeasureissimplyamappingofthepairoffeat urevectorstoapositive 26

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real-valuednumberyieldingameasureofthevisualsimilar itybetweenthetwoshapes as D :RN RN! R .Asuitablesimilarityfunctionshouldberobusttonoise, computationallyefcient,invarianttogeometrictransfo rmations,andfollowlocal linearity.Inaddition,itisdesirablethatsuchasimilari tymeasurebeametric.The similaritymeasuresthatsatisfythemetricpropertiesare called distance measures, whiletheothernon-metricsimilaritymeasuresarecalled divergence measures. Thechoiceofsimilaritymeasuredependsbothontheproblem andontheshape features.Inthissection,wereviewafewsimilaritymeasur eswiththefocusthat areappliedtopointsetandPDFrepresentations,buttheyco uldalsobeapplied toothershapefeatures.Westartwiththemostcommondistan cemeasure,the Euclideandistance.IfweconsiderthetwoshapesasXandYandassumethatthe correspondencebetweenthemisalreadyknown,the Euclideandistance isdenedas:d ( X Y ) =s XijX iY ij2 ,(2–1) whereX iandY iarethecorrespondingpointsinthetwosets.TheEuclideand istance ispartoftheL pMinkowskifamilyofdistancemeasures ,wheretheL pdistanceis denedas:d L p ( X Y ) = ps XijX iY ijp .(2–2) ManysimilaritymeasuresutilizetheL pdistances.Bottleneckdistancemeasuresthe minimumofthemaximumdistancesforallone-to-onecorresp ondencesbetweenthe twoshapesusingEuclideandistance.Hausdorffdistanceme asuresthemaximumofthe distanceofthepointsinXtoYandthepointsinYtoX[ 2 ],whichisdenedas:d ( X Y ) = maxmax x2X min y2Y d ( x y ), max y2Y min x2X d ( x y ),(2–3) whered ( x y )istheEuclideandistance.Choosingassociationweightsso asto minimizetheHausdorffdistanceandsummingthemupresults inMallowsdistance 27

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[ 41 ].Theearthmover'sdistance(EMD)[ 42 ]representsanotherdistancemeasureusing Euclideandistancebutwithsoftcorrespondence. Sincethecorrespondencebetweenpointsetsisusuallynotk nownapriori,point setsareusuallycomparedusingprobabilitydensityfuncti onrepresentation.InChapter 4 ,wewillcoverseveralmethodsthatusePDFstocompareshape s.TheL pMinkowski familyofdistancescanbedenedintermsofPDFsas:d L p ( P Q ) = ps Z jp ( x )q ( x )jp dx ,(2–4) wherep ( x )andq ( x )arethePDFsoftherandomvariablesXandY,orinourcasethe pointsetsofthetwoshapes. Anothercommonlyusedsimilaritymeasurecomputesthe intersection between twoPDFs[ 43 ]andisdenedas:d ( P Q ) =Zmin ( p ( x ), q ( x )) dx .(2–5) Manyvariationsoftheintersectionmeasureexistwhere( 2–5 )isnormalizedbyeither thesumorthedifferenceofthetwoPDFsoracombinationofth em[ 44 ].Mostof thesimilaritymeasuresutilizingtheintersectionmeasur ecanbetransformedintoL 1distancemeasuresbasedonthefollowingderivation:d ( P Q ) =Zmin ( p ( x ), q ( x )) dx = 11 2Z jp ( x )q ( x )jdx .(2–6) SomesimilaritymeasurescomparetwoPDFsbycomputingthe innerproduct betweenthem.Theinnerproductoftwovectorsmeasuresthei rproximitybasedon thecosineoftheanglebetweenthem.Thesameappliestofunc tions,andinourcase PDFs.Theinnerproductsimilaritymeasureisdenedas:d ( P Q ) = PQ =Zp ( x ) q ( x ) dx .(2–7) 28

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Avariationoftheinnerproductmeasure,whichweuseinourw orkinChapter 4 normalizestheinnerproductbythenormsofthetwoPDFsas:d ( P Q ) =Rp ( x ) q ( x ) dx q Rp ( x ) 2 dxq Rq ( x ) 2 dx .(2–8) Thismeasureisknownasthe cosinecoefcient becauseitmeasurestheangle betweenthetwoPDFs.Othervariationsoftheinnerproductm easureinclude normalizingthemeasureusingthesumofthetwoPDFs(harmon icmean),ortheir differencesquared,orsomeothercombination[ 45 ].Theinnerproductmeasuresare veryusefulandoftenusedintheeldofinformationretriev al[ 44 ]. Anotherfamilyofsimilaritymeasuresisbasedonsummingth e geometricmean of thetwoPDFswhichiscalledthedelitysimilarity[ 44 ]ortheBhattacharyyacoefcient [ 45 ]andisdenedasd ( P Q ) =Z p p ( x )q ( x ) dx .(2–9) TheBhattacharyyadistanceisdenedasthenegativelogari thmof( 2–9 )andsimilarto theinnerproductmethodsmeasurestheoverlapbetweenthet woPDFs.Othersimilarity measuresinthisgroupinvolvevariationsofthedelitymea sure. SeveralsimilaritymeasuresinvolvetheEuclideandistanc esquaredandarepartof the X2family .ThecornerstoneofthisgroupofmeasuresisthePearson X2divergence [ 46 ]whichisdenedas:d ( P Q ) =Z( p ( x )q ( x )) 2 p ( x ) dx .(2–10) Neyman X2divergencedividesthesquaredEuclideandistancebyQinstead,and boththesemethodsareasymmetric.Othervariationshavebe endesignedtoensure symmetrysuchasthesquared X2divergencewherethesumofbothPandQis usedasthenormalizationterm,orthesquaredEuclideandis tanceitselfwithoutany normalization. 29

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2.3InformationTheoreticSimilarityMeasures Thenextgroupofsimilaritymeasuresthatwereviewisthegr oundworkforthe measuresthatwedevelopinthisthesis.Thesemeasuresinvo lveinformationtheory, therefore,werstprovideabriefreviewofsomecoremathem aticalconceptsfrom informationtheorydevelopedbyClaudeShannonandAlfr edR enyi. 2.3.1Shannon'sEntropy Informationtheorywascreatedtohelpunderstandhowtoopt imallyencode messagesbasedontheirstatisticalstructuretotransmitt hroughnoisychannelswith minimumloss.Tounderstandhowtooptimallyencodeamessag e,Shannonrst neededtomeasuretheamountofinformationcarriedbytheme ssage.Basedon Hartley'sworkonmeasuringthe“amountofinformation”ofa noutcome[ 47 ],Shannon quantiedtheinformationcontentofanoutcomex i,whoseprobabilityisp ( x i ),asI ( x ) = log 1 p ( x i ) .(2–11) ToquantifytheinformationconveyedbytherandomvariableX,Shannonusedthe weightedsum(average)oftheinformationcontentoftheind ividualoutcomesasH ( P ) =Xx i p ( x i ) log 1 p ( x i ) = E [log p ( X )],(2–12) whichiscalledShannon'sentropyofrandomvariableXandisdenotedbyH.The entropyofthediscreterandomvariableXisameasureoftheuncertaintyassociated withthevalueofX.Theequationitselfprovidesinsightsofhowtheentropyof avariable ismeasured.Outcomesthatareveryrarecarryhighamountof information,butbecause theyareweighedbytheirprobability,theiroveralleffect isdiminished.Ontheother hand,outcomesthatareverycommonhavehighprobabilitybu tverylowinformation contentwhichagainreducestheireffectontheentropyvalu e.Thiscreatesa“balance” thathadnotbeenquantiedbeforeinprobabilisticreasoni ng[ 48 ]. 30

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Thedenitionofentropycanbeextendedtocontinuousrando mvariablesasH ( P ) = Zp ( x ) log p ( x ) dx(2–13) anditiscalleddifferentialentropy.Whentheentropyofad istributionismeasurewith respecttoanotherPDF,P,ratherthanthe“true”distribution,Q,wehaveaquantityH ( P ; Q ) = E P [log Q ]thatwerefertoascrossentropy. 2.3.2R enyi'sEntropy R enyidenedafamilyofentropiesas:H( P ) = 1 1 logXx i p( x i ),(2–14) where isafreeparameter.Renyi'sentropyisageneralizationofS hannon'sentropy [ 49 ].For !1R enyi'sentropytendstoShannon'sentropy[ 50 ].Themainadvantage ofR enyi'sdenitionofentropyisthatthelogarithmappearsou tsidethesumwhich separatestheeffectofthelogarithmfromtheargumentofen tropyandallowsthe estimationofentropynon-parametricallyfrompairwisesa mpledifferences.Forexample, for = 2wehaveR enyi'squadraticentropyofacontinuousrandomvariableXdened asH 2 ( P ) =logZp 2 ( x ) dx .(2–15) Kernel(Parzen)estimateofthePDF[ 51 ]denestheprobabilityp ( x )ofanarbitrary pointxusinganarbitrarykernelfunction ( )as^ p ( x ) = 1 NPN i =1 xx i ,where isthebandwidthparameter.IfweconsidertheGaussiankern el,G( x x i ) = 1 p 2exp kxx ik2 22 ,asthekernelofchoiceforthePDFestimateandsubstitutei tin theR enyi'squadraticentropy,wehaveaclosedformsolutioncom putedas: 31

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H 2 ( P ) =logZ 1 11 N NXi =1 G( x x i ) 1 N NXj =1 G( x x j ) =log 1 N 2 NXi =1 NXj =1Z 1 1G( x x i )G( x x j ) =log 1 N 2 NXi =1 NXj =1 Gp 2( x i x j ).(2–16) Inourwork,weutilizeR enyi'sentropy,specicallythesecondorderentropy,inst eadof Shannon'sentropybecausewecanestimateitnonparametric allyfrompairwisesample differences.Thereisnoapproximationinthisevaluatione xceptforthekerneldensity estimation.2.3.3Entropy-BasedSimilarityMeasures Kullback-Leiblerdivergence (KL)[ 52 ]measuresthedistancebetweentwo probabilitydensityfunctionsp ( x )andq ( x )andisdenedas:D KL ( PkQ ) =Z 1 1p ( x ) log p ( x ) q ( x ) dx .(2–17) TheKLdivergencecomputesthedifferencebetweenthecross entropyofPandQandtheentropyofPas:D KL ( PkQ ) = H ( P ; Q )H ( P ).Forourpurposes,itcanbe interpretedasmeasuringthesimilarityoftwovariablesby comparingtheirentropies withrespecttoaxeddistribution,P.KLdivergenceisasimilaritymeasurebutcannot beconsideredadistancebecauseitdoesnotsatisfythetria ngleinequalityproperty. Forthisreasonitiscalledadivergencemeasure.Italsodoe snotsatisfythesimilarity propertybutthatcaneasilybesolvedbyaveragingD KL ( PkQ )andD KL ( QkP )whichis knownastheJeffreysdivergence[ 53 ]. divergence extendsKLdivergenceto:D( PkQ ) =D KL ( Pk P + (1 ) Q ) + (1 ) D KL ( Qk P + (1 ) Q ) .(2–18) 32

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Thevalue = 0.5providesaspecialcase:D=0.5 ( PkQ ) = 1 2 D KLPkP + Q 2+ 1 2 D KLQkP + Q 2= HP + Q 2 1 2 ( H ( P ) + H ( Q ) ) ,(2–19) whereHisShannon'sentropy,andthisspecialcaseisknownastheJe nsen-Shannon divergence. Thegeneraldenitionof Jensen-Shannondivergence isderivedfromthe Jensen'sinequalitywhichstatesthatforanyPDFs,p 1 p 2 ..., p N,wehave: NXi =1i p i! NXi =1i( p i ),(2–20) providedthat isconvexovertherangeofp (), Pii = 1and i0.SinceShannon's entropyisstrictlyconcave[ 54 ],thereisasignreversal,andtheJensenShannon divergenceisdenedas:D JS ( p 1 p 2 ..., p N ) = H NXi =1i p i! NXi =1i H ( p i ).(2–21) Thisderivationcouldhavealsobeenreachedbyextending( 2–19 )toNPDFs. InChapters 4 and 5 wedescribeafewmoredivergencemeasuresbasedon R enyi'ssecondorderentropywhichwewillusetosimultaneou slycompareeitherpairs orgroupsofshapes.Thereareadvantagesanddisadvantages tothedifferentdistance functionsusedinsimilaritymeasures.Whileasimplemetho dmayleadtoveryefcient computation,itmaynotbescalabletorealworldapplicatio ns,thusmakingitineffective. Aswewillseeinthenextfewchapters,identifyingtheappro priatedistancemeasure dependsonthefeaturesextractedfromtheshapesandthecha llengesprovidedbythe applicationitwillbeused. 33

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CHAPTER3 POINTMATCHING-CORRENTROPY InChapter 2 ,wereviewedvariousshape-basedfeatures.Whilemanyelab orate featuresexisttorepresentshapes,thesefeaturesareusua llyapplicationdependentand oftencomputationallyexpensivetoextract.Inaddition,m anyofthesefeaturessufferin thepresenceofnoiseoroutliersinthedata.Points,ontheo therhand,arethebasic andmostgeneraltypeoffeature,wheretheshapeofanobject isexpressedinterms ofpointsrepresentinglandmarksintheboundaryandtheint erioroftheobject.Inour work,weutilizepointfeatures. Pointmatchingcomprisestwosubtasks:1)establishthecor respondencebetween thepointsinthetwoshapes/images,and2)retrievethespat ialtransformation/mapping thatalignsthetwoobjects.Eachofthesesubtasksiseasily solvedoncethesolution totheotherisknown.Thedifcultyariseswhenattemptingt osolvethemconcurrently. Forexample,determiningthecorrespondencebetweentwoob jectsusingsimplebrute forceisaverydauntingtaskasthecorrespondencespaceinc reasesexponentiallywith thecardinalityofthetwosets.Variousapproacheshavebee ntakentofacilitatesolving thesesubtasks.Belongieetal.in[ 27 ]andJainetal.in[ 55 ]provideshapedescriptors tofacilitatethecorrespondenceproblem.Shapecontextde scribesthedistributionofthe shapewithrespecttoeachpointontheshape.Findingthecor respondencebetween twoshapesisthenequivalenttondingthepointineachobje ctwithasimilarshape context. Theiterativeclosestpoint(ICP)algorithm[ 56 ]isthemostpopularmethodfor pointsetregistration,anditutilizesthenearest-neighb orrelationshiptoassignbinary correspondenceateachstep.Itthenndstheleastsquarest ransformationrelating thepointsetsbasedonthisestimateofthecorrespondence. Thealgorithmcontinues thisiterativeprocessuntilitreachesalocalminimum.The methodisverysimple, butbecauseitscostfunctionisnotdifferentiable,itexhi bitslocalconvergence.Asa 34

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result,ICPrequiresacloseinitialposeofthetwopointset s.Withanadequatesetof initialpositionsitcanperformanyrigidtransformation. Themethodalsosuffersinthe presenceofoutliersandisnotsuitablefornon-rigidtrans formations. Robustpointmatching(RPM)[ 57 ]improvesonICPbyemployingaglobal-to-local searchandsoftassignmentforthecorrespondence.Themeth odjointlydetermines thecorrespondenceandthetransformationbetweenthetwop oint-setsviadeterministic annealingandsoft-assignment.Thesoft-assignmentrelax estheone-to-onecorrespondencebetweenthetwopointsetsandthefuzzycorrespondence allowsforagradual improvementwithoutjumpsinthespaceofbinarycorrespond ences.Themethodisnot stableinthepresenceofoutliers.Inaddition,duetotheso ft-assignmentapproach,the complexityofthemethodishigh. In[ 58 ],ZhengandDoermannformulatepointmatchingasanoptimiz ationproblem topreservelocalneighborhoodstructures.Theirmethodis basedonthefactthatlocal relationshipamongneighboringpointsisveryimportantfo rnonrigidshapesandit isgenerallywellpreservedduetophysicalconstraints.Th eirapproachusesgraph matchingwheretwoneighboringpointsarerepresentedbyan edgebetweenthenodes andtheoptimalmatchbetweentwographsistheonethatmaxim izesthenumberof matchededges. Thesemethodsprovideveryinterestingapproachestosolvi ngthepointmatching problem,however,theyassumethatthepointfeaturesprovi deagoodrepresentation oftheirrespectiveobjects.Inourwork,wedealwithveryno isyobjectcontourswhere occlusionsandoutliersareverycommon.Ourgoalistodesig nalgorithmsthatwill berobusttothesetypesofnoise,specicallytooutliers.I nthischapter,wediscuss analgorithmwhichimprovesuponthesemethodsbyusinganon linearmeasure knownas correntropy [ 59 ].Correntropyisageneralizedcorrelationmeasureinvolv ing high-orderstatistics.Themethodwewilldescribeisrobus ttoimpulsivenoise,which behavesbetterinthepresenceofoutliers.However,simila rtoICP,itutilizesthe 35

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nearest-neighborrelationshiptoestimatethecorrespond enceofthepointsets.We willdiscussitsadvantagesoverlinearmeasures,andprovi detwoalternativestomitigate thecorrespondenceproblem. 3.1PointMatching SupposewehavetwopointsetsX =fx igN i =1andY =fy igN i =1,wherex i y i2 Rd.To alignpoint-setXtopoint-setY,point-setXhastoundergosometransformationfwhich mapsapointx iontoanewlocationx i = f ( x i ),wherey i = x i +i.Thisthenbecomesthe classicregressionproblemy i = f ( x i ) +i i = 1, ... n ,(3–1) wherefrepresentstheregressionmodeland iaretherandomerrors.Recovering thefunctionfthatcloselymatchesx itoy iisanill-posedproblembecausethereexists aninnitenumberoffunctionsfthatcouldsatisfyourgoal.Toselectaparticular solutionweneedaprioriknowledgeonthefunction.Weknowt hatthetransformation canbebrokendownintotwoparts:1) afne transformation,whichincludesthemajor differencesbetweenthetwopointsets,globalandlineartr ansformationssuchas rotation,scaling,shearandtranslation,and2) nonrigid transformation,whichincludes thelocaldeformationsthatcannotbeexpressedbytheafne transformation.Inaddition, weneedtoassumethatthefunctionis smooth whichensuresthattwosimilarinputs correspondtotwosimilaroutputs.Thisiscrucialwhendeal ingwiththenon-rigidpartof thetransformationaswewillseebelow.Wedenethesmoothn esstermbyintroducing aregularizationtermintoourproblem. AfneTransformationAbasictransformationfunctioninvolvingonlyrotationan dtranslationappliedtoa pointxcanbewrittenasfollowsf ( x ,, t ) = R () x + t ,whereR () =264cos ()sin () sin () cos ()375.(3–2) 36

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representtherotationangle,R ()representstherotationmatrixfor2-Ddatasamples, andtrepresentsthetranslationvector.Torepresentthetransl ationvectoraspartof theafnetransformationmatrix,weconsiderhomogeneousc oordinateswherethe originalvectorpointsareextendedto( d + 1)dimensionswiththelastelementbeing one(e.g.[ x 1 x 2 1] T).Inaddition,toaccountforanyafnetransformation,not just rotation,weconsideramatrixAof( d + 1)( d + 1)dimensionsandrepresenttheafne transformationfunctionasf ( x A ) = Ax. Non-RigidTransformationThemostpopularnon-rigidtransformationsaretheradialb asisfunctions(RBF), wherethetransformationisdenedasalinearcombinationo fthebasisfunctionsasf ( x W ) = NXi =1 w i(kxx ik),(3–3) wherew iareunknownparameters,and representthebasisfunctionswhich dependontheEuclideandistancebetweenapointxandthecontrolpointsx i.The mostcommonRBFsusedfornon-rigidapplicationsarethinpl atesplines(TPS)and GaussianRBFs.Thekeyadvantageofboththesefunctionsist hattheyapproachzero asymptotically,howevertheybothutilizeallcontrolpoin tswhichsignicantlyincreases theircomputationcomplexityasthenumberofcontrolpoint sincreases. RegularizationConstrainingthesolutionwitha`stabilizer'isanimporta ntpartofshapematching becauseitenforces`smoothness'onthetransformationfun ction.Sincewearedealing withanitesetofpoints,estimatingthenon-rigidtransfo rmationdoesnotprovide auniquesolution.Thereisaninnitenumberoftransformat ionsthatwouldmatch thecorrespondingpointsbutwillhaveverydifferentbehav iorintherestoftheshape contour.Imposingtherequirementofchoosingthesmoothes ttransformationfunction, whichiscontrolledbyaregularizationterm,willprovidea uniquesolution. 37

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TheregularizationtheoryoriginatesfromtheworkofTikho nov[ 60 ]wherethe existingoptimizationproblemisaugmentedwitharegulari zationterm.Inourcase,we have:min NXi =1 ( y if ( x i ) ) 2 +R ( f ),(3–4) where PN i =1 ( y if ( x i ) ) 2representsthe`delity'oftheapproximation,R ( f )isthe regularizationtermrepresentingtheconstraintonthe`sm oothness'offunctionf,and isafreeparameterthatrepresentsthetrade-offbetweenth eproximityofftothe solutionanditssmoothness.Wewillcovertheregularizati onterminmoredetailbelow. 3.2Correntropy MSEisthemostwidelyusedcostfunction.Mostoftheworktha twereviewedin thischapterutilizesMSEastheiroptimizationcostfuncti on.MSEisasecondorder statisticswhichfullyquantiesrandomvariablesthatare Gaussiandistributed.Fig. 3-1 (A)providesanillustrationoftheMSEcostfunctioninthej ointspaceofvariablesXandY.MSEisaquadraticfunctioninthejointspacewithavalleya longthex = yline.ThegureexplainsthebehaviorofMSEinthejointspac ewhereforvaluesofxclosetoyMSEtakessmallvalues,howeverforvaluesawayfromthex = yline MSEtakesvaluesthatincreasequadraticallyduetotheseco ndordermoment.The quadraticincreaseshowsthatMSEworkswellfordatawithsh ort-taildistributionssuch asGaussian.However,forlong-taildistributionssuchast hosecontainingoutliers,MSE isnotoptimalbecauseresultsfarawayfromthemeanwillhav eanampliedeffecton thetotalcost. CorrentropydoesnotsufferfromthelimitationofGaussian ityinherentincost functionsbasedonsecond-ordermoments.Correntropyquan tieshigherorder momentsofthePDFyieldingsolutionsthataremoreaccurate innon-Gaussianand nonlinearsignalprocessing.Correntropyisrobusttoimpu lsivenoise[ 61 ],whichis importantwhentheobjectcontoursmayhavepartiallyocclu dedareasortheboundaries 38

