Tensors in image processing and computer vision L. Cammoun, C.A. Castaño-Moraga, E. Muñoz-Moreno, D. Sosa-Cabrera, B. Acar (auth.), Assoc.Prof. Santiago Aja-Fernández, Assoc.Prof. Rodrigo de Luis García, Dr. Dacheng Tao, Dr. Xuelong Li (eds.) PDF
Preview Tensors in image processing and computer vision L. Cammoun, C.A. Castaño-Moraga, E. Muñoz-Moreno, D. Sosa-Cabrera, B. Acar (auth.), Assoc.Prof. Santiago Aja-Fernández, Assoc.Prof. Rodrigo de Luis García, Dr. Dacheng Tao, Dr. Xuelong Li (eds.)
Advances in Pattern Recognition Forfuthervolumes: http://www.springer.com/series/4205 · · Santiago Aja-Ferna´ndez Rodrigo de Luis Garc´ıa · Dacheng Tao Xuelong Li Editors Tensors in Image Processing and Computer Vision 123 Editors Asst.Prof.SantiagoAja-Ferna´ndez Asst.Prof.RodrigodeLuisGarc´ıa UniversidaddeValladolid UniversidaddeValladolid EscuelaTe´cnicaSuperiorde EscuelaTe´cnicaSuperiorde Ingenierosde Ingenierosde Telecomunicacio´n Telecomunicacio´n CampusMiguelDelibes CampusMiguelDelibes 47011Valladolid 47011Valladolid Spain Spain [email protected] [email protected] Dr.DachengTao Dr.XuelongLi BlkN4,02b-67 UniversityofLondon SchoolofComputerEngineering BirkbeckCollege NanyangTechnologicalUniversity SchoolofComputerScience& Singapore639798 InformationSystems [email protected] MaletStreet LondonWC1E7HX Serieseditor UnitedKingdom ProfessorSameerSingh,PhD xuelong [email protected] ResearchSchoolofInformatics LoughboroughUniversity Loughborough,UK ISSN1617-7916 ISBN978-1-84882-298-6 e-ISBN978-1-84882-299-3 DOI10.1007/978-1-84882-299-3 SpringerDordrechtHeidelbergLondonNewYork BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressControlNumber:2009926807 (cid:2)c Springer-VerlagLondonLimited2009 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicencesissued bytheCopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbe senttothepublishers. Theuseofregisterednames,trademarks,etc.,inthispublicationdoesnotimply,evenintheabsenceofa specificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefree forgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinformation containedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissions thatmaybemade. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Overthepastfewyears,tensorprocessingtoolshavebecomemoreandmorepopu- larinthefieldsofcomputervisionandimageprocessing,andtensor-valuedimage modalitieshavealsobeenmorecommonlyemployed,withtheremarkableexample ofDiffusionTensorMagneticResonanceImaging. However, tensor applications and tensor processing tools arise from very dif- ferent areas. This can prevent important advances from rapidly spreading over the scientificcommunity.Eventhoughnoveldiscoveriescangreatlybenefitmanyhet- erogeneous fields such as medical image processing or multilinear analysis, these advances are too often kept within the areas of knowledge where they were first employed. Given this fact, a compilation of some of the most recent advances in tensor processing can be a valuable tool for the scientific community, thus providing a usefulreferencestateoftheartfortensorprocessingapplicationsaswellasageneral insightontheareaswheretensorprocessingisbeingsuccessfullyapplied. The idea behind this book started after the Tensor Workshop held at the IEEE ComputerSocietyConferenceonComputerVisionandPatternRecognitionheldin Anchorage, Alaska, in June 2008. This workshop gathered experts from different fields working on tensor processing, and some of the ideas presented there were furtherdevelopedtoghetherwiththeresultsoftheTensorWorkshopheldearlierin Gran Canaria (Spain) in November 