Mathematical models for registration and applications to medical imaging PDF
Preview Mathematical models for registration and applications to medical imaging
MATHEMATICS IN INDUSTRY 10 Editors Hans-Georg Bock Frank de Hoog Avner Friedman Arvind Gupta Helmut Neunzert William R. Pulleyblank Torgeir Rusten Fadil Santosa Anna-Karin Tornberg THE EUROPEAN CONSORTIUM FOR MATHEMATICS IN INDUSTRY SUBSERIES Managing Editor Vincenzo Capasso Editors Robert Mattheij Helmut Neunzert Otmar Scherzer Otmar Scherzer Mathematical Models for Registration and Applications to Medical Imaging With 54 Figures, 12 in Color, and 12 Tables 123 Editor Otmar Scherzer .. Universitat Innsbruck .. Institut fur Informatik, Technikerstr. 21A A-- 6020 Innsbruck, Austria e-mail: [email protected] Library of Congress Control Number: 2006926829 Mathematics Subject Classification (2000): 65J15, 65F22, 94a08, 94J40, 94K24 ISBN-10 3-540-25029-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-25029-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset by the editors & SPI Publisher Services Production: LE-TE X Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: design & production GmbH, Heidelberg Printed on acid-free paper S P IN: 11397403 46/3100/SPI - 5 4 3 2 1 0 Preface Imageregistrationisanemergingtopicinimageprocessingwithmanyapplications inmedicalimaging,pictureandmovieprocessing.Theclassicalproblemofimage registration is concerned with finding an appropriate transformation between two datasets.Thisfuzzydefinitionofregistrationrequiresamathematicalmodelingand inparticularamathematicalspecificationofthetermsappropriatetransformations andcorrelationbetweendatasets.Dependingonthetypeofapplication,typically Euler, rigid, plastic, elastic deformations are considered. The variety of similarity measuresrangesfromasimpleLp distancebetweenthepixelvaluesofthedatato mutualinformationorentropydistances. Thisgoalofthisbookistohighlightbysomeexpertsinindustryandmedicine relevant and emerging image registration applications and to show new emerging mathematicaltechnologiesintheseareas. Currently,manyregistrationapplicationaresolvedbasedonvariationalprinci- ple requiring sophisticated analysis, such as calculus of variations and the theory ofpartialdifferentialequations,tonamebutafew.Duetothenumericalcomplex- ity of registration problems efficient numerical realization are required. Concepts like multi-level solver for partial differential equations, non-convex optimization, andsoonplayanimportantrole.Mathematicalandnumericalissuesintheareaof registrationarediscussedbysomeoftheexpertsinthisvolume. Moreover, the importance of registration for industry and medical imaging is discussedfromamedicaldoctorandfromamanufacturerpointofview. WewouldliketothankStephanieSchimkowitschforamarvelousjobintype- settingthismanuscript.Moreover,wewouldliketothankProf.VincenzoCapasso forthecontinuousencouragementandsupportofthisbookandIwouldliketoex- pressmythankstoUteMcCrory(Springer)forherpatienceduringthepreparation ofthemanuscript. The work of myself is supported by the FWF, Austria Science Foundation, ProjectsY-123INF,FSP9203-N12andFSP9207-N12.Withoutthesupportofthe FWFformyresearchthisvolumewouldnotbepossible. June,2005 OtmarScherzer(Innsbruck) Table of Contents PartI NumericalMethods AGeneralizedImageRegistrationFrameworkusingIncompleteImage Information–withApplicationstoLesionMapping StefanHenn,LarsHo¨mke,KristianWitsch............................. 3 MedicalImageRegistrationandInterpolationbyOpticalFlowwith MaximalRigidity StephenL.Keeling ............................................... 27 RegistrationofHistologicalSerialSectionings JanModersitzki,OliverSchmitt,andStefanWirtz ....................... 63 ComputationalMethodsforNonlinearImageRegistration UlrichClarenz,MarcDroske,StefanHenn,MartinRumpf,KristianWitsch... 81 ASurveyonVariationalOpticFlowMethodsforSmallDisplacements JoachimWeickert,Andre´sBruhn,ThomasBrox,andNilsPapenberg ........ 103 PartII Applications FastImageMatchingforGenerationofPanoramaUltrasound ArminSchoisswohl ............................................... 