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maynothavebeenextractedcorrectlyduetoimperfectionso ntheimage.Insuchcases, theseocclusionsorimperfectionsaretreatedasoutliers. Correntropyisageneralizedsimilaritymeasureinvolving high-orderstatistics whichmeasuresanonlinearsimilaritybetweenrandomvaria bles[ 59 ].Forourpurpose, correntropyisusefulbecauseitisdirectlyrelatedtothep robabilityofhowsimilartwo randomvariablesareinaneighborhoodofthejointspacecon trolledbythekernel bandwidth[ 62 ].Controllingthekernelbandwidthallowsustoadjustthe` observation window'inwhichsimilarityisassessedresultinginaneffe ctivemechanismtoeliminate thedetrimentaleffectofoutliers.Cross-correntropybet weentworandomvariablesXandYisdenedasv ( X Y ) = E XY [( X Y ) ] =ZZ ( x y ) p XY ( x y ) dxdy ,(3–5) wheretheexpectedvalueisoverthejointspaceand ( X Y )isanycontinuous, non-negativedenitekernel.Inourcase,theGaussiankern el,G( XY ) = 1 p 2exp kXYk2 22 ,isusedforitsproperties:symmetric,translationinvari ant,positive denite,approacheszeroaspointsmoveawayfromthecenter ,andthedecayingfactor canbecontrolledviathekernelbandwidth. Inpractice,thejointPDFisunknownandonlyanitenumbero fdatapoints f( x i y i )gN i =1isavailable,leadingtothesampleestimatorofcorrentrop yas^ v N ,( X Y ) = 1 N NXi =1 G( x iy i ).(3–6) Severalimportantpropertiesofcorrentropythatbecomeve ryusefulinourworkare: Forsymmetrickernels,correntropyis symmetric .Thus,theGaussiankernel providesasymmetricsimilaritymeasure. Correntropyis positive and bounded .ItreachesitsmaximumifandonlyifX = Y.Thisisimportantintheoptimizationprocess. 39

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FortheGaussiankernel,correntropyisaweighedsumofallt he evenmoments duetothenonlinearityofthekernel[ 59 ].Thisconrmsthestatementwemade previouslythatcorrentropyinvolveshigh-orderstatisti cs. Correntropyisasymmetric,positivedenitefunction,and therefore,it denesa reproducingkernelHilbertspace(RKHS) withcorrentropyasitsreproducing kernel[ 63 64 ]. Formoredetailsontheseandotherpropertiesofcorrentrop yreferto[ 48 ]. Thecross-correntropyestimatemeasurestheprobabilityo fhowsimilarthetwo randomvariablesareinaneighborhoodalongthelinex = ycontrolledbythekernel size.Fig. 3-1 (B)providesaplotofthecross-correntropyforaGaussiank ernel.The gureshowsthejointpdfandcross-correntropyofvariable sXandY.Itshowsthat correntropy,justlikeMSE,canbeusedasasimilaritymeasu reinthejointspace, butdifferentfromMSE,thecostemphasizesthebehavioralo ngthex = ylinewhile attenuatingcontributionsawayfromit,wheretheattenuat ioneffectdependsonthe kernelshapeandparameter.Wecanusecross-correntropyas anewcostfunctionfor shapematchingasmax NXi =1 G( y if ( x i ) ) R ( f ),(3–7) Notethatwehadtochangethesignoftheregularizationterm sincenowweare maximizingthecostfunctioninsteadofminimizingit. AMSE BCorrentropy Figure3-1.CostFunctionsinthejointspace(A)MSE(B)Corr entropy 40

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KernelBandwidthThecorrentropymeasurerequiresafreeparameter,kernelb andwidth,which isessentialintheoptimizationprocess.Thekernelsizeco ntrolsthe`observation window'inwhichsimilarityisassessed,andasaresult,itc ontrolstheshapeofthe performancesurface.Fig. 3-2 providesthreeexamplesoftheperformancesurface undervariousdegreesofrotationandscalingforthreediff erentkernelsizes.Forsmall kernelsizes,thereexistmanylocalmaximaintheperforman cesurface,whichwillcause thealgorithmtoconvergeatsuboptimalpointsandrelyheav ilyontheinitialconditions. Asthekernelsizeisincreased,theperformancesurfacebec omessmootherandmost ofthelocalmaximavanish.However,forverylargekernelsi zes,correntropybecomes equivalenttoMSEanditwillfacethesamedrawbacksthatwer ediscussedabove. Basedontheseobservations,wedeterminedthatthebestapp roachistoadaptthe kernelsizeduringtheoptimization.Basically,westartth ealgorithmwithalargekernel sizeandslowlydecreaseitsvaluethroughiterations,what isknownasthesimulated annealingprocess.Thisresultsinaperformancesurfaceth atchangesthroughiterations butprovidestwokeyadvantages:1)thealgorithmwillnotge tstuckinlocalmaxima,and 2)theconvergenceratewillbefaster(asthegradientofthe surfacewillincreasewith lowerkernelsizes). rotation angleradius -60 -40 -20 0 20 40 60 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 A = 0.01 rotation angleradius -60 -40 -20 0 20 40 60 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 B = 0.05 rotation angleradius -60 -40 -20 0 20 40 60 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 C = 0.10Figure3-2.Performancesurfacefordifferentkernelsizes :(A) = 0.01,(B) = 0.05, and(C) = 0.1041

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3.3Non-RigidTransformationandRegularization Thelastterminequation( 3–7 ),R ( f ),istheregularizationterm.Mostcommonly regularizationtermsareexpressedas kP fk2wherePisalinearoperator.Toregularize thenon-rigidtransformation,weconsiderRadialBasisFun ctions(RBF).ARBFis denedas:f ( x ) = NXi =1 w i(kxx ik),(3–8) wherew iarethewarpingcoefcients,and representtheradialbasisfunction.Forour work,weconsidertwodifferentRBFs:Thin-PlateSplines(T PS)andGuassians. 3.3.1Thin-PlateSpline ThinPlateSplinesaredenedas: In2D, i =kxx ik2 logkxx ik In3D, i =kxx ik In4Dorhigher,itdoesnotexist[ 65 ]. Thefunction isasolutiontothesquaredLaplacianequation 2 / 0,0[ 66 ],which meansthatitisproportionaltothe“generalizedfunction” 0,0meaningzeroeverywhere exceptattheoriginbutalsohavinganintegralequaltoone. TPStransformationresemblesthedeformationofthinmetal plate,whichallowsto approximatemanytypesofnaturaldeformations.Themainad vantageofTPSisthatit allowsforanexplicitdecompositionintolinearandnon-li nearparts[ 57 ].However,TPS providesglobalsupport,thatiseachcontrolpointhasaglo balinuenceontheoverall transformation,whichdoesnotallowtomodelcomplexlocal izedtransformations.For example,asmallperturbationononeofthecontrolpointsal waysaffectsthecoefcients correspondingtoalltheothercontrolpoints. Theregularizationterm,R ( f ),forthethin-platesplinesisthendenedas[ 66 67 ]:R ( f ) = 2X 1 ,d =1Z Rd @2 f @x 1 ...@x d2 dYi =1 dx i .(3–9) 42

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TPSpenalizesthesquaredsecondorderderivative,anditsn ullspaceincludesthe afnetransformations.Thecorrentropycostfunctionusin gTPSthenbecomes: J( f ) = NXi =1 G( y if ( x i ) ) Z Rd 2X 1 ,d =1 @2 f @x 1 ...@x d2 d x .(3–10) 3.3.2GaussianRadialBasisFunction ForGaussianRBFs,thefunction usesaGaussiankernel,whichduetoitskernel bandwidthlocalizesitssupport.Thisbecomesanadvantage ofGaussianRBFoverTPS aswecancontrolthelocalityofthespatialsmoothnessbych angingthevalueofthe kernelsize.Inaddition,GaussianRBFgeneralizestoanydi mensions,whereasTPSis denedonlyfortwoorthreedimensions.Thecorrentropycos tusingGaussianRBFis denedas: J( f ) = NXi =1 G( y if ( x i ) ) Z Rd1 Xm =0 c mkD m v ( x )k) 2 d x .(3–11) wherec mareparametersandDisaderivativeoperatorsuchthatD 2 l =r2 l v(scalar operator)andD 2 l +1 =r(r2 l v )(vectoroperator)[ 68 ].Thespectralanalogisoftheform: J( f ) = NXi =1 G( y if ( x i ) ) Z RDj~ f ( s )j2 ~ G ( s ) d s(3–12) whereGisapositivedenitefunction[ 69 70 ].TheGaussianfunctionwasselectedforGbecauseofitspropertiesofbeingsymmetricandpositivede nite,whichsimplify thesecalculations.Toselecta`smooth'function,weneedt osimplylookatthe frequencydomainderivation.Asmoothfunctionisonethath aslessenergyathigh frequencies.ThespectraldenitionoftheGaussianregula rizationtermcomputesthe powerateachfrequencybytakingtheL 2normandthenpassesitthroughahigh-pass lter1 ~ Gwhichattenuatesalllowfrequencies.Decreasingthevalue oftheregularization parameterR ( f )isthenequivalenttoreducingtheenergyathighfrequencie s.Notethat~ G ( s )isaGaussiandenedas~ G ( s ) = exp(ksk2 )where isaxedpositiveparameter. 43

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Parameter isdifferentfromthekernelsizeneededtocomputecorrentr opyandshould notbeconfused.Thevalueof determinesthelevelof`smoothness'. Tocontrolthesmoothness,thespatialregularizerspenali zefunctionderivatives whereasthespectralregularizerspenalizehighfrequenci es.Despitethechoiceof functional ,thesolutionstothecorrentropycostfunctionhavethesam eform: J( f ) = NXi =1 G( y if ( x i ) ) tr( W T W ).(3–13) 3.4PointMatchingUsingCorrentropy Again,supposewehavetwopointsetsX =fx igN i =1andY =fy igN i =1,wherex i y i2 Rd.Toalignpoint-setXtopoint-setY,point-setXhastoundergosome transformationfwhichcomprisesthetwotransformationmatrices:AandWasf ( x i A W ) = Ax i + W ( x i )(3–14) where A isa( d + 1)( d + 1)matrixrepresentingtheafnetransformation,andWis a( d + 1)Nwarpingcoefcientmatrixrepresentingthenon-rigidtran sformation,and ( x i )isaradialbasisfunctionofsizeN1.Tocontrolthewarpingeffect,wehaveto alsoaddaregularizationterm.Wecontrolthesmoothnessby constrainingthenonrigid transformationwithrespecttotheafnetransformation.T hecostfunctionbecomes J= v ( Y f ( x i A W )) kf ( x i A W )f ( x i A )k2 ,(3–15) ormorespecically J( A W ) = NXi =1 exp ky iAx iW( x i )k2 22 kW( x i )k2 .(3–16) Toprovideanintuitiveviewofthecostfunction,considert hemodelspaceforthe transformationfunctionfasanRKHS H (correntropyproperty4).Thefunctionfand themodelspace H canbedecomposedintotwofunctionsf = f 0 + f 1andsubspaces 44

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H=H0H1,wheref 0consistsoftheafnetransformationfunctionsand H0istheir space,andf 1consistsofthenon-rigidtransformationsand H1istheircorresponding subspace.Tocontrolthe`smoothness'ofthefunctionf,weneedtopenalizethe non-rigidpartofthefunction,f 1.So,theregularization(penalty)termcanbeconsidered asapenalizationoftheprojectionofthefunctionfontothe H1subspace,whichwe expressas kP 1 fk2whereP 1denotestheorthogonalprojectionoperatoronto H1.Thus, in( 3–16 ), kP 1 fk2 =kW( x i )k2,and isthesmoothingparameterwhichcontrolsthe balancebetweenthegoodness-of-tanddeparturefrom H0. SolvingforAandWBeforewesolve( 3–16 )forAandW,let'srstsimplifythenotation.ConsiderXandYasN( d + 1)matriceswhereeachrowrepresentsapointx iandy i.ConsiderKasaNNmatrixofradialbasisfunctionswhereeachrowrepresents ( x i ),thenthe warpingcoefcientmatrixWneedstoalsobetransposedtoaN( d + 1)matrix.The costfunction( 3–16 )thenbecomes J( A W ) = G ( YXAKW ) +trace ( W T KW )(3–17) TakingthederivativewithrespecttoAandWresultsinthefollowingequations:X T D Y ( G ) XA + X T D Y ( G ) KW = X T G T Y K D Y ( G ) XA + ( K D Y ( G ) +I ) KW = KG T Y ,(3–18) whereGistheG ( YXAKW )matrix,D Y ( G ) = diag ( 1 T YG ),andIisanidentitymatrix ofsizeNN.Wesimplifythissetofequationstothefollowing:D Y ( G ) XA + ( K D Y ( G ) +I ) W = G T Y XW = 0 .(3–19) 45

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Substituting( K D Y ( G ) +I )withnotationMforsimplication,wehavethefollowing solutionsforAandW:A = ( X T M1 D Y ( G ) X )1 X T M1 G T Y W = M1nID Y ( G ) XX T M1 D Y ( G ) X 1 X T M1oG T Y .(3–20) Tokeeptheafneandnon-rigidspacesseparatedsothatthep enalizationabovedoes notinterferewiththeafnecomponentwedecomposethematr ixXusingtheQR decompositionasX = [ Q 1 Q 2 ]264R 0375,(3–21) whereQ 1,Q 2,andRareN( d + 1),NN( d + 1),and( d + 1)( d + 1)matrices.Q = [ Q 1 Q 2 ]isanorthonormalmatrix,andRisanuppertriangularand invertiblematrix.TheQ 1andQ 2matricesallowustoconsidertheafneandnon-rigid transformationseparately.Forexample,consideringthes econdequationin( 3–19 ),we haveR T Q T 1 W = 0,whichmeansthatQ 1operatesinthenullspace H0,andsinceQ 1andQ 2areorthonormalmatrices,Q 2operatesin H1. SinceXW = 0,wehaveQ T 1 W = 0andQ 2 Q T 2 W = W.Multiplyingthe rstequationin( 3–19 )byQ T 2 D Y ( G )1andusingthefactthatQ T 2 X = 0,wehaveQ T 2 D Y ( G )1 MQ 2 Q T 2 W = Q T 2 D Y ( G )1 G T Y,whichresultsin:W = Q 2 ( Q T 2 D Y ( G )1 MQ 2 )1 Q T 2 D Y ( G )1 G T Y .(3–22) Multiplyingtherstequationin( 3–19 )byQ T 1 D Y ( G )1,wehaveRA = Q T 1 D Y ( G )1 ( YMW ),whichresultsin:A = R1 Q T 1 D Y ( G )1 ( G T YMW ).(3–23) parameter Thesolutionsfor A and W bothdependontheparameter whichcontrolsthe smoothnessoffunctionf.If islarge,thenon-rigidtransformationislimited.If 46

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issmall,thenon-rigidtransformationistooexiblewhich couldmakethealgorithm unstable.Tocontrolthelevelofrigidityandthusthestabi lityofthealgorithm,similar tothekernelsizeabove,weslowlyannealthe parameter.Ourgoal,andtheway wedescribethetransformations,isthatwewouldliketheal gorithmtorstmatchthe twopointsetsusingafnetransformation,thenwhenthesol utionhasreachedits optimumintheafnespacewewouldlikethealgorithmtomatc hthetwopointsets usingthenon-rigidtransformation.Bykeepingthevalueof parameterlargeduring therstiterationsofthealgorithm,werestrictthenon-ri gidtransformationandallow thealgorithmtoperformonlyglobal,afnetransformation s.Then,asthethevalueof parameterisgraduallydecreased,weslowlyintroducenonrigidtransformations. 3.5Analysis:DealingWithNoise/Outliers Correntropycomparestworandomvariablesintheneighborh oodofthejointspace controlledbythekernelbandwidth[ 62 ].Controllingthekernelbandwidthallowsusto adjustthe` observationwindow 'inwhichsimilarityisassessedresultinginaneffective mechanismtoeliminatethedetrimentaleffectofoutliers; thus,makingthistechnique robustagainstimpulsivenoise,whichmayoccurintheformo focclusionsorother imperfectionsontheobjectcontours.Below,weshowsevera lexamplesofnoiseand evaluatethebehaviorofbothMSEandcorrentropy.Notethat theonlytransformation appliedtotheobjectisa 45 orotation;asaresult,welimitthetransformationsthatare appliedtotheobjecttoonlyafnetransformation. GaussianNoiseFirst,wecomparetheperformanceofcorrentropyagainstMS EonwhiteGaussian noise.Anobjectisrotated 45 oandGaussiannoiseisaddedtoitsdatapointsas showninFig. 3-3 (A).Inthistypeofnoise,bothMSEandcorrentropyperformi nsimilar fashion.Theyndabalancebetweennoiselevelandapproach ingthetruelocations. Asmentionedabove,thealgorithmislimitedtoonlyafnetr ansformation.Ifnon-rigid 47

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transformationwereapplied,thetemplatedatapointswoul dhavebeenadaptedto accountforthelocalchangesinpointlocationsduetotheGa ussiannoise. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Ainitialpositions -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 BMSEalignment -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Ccorrentropyalignment Figure3-3.ComparisonofcorrentropyagainstMSEonwhiteG aussiannoise. ImpulsiveNoiseNext,wecomparethetwomethodsonimpulsivenoise.MSE,Fig 3-4 (B),attempts toaccommodateallthesamples.Asaresult,wenoticeasligh tcounterclockwise rotation,whichisthepullingforceappliedbytheimpulses amplesonthetop-rightand bottom-leftcornersofthegure.Correntropy,ontheother hand,worksonawindow size,whichactsasazoominglensclosetotheobjectcontour ,andignorespointsthat aredistant(e.g.outliers).ThisisclearlyshowninFig. 3-4 (C)wherebesidestheimpulse datapointstherestareverywellmatchedagainsttheorigin altemplate. Outliers/OcclusionsForthelastscenario,wewouldliketotesttheperformanceo fthetwomethods onoutliers/occlusionsintheobject.So,partoftheleftwi ngofthebatisremovedfrom theoriginaltemplate,andpartoftherightwingisremovedf romtherotatedcontour. MSE,again,attemptstoaccommodatethemissingpartsfromt hetwocontours,and asaresult,thenalalignmentofthetemplateagainsttheob jectisoff.Inthiscase,the templateisrotatedcounterclockwiseandshiftedslightly tothelefttoaccountforthe additionalsamplepointspresentintheobjectandalsotoma tchtheextrapointspresent 48

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-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Ainitialpositions -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 BMSEalignment -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Ccorrentropyalignment Figure3-4.ComparisonofcorrentropyagainstMSEonimpuls ivenoise. inthetemplatewhicharemissingintheobject(rightwing). Correntropy,againworking onawindowsize,ignoresthemissing/extrapartsoftheobje cts,andfocusesonthe partsthatareincommon. -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Ainitialpositions -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 BMSEalignment -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Ccorrentropyalignment Figure3-5.ComparisonofcorrentropyagainstMSEonoutlie rs/occlusions. KernelBandwidthEffectCorrentropyusesthekernelbandwidthtodeterminethewind owsize.Ifthe bandwidthislarge,correntropybehavesasMSE.Asthekerne lbandwidthdecreases, the`windowsize'decreases;thus,ignoringdatapointstha tarefarawayandpotential outliers.Fig. 3-6 showsanotherexampleofanobjectcorruptedbyimpulsiveno ise andthetemplatealignmentusing(A)MSEandcorrentropywit hthreedifferentkernel bandwidths:(B)large,(C)medium,and(D)small.Thealignm entusingcorrentropywith 49

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alargekernelsizelookssimilartotheMSEalignment.Asthe kernelsizeisreduced, theeffectoftheimpulsivenoiseisreducedandthecorrentr opyalignmentimproves.At thebeginningofthealignmentprocess,itisimportanttoha veawideviewing`window size'sothatthetemplatepointsareintheviewingrangeoft heobjectpointstoprovide anaccurateoverallinitialalignment.Asthetemplatecont ourispositionedcloserto theobjectcontour,itisimportanttonarrowtheviewing`wi ndowsize'sothatoutliers andimpulsivenoisedonotaffectthenalalignment.Basedo ntheseobservations,we decidetoannealthekernelbandwidthduringthealignmentp rocess. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 AMSEalignment -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Bcorrentropyalignment-bandwidth= large -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Ccorrentropyalignment-bandwidth= medium -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Dcorrentropyalignment-bandwidth= small Figure3-6.ComparisonofMSEagainstcorrentropyofdiffer entkernelbandwidths. 3.6TPSvsGaussianRBF Tounderstandtheeffectoftheradialbasisfunctionsonthe non-rigidtransformation, wecompareTPSandGaussianRBFonseveralshapeswithvariou sdegreesof non-rigidtransformation.Fig. 3-7 showsfourexperimentswheredifferentshapesare alignedusingbothmethods.BothTPSandGaussianRBFperfor mverywellonall experiments.Visually,thealignmentslookperfectinallc ases.Thedifference,which isshowninthetitleofeachsub-gure(MSEofthenalalignm ent),isverysmalland itshowsthatGaussianRBFperformsbetterthanTPS.Thisisd uetoTPShavinga globaleffectwhereasGaussianRBF'seffectislocalizedba sedonitskernelbandwidth 50

