Nonparametric Density Flowsfor MRI Intensity Normalisation

Nonparametric Density Flows
for MRI Intensity Normalisation

Daniel C. Castro(\Letter) Biomedical Image Analysis Group
Imperial College London, UK
1
   Ben Glocker Biomedical Image Analysis Group
Imperial College London, UK
1
1email: {dc315,b.glocker}@imperial.ac.uk
Abstract

Withtheadoptionofpowerfulmachinelearningmethodsinmedicalimageanalysis,itisbecomingincreasinglydesirabletoaggregatedatathatisacquiredacrossmultiplesites.However,theunderlyingassumptionofmanyanalysistechniquesthatcorrespondingtissueshaveconsistentintensitiesinallimagesisoftenviolatedinmulti-centredatabases.Weintroduceanovelintensitynormalisationschemebasedondensitymatching,whereinthehistogramsaremodelledasDirichletprocessGaussianmixtures.Thesourcemixturemodelistransformedtominimiseitsdivergencetowardsatargetmodel,thenthevoxelintensitiesaretransportedthroughamass-conservingflowtomaintainagreementwiththemovingdensity.Inamulti-centrestudywithbrainMRIdata,weshowthattheproposedtechniqueproducesexcellentcorrespondencebetweenthematcheddensitiesandhistograms.Wefurtherdemonstratethatourmethodmakestissueintensitystatisticssubstantiallymorecompatiblebetweenimagesthanabaselineaffinetransformationandiscomparabletostate-of-the-artwhileprovidingconsiderablysmoothertransformations.Finally,wevalidatethatnonlinearintensitynormalisationisasteptowardeffectiveimagingdataharmonisation.

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1 Introduction

Manymedicalimageanalysismethodsrelyonthehypothesisthatcorrespondinganatomicalstructurespresentsimilarintensityprofiles.Unlikecomputedtomography,magneticresonanceimagingdoesnotproducescansinanabsolutestandardscale,ingeneral.Evenwhenusingthesameimagingprotocols,therecanbesignificantvariationbetweendifferentscanners.Acquisitionparametershaveacomplexeffectontheluminanceoftheacquiredimages,thereforeasimplelinearrescalingofintensitiesisusuallyinsufficientforeffectivedataharmonisation[5].Therefore,acrucialfactorforenablingtheconstructionoflarge-scaleimagedatabasesfrommultiplesitesisaccuratenonlinearintensitynormalisation.Anumberofdifferentapproacheshavebeenintroducedforthistask(cf. [1]),themostwidely-adoptedofwhichisthatofNyúletal. [7].Theauthorsproposedtonormaliseintensitiesbymatchingasetofhistogramquantiles,usingtheseaslandmarksforapiecewiselineartransformation.Despiteitsapparentsimplicity,ithasprovenveryeffectiveinclinicalapplications[9].Ourproposedmethod,nonparametricdensityflows(NDFlow),isperhapsconceptuallyclosestto[5],whichinvolvesmatchingGaussianmixturemodels(GMMs)fittedtoapairofimagehistograms.Theauthorusedafinitemixturetorepresentapre-definedsetoffivetissuesclasses,whereasweproposetousenonparametricmixtures,focussingonaccuratelymodellingthedensityratherthandiscriminatingtissuetypes,andsidesteppingtheproblemofpre-selectingthenumberofcomponents.Afurtherdifferencetoourworkisthat,insteadofpolynomiallyinterpolatingbetweenthemeansofcorrespondingcomponents,webuildasmoothtransformationmodelbasedondensityflows.

2 Method

Webeginbyjustifyinganddescribingthedensitymodelusedtorepresenttheintensitydistributionstobematched.Wethenintroducethechosenobjectivefunctionwithitsgradientsforoptimisation.Finally,wepresentourflow-basedtransformationmodel,whichdeformsthedatasoitconformstothematcheddensitymodel.Notethatwefocushereonsingle-modalityintensitynormalisation,althoughtheentireformulationbelowextendsnaturallytothemultivariatecase.

Figure 1: ComparisonoftwoMRIscans,beforeandaftertheproposedNDFlownormalisation.Right:histograms(shaded)andfittedmixturemodels(dotted:likelihood,solid:mixturecomponents).

2.1 IntensityModel

Inordertobeabletomatchtheintensitydistributionsofapairofimages,asuitableprobabilitydensitymodelisrequired.Typically,finitemixturemodelsareconsideredforthistask[5, 8].However,awell-knownlimitationoftheseistherequirementtospecifyaprioriafixednumberofcomponents,whichmayinadditioncallforaniterativemodelselectionloop(e.g. [8]).Ontheoppositeendofthespectrum,anotherapproachistousekerneldensityestimation,whichiswidespreadforshaperegistration(e.g. [6, 4]).However,thisformulationwouldresultinanunwieldyoptimisationproblem,involvingthousandsormillionsofparametersandallpairwiseinteractions.Furthermore,thederivedtransformationwouldlikelynotbesatisfactorilysmoothwithoutadditionalregularisation.ToovercomebothissuesweproposetouseDirichletprocessGaussianmixturemodels(DPGMMs)[3].Insteadofspecifyingafixednumberofcomponents,theyrelyonavagueconcentrationparameter,whichregulatestheexpectedamountofclusteringfragmentationandenablesthemtoadapttheircomplexitytothedataathand.Byallowinganunboundednumberofcomponentsandsettingaversatileprioronthemixtureproportions,theyappearasaparsimoniousmiddlegroundforflexibilityandtractability.WefittheDPGMMstoeachimage’sintensitiesusingvariationalinference[2].Morespecifically,weimplementedanefficientweightedvarianttofitamixturedirectlytoeach1Dhistogram.

