模式识别英文经典论文Optimizationforlimitedangletomographyinmedicalimageprocessing.pdf
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模式识别英文经典论文Optimizationforlimitedangletomographyinmedicalimageprocessing.pdf
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OptimizationforlimitedangletomographyinmedicalimageprocessingXiaoqiangLua,YiSuna,YuanYuanb,naSchoolofElectronicandInformationEngineering,DalianUniversityofTechnology,ChinabCenterforOPTicalIMageryAnalysisandLearning(OPTIMAL),StateKeyLaboratoryofTransientOpticsandPhotonics,XianInstituteofOpticsandPrecisionMechanics,ChineseAcademyofSciences,Xian710119,Shaanxi,P.R.ChinaarticleinfoArticlehistory:
Received30July2010Receivedinrevisedform14December2010Accepted17December2010Availableonline30December2010Keywords:
LimitedangletomographyIll-posedinverseproblemTotalvariation(TV)abstractThispaperaimstoreducetheproblemsofincompletedataincomputedtomography,whichhappensfrequentlyinmedicalimageprocessandanalysis,e.g.,whenthehigh-densityregionofobjectscanonlybepenetratedbyX-raysatalimitedangularrange.Astheprojectiondataareavailableonlyinanangularrange,theincompletedataproblemcanbeattributedtothelimitedangleproblem,whichisanill-posedinverseproblem.Imagereconstructionbasedontotalvariation(TV)reducestheproblemandgivesbetterperformanceonedge-preservingreconstruction;however,theartificialparametercanonlybedeterminedthroughconsiderableexperimentation.Inthispaper,aneffectiveTVobjectivefunctionisproposedtoreducetheinverseprobleminthelimitedangletomography.Thisnovelobjectivefunctionprovidesarobustandeffectivereconstructionwithoutanyartificialparameterintheiterativeprocesses,usingtheTVasamultiplicativeconstraint.Theresultsdemonstratethatthisreconstructionstrategyoutperformssomepreviousones.&2011PublishedbyElsevierLtd.1.IntroductionComputedtomography(CT)technologyhasmadearevolu-tionaryimpactonmedicaldiagnosisandhasalsoprovensuccess-fulinindustrialnon-destructivetesting.However,itisnotalwayspossibletoacquireprojectiondatathroughacompleteangularrangeinmanyapplications.SomeexampleswouldbeX-raydoselimitations,ortimeconstraintswhenimagingamovingobject,orX-raysbeingobstructedwhenpassingthroughhigh-densityregionofobjects,anyofwhichcouldresultinlossofsomeprojections.Whenprojectiondataareonlyavailableinalimitedangularrange,asoccursinanumberofapplicationssuchasindentalradiology,surgicalimaging,thoracicimaging,mammogra-phy,etc.,thisleadstothenotoriousill-posedproblem.Theconventionalandmostcommonlyusedmethodforrecon-structionfromtomographicprojectionsisthestandardfilteredbackprojection(FBP)reconstructiontechnique,whichisnotsoadaptabletoincompleteprojectiondataandresultsinpoorreconstructionswithsevereartifactsinlimitedanglecases.Becauseoftheimportanceofthelimitedangleproblem,manystudiesaboutreconstructionalgorithmhavebeenintroducedoverthepasttwodecades.Inthepastdecade,withtherapiddevelopmentofcomputer,iterativereconstructionhasbecomeahotresearchtopicforlimitedangularrangereconstruction.Inlimitedangletomography,researchershavetriedseveraliterativeapproaches,suchasalgebraicreconstructionmethod,statisticaliterativemethod,totalvariationmethod,waveletmethod,andmanyothers.Natterer1andNattererandWubbeling2discussedmanymathematicalaspectsofformingreconstructionsfromlimiteddata,suchasuniquenessandstabilityofsolutions.Nassietal.3usedalinearleastsquareformulationinformulat-inganiterativereconstruction.Adaptivewavelet-Galerkinmethodisintroducedbytheauthorin4tosolvethelimitedangleproblem,andthereconstructionstrategyhasacomparableperformancewithasignificantreductionincomputationaltime.Singularvaluedecomposition-basedmethodwasgivenin5,6,whichwasthenusedtodevelopareconstructionalgorithm7,8.Statisticalinversionwithaprioriinformationforreconstructinginlimitedangletomographycouldbefoundin912.Grunbaum13madeadetailedstudyofsmallscalemodelsofthepracticalimplementationofsomeFouriermethodsforthereconstructionofobject.Relatedworkcouldbefoundin1418.Otheralgo-rithmsonlimitedangletomographyandfurtherreferencescouldbefoundin1923.Aclassicalmethodforsolvingill-posedproblemsisregular-izationwithaquadraticpenaltyfunction24.However,thisregularizationmethodhasatendencyofsmoothingthosesharpedgesinsolutionsthatoftencarryimportantinformation.Thisobservationexplainsaninterestinregularizationwithnon-quadraticpenalties,whichpreservesthediscontinuitiesinthesolution,themostpopularofwhichisprobablythetotalvariationpenalty.Therecentworksin25,26developedaniterativeimagereconstructionalgorithmbasedontheminimizationoftheimageTVthatwasappliedtolimitedangleproblemandgavegoodContentslistsavailableatScienceDirectjournalhomepage:
Recognition0031-3203/$-seefrontmatter&2011PublishedbyElsevierLtd.doi:
10.1016/j.patcog.2010.12.016nCorrespondingauthor.E-mailaddress:
(Y.Yuan).PatternRecognition44(2011)24272435edge-preservingreconstructions.ArecentpaperbyCand?