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.WhilethisisanadvantageoverTPS,itmayalsoprovideprob lemsbecausethereisa freeparameterthatneedstobecontrolled. 0 0.2 0.4 0.6 0.8 1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Aoriginalshapes 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 Boriginalshapes 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Coriginalshapes 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Doriginalshapes -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Error = 0.001368 ETPSalignment 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Error = 0.00076751 FTPSalignment -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.4 0.5 0.6 0.7 0.8 0.9 Error = 0.00013316 GTPSalignment 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Error = 8.2659e-005 HTPSalignment -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Error = 0.00049775 IGaussianalignment 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Error = 0.00025938 JGaussianalignment -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.4 0.5 0.6 0.7 0.8 0.9 Error = 2.0349e-005 KGaussianalignment 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Error = 2.3636e-005 LGaussianalignment Figure3-7.ComparisonofTPSandGaussianRBFonvariousnon -rigidtransformations. Fig. 3-8 demonstratestheproblemthatmightarisefromthekernelsi zeinthe GaussianRBFs.TheChinesecharactersabovearealignedusi ngthreevaluesof :0.1, 0.5,and1.0.Wecanseethatasthevalueof increases,thealignmentsuffers. Thisisduetotheincreaseintheeffectthatonecontrolpoin thasovertheothers.Inthe 51

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caseof = 1.0thecontrolisglobalanditisshownonthepooralignmentbet weenthe twoinstancesofthecharacter.Controllingthevalueof isaproblemspecicmatter thatneedstobeaddressedeverytime.Heuristicscouldbeus edtoassignavalueto byanalyzingthedistributionofthecontrolpoints. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Error = 2.3636e-005 A = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Error = 0.0061698 B = 0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Error = 0.0378 C = 1.0Figure3-8.Non-rigidtransformationusingGaussianRBFof varioussizes:(A)small,(B) medium,and(C)large 3.7PointCorrespondence Theiterativeclosestpointalgorithm[ 56 ]utilizesthenearest-neighborrelationship toassignabinarycorrespondencebetweenthetwopointsets .Thisestimateofthe correspondenceisthenusedtoupdatethetransformationfu nction,whichisthenused tore-estimatethecorrespondencebetweenthetwosets.Thi sisasimpleandquick approach,whichalwaysconvergestoalocalminimum.Theref ore,withanadequateset ofinitialposes,youcanalwaysndaglobalmatchforrigidt ransformations. Toimprovethecorrespondenceproblem,Belongieetal.[ 27 ]andJainetal.[ 55 ] usedshapedescriptorstoprovide context oftherestoftheshapewithrespecttoa givenpointintheshape.Findingcorrespondencesbetweent hepointsoftwoshapesis thenequivalenttomatchingpointswithsimilarshapeconte xt.Here,weintroduceanew descriptor, surprise ,thatcouldplayasimilarroleinshapematching. Aquantitativedenitionoftherelevanceasamplepointhas ontheoverallsetis requiredtosubjectivelymeasuretheinformationavailabl einthesamplepointbased 52

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onthecurrentknowledgeexpressedbytherestofthepoints. Surprisequantieshow muchinformationanewsamplecontainsrelativetoalearnin gsystemmodel[ 71 ].To understandthismeasure,weneedtoutilizeafewbasicdeni tionsfrominformation theorycoveredinChapter 2 .Ininformationtheory,informationmeasurestheuncertai nty orprobabilityofoccurrenceofanoutcome[ 54 ].Theinformationcontentofanoutcomex,whoseprobabilityisp ( x ),isdenedasI ( x ) = log 1 p ( x ) .Forarandomvariable(r.v.)X,theaverageinformationcontentisdenedasH ( X ) =Px p ( x ) log 1 p ( x )(Shannon's entropy).Thisclassicalinformationmeasureisbasedsole lyuponobjectiveprobabilities andcannotrepresentasystem'sindividualknowledgeordis criminatebetweensamples whichareimportanttoimprovesystem'sknowledgefromthos ewhichareirrelevant.To accountforthesubjectiveinformationwhichasamplexbringstoasystemwithitsown subjectivebelief,weneedtoconsideritssubjectiveproba bilityq ( x )[ 72 ].Thesubjective informationcontentofoutcomextothesystemwithsubjectiveprobabilityq ( x )isthen denedasI s ( x ) = log 1 q ( x ).Theaveragesubjectiveinformation,H s ( X ),isgivenbythe expectationvalueofthesubjectiveinformation,I s ( x ),takenwiththetrueprobabilitiesp ( x )asH s ( X ) =Px p ( x ) I s ( x ) =Px p ( x ) log 1 q ( x ).SinceH s ( X )measurestheuncertainty ofasystemthatdoesnotknowthecorrectprobabilities,its houldbelargerthanthe uncertaintyofanidealobserverthatknowsthetrueprobabi lities.Thus,wecanstate thatH s ( X )H ( X ),withequalitybeingtrueonlywhenthesystemhasfullknowl edgeof itsenvironment,thesubjectiveandobjectiveprobabiliti escoincide,q ( x ) = p ( x )[ 71 ].The differencebetweenH sandHisthendenedas:H m ( X ) = H s ( X )H ( X ) =Xx p ( x ) log p ( x ) q ( x ) ,(3–24) whichrepresentstheinformationthatthesystemismissing fromthedataandthatcan belearnedbyadjustingthesubjectiveprobabilitiesclose rtothetrueprobabilities.This iscalledthe informationgain andisthebasisforsurpriseasitmeasuresthesystem's 53

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ignorance[ 73 ].Thiscanbeunderstoodfromthefactthat( 3–24 )isnonnegativeand vanishesifandonlyifp ( x )andq ( x )coincide(Kullback-Leiblerdivergence)[ 74 ]. Thesurprisefactorismeasuredusingtheinformationgain( 3–24 )as:S ( x n ,M) = H m ( x n ) = KL ( P ( Mjx n ), P ( M ) ) .(3–25) Inessence,surprisemeasuresthedifferencebetweenthepr iorandtheposterior distributionsofthemodelbasedonthenewsamplepointx n.Iftheposterioristhesame astheprior,thedatapointcarriesnonewinformation,thus leavingknowledgeabout themodelunaffected.However,iftheposteriorsignicant lydiffersfromtheprior,the observationcarriesanelementofsurprisetothemodelindi catingitsimportance. 3.7.1SurpriseEffectonAfneandNon-RigidTransformatio ns Surprisebehaviorofcorrespondingpointsinanobjectdoes notchangewith respecttomosttransformations.Fig. 3-9 showsthreedifferentafneandonenon-rigid transformationswhereanobjectis(A)translated,(B)rota ted,(C)scaled,and(D) non-rigidlytransformed.Thecorrespondingsurprisegur esshownodifferenceat allforthetranslationandrotationcases,atranslationan dscalingforthescalecase, andslightdifferencesinthenon-rigidcase.Sincesurpris efactorforeachdatapoint iscomputedwithrespecttothepositionoftheotherpointsw ithrespecttothepoint ofinterest,translationandrotationdonotchangethat.Sc alingchangestherelative positiondistanceamongthepointsandthusthetranslation andscaleinthesurprise factor.ThisisalsoaffectedbythePDFestimationandsince weuseParzenwindow toestimateit,thekernelbandwidthhasaneffectonthesurp risefactor.Ifweupdate thekernelbandwidthtoaccountforthescalingoftheobject ,thenbothsurprisegraphs wouldmatch.Thenon-rigidtransformationdoeschangethes urprisevaluesofthe individualpointssincetheirrelativepositionischanged ,howevertheoverallshapeofthe surprisebeforeandafterthenon-rigidtransformationrem ainsverysimilar. 54

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-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Atranslation-shapes -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Brotation-shapes -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Cscale-shapes -1.5 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Dnon-rigid-shapes 0 10 20 30 40 50 60 70 0 1 x 10 -4 data pointssurprise factor Etranslation-surprise 0 10 20 30 40 50 60 70 0 1 x 10 -4 data pointssurprise factor Frotation-surprise 0 10 20 30 40 50 60 70 0 1 x 10 -4 data pointssurprise factor Gscale-surprise 0 10 20 30 40 50 60 70 80 90 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 x 10 -4 data pointssurprise factor Hnon-rigid-surprise Figure3-9.Effectofvariousafneandnonrigidtransforma tionsonsurprisefactor. 3.7.2SurpriseEffectonNoise Noise,ontheotherhand,doesaffectsurprisebehavior,but notdrastically.Fig. 3-10 showstwocaseswhereanobjectiscorruptedby(A)whiteGaus siannoise,and (C)impulsivenoise.Thecorrespondingsurpriseguressho wthedifferenceofsurprise valuesbeforeandaftertheadditionofnoise.Inthecaseoft heGaussiannoise(B), whilenoisehaschangedthesurprisevalues,theoverallsha peofthesurprisegraph remainsthesame.Inthecaseoftheimpulsivenoise(D),thes urprisevalueforthenoisy pointsisverydifferentfromtheoriginal,asexpected,how evertherestofthepointshave surprisevaluesveryclosetooriginals.Overall,theshape softhesurprisegraphsbefore andafternoisearesimilarforbothtypesofnoise. 55

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-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 AGaussian-shapes 0 10 20 30 40 50 60 70 0 1 x 10 -4 data pointssurprise factor BGaussian-surprise -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Cimpulsive-shapes 0 10 20 30 40 50 60 70 0 0.5 1 1.5 2 2.5 3 3.5 x 10 -4 data pointssurprise factor Dimpulsive-surprise Figure3-10.Changesinsurprisevalueduetonoiseaddedtot hedatapoints. 3.7.3PointCorrespondenceUsingSurprise Basedontheresultsabove,surpriseisagoodindicatorofth eshapeoftheobject. Here,wedemonstrateitscapabilitytocorrectlydetermine thecorrespondencebetween twopointsets.Intherstexample,werotateanobjectandad dwhiteGaussiannoise toitsdatapointsasshowninFig. 3-11 (A).Inaddition,wereorderthedatapointsso thattherst20datapointsareplacedattheend.Comparingthesurprisegra phsbefore andafterthetransformation/corruption,Fig. 3-11 (B),weseethatthetwographsare similarexceptofacircularshiftof20points.Todeterminethiscircularshift,weapply circularconvolutiontothetwographs.Themaximumpeakisa t20asshowninFig. 3-11 (C).Shiftingthedatapointsinthesecondcontourbythisam ountresultsinthe correctcorrespondenceshowninFig. 3-11 (D). InFig. 3-12 ,wedemonstratethesamethingbutonanobjectwhichisalso translatedandhasimpulsivenoise.Besidesthevisualdiff erencesbetweenthetwo surprisegraphs,Fig. 3-12 (B),thecircularconvolutionagainprovidesthecorrectsh ift index,20. InFig. 3-13 andFig. 3-14 ,weprovidetwomoreexampleswithnon-rigiddeformations. InFig. 3-13 ,thetemplatehasgonethroughnon-rigiddeformationandis corruptedby Gaussiannoise.Notethatthecorruptedpointsetlooksvery differentfromthetemplate, infactitisverydifculttorecognizethecharacter.Never theless,surpriseiscapableof 56

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-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Ainitialcorrespondence 0 10 20 30 40 50 60 70 0 1 x 10 -4 data pointssurprise factor Bsurprisesignatures 0 10 20 30 40 50 60 70 3.5 4 4.5 5 5.5 6 6.5 x 10-7 20 Cconvolution -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Dnalcorrespondence Figure3-11.Example1ofutilizingsurprisetodeterminesh apecorrespondence.The targetisarotatedversionofthetemplatecorruptedbyGaus siannoise. -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Ainitialcorrespondence 0 10 20 30 40 50 60 70 0 1 x 10 -4 data pointssurprise factor Bsurprisesignatures 0 10 20 30 40 50 60 70 4 4.5 5 5.5 6 6.5 x 10-7 20 Cconvolution -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Dnalcorrespondence Figure3-12.Example2ofutilizingsurprisetodeterminesh apecorrespondence.The targetisatranslatedandrotatedversionofthetemplateco rruptedby impulsivenoise. identifyingthesimilaritiesbetweenthetwosetsandsolvi ngfortheshiftindexasshown inFig. 3-13 (C)andthecorrectcorrespondenceinFig. 3-13 (D). InFig. 3-14 ,thetemplatehasgonethroughnon-rigiddeformation,rota tion,and scaling.Notehowdifferenttheirrespectivesurprisesign aturesvisuallylook.However, whencompared,eventhoughthetransformationshavechange dthetemplateso drastically,thesurpriseiscapableofidentifyingthecor respondecnebetweenthetwoset asshowninFig. 3-14 (D). 57

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-0.2 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 0.4 0.6 Ainitialcorrespondence 0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 x 10 -5 data pointssurprise factor Bsurprisesignatures 0 20 40 60 80 100 120 5.5 6 6.5 7 7.5 8 8.5 x 10 -9 Cconvolution -0.2 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 0.4 0.6 Dnalcorrespondence Figure3-13.Example3ofutilizingsurprisetodeterminesh apecorrespondence.The targetisanon-rigidlydeformedversionofthetemplatecor ruptedby Gaussiannoise. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ainitialcorrespondence 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -5 data pointssurprise factor Bsurprisesignatures 0 20 40 60 80 100 120 4 4.5 5 5.5 6 6.5 7 7.5 x 10 -10 Cconvolution 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Dnalcorrespondence Figure3-14.Example4ofutilizingsurprisetodeterminesh apecorrespondence.The targetisadeformedversionofthetemplate,wherethedefor mationisa combinationofnon-rigidtransformation+rotation+scali ng. 3.8BinaryvsSoftAssignment Ifthedatapointsarenotextractedinorderaroundtheconto ur,asshowninthe aboveexamples,thesurprisefactorcannotbeusedtondthe correspondence.We ranseveralexperimentswherethedataorderwaspermutedan dthendatapoints withsimilarsurprisefactorinbothshapecontoursweremat chedtogether,butthe correspondencewasincorrect.Fig. 3-15 providesandexampleofdatapoints(A) ordered,wherethereisa20pointshift,and(B)unordered,w heresamplesarerandomly selected.Surpriseisrendereduselessinsuchcasesbecaus ethestructureofthedata setsisdestroyedbythepermutation. 58

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Aordereddata Bunordereddata Figure3-15.Correspondenceusingsurprisewhendatapoint sare(A)orderedand (B)randomlypermuted. Insuchcases,our`context'featureisnotrobust.Toaccoun tfordatasetswith unorderedpoints,weturnbacktotheiterativeprocesswher eweestimatethe correspondencebetweenthetwosets,thenbasedonthatesti mationweupdate thetransformationparameters,andcontinuethisprocessu ntilthedatasetsare alignedorthealgorithmhasreachedalocalmaximum.Therea retwowaystoassign thecorrespondencebetweenthetwosets:1)binarycorrespo ndence,and2)soft assignment.Inbinarycorrespondence,eachpointinthers tshapeisassignedto apointinthesecondshape.Correspondenceisone-to-oneex ceptforoutliers.In softassignment,thebinarycorrespondenceisrelaxed,and apointintherstshape ispartiallymatchedwithmanyofthepointsinthesecondsha pe.Thisfuzzinessin assignmentmakestheresultingcostfunctionbehavebetter byimprovinggraduallyand continuouslywithoutjumpingintheperformancesurfaceas wouldbethecaseinbinary correspondence[ 75 ].Thisnotionoffuzzycorrespondencewasrstusedin[ 76 ]where theyaddanentropytermtotheoriginalbinaryassignmentpr ocess.Theycontrolthe fuzzinessbyintroducinga`temperature'parametertothei rentropytermandannealing it,whereathighertemperatures,theentropytermprovides softercorrespondences betweenthetwoobjectsallowingto“convexify”theobjecti vefunction. 59

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Inouralgorithm,correntropyindirectlyprovidesthister mrequiredtodetermine thefuzzinessofcorrespondence.Inourcase,thekernelban dwidthactsasthe temperatureparameter,whereatinitialstepsthebandwidt hislargeandthusproviding softassignmentamongalldatapoints.Asthebandwidthisre duced,theseassignments moveclosertobinaryassignments.Thisallowsforasmootht ransitioninthealignment/ correspondencespace.Theonlydifferenceisthatnowwenee dtoconsiderthefulljoint spaceofthetwovariables.Whilewithcorrentropyweworkin thejointspacealongthex = yline,inthecaseofsoftassignment,weneedtoutilizethefu lljointspaceand returntoinformationpotential(IP),IP ( X Y ) = 1 N 2 NXi =1 NXj =1 ( x iy j ).(3–26) ThisalsoincreasesthecomputationcomplexityfromO ( n )toO ( n 2 ). IPitselfisnotsufcienttoperformsoftassignment.Anadd itionalstepisneeded tonormalizewithrespecttoX.InChapter 4 ,wewilllookatanothersimilaritymeasure, Cauchy-Schwarzdivergence,whichwillprovidemoreinform ationonthelimitationsof IPitself.Herewebypassthatlimitationbytakingcareofth enormalizationandshowan exampleofsoft-assignmentusingcorrentropy. Fig. 3-16 providesanexampleofthealgorithmusingsoftassignmentt hrough informationpotential.Inthiscase,theoriginalobjectis rotated 45 oandGaussian andimpulsivenoisearealsoadded.Fig. 3-16 (A)showstheinitialalignmentand correspondencebetweenthepointsofthetwoobjects.Themo stsignicantcorrespondencesareshownincyanlines.Asexpected,attheearlystage softheprocess,a particularpointinoneshapeissoftlyassignedtoallthepo intsontheothershape.To visuallydescribethekernelbandwidth,`windowsize',eff ectonthecorrespondence, thecirclesrepresenttheradiusofthekernelbandwidtharo undthetemplatedatapoints. Thecyanlinesandthecirclesdrawnintherestofthesubgur es(B),(C),and(D) provideavisualexplanationoftheannealingprocesswhere atearlystagesthe 60

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kernelbandwidthislargeindicatingalargeviewingwindow whichinturnentailsmany signicantcorrespondencesamongthepointsinthetwoshap es.Asthealgorithm goesthroughiterations,theviewingwindowdecreasesandt henumberofsignicant correspondencesalsodecreases.Thisisshowninsubgures (B),(C),and(D),where atrst,mostoftheoutliershavesignicantcorrespondenc eswithmanyofthetemplate datapoints,butastheiterationscontinuethenumberofcor respondencesdecreases, andattheend,thereisnocorrespondencebetweentheoutlie rsandthetemplate. -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 1.5 Ainitialalignment -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 Brststep -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Cmiddlepoint -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Dnalalignment Figure3-16.AnexampleofIPalignmentonarotateddatasetc orruptedbyboth Gaussianandimpulsivenoise.(A)theinitialalignmentandcorrespondencebetweenthepointsofthetwoobjects,(B)th ealignment aftertherststep,(C)thealignmentatamiddlepointinthe process,and (D)thealignmentclosetotheend.Thecirclesrepresentara diusofthe kernelbandwidtharoundthetemplatedatapoints,indicati ngtheirviewing windowandthroughthesubgurestheannealingprocess.The cyanlines showthemostsignicantcorrespondencesattheparticular stage. 61

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3.9Algorithm Here,wesummarizethestepsneededtocomputeshapematchin gusing correntropy: Algorithm1 CorrentropyMatchingAlgorithm 1.Initializeparameters:Correntropybandwidth, ,regularizationparameter, ,and GaussianRBFbandwidth, 2.Initializeparameters:Afnetransformation,A,andnon-rigidtransformation,W. 3.BeginAlignment. (a)Computecorrespondence(Surprise,orIP).(b)UpdatetransformationparametersAandW. (c)Decreaseparametervaluesfor: ,and 4.Repeatsteps(a-c)untilENDCriterionisreached(alignm entoriterations). 62

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CHAPTER4 PDFMATCHING-CAUCHY-SCHWARZDIVERGENCE InChapter 3 ,weinvestigatedhowtoimprovethepointmatchingalgorith mforboth rigidandnon-rigidregistrationoftwopointsets.Themain problemthatwehadwith oursolution,thecorrentropymatchlter,wasthatthecorr espondenceofthepoints onthetwosetshastoeitherbepredeterminedoramethodtoco mputeitisrequired. Inthischapter,wedescribeanotheralgorithmwherethepoi ntsetsarerepresented asprobabilitydensityfunctionsandtheproblemofregiste ringpointsetsistreatedas aligningthetwodistributions.UsingthePDFsinsteadofth epointsmitigatestheneedto establishacorrespondencebetweenthepointsonthetwoset s.Inaddition,itprovidesa robustwayofdealingwithoutliersandnoise,whichiscruci alinourapplications. ThealgorithmoperatesonthedistancebetweenthePDFsofth etwopointsetsto recoverthespatialtransformationfunctionneededtoregi sterthetwosets.Thedistance measureusedistheCauchy-Schwarzdivergencewhichisderi vedfromtheinequality withthesamename.Thealgorithmisrobusttonoiseandoutli ers,anditperformsvery wellonvaryingdegreesofafneandnon-rigidtransformati ons. 4.1Background TsinandKanade[ 77 ]werethersttoproposeamethodwhicheliminatedthe pointcorrespondencerequirement.Theymodeledthetwopoi ntsetsaskerneldensity functionsandevaluatedtheirsimilarityandtransformati onupdatesusingthekernel correlationofthetwodensityestimates. Glaunesetal.[ 78 ]matchthetwopointsetsbyrepresentingthemasweighted sumsofDiracdeltafunctions.TheyuseGaussianfunctionst o`soften'theDiracdelta functionsandusediffeomorphictransformationstominimi zethedistancebetweenthe twodistributions. JianandVemuri[ 79 ]extendedthisapproachbyrepresentingthedensitiesas Gaussianmixturemodels(GMM).Theyderiveaclosed-formex pressiontocompute 63