2006, which was sponsored by the SIMILAR NetworkofExcellenceoftheEuropean6thFrameworkProgramme. Asthisbookcomprisestheoreticaladvancesandapplicationsregardingveryhet- erogeneous areas of image processing and computer vision, it has been organized into five parts, which are nevertheless preceeded by an introductory chapter about thedifferentapplicationsoftheuseoftensorsinsignalprocessing.PartIisdevoted to the use of tensors and tensor field processing in general. The processing tools describedinthesechapterscanbeappliedinanumberofdifferentapplications.In PartII,twotensortechniquesforimageprocessingarepresented.Later,PartIIIfo- cusesontheuseoftensorsincomputervisionapplications,suchascameramodels or multilinear applications. As medical imaging is one of the areas that has taken more advantage of the advances of tensor processing, Part IV is dedicated to this v vi Preface issue,collectingapplicationsfromDiffusionTensorMagneticResonanceImaging tostraintensorestimationincardiacanalysisorelastographyimaging.Finally,Part Visdevotedtostorage,visualizationandinterfaceswithtensors,anissueofconsid- erableimportancesincethisnewdatamodalitypresentsparticularitiesthatrequire newapproachestotheseotherwisetraditionalproblems. The preparation of this book has been an arduous and difficult task. We would like to thank all the authors for their great effort and dedication in preparing their contributions. Also, all the reviewers, and specially the members of the editorial boardsoftheTensorWorkshopsatCVPR’08andinGranCanariain2006,deserve ourutmostgratitude.Thanksalottoallofthem. Boston,MA RodrigodeLuisGarc´ıa Fall2008 SantiagoAjaFerna´ndez Contents Preface ........................................................ v AReviewofTensorsandTensorSignalProcessing.................... 1 L.Cammoun,C.A.Castan˜o-Moraga,E.Mun˜oz-Moreno,D.Sosa-Cabrera, B.Acar,M.A.Rodriguez-Florido,A.Brun,H.KnutssonandJ.P.Thiran PartI TensorsandTensorFieldProcessing SegmentationofTensorFields:RecentAdvancesandPerspectives ...... 35 RodrigodeLuis-Garc´ıa,CarlosAlberola-Lo´pezandCarl-FredrikWestin AVariationalApproachtotheRegistrationofTensor-ValuedImages .... 59 SebastianoBarbieri,MartinWelkandJoachimWeickert QualityAssessmentofTensorImages............................... 79 Emma Mun˜oz-Moreno, Santiago Aja-Ferna´ndez and Marcos Martin- Fernandez AlgorithmsforNonnegativeTensorFactorization .................... 105 StefanosZafeiriou PDE-basedMorphologyforMatrixFields:NumericalSolutionSchemes . 125 BernhardBurgeth,MichaelBreuß,StephanDidas,andJoachimWeickert PartII TensorsinImageProcessing SphericalTensorCalculusforLocalAdaptiveFiltering................ 153 MarcoReisertandHansBurkhardt OnGeometricTransformationsofLocalStructureTensors ............ 179 Bjo¨rnSvensson,AndersBrun,MatsAndersson,andHansKnutsson vii viii Contents PartIII TensorsinComputerVision Multi-ViewMatchingTensorsfromLinesforGeneralCameraModels... 197 SimoneGaspariniandPeterSturm BinocularFull-BodyPoseRecognitionandOrientationInferenceUsing MultilinearAnalysis ............................................. 215 PengandQian ApplicationsofMultiviewTensorsinHigherDimensions .............. 237 MarinaBertolini,GianMarioBesana,andCristinaTurrini ConstraintsfortheTrifocalTensor................................. 261 AlbertoAlzatiandAlfonsoTortora PartIV DiffusionTensorImagingandMedicalApplications ReviewofTechniquesforRegistrationofDiffusionTensorImaging ..... 273 Emma Mun˜oz-Moreno, Rube´n Ca´rdenes-Almeida and Marcos Martin-Fernandez PracticalandIntuitiveBasisforTensorFieldProcessingwithInvariant GradientsandRotationTangents .................................. 