Riemannian Computing in Computer Vision PDF
Preview Riemannian Computing in Computer Vision
Pavan K. Turaga · Anuj Srivastava Editors Riemannian Computing in Computer Vision Riemannian Computing in Computer Vision Pavan K. Turaga • Anuj Srivastava Editors Riemannian Computing in Computer Vision 123 Editors PavanK.Turaga AnujSrivastava ArizonaStateUniversity FloridaStateUniversity Tempe,AZ,USA Tallahassee,FL,USA ISBN978-3-319-22956-0 ISBN978-3-319-22957-7 (eBook) DOI10.1007/978-3-319-22957-7 LibraryofCongressControlNumber:2015954575 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia(www. springer.com) Contents 1 WelcometoRiemannianComputinginComputerVision............. 1 AnujSrivastavaandPavanK.Turaga PartI StatisticalComputingonManifolds 2 Recursive Computation of the Fréchet Mean on Non-positivelyCurvedRiemannianManifoldswithApplications.... 21 Guang Cheng, Jeffrey Ho, Hesamoddin Salehian, andBabaC.Vemuri 3 KernelsonRiemannianManifolds ....................................... 45 SadeepJayasumana,RichardHartley,andMathieuSalzmann 4 CanonicalCorrelationAnalysisonSPD.n/Manifolds................. 69 Hyunwoo J. Kim, Nagesh Adluru, Barbara B. Bendlin, SterlingC.Johnson,BabaC.Vemuri,andVikasSingh 5 Probabilistic Geodesic Models for Regression and DimensionalityReductiononRiemannianManifolds.................. 101 P.ThomasFletcherandMiaomiaoZhang PartII Color,Motion,andStereo 6 RobustEstimationforComputerVisionUsingGrassmann Manifolds.................................................................... 125 SaketAnand,SushilMittal,andPeterMeer 7 MotionAveragingin3DReconstructionProblems..................... 145 VenuMadhavGovindu 8 Lie-TheoreticMulti-RobotLocalization................................. 165 XiaoLiandGregoryS.Chirikjian v vi Contents PartIII Shapes,Surfaces,andTrajectories 9 CovarianceWeightedProcrustesAnalysis............................... 189 ChristopherJ.Brignell,IanL.Dryden,andWilliamJ.Browne 10 ElasticShapeAnalysisofFunctions,CurvesandTrajectories........ 211 ShantanuH.Joshi,JingyongSu,ZhengwuZhang, andBoulbabaBenAmor 11 WhyUseSobolevMetricsontheSpaceofCurves...................... 233 MartinBauer,MartinsBruveris,andPeterW.Michor 12 ElasticShapeAnalysisofSurfacesandImages ......................... 257 SebastianKurtek,IanH.Jermyn,QianXie,EricKlassen, andHamidLaga PartIV Objects,Humans,andActivity 13 DesigningaBoostedClassifieronRiemannianManifolds............. 281 FatihPorikli,OncelTuzel,andPeterMeer 14 A General Least Squares Regression Framework onMatrixManifoldsforComputerVision .............................. 303 YuiManLui 15 DomainAdaptationUsingtheGrassmannManifold................... 325 DavidA.ShawandRamaChellappa 16 Coordinate Coding on the Riemannian Manifold of SymmetricPositive-DefiniteMatricesforImageClassification....... 345 MehrtashHarandi,MinaBasirat,andBrianC.Lovell 17 SummarizationandSearchOverGeometricSpaces ................... 363 NiteshShroff,RushilAnirudh,andRamaChellappa Index............................................................................... 389 Chapter 1 Welcome to Riemannian Computing in Computer Vision AnujSrivastavaandPavanK.Turaga Abstract The computer vision community has registered a strong progress over the last few years due to: (1) improved sensor technology, (2) increased computa- tion power, and (3) sophisticated statistical tools. Another important innovation, albeit relatively less visible, has been the involvement of differential geometry in developing vision frameworks. Its importance stems from the fact that despite large sizes of vision data (images and videos), the actual interpretable variability lies on much lower-dimensional manifolds of observation spaces. Additionally, naturalconstraintsinmathematicalrepresentationsofvariablesanddesiredinvari- ances in vision-related problems also lead to inferences on relevant nonlinear manifolds. Riemannian computing in computer vision (RCCV) is the scientific area that integrates tools from Riemannian geometry and statistics to develop theoretical and computational solutions in computer vision. Tools from RCCV hasledtoimportantdevelopmentsinlow-levelfeatureextraction,mid-levelobject characterization, and high-level semantic interpretation of data. In this chapter we provide background material from differential geometry, examples of manifolds commonly encountered in vision applications, and a short summary of past and recentdevelopmentsinRCCV.Wealsosummarizeandcategorizecontributionsof theremainingchaptersinthisvolume. 1.1 Introductionand Motivation The computer vision community has witnessed a tremendous progress in research andapplicationsoverthelastdecadeorso.Besidesotherfactors,thisdevelopment can be attributed to: (1) improved sensor technology, (2) increased computation power, and (3) sophisticated statistical tools. An important source of innovation A.Srivastava((cid:2)) DepartmentofStatistics,FloridaStateUniversity,Tallahassee,FL,USA e-mail:[email protected] P.K.Turaga SchoolofArts,Media,Engineering&ElectricalEngineering, ArizonaStateUniversity,Tempe,AZ,USA e-mail:[email protected] ©SpringerInternationalPublishingSwitzerland2016 1 P.K.Turaga,A.Srivastava(eds.),RiemannianComputinginComputerVision, DOI10.1007/978-3-319-22957-7_1 2 A.SrivastavaandP.K.Turaga has also been the fact that, in addition to statistics, differential geometry has also emerged as a significant source of tools and resources to exploit knowledge and context in computer vision solutions. The usage of geometry in computer vision isrelativelyrecentandsparse.Thiscombinationofgeometryandstatisticsisvery fundamentallytiedtothenatureofvisiondata,characterizedbydominanceofobject structuresandtheirvariabilities.Computervisionismainlyaboutfindingstructures of interest, often representing physical objects being observed using cameras of different kinds. Since these structures are inherently variable—two images of the same object will exhibit pixel variability—one needs to take this variability into accountusingstatistics.Wefocusonideasthatborrowbasictoolsfromtraditionally disparate branches—statistics and geometry—and blend them together to reach novelframeworksandalgorithmsinvision. 