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theL 2distancebetweenthetwoGaussianmixturesandtheupdateme thodtoalignthe twopointsets.Theyusethethin-platesplinetoparameteri zethesmoothnon-linear transformation. Myronenkoetal.alsointroducedaprobabilisticmethodfor pointmatchingcalled coherentpointdrift(CPD)[ 70 ].AGMMisttooneofthepointsetsusingtheother dataset'spointsasinitialpositionfortheGaussiancentr oids.Inaddition,theydo notmakeanassumptionofthetransformationmodelasin[ 79 80 ]whereTPSwas considered.Instead,amotioncoherenceconstraintisimpo sedoverthevelocityeldof themotionoftheGaussiancentroids.Thesmoothnessontheu nderlyingtransformation isimposedbasedonthemotioncoherencetheory[ 81 ]. Inthischapter,weintroduceanewalgorithmwherethetwopo intsetsare representedasdensityfunctionsandthesimilaritybetwee nthemismeasuredusing aninformationtheoreticmeasure,theCauchy-Schwarzdive rgence[ 48 ].Thedivergence methodisexpressedintermsofinnerproductsofthetwoPDFs whichessentially estimatesthe crossinformationpotential ofthetwodensities.Thistermiscrucialin determiningthesimilaritybetweenthetwosetsbecauseitm easurestheinteraction oftheeldcreatedbyoneofthePDFsonthelocationsspecie dbytheother.The Cauchy-Schwarzinformationpotentialeldexertsinforma tionforcesonsamplesofthe secondpointsetforcingthemtomovetowardapaththatwillp rovidethemostsimilarity betweenthetwoPDFs. WeshowthattheCauchy-Schwarzdivergenceprovidesabette rsimilaritymeasure. Inaddition,weprovideclosedformsolutionsforthetransf ormationupdates.Thisis oneoftheadvantagesofourmethodcomparedtoothersimilar itymeasures,suchas Kullback-Leiblerdivergence,whicharedifculttoestima te.Tocomputethenon-rigid transformations,weutilizebothradialbasisfunctionsco veredinChapter 3 64

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4.2Cauchy-SchwarzDivergence Giventwodensityfunctionsfandg,theCauchy-Schwarzdivergence( D CS ) [ 48 ]measuresthesimilaritybetweentheminsimilarfashionto theKullback-Leibler divergence[ 74 ].ThemeasureisderivedfromtheCauchy-Schwarzinequalit y[ 82 ]: Zf ( x ) g ( x ) dx s Zf 2 ( x ) dxZg 2 ( x ) dx ,(4–1) where,forPDFs,theequalityholdsifandonlyiff ( x ) = Cg ( x )withC = 1.Tosimplify thecalculations,wetakethesquareof( 4–1 )andtheCauchy-Schwarzdivergenceoftwo PDFs[ 48 ]isdenedas D CS( fkg ) =log Zf ( x ) g ( x ) dx2 Zf 2 ( x ) dxZg 2 ( x ) dx .(4–2) D CS( fkg )isasymmetricmeasureand D CS( f g )0,wheretheequalityholdsifand onlyiff ( x ) = g ( x ).However,thetriangleinequalitypropertydoesnothold,s oitcannot beconsideredasametric. D CS( f g )canbebrokendowntoandrewrittenas: D CS( fkg ) =2 logZf ( x ) g ( x ) dx + logZf 2 ( x ) dx + logZg 2 ( x ) dx .(4–3) Theargumentoftherstterm, Rf ( x ) g ( x ) dx,estimatestheinteractionsonlocations withinthesupportoff ( x )whenexposedtothepotentialcreatedbyg ( x )(orviceversa). Thistermmeasuresthesimilarity(distance)betweenthetw oPDFs.Thetermitselfis Renyi'squadraticcrossentropy [ 48 ].Itcanalsobeinterpretedastheinformationgain fromobservinggwithrespecttothe“true”densityf.Theothertwoterms,logRf 2 ( x ) dxandlogRg 2 ( x ) dx,arethenegativeRenyi'squadraticentropiesoftherespec tivePDFs andareconsideredasnormalizingtermsthatactasregulari zers.(Recallthatthe informationpotential(IP)discussedinthesoft-assignme ntsectionofChapter 3 required normalizationwithrespecttoXtofunctionproperly.ThenormalizingtermlogRf 2 ( x ) dxhereprovidesthenormalizationneeded.)TheCauchy-Schwa rzdivergencecanthenbe 65

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interpretedas: D CS( fkg ) = 2H2 ( f ; g )H2 ( f )H2 ( g ).(4–4) 4.3MatchingAlgorithm SupposewehavetwopointsetsX = ( x 1 ..., x N ) TandY = ( y 1 ..., y M ) T,wherex i y j2 Rd.Thekernel(Parzen)estimateofthePDF[ 51 ]ofanarbitrarypointxusingan arbitrarykernelfunction ()isgivenby^ p ( x ) = 1 N NXi =1 xx i ,(4–5) where isthebandwidthparameter.TheGaussiankernel,G( x x i ) = 1 p 2exp kxx ik2 22 isconsideredasthekernelofchoiceforitsproperties:sym metric,positivedenite, approacheszeroaspointsmoveawayfromthecenter,andthed ecayingfactorcanbe controlledviathekernelbandwidth. SubstitutingtheGaussiankernelPDFestimatorintheCauch y-Schwarzdivergence ( 4–3 )andperformingsomestraightforwardmanipulations(thei ntegraloftheproduct oftwoGaussiansis exactlyevaluated asthevalueoftheGaussiancomputedatthe differenceoftheargumentsandwhosevarianceisthesumoft hevariancesofthetwo originalGaussianfunctions[ 48 ])resultsintheestimator: D CS( P ( Y )kP ( X )) =2log MXi =1 NXj =1 G ( y ix j )+log MXi =1 MXj =1 G ( y iy j )+log NXi =1 NXj =1 G ( x ix j ).(4–6) ToalignthetwomixturesP ( Y )andP ( X ),thedensityfunctionP ( X )needs toundergo afne and non-rigid transformationsinasimilarwayasdiscussedfor correntropyinChapter 3 .Thetransformationscanthenbeexpressedas:P ( X )A !PA( X )N R !PN R( X ) 66

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Torepresenttheafneandnon-rigidtransformations,wefo llowthesamestepsas withcorrentropy.Thetransformationfunctionfisdenedas:f ( x A W ) = Ax + W ( x ),(4–7) whereAisthe( d + 1)( d + 1)matrixrepresentingtheafnetransformation,Wisa( d +1)Nwarpingcoefcientmatrixrepresentingthenon-rigidtran sformation,and ( x )istheRBFwithsizeN1. Thesecondtermin( 4–6 ),logPM i =1PM j =1 G ( y iy j ),dependsonlyonthedesired pointsetanditsvalueneverchangesnordoesitaffectthetr ansformationfunctionf, thusweremoveitfromfutureconsiderationsofthecostfunc tion.Substitutingxwiththe transformedf ( x ),theupdatedcostfunctionbecomes: D CS( P ( Y )kPN R( X )) =2 log MXi =1 NXj =1 G ( y iAx jWk j ) + log NXi =1 NXj =1 G ( A ( x ix j ) + W ( k ik j )) + C .(4–8) Tocontrolthe`smoothness'ofthenon-rigidfunction,wene edtoagainincludea regularizationparameter.Thecostfunctionthenbecomes: J=D CS( P ( Y )kPN R( X )) + tr( W T KW ).(4–9) whereKistheNNsymmetricGrammatrixofinitialpointsetswithelementsk ij =kx 0 ix 0 jk2 logkx 0 ix 0 jk fortheTPS,andk ij = exp(1 2kx 0 ix 0 j k) 2fortheGaussianRBF. SolvingforAandWBeforewesolve( 4–9 )forAandWlet'ssimplifythenotation.ConsiderG ( Y X T )equivalenttoG ( y iAx jWk j ),8fy igM i =1 ,fx jgN j =1.Similarly,considerG ( X X T )equivalenttoG ( A ( x ix j ) + W ( k ik j )),8fx igN i =1 ,fx jgN i =1.Then,( 4–9 )canbewritten 67

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as J( A W ) =2 log1 T Y G ( Y X T ) 1 X+ log1 T X G ( X X T ) 1 X+trace ( W T KW ).(4–10) TakingthederivativewithrespecttoAandWresultsinthefollowingequations:X T ( DDG ) ( XA + KW ) = X T ~ G ( Y X T ) T Y K ( DDG ) ( XA + KW ) +KW = K ~ G ( Y X T ) T Y ,(4–11) whereDDG = D Yf~ G ( Y X T )gD Xf~ G ( X X T )g+ ~ G ( X X T ),~ G = G 1 TG1toaccountforthe derivativeoflog(),andD @fGg= diag (1 TG ).ThesolutionsforAandWare:A =X T M1 ( DDG ) X 1 X T M1 ~ G ( Y X T ) T Y W = M1nI( DDG ) XX T M1 ( DDG ) X 1 X T M1o~ G ( Y X T ) T Y(4–12) whereM = K ( DDG ) +I.UtilizingtheQRdecompositionofXwecanseparatethe computationofAandWas:W = Q 2 ( Q T 2 ( DDG )1 MQ 2 )1 Q T 2 ( DDG )1 ~ G ( Y X T ) T Y .(4–13)A = R1 Q T 1 ( DDG )1 ( ~ G ( Y X T ) T YMW ),(4–14) Thesesolutions,eitherdirectorthroughtheQRdecomposit ion,arenotverystable. Bothupdatesdependontheparameter ,whichmaycauseinstability.Therefore,the algorithmrequiresaveryslowannealingprocesstoreachth eoptimalsolution.To avoidtheinstabilityandspeeduptheconvergence,weprovi deanothersetofsolutions basedonxedpointupdateruleswherethenewvaluesforAandWarecalculatedwith respecttotheiroldvalues. ThexedpointupdateequationsforAandWare:A=X T D Yf~ G ( Y X T )gX 1nX T ~ G ( Y X T ) T Y + X TD Xf~ G ( X X T )g~ G ( X X T )XA + X TD Xf~ G ( X X T )g~ G ( X X T )D Yf~ G ( Y X T )g KWo (4–15) 68

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W=K D Yf~ G ( Y X T )gK +I 1nK ~ G ( Y X T ) T Y + KD Xf~ G ( X X T )g~ G ( X X T )KW + KD Xf~ G ( X X T )g~ G ( X X T )D Yf~ G ( Y X T )g XAo (4–16) NoticethatthetransformationmatricesAandWarepresentinbothsidesofthexed pointupdateequations.Also,whilethereexistmanywaysof performingthexedpoint update,thisparticularsolutionsetwaschosenbecauseofi tssmoothbehavioronbothAandWacrossiterations. 4.4ExperimentalResults First,weanalyzedtheeffectofkernelbandwidth, ,usedforthePDFestimations ontheconvergencerate.Asimpleexperimentofmatchingtwo shapesagainstawide rangeofkernelsizeswasrunasshowninFig. 4-1 .Theresultsdemonstratedtwo things:thekernelbandwidthisassociatedwiththedynamic rangeofthedata,andthe fastestconvergencecanbeachievedbycontrolling(anneal ing)thekernelsizethrough theiterations.Asaresult,allthedatausedintheexperime ntswasnormalizedtohavea varianceof1andthekernelsizeissetto1andgraduallyannealedwith = 0.95.Other parametersusedinthealgorithm:theRBFbandwidth issetto5,andthestopping conditionissettoeither200iterationsorwhenthechangeinparametersdropsbelowa thresholdof106.Ingeneral,thealgorithmconvergeinlessiterationswith theexception ofafewdifcultcases.4.4.1Analysison2DFishData Non-rigidEvaluation :Inthisexperiment,wewarpthetemplatethroughaseries ofnon-rigidtransformationstotestthealgorithm'sperfo rmanceonsolvingdifferent degreesofdeformation.Fig. 4-2 showsanexamplewherethealgorithmwastested againsttwocleanpointsets.TheresultshowsthatCS-Divpr oducesaccurateresultsfor non-rigidregistrationoncleanpointsets. 69

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0.5 1 1.5 2 2.5 3 3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Astartingposition 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 iterationserrorConvergence rate for different kernel bandwidths s = 1 s = 5 s = 10 0.80*(10 s ) Bconvergencerate Figure4-1.Theeffectofthekernelbandwidthontheconverg encerate. Ainitialposition Bnalalignment Cunderlyingdeformation Dinitial-nallinks Figure4-2.Exampleofnon-rigidregistrationfortheCS-Di valgorithm.Therstcolumn showsthetemplate( )andreference(+)pointsets.Thesecondcolumn showstheregisteredpositionofthetemplatesetsuperimpo sedoverthe referenceset.Thethirdcolumnrepresentstherecoveredun derlying deformation.And,thelastcolumnshowsthelinkbetweenthe initialandnal templatepointpositions. Noiseandoutlierpresence :Inthisexperiment,theregistrationproblembecomes morechallengingwiththeadditionofrandomnoisetotheref erencepointset.In addition,theheadoftheshinthereferencepointsetandth etailoftheshinthe templatepointsetareremoved.Fig. 4-3 showstheresultsofoneoftheexperiments. TheresultsshowthatCS-Divisrobustagainstbothmissingp oints(occlusion)and corrupteddata(noise). Afnetransformation :Inthisexperiment,bothafneandnon-rigidtransformati ons areappliedtothereferencepointsettotestthealgorithms capabilitytoapproximate boththeglobalandlocaltransformations.Thealgorithmpe rformfairlywellwithvarious 70

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Ainitialposition Bnalalignment Cunderlyingdeformation Dinitial-nallinks Figure4-3.Amoredifcultexample,wherethereferencepoi ntsetiscorruptedtomake theregistrationmorechallenging.Noiseisaddedandthes hheadis removedinthereferencepointset.Thetailisalsoremovedi nthetemplate pointset. degreesofperformancedependingonthedifcultyoftherot ation.Fig. 4-4 showsan exampleofrotationof35 andsometranslationalongwiththenon-rigidtransformati on. Inthiscase,thealgorithmperformwell,butthemappingisn otasaccurateasthatof experiment1. Whentherotationangleisabove60 ,thealgorithmperformspoorly.Asimple waytoovercomethisproblemwouldbetohavemultipleinitia lorientationsforthe templatepointset.Wealsodesignedamorespecicupdateal gorithmwheretheafne transformationmatrix,A,wasassumedtoberotation+translation.Inthiscase,rath er thansolvingforeachelementofthetransformationmatrix, wesolvedfortherotation angle, .Settingthisconstraintontheelementsofthetransformat ionmatrixallowedfor afasterandmoreaccurateconvergenceupto 85 .Higherangleswouldresultinaip ofthetemplate.4.4.2QuantitativeEvaluation Weevaluatetherobustnessofbothalgorithms:Correntropy andCauchy-Schwarz Divergenceonvariouslevelsof:deformation,noise,outli ers,rotation,andocclusion onsyntheticdata.Thedatasetsusedintheseexperimentswe resynthesizedfrom [ 58 76 ].Eachdatasetcontainstwoshapes,aChinesecharacterand ash,whichhave undergonedifferentdegreesofdistortion.Inthedeformat iondataset,differentlevelsof 71

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Ainitialposition Bnalalignment Cunderlyingdeformation Dinitial-nallinks Figure4-4.Exampleofafneandnon-rigidtransformation. Thereferencepointsetis rotatedandtranslatedinadditiontothenon-rigidtransfo rmationtoevaluate thecapabilityofthealgorithmstohandleglobalandlocalt ransformations. deformationareappliedtothetemplatepointsettocreatea targetpointset.Intherest ofthedatasets,rstaslightdeformationhasbeenappliedt othetemplatepointset,and thenthedeformedtemplateundergoestherespectivedistor tion.Atotalof100pointsets aregeneratedateachlevelofdistortion. Ouralgorithmsalongwith:coherentpointdrift(CPD)[ 70 ],diffeomorphicmatching (Diff)[ 78 ],thin-platesplinesrobustpointmatching(TPS-RPM)[ 76 ],robustpoint matchingbypreservinglocalneighborhoodstructures(PLN S),andregistrationusing Gaussianmixturemodels(GMM)[ 79 ]weretestedineachdataset.Allalgorithmswere updatedfor300iterations.Theaccuracyofthematchisquantiedusingroo tmean squareerror(RMSE).Theevaluationmetricwasselectedfol lowingtheworkof[ 58 ]on thesamedataset.Thestatisticalresults:errormeansands tandarddeviationsforeach datasetwillbeshownontheresultsbelow.4.4.2.1Deformation First,wetestthealgorithmsonvariouslevelsofdeformati on.Fig. 4-5 showsthe templatesfortheChinesecharacterandtheshontherstco lumn,anexampleof thedeformationthatthetemplateshaveundergoneonthesec ondcolumn,andthe performanceofcorrentropyandCauchy-Schwarzdivergence (CS-Div)onthethirdand fourthcolumns.Fig. 4-6 showstheperformanceofeachalgorithmonthedifferentlev els ofdeformation.Theperformanceisevaluatedusingrootmea nsquareerror.Theerror 72

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barsindicatethemeanandstandarddeviationoftheRMSEove rthe100pointsets.In thisexperiment,thecorrentropymethoduses`surprise'to computethecorrespondence betweenthetemplateandthedeformedshape.Inaddition,bo thcorrentropyand CS-Divusethin-platesplinesforthenon-rigidtransforma tion.TPSwasselectedinstead ofGaussianRBFtokeepasmanyoftheparametersthesameinal lalgorithmsas possible. Asshownfromtheresults,correntropyoutperformsalltheo thermethods.This alsoindicatesthatsurpriseisrobusttodeformationsandi sagoodmethodtoobtain correspondenceundertheseconditions.CS-Divperformanc eiscomparabletotheother algorithms.TheCS-Divmeanerrorislowerthantherestofth ealgorithmsonmost ofthedeformationlevelsespeciallyonthehighlevel.Howe ver,fewofthem(Diffand PLNS)fallinsidetheCS-DIVstandarddeviationonallthele vels,whichmeansthatwe cannotstatisticallyclaimthatCS-DivisbetterthanDiffo rPLNS. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Acharacter 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Bdeformed 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 RMSE = 0.0012539 CCorrentropy 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 RMSE = 0.0075741 DCS-Div 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Esh 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Fdeformed 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 RMSE = 0.00071361 GCorrentropy 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 RMSE = 0.0029837 HCS-Div Figure4-5.Anexampleofthedegreeofdeformationthatthet woshapeshave undergone,andshapematchingperformanceofCorrentropya ndCS-Div. 73

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0.02 0.035 0.05 0.065 0.08 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Correntropy CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0.02 0.035 0.05 0.065 0.08 0 0.01 0.02 0.03 0.04 0.05 0.06 Correntropy CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-6.Matchingperformancecomparisonundervarious degreesofnon-rigid warping.Algorithmsused:Correntropy,CS-Div,CPD,Diff, TPS-RPM,PLNS, andGMM.Datasets:(A)thecharacterand(B)shshapes. InvestigatingtheresultsinFig. 4-6 ,wenoticedthatthestandarddeviationof mostalgorithmsathighlevelsofdeformationbecomesveryl arge.Thesestandard deviationbarsdonotprovidemuchinformationaboutthealg orithm.Thishappens becauseathigherlevelsofdeformation,incertaintrials, themethodscompletelyfail tondthedeformation.Thisresultsinhigherrorvalues,wh ichinturncausesthe standarddeviationtoexplode.Therefore,wedeterminedth atinadditiontothemean andstandarddeviationresultswewillalsoshowtheresults usingthemedian.Fig. 4-7 showsthemedianresultsofallthemethodsforthetwodatase ts.Thisprovidesaclearer pictureofwhenamethodperformswellandwhenitfails.4.4.2.2Noise Next,wecomparethealgorithmsonvariouslevelsofnoise.F ig. 4-8 showsan exampleofthenoisedistortionthatthetemplateshaveunde rgoneandthetemplate alignmentsbasedonourtwoalgorithmsalongwiththeirRMSE .Fig. 4-9 and 4-10 showtheperformanceofeachalgorithmonthedifferentleve lsofnoise.Asinthe deformationcase,Fig. 4-9 showsthemeanRMSEandstandarddeviationresults,and Fig. 4-10 showsthemedianRMSEresults.Inthisexperiment,weinclud etwoversions 74

