2008-Prior image constrained compressed sensing (P

Priorimageconstrainedcompressedsensing(PICCS):AmethodtoaccuratelyreconstructdynamicCTimagesfromhighlyundersampledprojectiondatasetsGuang-HongChena,JieTang,andShuaiLengDepartmentofMedicalPhysicsandDepartmentofRadiology,UniversityofWisconsininMadison,600HighlandAvenue,Madison,Wisconsin53792−1590AbstractWhenthenumberofprojectionsdoesnotsatisfytheShannon/Nyquistsamplingrequirement,streakingartifactsareinevitableinx-raycomputedtomography(CT)imagesreconstructedusingfilteredbackprojectionalgorithms.Inthisletter,thespatial-temporalcorrelationsindynamicCTimaginghavebeenexploitedtosparsifydynamicCTimagesequencesandthenewlyproposedcompressedsensing(CS)reconstructionmethodisappliedtoreconstructthetargetimagesequences.ApriorimagereconstructedfromtheunionofinterleaveddynamicaldatasetsisutilizedtoconstraintheCSimagereconstructionfortheindividualtimeframes.Thismethodisreferredtoaspriorimageconstrainedcompressedsensing(PICCS).InvivoexperimentalanimalstudieswereconductedtovalidatethePICCSalgorithm,andtheresultsindicatethatPICCSenablesaccuratereconstructionofdynamicCTimagesusingabout20viewangles,whichcorrespondstoanundersamplingfactorof32.Thisundersamplingfactorimpliesapotentialradiationdosereductionbyafactorof32inmyocardialCTperfusionimaging.KeywordsdynamicCT;imagereconstruction;compressedsensingAccordingtothestandardimagereconstructiontheoryinmedicalimaging,inordertoavoidviewaliasingartifacts,thesamplingrateoftheviewanglesmustsatisfytheShannon/Nyquistsamplingtheorem.TheuniversalapplicabilityoftheShannon/Nyquistsamplingtheoremliesinthefactthatnospecificpriorinformationabouttheimageisassumed.However,inpractice,somepriorinformationabouttheimageistypicallyavailable.Whentheavailablepriorinformationisappropriatelyincorporatedintotheimagereconstructionprocedure,animagemaybeaccuratelyreconstructedeveniftheShannon/Nyquistsamplingrequirementissignificantlyviolated.Forexample,ifoneknowsatargetcomputedtomography(CT)imageiscircularlysymmetricandspatiallyuniform,onlyoneviewofparallel-beamprojectionsisneededtoaccuratelyreconstructthelinearattenuationcoefficientoftheobject.Anotherexampleisthatifoneknowsthatatargetimageconsistsofonlyasinglepoint,thenonlytwoorthogonalprojectionsaresufficienttoaccuratelyreconstructtheimagepoint.Alongthesamelineofreasoning,ifatargetimageisknowntobeasetofsparselydistributedpoints,onecanimaginethattheimagemaybereconstructedwithoutsatisfyingtheShannon/Nyquistsamplingrequirements.Ofcourse,itisahighlynontrivialtasktoformulatearigorousimagereconstructiontheorytoexploitthesparsityhiddenintheaboveextremalexamples.Fortunately,anewimagereconstructiontheory,compressedsensing(CS),wasrigorouslyaAuthortowhomcorrespondenceshouldbeaddressed.Electronicmail:[e-mail];availableinPMC2009March13.Publishedinfinaleditedformas:MedPhys.2008February;35(2):660–663.NIH-PAAuthorManuscriptNIH-PAAuthorManuscriptNIH-PAAuthorManuscriptformulatedtosystematicallyandaccuratelyreconstructasparseimagefromanundersampleddataset.1,2IthasbeenmathematicallyproventhatanN×NimagecanbeaccuratelyreconstructedusingontheorderofSlnNsamplesprovidedthatthereareonlySsignificantpixelsintheimage.AlthoughthemathematicalframeworkofCSiselegant,therelevanceinmedicalimagingcriticallyreliesontheanswerstothequestions:(1)Aremedicalimagessparse?(2)Ifamedicalimageisnotsparse,canweusesometransformstomakeitsparse?Infact,arealmedicalimageisoftennotsparseintheoriginalpixelrepresentation.Forexample,inacontrastenhancedCTexam,imagesbeforecontrastinjectionoraftercontrastenhancementarenotsparseasshownbythehistogramsofthepixelvaluesinFig.1(a).However,medicalimagingphysicistsandclinicianshaveknownforalongtimethatasubtractionoperationcanmaketheresultantimagesignificantlysparser.InthenewlyproposedCSreconstructiontheory,mathematicaltransformshavebeenappliedtoasingleimagetosparsifytheimage.Thesetransformsarereferredtoassparsifyingtransforms.Forexample,theimageinFig.1(a)canbesparsifiedbyapplyingadiscretegradientoperationwhichisdefinedas(1)whereX(m,n)istheimagevalueatthepixel(m,n)andDxX=X(m+l,n)−X(m,n)andDyX=X(m,n+1)−X(m,n).Thisimagespecifiedby▽m,nX(m,n)isreferredtoasthegradientimage.AsshowninFig.1,thediscretegradientimageissignificantlysparser.Tobemorequantitative,ifwedefineasignificantpixelastheonewithmorethan10%ofthehighestpixelvalueinanimage,thenwecanuseahistogramtodemonstratethedistributionofthenumberofsignificantpixelsintheoriginalimageandthediscretegradientimage(Fig.1).Thediscreregradientimageisthreetimessparserthantheoriginalimage.Thisexampledemonstratesthatamedicalimagecanbemadesparseeveniftheoriginalimageisnotverysparse.Thebasicideaincompressedsensing(CS)imagereconstructiontheorycanbesummarizedasfollows:insteadofdirectlyreconstructingatargetimage,thesparsifiedversionoftheimageisreconstructed.Inthesparsifiedimage,significantlyfewerimagepixelshavesignificantimagevalues.Thus,itispossibletoreconstructthesparsifiedimagefromanundersampleddatasetwithoutstreakingartifacts.Afterthesparsifiedimageisreconstructed,an“inverse”sparsif