1.1.1 WhatIsRCCV? A large variety of computer vision solutions can be expressed as either inferences under appropriate statistical models or minimizations of certain energy functions. In some cases, the solutions are naturally constrained to lie in subsets that are not linear, i.e., they are neither subspaces nor affine subspaces (translations of subspaces). In fact, in high-level computer vision, it is not difficult to realize that the set of objects of most common kinds do not follow the rules of Euclidean geometry, e.g., adding or subtracting two images of an automobile does not result inavalidimageofanautomobile.Thus,thesetofautomobilepicturesisnoteasy tocharacterizeattheraw-pixellevel.Insomespecialcases,undercertainenabling assumptions,aswillbediscussedinthelaterchapters,onecanarriveatwell-defined characterizations ofobjectsinimages.Asimpleexampleofimposingaconstraint inrepresentationiswhenavectorisforcedtohaveaunitnorm,inordertopursuea scale-invariantsolution,andthedesirablespaceisaunitsphere.Thenonlinearityof suchspacesmakesitdifficulttoapplytraditionaltechniquesthatprimarilyrelyon Euclideananalysisandmultivariatestatistics.Considerthechallengeofdeveloping statistical tools when even addition or subtraction is not a valid operation on that domain!Thus,evensimplestatistics,suchassamplemeansandcovariances,areno longerdefinedinastraightforwardway.Oneneedsawholenewapproachtodefine and compute statistical inferences and optimization solutions. This motivates the needforRCCV. Definition1.1(Riemannian Computing in Computer Vision—RCCV). Rie- mannian Computing in Computer Vision, or RCCV, is the scientific area that integratestoolsfromRiemanniangeometryandstatisticstodeveloptheoreticaland computational solutions for optimization and statistical inference problems driven byapplicationsincomputervision. Inthecontextofnonlinearspaces,theanalysisreliesonelementsofdifferential geometry. Here, the chosen space is established as a differentiable manifold and 1 WelcometoRiemannianComputinginComputerVision 3 endowed with a smooth metric structure, termed Riemannian structure, to enable computation of distances and averages. A Riemannian structure, in principle, is capableofprovidingmostofthetoolsforcalculus—averages,derivatives,integrals, path lengths, and so on. In some cases, the spaces of interest enjoy an additional structure, such as a group structure, that is useful in the analysis. A group implies presence of a binary operation that allows one to compose and invert elements of the set. Such groups, manifolds with a group structure, are called Lie groups and play a very important role in RCCV. The examples of Lie groups include the sets of rigid and certain nonrigid transformations on objects and images. Sometimes Liegroupsarealsousedtorepresentnuisancevariabilityinsituationswheresome transformationsarenotdeemedrelevant.Forinstance,inshapeanalysisofobjects, theirposerelativetothecameraistreatedasanuisanceparameter. Next, we provide some basic background material for an uninitiated reader. Thosewithaworkingknowledgeofdifferentialgeometrycanskipthissection. 1.2 BasicToolsfromRiemannianGeometry Inthissectionweintroducesomenotationandbasicconceptsthatareneededinthe development of RCCV. Some of the following chapters define additional notation andideasthatarespecifictotheirowntreatments. The key idea in RCCV is that the underlying space of interest is nonlinear. Let M represent the set of allowable feature values. Being nonlinear implies that for arbitrary points p1 and p2 in M, and arbitrary scalars ˛1 and ˛2, the combination ˛1p1 C˛2p2 may not be in M. In fact, even the operation “C” may only be interpretable when we embed M inside a larger vector space V. While such embeddings are possible in most cases, the issue is that there are often many embeddings available and selecting one is a nontrivial problem. Given an embedding,thequantity˛1p1C˛2p2willlieinV (sinceitisavectorspace)butnot necessarilyinM.Sinceaddition(andsubtraction)arenotvalidoperationsinM,one needsdifferentsetsoftoolstohandleM. 1.2.1 TangentSpaces,ExponentialMap,andItsInverse One starts by establishing M as a “differentiable manifold.” Loosely speaking, it implies that small neighborhoods can be smoothly (diffeomorphically) mapped to open sets of Euclidean spaces, and such mappings can be composed smoothly. We will avoid precise definitions here and refer the reader to any introductory textbookondifferentialgeometry,suchas[7,55].Givenamanifold,thenextstepis toimposeametriconitusingaRiemannianstructure(developedusingasequence ofstepsasfollows).OnedenotesthesetofallvectorsthataretangenttoMatapoint p 2 M denote it by T .M/. (A tangent vector is the derivative of a differentiable p
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