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0.02 0.035 0.05 0.065 0.08 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Correntropy CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0.02 0.035 0.05 0.065 0.08 -0.005 0 0.005 0.01 0.015 0.02 0.025 Correntropy CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-7.MedianRMSEresultsforvariousdegreesofnon-r igidwarping.Algorithms used:Correntropy,CS-Div,CPD,Diff,TPS-RPM,PLNS,andGM M. Datasets:(A)thecharacterand(B)shshapes. ofcorrentropy:usingTPS(Corr-TPS)andusingGaussianRBF (Corr-Gauss)toshow theadvantagesofeachfunction.Corr-Gaussperformsbette rathigherlevelsofnoise. ThisisduetotheadvantagethatGaussianRBFpresentincomp arisontoTPSwhere theyactonalocalizedareawhereasTPShavemoreofaglobale ffect.Thisglobal effectisnoticedinhighlevelsofnoisewherechangesincer taincontrolpointsresonate overthewholetemplate.WhileCorr-Gaussperformsbetteri nhigherlevelsofnoise,it isoutperformedbyCorr-TPSinlowerlevels.Thisrevealsth eweaknessofGaussian RBFwhichishavingtodealwithanadditionalfreeparameter .Theresultsshowthatthe kernelbandwidthselectedinthiscasewasnotoptimalforth edataset.Adifferentkernel sizewouldhaveprobablyprovidedbetterresults.Also,wew ouldliketopointoutthat correntropyagainuses`surprise'tocomputethecorrespon dence.Thisshowsagainthat surpriseisrobusttonoiseinthedata. TheCS-Divalsoperformswellinhighlevelsofnoise,howeve ritsperformance ispoorinlowerlevels.Inthehighnoiselevels,PLNShasabe ttermeanerrorthan CS-DivandCorr-TPS,anditisveryclosetoCorr-Gauss,howe veritsstandarddeviation 75

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encompassesbothourmethods,soagain,statisticallyweca nnotclaimwhichmethodis thebest. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Acharacter 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Bnoise 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RMSE = 0.01261 CCorrentropy 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RMSE = 0.015286 DCS-Div 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Esh -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Fnoise -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 RMSE = 0.027512 GCorrentropy -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 RMSE = 0.033637 HCS-Div Figure4-8.Anexampleofthelevelofnoisecorruptionthatt hetwoshapeshave undergone,andshapematchingperformanceofCorrentropya ndCS-Div. 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Corr-TPS Corr-Gauss CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Corr-TPS Corr-Gauss CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-9.Matchingperformancecomparisonundervarious degreesofnoise. Algorithmsused:Corr-TPS,Corr-Gauss,CS-Div,CPD,Diff, TPS-RPM, PLNS,andGMM.Datasets:(A)thecharacterand(B)shshapes 76

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0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0.06 Corr-TPS Corr-Gauss CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0.06 Corr-TPS Corr-Gauss CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-10.MedianRMSEresultsforvariousdegreesofnois e.Algorithmsused: Corr-TPS,Corr-Gauss,CS-Div,CPD,Diff,TPS-RPM,PLNS,an dGMM. Datasets:(A)thecharacterand(B)shshapes. 4.4.2.3Outliers Wethencomparethealgorithmsonvariouslevelsofoutliers .Fig. 4-11 shows anexampleoftheoutliercorruptionforthetwotemplatesal ongwiththetemplate alignmentsbasedonourtwoalgorithms.Inthisexperiment, thesurprisecannotbe usedtoidentifythecorrespondencebetweenthetemplatean dthecorruptedinstance, becausetheoutliersareaddedinadditiontotheactualshap e,whichchangesthe surprise`signature'oftheshape.Therefore,weuseneares t-neighborasthemethod todeterminethecorrespondencebetweenthetwoshapes.Not icethatbecauseof thisinaccuratemeansofdeterminingthecorrespondence,t hecorrentropyalgorithm performspoorlyonthecharactershape. Fig. 4-12 and 4-13 showtheperformanceofeachalgorithmonthedifferentleve ls ofoutliercorruption,providingresultsforthemeanandst .d.inFig. 4-12 andmedianin Fig. 4-13 .Again,becausethecorrespondencemethodisnotveryaccur ate,correntropy doesnotperformaswellasinpreviouscases.Noticethatint heshdataset,where theoutliercorruptionisnotthatoverwhelming,CorrandCS -Divperformveryclose tooneanother.TheCS-Div,performsverywellinbothdatase ts,especiallyonhigh 77

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outlier-to-dataratios,whichshowsthatthemethodisrobu stagainstoutliers.However, wenoticethatPLNSperformsmuchbetterthanCS-Divinthes hdataset.Thereason forthisperformancemightbebecausePLNSistheonlymethod thatwedonotcontrol theiterationsnumber.Rememberthatwhenwedescribedthee xperimentssetupwe indicatedthatallmethodsarerestrictedtoonly300iterations.ForthePLNSalgorithm, wedonothavethesourcecode,justanexecutable,thuswedon otknowwhenorhow manyiterationsthealgorithmhasgonethrough.Toshowthat ourmethodsperform betterthantheresultsshowninFig. 4-12 andarecomparabletothePLNS,were-ran ourmethodsontheshdatasetfor500iterations.Fig. 4-14 showstheresultsof500iterations.Comparingthetwoguresshowsthatthealgorit hmperformanceimproved andinthiscaseCS-DivperformsmuchclosertoPLNS.Inthela stlevel,twooutliers perdatasample,PLNSmeanerrorislowerthanCS-Div,howeve rCS-Divstillfallinside PLNSstandarddeviation,sothemethodsarestillconsidere dofsimilarperfomrance. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Acharacter 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Boutliers 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 RMSE = 0.075893 CCorrentropy 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 RMSE = 0.038707 DCS-Div 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Esh -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Foutliers -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 RMSE = 0.038295 GCorrentropy -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 RMSE = 0.034484 HCS-Div Figure4-11.Anexampleofthenumberofoutlierspointsthat havebeeninsertedinthe twoshapes,andshapematchingperformanceofCorrentropya ndCS-Div. 78

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0 0.5 1 1.5 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Correntropy CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 Correntropy CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-12.Matchingperformancecomparisonundervariou slevelsofoutliers. Algorithmsused:Correntropy,CS-Div,CPD,Diff,TPS-RPM, PLNS,and GMM.Datasets:(A)thecharacterand(B)shshapes.Theleve lofoutliers (x-axis)ismeasuredasoutlier-to-dataratio. 0 0.5 1 1.5 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Correntropy CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 Correntropy CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-13.MedianRMSEresultsforvariouslevelsofoutli ers.Algorithmsused: Correntropy,CS-Div,CPD,Diff,TPS-RPM,PLNS,andGMM.Dat asets:(A) thecharacterand(B)shshapes.Thelevelofoutliers(x-ax is)ismeasured asoutlier-to-dataratio. 4.4.2.4Rotation Next,wecomparethealgorithmsonvariousrotationangles. Fig. 4-15 shows twodifferentangletransformationsforthetwoshapes.The characterisrotated60 whereastheshisrotated180 .Noticethatcorrentropyperformsverywellinboth cases.Thisisduetothefactthatcorrentropyutilizes`sur prise'todeterminethe 79

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0 0.5 1 1.5 2 -0.02 0 0.02 0.04 0.06 0.08 0.1 Correntropy CS-Div PLNS Ash Figure4-14.ComparisonofCorrentropy,CS-Div,andPLNSun dervariouslevelsof outliersfortheshshapes.CorrentropyandCS-Divwereall owedtorunfor500iterations.Thelevelofcorruption(x-axis)istheoutlier -to-dataratio. correspondencebetweentheshapes,andasweshowedinChapt er 3 ,surpriseis robusttoanyrotationtransformation.Ifweweretousenear est-neighbormethodto determinethecorrespondence,correntropywouldhaveprob ablygottenstuckatalocal minimum(asshownbelow).CS-Divperformswellforanyangle sbelow45 .Forangles abovethat,whiletherearecasesthatitmayndthecorrectr otation,suchasinthecase of60 forthecharactershapeinthisexample,correctalignmenti snotguaranteed.We canseethatinthecaseof180 ,CS-Divconvergedatalocalminimum. Fig. 4-16 and 4-17 showthemeanandmedianresultsofeachalgorithmonthe differentangletransformations.Inthisexperiment,weus etwomethodstodeterminethe correspondenceforcorrentropy,surprise(Corr-Surp)and nearest-neighbor(Corr-NN). WecanseethatCorr-Surpoutperformsallothermethods.Thi sisduetotherobustness ofsurpriseagainstrotationtransformations.Corr-NNper formscomparablywellfor transformationanglesupto30 ,andfailsforanylargerangles(willgetstuckinalocal minimum).CS-Divalsowillperformverywellfortransforma tionanglesupto45 .For anglesupto60 ,dependingontheshape,CS-Divmaycorrectlydeterminethe angle. NoticethatthestandarddeviationforCS-Divat60 reaches0indicatingthatincertain 80

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casesthealgorithmwasabletondthecorrecttransformati on.Thisactuallyisbetter illustratedinFig. 4-17 (B)whereformanyofthetrialsontheshdataset,CS-Divisa ble todeterminethetransformationupto60 .Beyond60 ,CS-Divwillnotbeabletond thecorrecttransformationandwillconvergeatalocalmini mum.Thisisthecasefor mostofthealgorithmsthatdonotutilizeanyspecialfeatur esabouttheshapes.PLNS, ontheotherhand,similartoCorr-Surp,usesinformationab outthestructureofthe shapetodeterminethecorrespondencebetweenthetwopoint sets,andasaresult,its performancedoesnotdeterioratefromthedifferentangler otations. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Acharacter 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 B60 rotation 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 RMSE = 0.0007519 CCorrentropy 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 RMSE = 0.013159 DCS-Div 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Esh -0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 F180 rotation -0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 RMSE = 0.00047936 GCorrentropy -0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 RMSE = 0.28173 HCS-Div Figure4-15.Anexampleofapplyingtwodifferentrotationa nglestothetwoshapesand theshapematchingperformanceofCorrentropyandCS-Div.T hecharacter isrotatedby60 ,noticethatinthiscaseCS-Divisabletocorrectlyndthe rotationtransformation.Theshisrotated180 ,thistransformationis impossiblefortheCS-Divalgorithmtocorrectlydetermine ,insteadthe algorithmsconvergesatalocalminimum.Correntropyperfo rmsverywellin anyanglebecausethe`surprise'isabletocorrectlydeterm inethe correspondencebetweenthetwoshapes. 4.4.2.5Occlusion Finally,wecomparethealgorithmsonvariouslevelsofoccl usion.Fig. 4-18 shows thetwoshapeswith30%occlusionandthenalshapealignmentofbothalgorithms. 81

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0 30 60 90 120 180 0 0.1 0.2 0.3 0.4 0.5 0.6 Corr-Surp Corr-NN CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0 30 60 90 120 180 0 0.1 0.2 0.3 0.4 0.5 Corr-Surp Corr-NN CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-16.Matchingperformancecomparisonundervariou srotationangles. Algorithmsused:Corr-Surp,Corr-NN,CS-Div,CPD,Diff,TP S-RPM,PLNS, andGMM.Datasets:(A)thecharacterand(B)shshapes.Ther otation angles(x-axis)rangefrom0 to180 0 30 60 90 120 180 0 0.1 0.2 0.3 0.4 0.5 0.6 Corr-Surp Corr-NN CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0 30 60 90 120 180 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Corr-Surp Corr-NN CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-17.MedianRMSEresultsforvariousrotationangle s.Algorithmsused: Corr-Surp,Corr-NN,CS-Div,CPD,Diff,TPS-RPM,PLNS,andG MM. Datasets:(A)thecharacterand(B)shshapes.Therotation angles (x-axis)rangefrom0 to180 Correntropyusessurprisetodeterminethecorrespondence betweenthetemplateand theoccludedtarget.Inbothshapes,correntropyalignsver ywellwiththetargetexcept fortheareawhereocclusionoccurs.Thishappensbecauseth osetemplatepoints donothaveanycorrespondingpointsonthetarget,andasare sult,theirnon-rigid transformationweightsaremissing.Thenaltemplatealig nmentforthosepoints 82

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dependsonlyontheafnetransformation.CS-Div,ontheoth erhand,performsverywell evenontheoccludedareasbecauseCS-Divisbasedonthefull structureoftheshape (itsPDF)ratherthanonindividualcorrespondence. Fig. 4-19 and 4-20 showtheperformanceofeachalgorithmonthedifferent occlusionratios.Wedemonstratethecorrentropycapabili tyusingbothsurprise andnearestneighbor.Noticethattheperformanceofcorren tropyusingsurprise changesveryslightlythroughthedifferentlevelsofocclu sion.Thisisduetothe accurateassignmentofthepointcorrespondencebetweenth eshapes.Correntropy usingnearest-neighborperformsworse.Thisisduetothene arest-neighborapproach assigningthewrongcorrespondencebetweenthetwoshapes, resultinginhigherfailure rateastheocclusionrateincreases.CS-Divperformswello nlevelsoflowocclusion, butitsperformancestartsdegradingonhigherocclusionle velsandcannotbecompared tothestate-of-the-artmethods.Therearetworeasonsfort hepoorperformance: theocclusionareaisverylargecomparedtotherestofthesh ape,andthenon-rigid transformationcausesconfusionintheshapealignment.If theoccludedtargetshapes hadonlyafnetransformations,theperformancewouldhave beenbetter,becausethe overallshapestructurewouldhaveremainedintact.4.4.3Analysison3DBunnyData Theextensionofthealgorithmto3Disstraightforward.Fig 4-21 (left)shows twoobjects:bun000andbun045borrowedfromtheStanfordbu nnydataset[ 83 ]. Thealgorithmproducesavisuallycorrectregistrationass hownonFig. 4-21 (right). Infact,similartothe2Dcase,thealgorithmperformswellf orrotationsupto60 as showninFig. 4-22 (A)wheretheerroriszeroforallrotationsupto60 ineachaxis. Thecombinationofrotationsonmultipleaxesbecomesdifc ulttoidentifyalimitation boundaryasthecombinationofdifferentangleswillhavedi fferenteffectsdepending onthestructureoftheobjectandalsotheorderinwhichther otationsaretaken.InFig. 4-22 (B),theperformanceresultsforrotationsonallthreeaxes forthebunnyareshown 83

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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Acharacter 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 0.2 0.4 0.6 0.8 1 B30%occlusion 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 0.2 0.4 0.6 0.8 1 RMSE = 0.00098198 CCorrentropy 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 0.2 0.4 0.6 0.8 1 RMSE = 0.00038926 DCS-Div 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Esh 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F30%occlusion 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RMSE = 0.00071454 GCorrentropy 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RMSE = 0.00075644 HCS-Div Figure4-18.Anexampleofocclusionwhere30%ofeachpointsetisoccluded,and shapematchingperformanceofCorrentropyandCS-Div. withtherotationorder:rotX*rotY*rotZ.Toaccountforhig herrotationangles,weneedto placethisrequirementontheafnetransformationmatrixb yconstrainingitselementsto bepronetorotationbehaviorrstandthenscalingsothatth esystemdoesnotgetstuck intoalocalminimum.4.4.4EvaluationonMPEG-7ShapeDatabase ThenalexperimentinvolvedtheMPEG-7shapedatabase,spe cicallythe CE-Shape-1partB[ 84 ].MPEG-7isastandarddatabaseusedtotestshapeclassica tion andretrievalmethods.Thedatabaseconsistsof1400silhouetteimages:70categories with20examplespercategory.Theperformanceismeasuredusingth e“bull'seyetest” whereeveryshapeinthedatabaseiscomparedtoallothersha pesandthenumberof shapesfromthesamecategoryamongthe40mostsimilarshapesisreported.Since someofthecategorieshaveshapesthatappearippedhorizo ntallyorverticallyand ouralgorithmcannotaddressthesetypesofvariationsinth eshape,wemodiedthe costfunctiontoincludecomparisonsagainsttheoriginals hape,ippedhorizontally,and 84

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0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 Corr-Surp Corr-NN CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 Corr-Surp Corr-NN CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-19.Matchingperformancecomparisonundervariou slevelsofocclusion. Algorithmsused:Correntropy,CS-Div,CPD,Diff,TPS-RPM, PLNS,and GMM.Datasets:(A)thecharacterand(B)shshapes.Theoccl usionratio (x-axis)rangesfrom0to50%oftheshape. 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 Corr-Surp Corr-NN CS-Div CPD Diff TPS-RPM PLNS GMM Acharacter 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 Corr-Surp Corr-NN CS-Div CPD Diff TPS-RPM PLNS GMM Bsh Figure4-20.MedianRMSEresultsforvariouslevelsofocclu sion.Algorithmsused: Correntropy,CS-Div,CPD,Diff,TPS-RPM,PLNS,andGMM.Dat asets:(A) thecharacterand(B)shshapes.Theocclusionratio(x-axi s)rangesfrom0to50%oftheshape. ippedvertically: D CS( YkX ) = minfD( YkX ),D( YkX h ),D( YkX v )g.(4–17) Theresultsthatweobtainhavea83%retrievalrate.Thereasonforthelowperformance isthedifcultyofdatasampleswithinaclass,whichisdemo nstratedinFig. 4-23 (A). 85

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-0.05 0 0.05 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 -0.05 0 0.05 Initial Alignment Ainitialalignment -0.05 0 0.05 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 -0.05 0 0.05 Final Alignment Bnalalignment Figure4-21.3Dregistration:initialalignmentisoffby45 0 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 rotation (deg)error rot X dir rot Y dir rot Z dir Arotationonindividualaxis 0 50 100 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 90 Z Y X Brotationonallaxes Figure4-22.ErrorrateofCauchy-Schwarzdivergenceonthe 3Dbunnyrotatedat variousdegreeson(A)eachofthethreeaxis,and(B)onallth reeaxes. Inaddition,thereisdifcultyduetothesimilarityofsamp lesamongdifferentclasses, whichisdemonstratedinFig. 4-23 (B).Fig. 4-23 (A)showsafewexamplesfromthe “device9”classwherethedetailsinsidetheobjectmakeitd ifculttoobtainlowerror rateforobjectsofthesameclass.Fig. 4-23 (B)showsexamplesfromtwodifferent classes,onthetoprowweshowseveralexamplesofthe“guita r”classandonthe bottomrowweshowafewexamplesofthe“spoon”class.Eventh oughthesearetwo 86

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differentobjects,theirdatasampleshavesimilarshapesw hichresultsinmixupsduring retrieval. Adifcultywithinclass Bdifcultyamongclasses Figure4-23.ExamplesofdifcultiesontheMPEG-7dataset. (A)Exampleofvariation difcultywithinaclass.(B)Exampleofsimilaritydifcul tyamongsamples ofdifferentclasses.Toprowshowssamplesfromtheguitarc ategory, bottomrowshowssamplesfromthespooncategory. 87

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CHAPTER5 GROUPWISESHAPEMATCHING InChapter 4 ,wedescribedtheCauchy-Schwarzdivergence( D CS )tomeasurethe similaritybetweentwodensityfunctions.Thismethodandt hemethodswecompared itagainstarelimitedtoregisteringonlyapairofpointset s.Incertainapplications, suchasmedicalimageregistration,thereisaneedtosimult aneouslyregisteragroup ofpointsets.Noneofthesemethodsisdirectlyextendiblet ogroupwisealignmentof multiplepointsets.Inaddition,thesemethodsareallbias ed(unidirectionalupdate) whereonepointsetactsasthetargetandtheotherpointseti stransformedtoalign withit.Inapplicationswheregroupwiseregistrationisre quired,theremaynotbean actualtemplatetomatchagainst(e.g.medicalimages).The refore,thereisaneed todetermineamiddlecommonorientation/positiontoalign thepointsets.Estimating ameaningfulaverageshapefromasetofunlabeledshapesisa keychallengein deformableshapemodeling.Inthischapter,weextendourpr eviousworktogroupwise registrationwhereallthepointsetsaresimultaneouslytr ansformedtowardanaverage shape,whichisindirectlyestimatedthroughthetransform ationprocess. 5.1Background Chuietal.[ 85 ]presentedajointclusteringandmatchingalgorithmthatw ouldnd ameanshapefrommultipleshapesrepresentedbyunlabeledp ointsets.Theirprocess followssimilarapproachtotheirpreviousworkin[ 76 ]whereexplicitcorrespondence needstorstbedetermined.Inaddition,themethodisnotro busttooutliers,sostability isnotalwaysguaranteed. Wangetal.[ 86 ]proposedanothermethodforgroupwiseregistrationwhere the pointsetsarerepresentedasdensityfunctions.Basedonth esameprinciplesthat wediscussedinChapter 4 ,theiralgorithmsimultaneouslyregistersthedataand determinesthemeanshapewithoutchoosinganyspecicpoin tsetasareferenceor solvingforcorrespondences.Intheirapproach,theyminim izetheJensen-Shannon 88

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divergenceamongcumulativedistributionfunctions(CDFs )estimatedfromthepoint setsanduseTPSforthenon-rigidtransformation.Theymove dfromPDFstoCDFs becauseCDFsaremoreimmunetonoiseandarealsowelldened sinceCDFisan integralmeasure.However,theCDFestimationiscomputati onallyveryexpensiveand therearenoclosed-formsolutionstotheirupdates. Chenetal.[ 87 ]developedanothergroupwiseregistrationmethodbasedon the Havrda-Charv atdivergenceforCDFs.Similarto[ 86 ],theyusethecumulativeresidua entropytorepresenttheCDFs.Theirmethod,CDF-HC,genera lizestheCDF-JSbutis muchsimplertoimplementandcomputationallymoreefcien t. Inthischapter,weextendourCauchy-Schwarzdivergenceme thodtoaccountfor unbiasedgroup-wiseregistration.ThemethodisbasedonH ¨ older'sinequalitywhich generalizestheCauchy-Schwarzinequality.Weprovideacl osedformsolutionforthe updatesusingH ¨ older'sdivergence,howevertheseupdatesarecomputation allyvery expensive.Anapproximation,ratherthantheclosedformso lution,oftheH ¨ older's divergenceisalsoprovidedwhichisequivalentincomplexi tytothemethodsmentioned above. WealsodevelopanothermethodbasedonR enyi'squadraticentropy.Themethod ismuchsimplerandfastertocompute,andtheresultsarever yaccurate.Wecompare bothmethodsagainstCDF-JSandCDF-HConvariousdatasets. 5.2H ¨ older'sDivergence Cauchy-SchwarzdivergenceisderivedfromtheCauchy-Schw arzinequality ( 4–1 )whichcomparestheinnerproductoftwofunctionsagainstt heirmarginals. Cauchy-SchwarzInequalitycanbeextendedtothreeormored ensityfunctionsas: Zf ( x ) g ( x ) h ( x ) dx2 Zf 2 ( x ) dxZg 2 ( x ) dxZh 2 ( x ) dx .(5–1) However,usingthisderivation,wecannotdeterminewheneq ualityholds.Evenifthe PDFsareidentical,itdoesnotguaranteeequality.Theuppe rboundaryisveryloose. 89