139 InpaintingofMoviesUsingOpticalFlow HaraldGrossauer ................................................ 151 PartIII MedicalApplications MultimodalityRegistrationinDailyClinicalPractice RetoBale....................................................... 165 ColourImages Clarenzetal.,Hennetal.,Weickertetal.,Bale......................... 185 List of Contributors OtmarScherzer 40225Du¨sseldorf,Germany UniversityofInnsbruck [email protected] InstituteofComputerScience Technikerstraße21a LarsHo¨mke 6020Innsbruck,Austria ForschungszentrumJu¨lichGmbH [email protected] Institutfu¨rMedizin StreetNo. ArminSchoisswohl 52425Ju¨lich,Germany GEMedicalSystems [email protected] KretzUltrasound Tiefenbach15 KristianWitsch 4871Zipf,Austria Heinrich-Heine University of [email protected] Du¨sseldorf Lehrstuhlfu¨rAngewandteMathematik RetoBale MathematischesInstitut Universita¨tsklinikfu¨rRadiodiagnostik Universita¨tsstraße1 SIP-Labor 40225Du¨sseldorf,Germany Anichstraße35 [email protected] 6020Innsbruck,Austria [email protected] StephenL.Keeling Karl-FranzensUniversityofGraz HaraldGrossauer InstituteofMathematics UniversityofInnsbruck Heinrichstraße36 InstituteofComputerScience 8010Graz,Austria Technikerstraße21a [email protected] 6020Innsbruck,Austria [email protected] JanModersitzki UniversityofLu¨beck StefanHenn InstituteofMathematics Heinrich-Heine University of Wallstraße40 Du¨sseldorf D-23560Lu¨beck Lehrstuhl fu¨r Mathematische Opti- [email protected] mierung MathematischesInstitut OliverSchmitt Universita¨tsstraße1 UniversityofRostock X ListofContributors InstituteofAnatomy KristianWitsch Gertrudenstraße9 Heinrich-Heine University of D-18055Rostock,Germany Du¨sseldorf [email protected] Lehrstuhlfu¨rAngewandteMathematik Universita¨tsstraße1 StefanWirtz 40225Du¨sseldorf,Germany UniversityofLu¨beck [email protected] InstituteofMathematics Wallstraße40 D-23560Lu¨beck JoachimWeickert [email protected] MathematicalImageAnalysisGroup, FacultyofMathematicsandComputer UlrichClarenz Science, Gerhard-Mercator University of SaarlandUniversity,Building27, Duisburg 66041Saarbru¨cken,Germany. InstituteofMathematics [email protected]. Lotharstraße63/65, 47048Duisburg,Germany [email protected] Andre´sBruhn MathematicalImageAnalysisGroup, MarcDroske FacultyofMathematicsandComputer UniversityofCalifornia Science, MathSciencesDepartment SaarlandUniversity,Building27, 520PortolaPlaza, 66041Saarbru¨cken,Germany. LosAngeles,CA,90055,USA [email protected]. [email protected] StefanHenn NilsPapenberg Heinrich-Heine University of MathematicalImageAnalysisGroup, Du¨sseldorf FacultyofMathematicsandComputer Lehrstuhl fu¨r Mathematische Opti- Science, mierung SaarlandUniversity,Building27, Universita¨tsstraße1 66041Saarbru¨cken,Germany. 40225Du¨sseldorf,Germany [email protected]. [email protected] MartinRumpf ThomasBrox Rheinische Friedrich-Wilhelms- MathematicalImageAnalysisGroup, Universita¨tBonn FacultyofMathematicsandComputer Institutfu¨rNumerischeSimulation Science, Nussallee15, SaarlandUniversity,Building27, 53115Bonn,Germany 66041Saarbru¨cken,Germany. [email protected] [email protected]. Part I Numerical Methods A Generalized Image Registration Framework using Incomplete Image Information – with Applications to Lesion Mapping StefanHenn1,LarsHo¨mke2,andKristianWitsch3 1 Lehrstuhlfu¨rMathematischeOptimierung,MathematischesInstitut,Heinrich-Heine Universita¨tDu¨sseldorf,Universita¨tsstraße1,D-40225Du¨sseldorf,Germany. [email protected] 2 Institutfu¨rMedizin,ForschungszentrumJu¨lichGmbH, D-52425Ju¨lich,[email protected] 3 Lehrstuhlfu¨rAngewandteMathematik,MathematischesInstitut,Heinrich-Heine Universita¨tDu¨sseldorf,Universita¨tsstraße1,D-40225Du¨sseldorf,Germany. [email protected] Abstract Thispaperpresentsanovelvariationalapproachtoobtainad-dimensional displacementfieldu = (u ,··· ,u )t,whichmatchestwoimageswithincomplete 1 d information. A suitable energy, which effectively measures the similarity between theimagesisproposed.Analgorithm,whichefficientlyfindsthedisplacementfield by minimizing the associated energy is presented. In order to compensate the ab- senceofimageinformation,theapproach