Mathematical models for registration and applications to medical imaging

191 Pages·2006·5.158 MB·English

by  Scherzer O.

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
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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.