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ToprovideabetterextensiontomorethantwoPDFs,weturnto H ¨ older'sinequality: Zf ( x ) g ( x ) h ( x ) dx3 Zf 3 ( x ) dxZg 3 ( x ) dxZh 3 ( x ) dx .(5–2) where,forPDFs,theequalityholdsifandonlyiff ( x ) = g ( x ) = h ( x ).Ageneralizedform ofH ¨ older'sinequalityformorethanthreetermsiswrittenas ZnYk =1 f k ( x ) dx!nnYk =1Zf n k ( x ) dx .(5–3) SimilartoChauchy-Schwarzdivergence,wederivetheH ¨ older'sdivergenceas: D H( f k ) =log ZnYk =1 f k ( x ) dx!n nYk =1Zf n k ( x ) dx .(5–4) Thedivergence D H( f k )canbebrokendownintopartsandrewrittenas: D H( f k ) =n logZnYk =1 f k ( x ) dx + nXk =1 logZf n k ( x ) dx .(5–5) SupposewehaveNpointsetsX k,k21, ..., N.EachpointsetX k =fx k 1 ..., x k Mg wherex k i ,2 Rd.Thekernel(Parzen)estimateofthePDF[ 51 ]ofanarbitrarypointxusinganarbitrarykernelfunction ()isgivenas^ f ( x ; x k ) = 1 k M k MXk i =1 xx k i ,(5–6) where isthebandwidthparameter.TheGaussiankernel,G( x x k ) = 1 p 2exp kxx kk2 22 isagainconsideredasthekernelofchoiceforitspropertie s:symmetric,positive denite,approacheszeroaspointsmoveawayfromthecenter ,andthedecayingfactor canbecontrolledviathekernelbandwidth.Consideringthe argumentofthersttermin ( 5–5 ),substitutingtheGaussiankernelandperformingsomeman ipulations,resultsin thefollowingestimator: 90

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ZNYk =1 f k ( x ) dx=ZNYk =1 k MXk i =1 G( xx k i ) dx = NYk =1 k MXk i =1ZG( xx k i ) dx = 1 MX1 i =1N MXN i =1 N1Yp =1 NYq = p +1 Gp N( x i px i q )(5–7) Theargumentofthesecondtermcanbeestimatedinasimilarw ay.Theclosedform solutiontotheH ¨ older'sdivergenceisthenwrittenas: D H( f k ) =N log 1 MX1 i =1N MXN i =1 N1Yp =1 NYq = p +1 Gp N( x i px i q ) + NXk =1 log k MXk i =1k MXk i =1 N1Yp =1 NYq = p +1 Gp N( x i kx i k ).(5–8) 5.3MatchingAlgorithm Similartobeforewerepresentthetransformationfofdatapointxintermsofthe afneandnon-rigidmatricesAandW.Differentfrombefore,wedonothaveaxed pointsetY.Inthegroup-wiseregistrationcase,allpointsets fX kgN k =1willundergosome afneandnon-rigidtransformationtoreachanaveragetran sformation.Foreachpoint setwehavethetransformationmatricesA k W k ,fork = 1N.Thetransformation functionfthatadatapointx khastoundergotoreachtheaverageshapeiswrittenas:f ( x k ; A k W k ) = A kx k + W k ( x k ).(5–9) SolvingforAandWBeforewesolveforAandWlet'ssimplifythenotation.Tounderstandsomeofthe termsthatwillbeusedintheupdates,weprovideavisualexp lanation.Fig. 5-1 shows acubegeneratedfromthreerandomvariablesX Y and Z.Thedimensionsofthe cubewilldependonthepointsetsthatwewouldliketoalign. Thecubeisgenerated 91

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bytheproducttermsin( 5–8 ).Forexample,inthecaseofthreevariables,thecubeis generatedbyrstreplicatingtheGaussianmatricesG ( Y X ), G ( Y Z ), and G ( Z X )into themissingdimension(s)andthenmultiplyingthemusingth eHadamardproduct. Figure5-1.CubeillustrationtoexplainthetermsofH ¨ older'sdivergenceupdate. TocomputeA kandW k,wedealonlywiththersttermin( 5–8 )andonlyoneof thelogarithmsinthesecondterm,theonethatnormalizesX k.So,wewillnotatethe twohypercubesassimplyG ( XYZ )andG ( XXX ).Intheupdateequations,weneedto sumthehypercubeoverallbuttwodimensions,basicallypro jecteverythingontoone ofthecubefaces.Weindicatethesummationofthehypercube byprovidingtheface dimensionsrst,followedbyasemi-colon,followedbythes ummedoverdimensions, e.g.G ( XY ; Z )indicatesthesummationoccurredoverdimensionZ.Inaddition,to accountforthederivativeofthelogarithm,alltheseterms arenormalizedbytheoverall sumoftherespectivehypercube. Tokeeptheequationssimple,intheseupdates,wewillprovi deonlytheimportant terms(notthecross-terms,e.g.X T GK ).TheupdatesforA kandW kare:A k= N NXl6= k X k T D l ( G ( X k X l ; @)) X k! 1(N NXl6= k X k T G ( X k X l ; @) X l A l + 2 NXl6= k X k T ( D k ( G ( X k X k ; @))G ( X k X k ; @) ) X k A k!),(5–10) 92

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W k= N NXl6= k K k D l ( G ( X k X l ; @)) K k! 1(N NXl6= k K k G ( X k X l ; @) K l W l + 2 NXl6= k K k ( D k ( G ( X k X k ; @))G ( X k X k ; @) ) K k W k!),(5–11) where@denotestherestofthevariables,andD l ( G ) = diag ( G1 l ).NotethatA kandW karepresentinbothsidesoftheupdateequations.Thiskeeps thesolutionstable. 5.4ApproximationofH ¨ older'sDivergence TheA k( 5–10 )andW k( 5–11 )updatesusingH ¨ older'sdivergenceclosedform solutionhaveveryhighcomplexity,whichmakesH ¨ older'sdivergenceinfeasiblefor aligningmanyshapeswithlargenumberofdatapoints.Tosim plifythecalculations,here weprovidetwodifferentestimatesofH ¨ older'sDivergence.Therstisbasedonsimple math,andthesecondisbasedonintuition.5.4.1IntegratingOverOneoftheTerms Considertheintegral Rf ( x ) g ( y ) h ( z ) dx.Wecancomputethisintegralwith respecttoxorwithrespecttooneofthePDFs,e.g.h ( z ).So, Rf ( x ) g ( y ) h ( z ) dx = E z [ f ( z ; x ) g ( z ; y )].UsingakerneldensityestimateofthePDF,theintegralise quivalent to: Zf ( x ) g ( y ) h ( z ) dx = 1 M z M zXi"1 M x M xXj G ( z ix j )1 M y M yXl G ( z iy l )# (5–12) Wecanthenre-write( 5–5 )as: D H( f k ) =N log 1 M N M NXi N1Yk =1 1 M k M kXj G ( x N ix k j )!+ NXk =124log 1 M k M kXi 1 M k M kXj G ( x k ix k j )!N135.(5–13) ToupdateA kandW k,weneedtotakethederivativeofaproductterm.Therefore, wedeneF ( X N X k ) =QN1 i6= kfD X i (X N ,X i )gG ( X N X k ).Also,whenupdatingA korW k,we taketheexpectationwithrespecttovariableX k.Thetransformationupdatesarethen computedas: 93

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A k= N NXl6= k X k T D l ( F ( X k X l )) X k! 1(N NXl6= k X k T F ( X k X l ) X l A l + 2 NXl6= k X k T ( D k ( F ( X k X k ))F ( X k X k ) ) X k A k!),(5–14)W k= N NXl6= k K k D l ( F ( X k X l )) K k! 1(N NXl6= k K k F ( X k X l ) K l W l + 2 NXl6= k K k ( D k ( F ( X k X k ))F ( X k X k ) ) K k W k!),(5–15) Thecomputationcomplexityoftheestimatedupdatesismuch smallerthantheclosed formsolutions.Theupdateslooksimilar,infact,theonlyd ifferenceistheinteraction matrix.However,computingthismatrixprovestobeverytim econsumingfortheclosed formsolution.5.4.2ReducingtheHypercubetoJustOneFace LookingbackatH ¨ older'supdateequations( 5–10 )and( 5–11 ),wenoticethatat anystepintheupdatethewholecubeisprojectedontooneofi tsfaces.Ratherthan summingoveralltheotherdimensionstoreachtooneofthefa ces,hereweconsider onlythefaceitself,G ( X Y ).Thisreducesthecomputationtremendouslybecausewe donotneedtocalculatethehypercubesanylonger,whichwer ethemostcomputational partofthealgorithm.ThenewA kandW kupdatefunctionsare:A k= N NXl6= k X k T D l ( G ( X k X l )) X k! 1(N NXl6= k X k T G ( X k X l ) X l A l + 2 NXl6= k X k T ( D k ( G ( X k X k ))G ( X k X k ) ) X k A k!),(5–16) 94

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W k= N NXl6= k K k D l ( G ( X k X l )) K k! 1(N NXl6= k K k G ( X k X l ) K l W l + 2 NXl6= k K k ( D k ( G ( X k X k ))G ( X k X k ) ) K k W k!),(5–17) whereG ( X k X l ) = G ( f ( x k i )f ( x l j )),8fx k igk M i =1 ,fx l jgl M j =1.Notethathereagainweare notshowingthecrossterms. Noticethattheupdateequations( 5–16 )and( 5–17 )forpointsetX ksimplycompare pointsetX kwithalltheotherpointsetsinapair-wisefashion,andthei nteractionsare summedtogether.ThisisequivalenttocomputingtheCauchy -Schwarzdivergence amongallthepointsetsinapair-wisefashion.Theapproxim atedH ¨ older'sdivergence canbere-writtenas: D H( f k ) =N log N1Xp =1 NXq = p +1 G ( X p X q ) + NXk =1 log N1Xp =1 NXq = p +1 G ( X k X k ).(5–18) Basicallythisestimatorreplacedtheproductoperatorsin ( 5–8 )withsummation operators. 5.5GroupwiseAlignmentUsingR enyi'sSecondOrderEntropy Toprovideaclosedformbutalsoacomputationallyfeasible solution,welookata differentwayofapproachingourproblem.In[ 86 ],agroupwiseregistrationmethodwas proposedusingJensen-Shannondivergencewhichisasymmet rizedandsmoothed versionoftheKullback-Leiblerdivergence.Foragivenase tofdistributions fP 1 ..., P ng Jensen-Shannondivergenceisdenedas: D JS( P 1 ..., P n ) = H nXi =1i P i! nXi =1i H ( P i ),(5–19) where iaretheweightsfortheprobabilitydistributions.Intheca seofshapealignment, theseweightsareallconsideredequal, i = 1 n. TheJensen-ShannondivergenceisderivedfromtheJensen's inequalitywhich statesthatforanyPDFs,p 1 p 2 ..., p N,wehave PN i =1i p i PN i =1i( p i ),provided 95

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that is convex overtherangeofp (), Pii = 1,and i0.SinceShannon'sentropy, whichforarandomvariableXisestimatedasH ( X ) = Pi p ( x i ) log p ( x i ),isdifcultto computebecausethelogarithmisfoundinsidethesummation /integral,wewouldliketo consideranotherentropydenitionwhichhasaclosedforms olutiontoitsestimate. R enyi'sentropyisageneralizationofShannon'sentropy[ 49 ]andmitigatesthe logarithmproblembymovingthelogarithmoutsidethesumma tion.R enyi'sentropyis estimatedas:H( X ) =1 1 logXi p( x i ).(5–20) For !1R enyi'sentropytendstoShannon'sentropy[ 50 ].Inaddition,for 2(0, 1)R enyi'sentropyisconcave.Therefore,similartoJensen-Sh annondivergence( 5–21 ), wecandenetheJensen-R enyidivergenceas: D JR( P 1 ..., P n ) = H nXi =1i P i! nXi =1i H( P i ),(5–21) where i'sareagaintheweightsfortheprobabilitydistributions, and0<<1.The Jensen-R enyidivergenceisnon-negative,symmetric,andvanishesi fandonlyifthe probabilitydistributionsP iareallequal.Also,similartotheentropydenition,when !1,Jensen-R enyidivergenceisexactlyJensen-Shannondivergence[ 88 89 ]. Forourpurposes,wewouldliketousethesecondorderR enyi'sentropy( = 2), sincethereexistsaclosedformsolution:H 2 ( X ) =logZp 2 ( x ) dx =logZ 1 1 1 N NXi =1 G( xx i )!2 dx =1 N 2 log NXi =1 NXj =1 Gp 2( x jx i ).(5–22) However,Jensen'sinequalityrequiresthatentropybeacon vexfunction(orconcavefor areversaloftheinequality).R enyi'sentropyfor >1isneitherpureconvexnorpure concave[ 48 50 ].Therefore,wecannotdirectlyapplyR enyi'squadraticentropytothe Jensen'sinequality. 96

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5.5.1ProblemFormulation Weapproachtheprobleminadifferentway.Let~ Xrepresentadistribution,and let~ X 1 ..., ~ X NbeNindependentidenticallydistributed(i.i.d.)randomvari ables(r.v.) fromthedistribution~ X.Eachr.v.~ X k =f~x k 1 ..., ~x k Mg ,where~x k i ,2 Rd.LetX 1 ..., X Nbe corruptedversionsof~ X 1 ..., ~ X N,where~ X khasundergonesometransformationg ksuch thatX k = g k ( ~ X k ).Ourgoalistondtheinverseofeachfunctiong ksoastoretrieve theuncorruptedr.v.~ X k = g1 k ( X k ).Fornotationsimplicity,wewillconsiderthefunctionf k = g1 k. WeestimateR enyi'sentropyofar.v.~ X kasH 2 ( ~ X k ) =logRf 2 k ( x k ) d x.The argumentofthelogarithm, Rf 2 k ( x k ) d x,isknownastheinformationpotential(IP)of therandomvariable.WeknowthattheL 2normoffunctionf, R kf ( x )k2 dxisaconvex function(trueforanyp1).Inaddition,thesquareofaconvexfunctionisalsoconvex asaresult,theinformationpotentialisaconvexfunction. Thedenitionofconvexitystatesthatf (x 1 + (1 ) x 2 ) f ( x 1 ) + (1 ) f ( x 2 ). Jensen'sinequalitygeneralizestheconvexitytomultiple variablesas: PN k =1k x k PN k =1k( x k ),where isaconvexfunctionand PN k =1k = 1.SubstitutingtheIPfor andusingourrandomvariables~ X k,wehave:IP N[k =1k ~ X k! NXk =1k IP ( ~ X k ).(5–23) Theinformationpotentialofarandomvariable~ X kisestimatedas:^ IP ( ~ X k ) =Zf 2 k ( x k ) d x =Z 1 1k MXk i =1 G( xx k ) 2 d x .(5–24) whereGisakerneldensityfunction,inourcasetheGaussianfuncti on.Ifweconsider allweights kthesame, k = 1 N,wecanexpandtheIPtermsas: Z 1 N NXk =1 f k ( x k ) d x!21 N NXk =1Zf 2 k ( x k ) d x .(5–25) 97

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Expandingthelefthandside,wehave1 N 2 NXk =1Zf 2 k ( x k ) d x + NXi =1 NXj =1Zf i ( x i ) f j ( x j ) d x! 1 N NXk =1Zf 2 k ( x k ) d x .(5–26) Dealingwithoner.v.atatime,wehaveNinequalitiesas: (1 N Zf 2 k ( x k ) d x + NXi6= kZf k ( x k ) f i ( x i ) d x! Zf 2 k ( x k ) d x)N k =1 .(5–27) Cauchy-Schwarzinequalitystatesthat Rf k ( x k ) f i ( x i ) d x R p kf k ( x k )k p kf i ( x i )kd x.Since Rf k ( x k ) d xisjustatranslatedversionof Rf i ( x i ) d xandthustheirnormsareequivalent, wehave Rf k ( x k ) f i ( x i ) d x Rf 2 k ( x k ) d x.Thisshowsthatalltheinequalitiesin( 5–27 )are valid,andtheequalitiesholdifandonlyifallf k ( X k )areequivalent. Thisindicatesthattheweighedsumoftheinformationpoten tialoftherandom variablesisgreaterthantheinformationpotentialofthei rweighedunion,andequality isvalidifandonlyifalltherandomvariablesareequivalen t.Sincethelogarithmisa strictlyincreasingfunction,applyingthelogarithmonbo thsidesof( 5–23 )doesnot changetheinequality.Inaddition,ourgoalistondthefun ctionsf kthatwillbringeach corruptedr.vX ktoitsinitialstate~ X k.Ifwesubstitute~ X kwithf k ( X k )andapplylogto ( 5–23 ),theequalitywillholdonlyifallf k ( X k )areequivalent.So,ournewcostfunction andsimilaritymeasureisminimizingthedifferenceofthet wotermsintheinequalityas: J= 1 N NXk =1 log IP ( f k ( X k ))log IP 1 N N[k =1 f k ( X k )!.(5–28) Thetwotermsin( 5–28 )canbeinterpretedasR enyi'squadraticentropiesandthecost functioncanbere-writtenas: J= H 2 1 N N[k =1 f k ( X k )! 1 N NXk =1 H 2 ( f k ( X k )).(5–29) Theaimofthenewsimilaritymeasureistotransformthevari ablesX ksothattheir averageentropyisequivalenttotheentropyoftheoverall SX k. 98

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SolvingforAandWH 2 (SX k )canbeexpressedas:H 2 [X k=log"NXi =1 NXj =1 G ( X i X j T )# (5–30) whereG ( X i X j T )isamatrixgeneratedusingvariablesX iandX jwhoseelementsG m n = exp kX i mX j nk2 .Therefore,takingthederivativewithrespecttoA korW kinvolvesonlytheG ( X k X @ T ),G ( X @ X k T )matricesfromH 2 ( ~ X ),andH 2 ( X k ). TheupdateequationsforA kandW kare:A k= NXl6= k X k T D l ( G ( X k X l )) X k! 1(NXl6= k X k T G ( X k X l ) X l A lX k T ( D k ( G ( X k X k ))G ( X k X k ) ) X k A k + 1 NX k T ( D k ( G ( X k X k ))G ( X k X k ) ) X k A k ,(5–31)W k= NXl6= k K k D l ( G ( X k X l )) K k! 1(NXl6= k K k G ( X k X l ) K l W lK k ( D k ( G ( X k X k ))G ( X k X k ) ) K k W k + 1 N ( K k ( D k ( G ( X k X k ))G ( X k X k ) ) K k W k ).(5–32) Twoimportantthingstorememberabouttheseupdateequatio ns:1)Toreducecluttering oftheupdateequations,wedonotshowherebutallthematric esarenormalizedby theirrespectiveentropies.Thisaccountsforthelogarith mderivativeoftheentropies.It willalsoexplainwhythethirdandfourthtermineachupdate equationlooksimilar.The differenceisthattheirrespectiveG ( X k X k )arenormalizedbydifferententropies,inthe thirdtermbyH 2 ( ~ X )andinthefourthbyH 2 ( X k ).2)Here,againtoreduceclutter,weare notincludingthecrossterms.However,thoseneedtobeincl udedintheimplementation code. 99

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5.5.2ScaleInvariance Entropyisnotscaleinvariant,sousingthedivergencemeas ureaspresentedabove toalignthepointsetsprovidesasolutionatasaddlepointi ntheperformancesurface. Theglobalminimumwouldbereachedwhenallthepointsetsar ecollapsedtoasingle point.Asaresult,dependingonthedataset,wemayendupwit hasolutionwhere allthepointsetsareeithercollapsedtoasinglepointorar ealignedtoascaled-down versionofthedesiredalignment.Toensurethatsuchacolla psedoesnotoccur,we couldintroducearegularizertothecostfunctiontoconne thesolutionspacesoas toreducetheoverallentropyofthepointsetsthroughalign mentwithoutreducingthe entropyoftheindividualsetsthrowscalingdown.However, thisconstraintwouldalso introduceanewfreeparametertothemodel. Instead,werelyonthefactthattheentropyofarandomvaria bledependsonits standarddeviation.Thus,topreventunwanted,scaled-dow nsolutions,wemodifythe costfunctionin( 5–29 )bydivingtheentropiesbytheirstandarddeviationwhichh as beenproventobescaleinvariant[ 90 91 ].Themodiedcostfunctionisscale-invariant andthecollapse-to-a-singlepointisnottheglobalminimu manylonger.Thenewcost functionis: J=(H 2 [~ X k N! 1 2 log"Var [~ X k N!#) (1 N NXk =1H 2 ( ~ X k )1 2 loghVar ( ~ X k )i ).(5–33) TosolvethenewcostfunctionforAandW,hereweprovideonlytheadditional termsduetothestandarddeviation:A k= 21 N X k T X k 1(1 N X k T NXl =1 X l A l X k T X k A kS ( X k T ) S ( X k A k ) ) (5–34)W k= 21 N K k K k 1(1 N K k NXl =1 K l W l( K k K k W kS ( K k ) S ( K k W k ) )) (5–35) 100