isbasedonanenergyminimizinginter- polationofthedisplacementfieldintotheholesofmissingimagedata.Thisinter- polationiscomputedviaagradientdescentflowwithrespecttoanauxiliaryenergy norm.Thisincorporatessmoothnessconstraintsintothedisplacementfield.Appli- cationsofthepresentedtechniqueincludetheregistrationofdamagedhistological sectionsandregistrationofbrainlesionstoareferenceatlas.Weconcludethepaper byanumberofexamplesoftheseapplications. Keywords imageregistration,inpainting,functionalminimization,finitedifference discretization,regularization,multi-scale 1 Introduction. Deformableimageregistrationofbrainimageshasbeenanactivetopicofresearch inrecentyears.Drivenbyevermorepowerfulcomputers,imageregistrationalgo- rithmshavebecomeimportanttools,e.g.in – guidanceofsurgery, – diagnostics, – quantitative analysis of brain structures (interhemispheric, interareal and in- terindividual), – ontogeneticdifferencesbetweencorticalareas, – interindividualbrainstudies. 4 StefanHenn,LarsHo¨mke,andKristianWitsch The need for registration in interindividual brain studies arises from the fact thatthehumanbrainexhibitsahighinterindividualvariability.Whilethetopology is stable on the level of primary structures, not only the general shape, but also the spatial localization of brain structures varies considerably across brains. That renders a direct comparison impossible. Hence, brains have to be registered to a common“referencespace”,i.e.theyareregisteredtoareferencebrain.Oftenthere arealso,so-calledmaps,thatresideinthesamereferencespace.Insocalledbrain atlases there are additional maps that contain different kinds of information about the reference brain, such as labeled cortical regions. Once an individual brain has beenregisteredtothereferencebrainthemapscanbetransferredtotheregistered brain. It is not only that obtaining the information from the individual brain itself is often more intricate than registering it to a reference, in some cases it is also impossible.Forinstance,themicrostructureofthebraincannotbeanalyzedinvivo, sincetheresolutionofinvivoimagingmethods,suchasMRIandPET,istoolow. Registrationcanalsobeameansofcreatingsuchmaps,bytransferringinformation fromdifferentbrainsintoareferencespace. Inthelastdecadecomputationalalgorithmshavebeendevelopedinordertomap twoimages,i.e.todeterminea“bestfit”betweenthem.Althoughthesetechniques havebeenappliedverysuccessfullyforboththeuni-andthemulti-modalcase(e.g. see[1,2,7,8,10,11,13,19,21,22,25])thesetechniquesmaybelessappropriate for studies using brain-damaged subjects, since there is no compensation for the structural distortion introduced by a lesion (e.g. a tumor, ventricular enlargement, largeregionsofatypicalpixelintensityvalues,etc.). Generally the computed solution cannot be trusted in the area of a lesion. The magnitudeoftheeffectonthesolutiondependsonthecharacteroftheregistration scheme employed. It is not only that these effects are undesirable, but also that in some cases one is especially interested in where the lesion would be in the other image.If,forinstance,wewanttoknowwhichfunctionisusuallyperformedbythe damagedarea,wecouldregisterthelesionedbraintoanatlasandmapthelesionto functionaldatawithinthereferencespace. In more general terms the problem can be phrased as follows. Given are two imagesandadomainGincludingasegmentationofthelesions.Theaimofthepro- posedimageregistrationalgorithmistofinda“smooth”displacementfield,which – minimizesagivensimilarityfunctionaland – conservethelesioninthetransformedtemplateimage. There have been approaches to register lesions manually[12]. In this paper we present a novel automatically image registration approach for human brain vol- umes with structural distortions (e.g a lesion). The main idea is to define a suit- able matching energy, which effectively measures the similarity between the im- ages. Since the minimization solely the matching energy is an ill-posed problem weminimizetheenergybyagradientdescentflowwithrespecttoaregularityen- ergyborrowedfromlinearelasticitytheory.Theregularizationenergyincorporates smoothnessconstraintsintothedisplacementfieldduringtheiteration.
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