esetal.27proposedaTValgorithmandprovedanimpressiveresultaboutthepossibilityofperfectreconstruction,givensmallamountsofdata.Theauthorsin2832conductedalotofworkonimagereconstructionfromincompleteprojections.AndSidkyetal.28andSidkyandPan29appliedTValgorithmforimagereconstructionfromdivergent-beamprojectionsapplicabletobothfan-beamandcone-beamCTimaging.Inthoseworks,theauthorspresentedaTViterativealgorithmthatcanreconstructperfectimagesfrominsufficientdatathatmayoccurduetopracticalissuesofCTscanning.AndtheminimizationoftheimageTVisperformedbythegradientdescentmethodintheTValgorithm.ThealgorithmbasedonTVismoresuitableforlimitedangletomographybecauseitcanreconstructsharpdiscontinu-itiesoredgeswithsparseorinsufficientdatathatmayoccurduetopracticalissuesofCTscanning.TheeffectivenessoftheTValgorithmreliesonthefactthattheimagedobjecthasarelativelysparsegradientimage.Theadditionofthetotalvariationoftheobjectivefunctionhasaverypositiveeffectonthequalityofreconstructionforpiecewisesmoothnessofanunknownimage.Despiteofthesuccessofreconstructionsinlimitedangletomo-graphybasedontotalvariationmethod,adrawbackisthepresenceofaninitiallyartificialparameterintheminimizationoftheTVnorm,whichcanonlybedeterminedbyconsiderablenumericalexperimentationsandaprioriinformationofthedesiredreconstruction.Thisisoftenverytime-consuming.Onemaysavelotsoftimeandcomputationalcostifthereisaniterativemethodthatcangiveusareasonableparameterwithinapracticallyacceptablenumberofiterations.Thispaperintendstomakesomefurthercontributiontothesubjectinreconstructinganimagefromalimitedangle.Wefirstdefinedanewobjectivefunction,whichistheproductofthenormalizederrorsintheprojectionequationsandtheTV-factor.ThenwemadeuseofaconjugategradientmethodtoimplementtheminimizationoftheimageTV,followedbyprojectiononconvexsetsforenforcingtheprojectiondataconstraints.Inthismethod,itisnotnecessarytodeterminetheartificialparameter.Andtheartificialparameterisdeterminedbytheiterativeprocesses.Hence,theproposedalgorithmcanavoidtheinitialartificialparameterinTValgo-rithmthatimplementstheminimizationoftheimageTV.Wedemonstratethefeasibilityofourmethodbynumericalexamplesusingprojectiondatainalimitedangularrange.Inthenumericalexamples,wefoundthatthequalityofreconstructedimagebytheproposedmethodisimprovedcomparedtotheTVrecon-structionsreportedin29.Thispaperisorganizedasfollows.InSection2,wedemon-stratediscretedatamodelforcircularcone-beamCT.InSection3,wedescribethenewoptimizationprogramforthelimitedangletomography.NumericalresultsarepresentedinSection4,andconclusionsarepresentedinSection5.2.Datemodelincircularcone-beamCTIncircularcone-beamCT,thetaskinimagereconstructionistorecoverthedensityofanobjectf(x)underexaminationprovidedbyasetoflineintegrals.WhentherotationangleoftheX-raysourceisdefinedasa,thecone-beamprojectionoftheobjectfunctionf(x)atapoint(u,v)onthedetectorcanbeexpressedaspu,v,aZ10fslya,u,vdl1wheresourcesisdefinedasssaRcosa,Rsina,02whereRdenotesthedistancefromthesourcepointtotherotationaxis.Andtheraydirectionvectory(a,u,v)indicatesthedirectionoftheraystartingfromthesourcepoints(a)andpassingthroughthepoint(u,v)onthedetector.Bydiscretizingtheprojectionacquisitionmodel,Eq.
(1)canbeapproximatedbyfollowingthediscretelinearsystem:
PKf3TheprojectionvectorPconsistsofM-lengthmeasuredprojec-tionrays.ThevectorfhasN-lengthpixelvaluesthatrepresenttheobjectfunction.ThesystemmatrixKisadiscretemodelfortheintegrationinEq.
(1).Hence,theprojectionacquisitionmodelcanbewritteninanexpandedformasasetoflinearequationswhenconsideringtheerrorofmeasurementornoise:
k11f1k12f2k13f3?
k1NfNn1p1k21f1k22f2k23f3?
k2NfNn2p2.kM1f1kM2f2kM3f3?
kMNfNnMpM4whereMisthenumberofX-rayintegrals,Nisthenumberofpixelsintheobject,nii1,2,.,Mistheadditivenoiseasso-ciatedwiththemeasurement,kijissystemweightsthataredeterminedbytheintersectionlengthoftheithraythroughthejthpixel,andfjrepresentsthejthpixelvalue.3.TheoptimizationprogramforlimitedangletomographyInthissection,webrieflyreviewtheminimizationoftheimagetotalvariationthatappliestolimitedangletomography,followedbysomedrawbacksofthemethod.Afterthat,weproposeanewoptimizationprogramforlimitedangletomography.3.1.ReviewTosimplifynotations,theprojectionacquisitionmodelinEq.(4)isdescribedbythefollowingformula:
PKfn5w
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