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whereS ( X k ) =Pk M k i =1 X k i,andallthesamplesX k iarerstnormalizedbythenumberof samplesinX k,suchasX k i = X k i k M.Again,needtorememberthattoreduceclutterinthe equations,thecrosstermsarenotincluded.Theoverallupd ateequationsforAandWwouldbeaddingtherespectivepartsof( 5–31 )and( 5–34 )forA,and( 5–32 )and( 5–35 ) forW. 5.6ExperimentalResults Westartwithaverysimpleexampletodemonstratethegroupw iseregistration capabilitiesofourmethods.Fig. 5-2 showsanexampleofmultipleshapesofthesame objectrotated,translatedandscaledatdifferentsizes.F ig. 5-2 (A)showstheinitial positionoftheobjects.Therestofthesubguresshowsteps throughthealignment process.Notethattherearenojumpsoccurringduringthere gistrationprocess,which isnotalwaysthecasewithmanyothermethods.Thenalalign mentisanaverage ofrotationandscalingofalltheindividualshapes.Itdoes notmatchwithanyofthe originalshapes,butitisthebestcompromiseinorientatio nandsizeofthedifferent shapes.5.6.1GroupwiseRegistrationofAtlasConstruction Wenowshowademonstrativeexampleofgroupwiseregistrati onusingbothafne andnon-rigidtransformationsonadatasetborrowedfrom[ 87 ].Thedatasetcontains pointsextractedfromtheoutercontourofthecorpuscallos um(CC)ofsevensubjects. Inthisexperiment,wedemonstratetheabilityofourtwoalg orithmsfor unbiased 2D atlasconstruction.Weshowthatbothouralgorithms:H ¨ older'sdivergenceandEntropy areabletosimultaneouslyalignmultipleshapesintoamean shape.Fig. 5-3 shows theregistrationresultsforthetwoalgorithmsandalsofor CDF-HC[ 87 ].Therstseven images[A-G]showthedeformationofeachpointsettotheatl asgeneratedbyoneof thethreemethods.Thecolorschemeusedintherstsevenima gesisasfollows:the initialpointsetisdenotedwithblue`+',thedeformedpoin tssetsaredenotedwithcircles whichareincolorgreen,black,redandmagentacorrespondi ngtoCDF-HC,Holder's, 101

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-1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 1.5 Ainitialposition -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Bstep5 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Cstep20 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Dstep50 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Estep80 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Fnalalignment Figure5-2.ExampleofmultipleshapesalignedusingtheEnt ropycostfunction. Entropy,andConstrainedEntropyalgorithms.Image[H]sho wsthesuperimposed pointsetsbeforetheregistration.Images[I-L]showthesu perimposedpointsetsafter registrationforeachmethod.NoticethatCDF-HCfollowsmo recloselypointset7 [G]anditsnalregistrationissmallerthantherestofthep ointsets.Holder'snal registrationhastheclosestalignmentwiththeinitialpos itionsofeachindividualpoint set.Entropy,similartoCDF-HC,providesanalregistrati onclosesttopointset7,and isscaleddowninthey-axis.Thisisbecausewedidnotinclud etheconstraintterm in( 5–33 ).Image[L]showstheresultsoftheEntropytermwiththecon straint.This demonstratestheimportanceoftheconstraintwhenweminim izeentropies.Thenal noteonthisexperimentisthatourmethodsnotonlyprovidea nalregistrationthat resemblestheaverageshapemoreclosely,buttheyalsoprov ideabettertofthenal pointsets. 102

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-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 APointSet1 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 BPointSet2 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 CPointSet3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 DPointSet4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 EPointSet5 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 FPointSet6 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 GPointSet7 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 HBeforeRegistration -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 ICDF-HC -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 JHolder's -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 KEntroy -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 LEntropy+Constraint Figure5-3.Exampleofunbiasedgroupwisenon-rigidregist rationonrealCCdatasets. PerformancecomparisonusingCDF-HC,Holder's,Entropy,a nd Scale-InvariantEntropymethods. Next,weaddafewoutlierpointsinoneofthepointsets,poin tset7,andcompare theregistrationcapabilitiesofthemethodsagainstnoise .Fig. 5-4 [A]showsthe pointsetsbeforeregistrationplustheoutliersampleswhi chareshownindarkstars. Sub-gures[B-F]showthenalregistrationforthethreeme thods.Inadditiontothe sevenpointsetsafterregistration,eachsub-gurealsoco ntainspointset7fromthe 103

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registrationwithouttheoutliersamplestoshowthedeviat ionsthatmayhaveoccurred inthenalregistrationduetotheoutliers.Theresultssho wthatourmethodssimilarto CDF-HCarerobusttooutliers.However,thenalregistrati onofourmethodsisbetter thanthatofCDF-HC.Whencomparedagainstpointset7ofther egistrationwithout outliers,ourmethodshaveverylittledeviation,whereasC DF-HCshowsamuchlarge deviation. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 ABeforeRegistration -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 BCDF-HCRegistration C -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 DHolder's -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 EEntropy -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.15 -0.1 -0.05 0 0.05 0.1 FEntropy+Constraint Figure5-4.Exampleofunbiasedgroupwiseno-rigidregistr ationonoutliernoise. Comparethethreemethods:Holder's,Entropy,andCDF-HC. Weneedtopointoutthateventhoughourmethodsperformvery well,wehave twofreeparametersinthecaseswhenTPSisusedandthreepar ameterswhen GaussianRBFisused;basically,kernelsize requiredtoestimatePDFs,regularization parameter thatcontrolsthe`smoothness'ofthefunction,andkernelb andwidth thatdeterminesthelocalityinuenceofcontrolpointsinG aussianRBF.Determining theseparametersandestablishinganoptimalannealingrat eisdifcultandproblem 104

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dependent.Inourexamples,weinitialize = 1, = 1,and = 0.1anddecrease at97%rateand at95%rate.However,fordifferentannealingrates,wewould getdifferentnalregistrations.Fig. 5-5 showsthreeexamplesofdifferentannealing ratesfor whilekeepingtheannealingratefor thesame,97%.Noticethatatslow annealingrates,thepointsetsareveryrigidandtheregist rationfails.However,atvery lowannealingratesweloosepointsetstructure.Therefore ,theannealingratesneedto bebalancedcarefully. 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.05 0 0.05 0.1 0.15 0.2 A97% 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.05 0 0.05 0.1 0.15 0.2 B95% 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.05 0 0.05 0.1 0.15 0.2 C93%Figure5-5.Registrationresultsatdifferent annealingrates. 5.6.2GroupwiseRegistrationofBiasedDatasets Todemonstratetheaccuracyandrobustnessagainstnoise,w eborrowedtwo data-setsfrom[ 87 ]:olympiclogoandnoisysh.Thesedatasetsweresynthetic ally generatedtobebiased.Therstpointsetistheoriginalset andtheothersixwere generatedbypassingthepointsetthroughvariousnon-rigi dtransformationsusing thin-platesplines.Fortheseconddataset,thesh,inaddi tiontothetransformation,ten randomlygeneratedjitterpointswerealsoinsertedintoea chpointset.Theinitialpoint setpositionsandthenalregistrationresultsforCDF-HC, Holder's,andEntropyare showninFig. 5-6 Toanalyticallymeasuretheaccuracyoftheregistration,b asicallythesimilarity betweenthenalpointsets,wecomputetheKolmogorov-Smir nov(KS)statistic[ 92 ] betweenthegroundtruthpointset,whichistherstpointse tineachcase,andthenal, 105

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 ABeforeRegistration -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 BCDF-HC -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 CHolder's -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 DEntropy -3 -2 -1 0 1 2 3 4 -1.5 -1 -0.5 0 0.5 1 1.5 EBeforeRegistration -3 -2 -1 0 1 2 3 4 -1.5 -1 -0.5 0 0.5 1 1.5 FCDF-HC -3 -2 -1 0 1 2 3 4 -1.5 -1 -0.5 0 0.5 1 1.5 GHolder's -3 -2 -1 0 1 2 3 4 -1.5 -1 -0.5 0 0.5 1 1.5 HEntropy Figure5-6.Exampleofbiasedgroupwiseno-rigidregistrat ion.Comparethethree methods:Holder's,Entropy,andCDF-HContheolympiclogoa ndnoisysh datasets. registeredpointsets.TheaverageKS-statisticresultsfo rtheolympiclogandthenoisy shdatasetsareshowninTable 5-1 .Theresults,similartothegures,showthatour methodsperformmuchbetterthanCDF-HC. Table5-1.KSstatistic CDF-HCHolderEntropy olympiclogo0.02650.00880.0103Fish+outliers0.06490.03950.0387 Ourmethodsperformnotonlywellonanytransformationstyp esandnoise,but theyarealsoveryfasttocompute.Table 5-2 showstheaveragecomputationtimeof ourtwomethodsandCDF-HConthreedifferentsetsofdatawhe reallthreemethods aresettorunfor300epochs.Theresultsshowthatourmethod sarecomputationally lessexpensive.Thelastrowinthetable,showsthatthecomp utationtimeforCDF-HCis 106

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closetoourmethods.ThereasonforthisisthatCDF-HCstops earlybecauseofgetting stuckinalocalminimum,andasaresult,doesnotgothrougha lltheiterations(the averagestoptimeforthisdatasetwasat72iterations).Nev ertheless,bothourmethods performmuchfasterthanCDF-HC. Table5-2.Runtime CDF-HCHolderEntropy corpuscallosum123s23s21solympiclogo350s58s49sFish+outliers197s99s78s 107

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CHAPTER6 APPLICATIONS Inthischapter,wediscusstwoapplicationswheretheshape matchingalgorithms coveredinthepreviouschaptersareutilizedtosolveparto forthewholeproblem.In therstapplication,shapematchingisutilizedasafeatur etodescribetheshapes foundinimages.Inthesecondapplication,theshapematchi ngalgorithmswereusedto determinethedimensionsofanobject. 6.1Side-ScanSonarImageryClassication Improvementsinsonartechnologyhaveallowedforlargepor tionsoftheseabottom tobescannedquickly.Nevertheless,identifyingandclass ifyingobjectslyingonthe sea-bedisstilladauntingtaskforhumanoperatorsespecia llywhentheamountofdata toinspectislargeandthepossibletargetsthatneedtobeid entiedaresmallinsizesuchasmines.Specialinterestisshowninautomatingthese techniquesintheeldof mine-countermeasureswheretheprocessofmine-huntingin volvestwomajorsteps: rst,identifyingpossiblemine-likeobjects,andsecond, classifyingthemasminesornot andalsothetypeofmine. Ourworkinvolvesanadditionalstepoffusingimagescoming fromthesame object.Duringthesurveyoftheseabottom,thesensor(s)ma ydetectthesameobject multipletimesduetooverlappingtracks-thezig-zagnatur eofthesurveys.Thus,over thefullcourse,thesameobjectmaybedetectedmanytimesre sultinginnumerous observationsthatneedtobefusedintoasingletargetlocat ionlabel.Inourwork,we focusonlyontheclassicationandfusionoftheobjectsass umingthattheyarealready detectedandprovidedintheformofsnippets. Objecthighlightandshadowarerstextractedfromeachsni ppet.Next,template matchingisappliedtoeachsetofextractedcontours.Based onpriorknowledgeabout theobjectsinthedataset,asetofprototypetemplateswere designedtorepresent thepossibleminecategoriesfoundinthedataset.Inadditi on,thetransformation 108

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spaceislimitedonlytoasetofpossibletransformationsth atatemplateneedstotake toaccommodatefortheshapevariabilityinthesnippetsdue toobjectpositionand orientationwithrespecttothesonarsensor.Theclassica tionusesournon-linear matchlterssincethesonardatahasveryhighnoiselevel,t heobjectcontoursmay havepartiallyoccludedareas,andtheboundariesmaynotha vebeenextracted correctlyduetothestructureoftheseabottomorotherimpe rfectionsfromthesonar sensor.Wetreatanyocclusionsorimperfectionsasoutlier s.Intheresultssectionwe willshowtheaccuracyoftemplatematchinganditscontribu tiontoourapplicationasan objectfeature.6.1.1DataDescription Theoriginalsonarimageshavealreadybeensegmentedintos nippetscontaining anobjecthighlightanditsshadow,whicharisefromthepres enceorlackofacoustic reverberations,alongwithsomeofthebackground.Therear e360imagesbelongingto oneofthethreeclasses:cube,cylinder,andcone.Theimage saretakeninarangeof10to106meters,aspectangleof0to360degrees,andtheyare8bitgreylevel.Fig. 6-1 showsseveralexamplesofthesnippetsprovided.Here,weha vefourdifferentexamples ofthesameobjectviewedfromdifferentanglesandranges. Figure6-1.Examplesofthesameobjectviewedfromfourdiff erentanglesandranges. 109

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6.1.2TemplateGeneration Simpletemplatematchingtechniquesdonotworkinthisscen ario.Extracting thecontoursoftheobjecthighlightandshadowfromdiffere ntsnippetsandmatching themisnotfeasible.Therearenoafnetransformationstha tcouldmatchtheshadow contoursoftwodifferentsnippets.Thisisduetotheshadow beingcreatedfroma3D objectwhereasthecontoursarerepresentationsin2D.Toov ercomethisproblem, wedesigned3Dmodelsforeachofthecategories.Theshadowc ontourwasthen generatedbyprojectingtheobjectontothegroundbasedona pointofview.Thepoint ofviewprojectionfollowstheactualsynthesisofsonarima ges,whereaside-scansonar imageissynthesizedonelineatatimeasthesonarshfollow sthepath.Similarly, weusealineratherthanasinglepointasourprojectionview point.Fig. 6-2 showsthe 3Dtemplate,inthiscaseacube,inlightgreenandthepointo fview(lineofview)in darkgreen.Thehighlightandshadowcontoursextractedfro mthesnippetareshown inblueandblackcolor(noticeoutliersinthehighlightdue tosonarimperfections).The 3Dtemplatehighlightandshadowprojectionsareshowninma gentaandredcolor (noticethattheshadowprojectionfromthe3Dtemplatecann otbeachievedbysimple transformationsofasquareshape).Sinceasnippetcontain sinformationfromboththe objectanditsshadow,weusebothsetsofcontours,highligh tandshadow,duringthe templatematching.6.1.3ShapeMatchingAdaptationtotheApplication HighlightIntheside-scansonarimages,anobjectisviewedfromdiffe rentanglesand ranges,thustheonlytransformationsthathavebeenapplie dtotheobjectarerotation, scale,andtranslation.Therefore,webreakdownthetransf ormationmatrixtothethree fundamentalafnetransformations.Focusingontheattrib utesoftheseoperations helpsusconstrainthetransformationmatrixspace,andind oingso,speedupthe convergencerate. 110

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-60 -40 -20 0 20 40 60 -50 0 50 0 20 40 60 80 highlight highlt tmplt shadow shdw tmplt 3D template pointofview Figure6-2.Overlayofa3Dtemplatemodelanditsshadowproj ectionontheobject highlightandshadowcontoursextractedfromasnippet. Weconsidertheobjectofinterestasarotated,scaledandtr anslatedversionof oneoftheexistingpatterns.Throughaxednumberofparame terupdates,givenobjectYwecanidentifyiftheXpatternispresent.Inourapplication,Xisaperpendicular projectionofthe3Dobjectinto2D.Theparametersareupdat edbasedonmaximizing thecorrentropybetweentheobjectofinterestandthe`tran sformed'patternusingasa costfunctionJ ( A d ) = NXi =1 exp ky iAx idk2 22!,(6–1) wherey iandx iarethei thcolumnentriesofYandXrespectively.Thefunctionis maximizedwithrespecttothescalingandrotationmatrix:A =264ab b a375,(6–2) andthetranslationvectord.InordertoupdatethematrixA,wetakeintoconsideration therepetitionoftheelementsaandb,whicharercos andrsin whereristhe scalingfactorand istherotationangle.There,wedifferentiatewithrespect tothe 111

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vector[ a b ] Tinstead.Differentiatingwithrespecttothisvectorandeq uatingthegradient tozeroresultsinthefollowingxedpointupdaterule: 264a b375 NXi =1 z i exp ky iAx idk2 22! NXi =1kx ik2 exp ky iAx idk2 22!,(6–3) withz i =264x i 1 y i 1 + x i 2 y i 2 x i 1 y i 2 + x i 2 y i 1375,(6–4) wherex ikandy ikarethek thcomponentsofthex iandy irespectively.Similarlyby differentiatingwithrespecttodandequatingthegradienttozerowegettheupdaterule ford:d NXi =1 ( y iAx i ) exp ky iAx idk2 22! NXi =1 exp ky iAx idk2 22!.(6–5) ShadowTheshadowcontourmatchingismoredifcultthanhighlight matchingbecause inordertocreatethecorrectshadowshapeweneedtoconside r3Dobjecttemplates andprojecttheminto2Dbasedonapointofviewrepresenting thesonarsensor hittingtheobject.Tosimplifytheproblem,weconsidersep aratecostfunctionsfor updatingtherotationmatrixandthepointofview.Toupdate therotation/scaling matrix,thecorrentropycostfunctioniscomputedintermso ftheshadowcontoursh = [ sh x sh y ] Tandthetemplateprojectedshadow,whichisafunctionofthe pointof viewp = [ p x p y p z ] T,templateobjectc = [ c x c y c z ] T,andtherotation/scalingmatrixAasshownbelow:J ( A ) = NXi =1 e ksh if ( p i Ac i )k2 22 .(6–6) 112

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Thescaling/rotationmatrixAbecomesA =266664ab 0 b a 0 0 0 r377775,(6–7) wherewehaveanadditionalrow/columncountingforthethir ddimension.Theelementrrepresentsthescalingfactorinaandb.Again,differentiatingwithrespecttothevector[ a b ] Tandequatingthegradienttozeroresultsinthefollowingx edpointupdaterule: 264a b375 NXi =1 z i e ksh if ( p i Ac i )k2 22 NXi =1 p zi ( c 2 xi + c 2 yi ) e ksh if ( p i Ac i )k2 22 ,(6–8) withz i =264( sh x c x + sh y c y )( p zr c z ) + r c z ( p x c x + p y c y ) ( sh x c xsh y c y )( p zr c z ) + r c z ( p x c xp y c y )375.(6–9) Therisupdatedusing p a 2 + b 2. Toupdatethepointofview,thecostfunctionthenbecomesJ ( A d ) = NXi =1 e ksh iMc idk2 22 ,(6–10) wheresh i = [ sh x sh y ] T iandc i = [ c x c y c z ] T iarethei thcolumnentriesoftheshadow contourandobjecttemplaterespectively.Noticehowthetw ovectorshavedifferent dimensionsrepresentingthetwodifferentspaces.Inaddit ion,thevectorc ihasafourth dimensionwhichisneededtohomogenizethecoordinates.Th efunctionismaximized 113

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withrespecttotheprojectionmatrixM =266666664 p z 0 p x 0 0p z p y 0 0 0 0 0 0 0 1p z377777775,(6–11) resultinginthexedpointupdatep NXi =1 z i e ksh iMc idk2 22 ,(6–12) wherez i =2666664( c x p zsh x ( p zc z ) ) c z ( c y p zsh y ( p zc z ) ) c z c z( p xc x )( sh xp z ) + ( p yc y )( sh yp y ) ( p xc x )( sh xc z ) + ( p yc y )( sh yc y )3777775.(6–13) 6.1.4Results Werstextractthehighlightandshadowcontoursfromeachs nippetusingsimple imageprocessingmethods.Fig. 6-3 (A)showstheextractedhighlightandshadow contours.Theobjecthighlightcomprisesthebrightpixels inthesnippet,andasshown intheFig. 6-3 ,itisnotjustacontouraroundtheobject.Thisisduetothed ifculty ofidentifyingtheactualobject.Sincethehighlightconta insabandofpixelswhere theobjectboundaryisexpectedtobe,thesepixelsareindiv iduallyweighedbasedon theirbrightnesswhenusedaspointsonthehighlightcontou r.Also,notethatdueto imperfectionswiththesonarmethoditself,oftenthehighl ightwillcontainpixelclusters thatmightbefarfromtheactualobject,asshowninblueclus tersinFig. 6-3 (B).Another pre-processingstepcouldbetakentoremovetheseclusters fromthehighlight,but sincecorrentropyisrobustagainstimpulsivenoise(outli ers)andtoavoidremoving usefulclusterssuchastheoneonthebottomleftofthehighl ight,theyareincludedin thehighlightcontour. 114

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Theobjectshadowissimplertoidentifysincetheshadowisa darkspotonthe imagebehindtheobjectlocation,theareawherethereisnor eturnfromthesonar beam.Eventhoughtheshadowcontourissimplertoextractco mparedtothehighlight,it isnotalwaysaccuratelyextractedbecausethepixelintens itydifferenceontheshadow edgesvariesdependingonthetextureoftheterrainwhereth eobjectislocated.Thatis whywedonotalwaysgetstraightlinesaroundtheshadowcont our. Asnippet Btemplatematching Figure6-3.Anexampleofasnippetwheretheobjecthighligh tandshadowcontoursare extracted,andthe3Dtemplatematching. Oncetheobjecthighlightandshadowareextracted,weapply the3Dtemplate matchingasshowninFig. 6-3 (B).Theobjecthighlight(blue)iscomparedagainstthe templatehighlight(magenta).Toaccountforthenot-so-go odsegmentationprocess, eachpixelontheobjecthighlightisweighedbasedonitsint ensity.Inthismanner,the pixelsthatdonotbelongtotheobjecthighlightwillhavelo wintensityvalues,andasa resultlittleeffectonthetemplatetransformation.Inadd ition,thepixelclustersthatare locatedfarfromthetemplatecontour,donotfallunderthe` observationwindow',andas aresultareconsideredoutliersandsimplyignored. 115