299 GordonL.KindlmannandCarl-FredrikWestin FromSecondtoHigherOrderTensorsinDiffusion-MRI .............. 315 AurobrataGhoshandRachidDeriche DT-MRIConnectivityand/orTractography?:TwoNewAlgorithms ..... 335 BurakAcarandErdemYo¨ru¨k StrainRateTensorEstimationinCineCardiacMRIBasedonElastic ImageRegistration .............................................. 355 GonzaloVegasSa´nchez-Ferrero,AntonioTrista´nVega,LucilioCordero Grande,PabloCasasecadelaHiguera,SantiagoAjaFerna´ndez,Marcos Mart´ınFerna´ndezandCarlosAlberolaLo´pez StrainTensorElastography:2Dand3DVisualizations ................ 381 Dar´ıo Sosa-Cabrera, Karl Krissian, Javier Gonza´lez-Ferna´ndez, Luis Go´mez-De´niz, Eduardo Rovaris, Carlos Castan˜o-Moraga and Juan Ruiz-Alzola PartV Storage,VisualizationandInterfaces SimilarTensorArrays–AFrameworkforStorageofTensorArrayData 407 Anders Brun, Marcos Martin-Fernandez, Burak Acar, Emma Munoz- Moreno, Leila Cammoun, Andreas Sigfridsson, Dario Sosa-Cabrera, Bjo¨rnSvensson,MagnusHerberthsonandHansKnutsson Contents ix UserInterfacestoInteractwithTensorFields........................ 429 SusanaMerino-CaviedesandMarcosMart´ın-Ferna´ndez T-flash:TensorVisualizationinMedicalStudio ...................... 455 J.Wiklund,V.Nicolas,P.Rondao,M.AnderssonandH.Knutsson Index .............................................................467 A Review of Tensors and Tensor Signal Processing L.Cammoun1,C.A.Castan˜o-Moraga2,E.Mun˜oz-Moreno4,D.Sosa-Cabrera2, B.Acar5,M.A.Rodriguez-Florido2,3,A.Brun6,H.Knutsson6,J.P.Thiran1 Abstract Tensorshavebeenbroadlyusedinmathematicsandphysics,sincethey areageneralizationofscalarsorvectorsandallowtorepresentmorecomplexprop- erties.Inthischapterwepresentanoverviewofsometensorapplications,especially thosefocusedontheimageprocessingfield.Fromamathematicalpointofview,a lotofworkhasbeendevelopedabouttensorcalculus,whichobviouslyismorecom- plexthanscalarorvectorialcalculus.Moreover,tensorscanrepresentthemetricof avectorspace,whichisveryusefulinthefieldofdifferentialgeometry.Inphysics, tensors have been used to describe several magnitudes, such as the strain or stress ofmaterials.Insolidmechanics,tensorsareusedtodefinethegeneralizedHooke’s law, where a fourth order tensor relates the strain and stress tensors. In fluid dy- namics, the velocity gradient tensor provides information about the vorticity and the strain of the fluids. Also an electromagnetic tensor is defined, that simplifies the notation of the Maxwell equations. But tensors are not constrained to physics andmathematics.Theyhavebeenused,forinstance,inmedicalimaging,wherewe can highlight two applications: the diffusion tensor image, which represents how moleculesdiffuseinsidethetissuesandisbroadlyusedforbrainimaging;andthe tensorial elastography, which computes the strain and vorticity tensor to analyze the tissues properties. Tensors have also been used in computer vision to provide informationaboutthelocalstructureortodefineanisotropicimagefilters. 1SignalProcessingInstituteEcolePolytechniqueFe´de´raledeLausanneSwitzerland Email:{leila.cammoun,JP.Thiran}@epfl.ch 2CenterforTechnologyinMedicine.DepartmentofSignalsandCommunciations UniversityofLasPalmasdeGranCanaria. 3CanaryIslandsInstituteofTechnology Email:{ccasmor,dario,marf}@ctm.ulpgc.es 4LaboratoryofImageProcessing.Univ.deValladolid(Spain) Email:[email protected] 5Electrical-ElectronicsEng.Dept.BogaziciUniversityIstanbulTurkey Email:[email protected] 6BiomedicalEngineeringDept.CMIVLinko¨pingsUniversitetSweden Email:{andbr,knutte}@imt.liu.se 1
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