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Theobjectshadow(black)iscomparedagainsttheprojected shadowofthe template(red).Wewouldlikethealgorithmtofocusprimari lyonthebottomofthe shadow,becausethatisthepartthatcontainsmostofthesha peinformation.The shadowpartclosetotheobjecthighlightwillusuallyhaved ifferent,irregularshapes dependingonthehighlightandotherimperfectionsinthatp artofthesnippet,asshown inFig. 6-3 (B).Thesidesusuallywillbelongstraightlinesreecting theprojectionof theobject.Therefore,mostoftheinformationabouttheobj ectiscontainedatthevery bottomoftheshadow.Asaresult,weweightheshadowcontour pointsbasedontheir distancefromthebottom. Thehighlightandshadowoftheobjectinthesnippetarethen comparedagainstall 3Dmodels.Fig. 6-4 showstheresultsofcomparinganobjectagainsttwotemplat es: (A)againstthecorrecttemplate,and(B)againstawrongtem plate.Theincorrect templatedeformstoadjusttotheshadowshapebutduetoitss hapeconstraints,it canmatchwellonlyonthesides.Thisreiterateswhatweexpl ainedaboveaboutthe shadowmatching.Theimportantinformationabouttheobjec tiscontainedonlyatthe bottompartoftheshadow,therestisjustanelongationofth eshadowdependingon therelativeheightanddistanceofthesonarsensorfromthe objectofinterest.Thus,the similaritymeasureshouldweighthesedifferentpartsofth eshadowdifferentlybyputting emphasisonthebottompartandsimplyignoringthetoppart. Similartothebottom oftheshadow,thepoormatchisalsonoticedonthehighlight wheretoaccommodate asmuchoftheshapeaspossiblethetemplaterotatesintoadi agonalresultingina suboptimalsolution. Table 6-1 showstheclassicationresultsofthedatasetbasedontemp late matching.Thehighestcorrectclassicationwasforthecyl indershapessincetheir highlightsandshadowswereverydifferentfromthoseofobj ectsofcubeorconetype. Thelowestcorrectclassicationratewasfortheconeshape sbecausetheirhighlight wasverydifculttocorrectlyclassify.Manyoftheconeins tancesresembledacube 116

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Acorrect Bwrong Figure6-4.Finalresultsof3Dtemplatematchingonthe(A)c orrecttemplateand(B)an incorrecttemplate.Objecthighlightandshadowareshowni nblueandcyan whiletemplatehighlightandshadowandareshowninmagenta andred. observedata45 angle.Theresultsalsoshowthatmostoftheconfusionforth econe shapesoccurredonthecubetemplate. Table6-1.Confusionmatrix True n Predicted cubecylindercone cube90.00%4.17%5.83%cylinder2.50%96.67%0.83%cone17.50%3.33%79.17% Theshapesimilarityresultswerenallyprovidedasobject featurestoadynamic tree(DT)graphicalmodel[ 93 94 ].TheDTmodelfusedtheinformationfrommultiple featuregroups:location,texture,andshape,intoanon-re dundantrepresentation. Table 6-2 providesasummaryoftheresultsofthiswork(formoreinfor mationonthe specicsofthisworkread[ 95 96 ]).Thetablesummarizestheresultsof2700scenarios ofvariousobjecttypesunderdifferentbackgroundsandloc ationdistributions.Therst groupofcolumnsprovidesresultsfordatasetscomposedofa singleobjecttype,thus ourshapefeaturesarerendereduselessinthesescenarios. However,thesecondgroup ofcolumnsprovidesresultsfordatasetswithdifferenttyp esofobjects.Noticethejump intheclassicationaccuracywhenthedatasetcomprisesdi fferentobjecttypes. 117

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Table6-2.AsummaryoftheDTresults ObjectSameTypeDifferentType BackgroundCorrectIncorrectUnasgndCorrectIncorrectUn asgnd SameType87.52%8.85%3.63%95.21%3.19%1.61%SimilarType89.90%6.95%3.15%95.33%3.30%1.37%DifferentType91.17%6.10%2.74%95.74%2.97%1.29% 6.2RemoteContactlessStereoscopicMassEstimationSyste m Therearemanyapplications,whereacontactlesssystemand methodfor estimatingthemassorweightofatargetobjectisrequired. Inourapplication, measuringtheweightofanimalsissignicanttoanimalheal thingeneral,development andyieldincattleranches,statisticalanalysisinzoolog y,etc...Inmanyofthese applications,weighingtheanimalisdifcultorevenimpos sible(i.e.animalinthewild). Evenincaseswhereitispossible,theprocessoftakingthea nimaltoascaleisvery stressfultotheanimalandoftenresultsinweightloss.For suchcases,wedesigned amethodwheretheanimal,thetargetobject,isimagedandas patialrepresentation ofthetargetanimalisderivedfromtheimages.Avirtualspa tialmodelisprovidedof atemplateobjectofthesamecategoryastheanimal.Thetemp latemodelisthen transformedtooptimallytthespatialrepresentationoft heindividualanimal.The massorweightofthetargetobjectisthenestimatedasafunc tionofshapevariables characterizingthetransformedtemplateobject.6.2.1DataDescription Astereo-visioncamerasystemisusedtocapturetwoimageso ftheanimalfrom differentpointsofview,asshowninFig. 6-5 .Theseimagesarethenusedtogenerate astereoscopicimage.Thisstereoimageincludesdepthdata correspondingtoan estimateddistancefromtheplanetoeachpointontheobject spresentintheimage. Thisdepthinformationisthenusedtogeneratea3Dpointclo udofthe`scene'(allthe objectspresentintheimage),asshowninFig. 6-6 .Next,the3Dpointcloudiscropped tosubstantiallyremoveunwantedpartsofthe`scene'andle aveonlythedatapoints 118

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representingtheobjectofinterest,theanimalatthecente rofthe3Dpointcloud.Fig. 6-7 showsthecroppedanimal.Notethatthecroppingmethoddoes notcompletely removethegroundorallthenoisesurroundingtheanimal. Aleftimage Brightimage Figure6-5.Exampleofthetwoimagesprovidedbythestereos copiccamera. Figure6-6.The3Dpointcloudextractedfromthestereoscop icimage.Cowispresent onthebottom-centerportionofthepointcloud. 119

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Figure6-7.Thecroppedcowfromthe3Dpointcloud. 6.2.2Algorithm SincethecroppingmethodisnotalwaysrobustasshowninFig 6-7 ,somemore preprocessingmightberequiredtoremovethegroundandoth erclutteraroundthe animal.Mostofthesparsecluttercanbeignoredbythealgor ithmitself,byconsidering themasoutliers,howeverwhentheseunwantedsegmentsarea signicantpartofthe object,suchasthegroundinthiscase,theywillaffectthe nalalignmentiftheyarenot removed. Fig. 6-8 (a)showstheanimal3Dpointsetafterallthepreprocessing .Notethat thereisstillsomeclutterleft,butthealgorithmshouldbe abletoavoidthemby consideringthemasnoise/outliers.The3Dpointsetisthen comparedagainsta3D templatemodeloftheanimalcategory.Fig. 6-8 (b)showsthemodelthatweused. Standardrigidandnon-rigidtransformationareappliedto thetemplatemodeltot the3Dobjectpointcloud.Fig. 6-8 (c)showsthenalalignmentofthetemplatemodel againstthe3Dpointset.Notethatthecowpointsetisonlyha lfacow,ascanbeseen fromthetopviewinFig. 6-8 (d),whichmakesitdifculttogetaverygoodmatch.In 120

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addition,thistemplatemodelcannotaccountformovements oftheheadorthelegs ofthecow.Thenon-rigidtransformationaccountsforparto fthismotions,butcannot completelytakecareofit. -0.5 0 0.5 1 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 A3Dpointcloud -0.5 0 0.5 -0.1 0 0.1 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 B3Dtemplate -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.2 0 0.2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Cnalalignment -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.2 0 0.2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Dtopview Figure6-8.Exampleofalignmentofthe3Dpointcloudagains tatemplate. 6.2.3Results Thealgorithmwastestedwithagroupofthirty-onecattleth atrangedinweight frombetween174and523pounds.Forevaluationandmodel-buildingpurposes, theweightofeachindividualanimalwasmeasuredusingacon ventionallivestock scale.Thedifferencebetweenthemeasuredweightoftheind ividualanimalandthe estimatedweightoftheindividualanimalwascalculated.T heaverageabsoluteerror was26.09pounds,whichcomparestoanaveragecattleweightof366.4pounds.The 121

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testsurprisinglydemonstratedthatevenwithasinglecrud evirtualmodelappliedto cattlewithawiderangeofweights,thesystemaccuratelypr edictedtheweightofthe cattlewithin30pounds,anerrorrateoflessthan10%whichisthedesiredstandard. Inaddition,wetestedonanotherheaviersetofanimalswher emultipleimages weretakenofeachanimal.Fig. 6-9 showsanexampleofsixdifferentpointcloudsof thesameanimaltakenfromslightlydifferentperspectives .Noticethedifferencesin eachpointset.Weevaluatedthepointcloudsfromeachanima landthesummarized resultsareshowninFig. 6-10 .Thegureshowstheactualweightofthecowinaddition totheaverageweightestimatedfromthealgorithmandthest andarddeviation.We noticethatinmanycasestheestimatedaverageweightishig herthantheactual weightoftheanimal.Thisisduetotheequationthatisusedt ocomputetheweight. Theequationdependsonthedimensionsoftheanimal:length ,height,andwidthbut doesnottakeintoconsiderationanythingelseabouttheani malsuchasitsdensity, muscle-to-fatratio,etc...whichplaycriticalrolesinth eweight.Duetothisassumption, theestimatedweightsforthisparticularsetofanimalsten dtobelargerthantheactual weight.However,lookingatthestandarddeviation,wenoti cethatoveralltheweightsfall withinthesamerange.And,thisrangeiswithin10%oftheactualweightoftheanimal. Thisshowsthatthealgorithmitselfworkswelltoalignthet emplateagainsttheextracted pointclouds,andalltheinstancesofthesameanimalprovid esimilardimensions. LookingbackatthepointcloudsinFig. 6-9 wenoticethatthereisalotofvariation amongsomeofthepointsets.Thisvariationwillaffectthe nalalignmentofthe templatewiththepointcloud,andasaresult,itwillaffect theweightestimation.To mitigatethevariationaleffectandtoalsoremovesomeofth enoise,werananother setofexperimentswherewerstgroupedthepointsetstoget herusingourgroupwise alignmentalgorithms.Fig. 6-11 showsthesixpointcloudsinFig. 6-9 plusanadditional oneandtheirnalunbiasedalignment.Fig. 6-11 (A)-(G)showthepointcloudsinitialand 122

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Apointcloud1 Bpointcloud2 Cpointcloud3 Dpointcloud4 Epointcloud5 Fpointcloud6 Figure6-9.Examplesofdifferentpointcloudsofthesamean imal. nalpositions.Fig. 6-11 (H)showstheinitialalignmentofallthepointcloudsandFi g. 6-11 (I)showsthenalalignment. Theactualweightofthiscowwas576pounds.Theestimatedweightsusingeach pointcloudbeforeandafterthealignmentareshowninTable 6-3 .Theestimatedweight ofthealignedpointcloudsisclosertotheactualweighttha nthatbeforealignment. However,wenoticedthattheestimatedweightsbeforetheal ignmenttendtobehigher thantheactualweight,whereastheestimatedweightsafter thealignmenttendtobe lower.Oneofthereasonsforthisdifferenceisduetothenoi selevelwhichisreduced afterthealignment.Anotherreasonisduetotheequationus edtoestimatetheweight. Thedepth(width)oftheanimalisthemostsensitivedimensi ontoestimatebecauseof thetechnologyusedtogeneratethepointclouds.Toensuret hatouralgorithmwasnot shrinkingthepointsets,eventhoughweusedthescale-inva riantEntropyalgorithm,we 123

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 400 500 600 700 800 900 1000 1100 1200 weight actual weight estimated Figure6-10.Resultsoftheseconddataset.Graphshowsthea ctualweightsandthe averageestimatedweightswiththestandarddeviation. showthepointcloudsfromatopviewwherethewidthoftheani malisclearlynoticed. Fig. 6-12 redisplaysthepointcloudsviewedfromthetop.Thealigned pointclouds,in certaincases(i.e.pc4),displaysomeshrinkageinthewidt hdimension,butwecannot determineifthisshrinkageisduetothevariationsinthesh apeofthepointcloudsinthis dimensionorduetothealgorithmwehaveused. Table6-3.Estimatedweights PC1PC2PC3PC4PC5PC6PC7Average before664623657611709688636655 35 after506502504502545508511511 15 Ourgoalwastoprovideanestimateoftheweightwithinaminu teoftakingthe stereoimage.Toaccomplishthis,thealgorithmwassimpli edtremendously.The afnetransformationmatrixwasbrokendowntothethreetra nsformation:rotation, scale,andtranslation.Solvingforeachtransformationin dividuallynotonlyspedupthe convergenceratebyconningthetransformationspace,but alsoprovidedameansof calculatingtheweightoftheanimal.Thenon-rigidtransfo rmationwasremovedfromthe onlineprocess. 124

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Apointcloud1 Bpointcloud2 Cpointcloud3 Dpointcloud4 Epointcloud5 Fpointcloud6 Gpointcloud7 Hinitialposition Inalalignment Figure6-11.Exampleofunbiasedgroupwisenon-rigidalign mentofthe3Dpointclouds. Becauseoftheunreliabilityofanindividualweightestima te,wedecidedtotake multipleshotsoftheanimalandaverageoftheindividuales timates.Wewantedtoapply thegroupwisealignmenttorstcomewithanaveragepointcl oudbeforethealignment, howeverthegroupwisealignmentprocesswasverylongcompa redtothesumof theindividualalignmentsagainstthetemplate.Forthisre ason,wedecidedtosimply averageovertheindividualestimatesinsteadofgroupalig nment.Ournalalgorithm 125

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 3 3.1 3.2 3.3 3.4 3.5 Apointcloud1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 2.9 3 3.1 3.2 3.3 3.4 3.5 Bpointcloud2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4 3.45 3.5 Cpointcloud3 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4 3.45 3.5 Dpointcloud4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 Epointcloud5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 3 3.1 3.2 3.3 3.4 3.5 3.6 Fpointcloud6 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 3 3.1 3.2 3.3 3.4 3.5 3.6 Gpointcloud7 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 Hinitialposition -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 3 3.1 3.2 3.3 3.4 3.5 Inalalignment Figure6-12.Exampleofunbiasedgroupwisenon-rigidalign mentofthe3Dpointclouds viewedfromthetop. wasableofperformingunderoneminute(fromthemomentofta kingthepicturetothe nalweightestimate),andtheresultswerewithin10%oftheanimalweight. 126

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CHAPTER7 CONCLUSIONSANDFUTUREDIRECTIONS 7.1Summary Inthisthesis,wehavecontributedseveralnewsimilaritym easuresbasedon informationtheorytocompareeitherapairorgroupofshape s.Ouralgorithmsrelyon featuresextractedfromtheshapes,andinparticularpoint features.Theyareapplied eitherdirectlytothepointsets,correntropy,ortoPDFrep resentations,thedivergence measures. Therstalgorithmwasbasedonanon-linearsimilaritymeas ureknownas correntropy.Correntropymeasuresthesimilarityoftwopo intsetsinaneighborhood ofthejointspacecontrolledbythekernelbandwidth.Theke rnelinducesnon-linearity tothemeasurewhichprovidesinformationabouthigherorde rmomentsofthejoint PDF.Thisinturnyieldssolutionsthataremoreaccuratetha nmostoftheothermethods whichrelyonMSEasthecostfunction. Correntropyisapoint-basedsimilaritymeasurewhichmeas uresthesimilarity alongthelinex = yinthejointspace.Consequently,itdependsonpredetermin ed correspondencebetweenthetwopointsets.Sincethecorres pondencebetween pointsetsisusuallynotknownapriori,weintroducedanewm ethodtodeterminethe correspondence.Themeasureiscalled`surprise'andprovi dessubjectiveinformation abouteachpointontheset.Itmeasuretheinformationgainf romthesamplewhichwe callthesurprisefactor.Thisisanadditionalfeatureadde dtothepointswhichsimilarto othercontext-basedshapefeaturesprovidesawayofdeterm iningthecorrespondence betweenthepointsrepresentingtheshapecontour. Tocompletelymitigatethecorrespondenceproblem,wedesi gnedanotheralgorithm wherepointsetsarerepresentedasPDFs.Thesimilaritymea sure,knownasthe Cauchy-Schwarzdivergence,measuresthedistortionbetwe enthereferencepoint 127

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setandthetemplatethroughthecross-entropybetweenthet wosets.Thealgorithm indirectlyemploysaglobal-to-localsearchandsoftassig nmentforthecorrespondence. Toaccommodateforgroupwiseregistration,wedesignedtwo morealgorithms thatcomparemultiplepointsets(PDFs)simultaneously.Th erstalgorithmisbased onH ¨ older'sinequality.ItisadirectextensionoftheCauchy-S chwarzdivergence.This methodprovedtobeverycomputationallyexpensive.Afewap proximationtechniques wereprovidedwhichsimplifythecalculationsandstillpro videveryaccurateresults. ThesecondalgorithmisbasedonR enyi'ssecondorderentropyandJensen's divergence.Itcomparestheoverallentropyoftheunionofs hapesagainsttheweighed sumoftheindividualentropies.Thismethodprovedtobecom putationallyverysimple andalsoprovidesveryaccurateresults.Wealsoprovidedam orerobustextension whichisscaleinvariant.Thescaleinvarianceisreachedby dividingtheentropybythe standarddeviation. Allourmethodshaveafreeparameter,thekernelbandwidth. Weuseddeterministic annealingtoslowlyreducethevalueofthekernelbandwidth .Thispreventsthe algorithmsfromgettingstuckinlocalextrema.Inaddition ,itprovidesafasterconvergence rate.Theannealingalsomakesthemethodsrobustagainstno iseandoutliers/ occlusion. Tocomputethenon-rigidtransformations,weprovidedtwoa lternatives:using thin-platesplinesandGaussianRBFs.TPSprovidesmoreofa globaleffecton thenon-rigidity,andasaresult,incertaincasesitperfor msworsethanGaussian RBFwhichprovidesalocalizedeffect.However,theGaussia nRBFreliesonafree parameterthatcontrolsthelocalization.Determiningthe valueofthisparameteris challengingandproblemdependent. Inaddition,weneedtoconstrainthenon-rigidtransformat ionsothatthestructural integrityofthenaltransformedshapesisnotdestroyed.T hepenalizationtermrequires anotherparameter,wherehighvalueswouldrestrictthenon -rigidtransformationand 128

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lowvalueswouldallowittobetooexible.Tocontroltherig idityofthetransformation, similartothekernelbandwidth,weslowlyreduceitsvalues othatatthebeginningwe areprimarilyfocusedonglobaltransformationsandlaterw eslowlyintroducenon-rigid transformationstorenethemapping. Forallouralgorithmswehaveprovidedclosed-formsolutio ns,whichmakesthem easytocomputeandfeasibleformanyreal-worldapplicatio ns.Inthisthesis,the algorithmswereappliedtotwoapplications:1)extracting objectfeaturesfromsonar imagestogroupsimilarobjects,and2)estimatingtheweigh tofanobjectbydetermining itsdimensions. 7.2FutureDirections Inthegroupwisematchingexperiments,wecomparedagainst severalmeasures thatusecumulativedistributionfunction(CDF)insteadof PDF.TheclaimisthatPDF maynotbedenedeverywhere,whereasCDFbeinganintegralm easure,ismore robust.In[ 87 ],Chenetal.demonstratedthatCDFprovidesmoreaccurater esultsthan PDF.However,oneoftheproblemsofusingCDFinsteadofPDFi stheincreasein computationalcostandimplementationcomplexity.Nevert heless,itwouldbeinteresting toimplementourmethodsusingCDFandevaluatetheperforma nceimprovement. Then,basedontheimprovementlevelandcomputationalcomp lexitywecandetermine ifusingCDFisbenecialornot.Inaddition,Wangetal.[ 86 ]usedanalternative measureofinformationknownascumulativeresidualentrop y(CRE)insteadofCDF.It wouldalsobeinterestingtoseetheimprovementinperforma nceduetoCRE. Finally,wewouldliketoapplythesemethodstomedicalimag eregistration.Several ofthemethodsthatwecomparedagainsthavealreadybeenapp liedtomedicalimage registrationwithverypromisingresults.Sinceourmethod soutperformedthem,itwould beinterestingtoapplyouralgorithmstoimageregistratio nandseeifthereisany signicantcontributionovertheexistingmethods. 129

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BIOGRAPHICALSKETCH ErionHasanbelliureceivedaBachelorofScienceandaMaste rofSciencedegree inComputerSciencefromJacksonvilleStateUniversityin2 002and2004,respectively. HethenreceivedaMasterofScienceandaDoctorofPhilosoph ydegreeinElectrical andComputerEngineeringfromUniversityofFloridain2008 and2012,respectively. Hisresearchinterestsincludesignalprocessing,machine learning,informationtheoretic learning,computervisionandpatternrecognition. 137