2.2 DensityMatching

Thefirststepistoperformacoarseaffinealignmentbymatchingthemovingdensity’sfirstandsecondmomentstothetarget’s,accountingforarbitrarytranslationandrescalingofthevalues.Thissameaffinetransformationisthenalsoappliedtothedatabeforethenonlinearwarpingtakesplace.Wequantifythedisagreementbetweentwoprobabilitydensityfunctionsandonaprobabilityspacebymeansofthedivergence:

(1)

whereistheinnerproductandisitsinducednorm.Asidefrombeingsymmetric,thisquantityispositiveandreacheszeroiff.Crucially,unliketheusualKullback–Leiblerdivergence,itisexpressibleinclosedformforGaussianmixturedensities.LetanddenotetwoGaussianmixtures,withcomponentsand.Equation 1hastractablegradientsw.r.t. theparametersof(Appendix 0.A),whichweusetooptimiseitscomponents’meansandprecisions.Wehavefound,inpractice,thatitislargelyunnecessarytoadaptthemixingproportions,,togetanexcellentagreementbetweenmixturedensities.Infact,changingthemixtureweightswouldrequiretransferringsamplesbetweenmixturecomponents.Althoughsurelypossible,wepointoutthatinthecontextofhistogrammatchingthiswouldimplyalteringtheirsemanticvalue(e.g. consideramixtureoftwowell-separatedcomponentsrepresentingdifferenttissuetypes).

2.3 Warping

AftermatchingoneGMMtoanother,wealsoneedawaytotransformthedatamodelledbythatGMMsoitmatchesthetargetdata.Tothisend,wedrawinspirationfromfluidmechanicsanddefinethewarpingtransformation,,asthetrajectoriesofparticlesundertheeffectofavelocityfieldovertime,takingtheprobabilitydensityforthemechanicalmassdensity.Thekeypropertythatsuchflowmustsatisfyisconservationofmass:,whereisspecifieddirectlyfromthedensitymatching.Letusfirstconsiderthecaseofwarpingasinglemixturecomponent.ArandomvariablecanbeexpressedviaadiffeomorphicreparametrisationofastandardGaussian,withand.Assumingitsmeanandprecisionarechangingwithratesand,respectively,wecanintroduceavelocityfieldforitssamplessothattheyagreewiththisevolvingdensity.Theinstantaneousvelocityat‘time’isthusgivenby

(2)

Inthecaseofamixturewithconstantweights,wecanconstructasmooth,mass-conservingglobalvelocityfieldas

(3)

whichissimplyapoint-wiseconvexcombinationofeachcomponent’svelocityfield,,weightedbythecorrespondingposteriorassignmentprobabilities.Finally,thewarpingtransformationisgivenbythesolutiontothefollowingordinarydifferentialequation(ODE):

(4)

Withdefinedasabove,wecanprovethatisindeedthedensityofsamplesfromtransformedthrough,i.e. (Appendix 0.B).Crucially,thetruesolutiontoEq. 4isdiffeomorphicbyconstruction,andcanbenumericallyapproximated(andinverted)witharbitraryprecision.Inparticular,weemploytheclassicfourth-orderRunge–KuttaODEsolver(RK4).Nowassumeweobtainoptimalparametervaluesandaftermatchingto.Wecanthenwarpthedatausingtheaboveapproach,forexamplelinearlyinterpolatingtheintermediateparametervalues,and,hencesettingtheratesinEq. 2toconstantvalues,and,andintegratingEq. 4for.

2.4 PracticalConsiderations

Sinceeachmedicalimageinadatasetcanhavemillionsofvoxels,computingtheposteriorsandflowsforeveryvoxelindividuallycanbetooexpensiveforbatchprocessing.Tomitigatethisissue,wecancomputetheend-to-endtransformationonameshintherangeofinterest,whichistheninterpolatedfortheintensitiesintheentirevolume.Inthereportedexperiments,wehaveusedauniformly-spacedmeshof200points,whichhasprovenaccurateenoughfornormalisationpurposes.Notethatthetransformationcouldalsobecomputedonthehistogramofdiscreteintensityvaluesandbuiltintoalook-uptable.However,thiswouldnotscalewelltotwoormoredimensionsformulti-modalintensitynormalisation,whereasameshwouldnotneedtobeveryfinenorrequirearegulargridlayout.

3 Experiments

3.1 Dataset

Ourexperimentswererunon581T1-weightedMRIscansfromtheIXIdatabase,collectedfromthreeimagingcentreswithdifferentscanners.111http://brain-development.org/ixi-dataset/Eachscanwasbiasfield-correctedusingSPM12222http://www.fil.ion.ucl.ac.uk/spm/software/spm12/withdefaultsettingsandrigidlyregisteredtoMNIspace.SPM12wasfurtherusedtoproducegreymatter(GM),whitematter(WM)andcerebrospinalfluid(CSF)tissueprobabilitymaps.Weobtainedbrainmasksbyaddingthethreeprobabilitymapsandthresholdingat.Thestatisticsreportedbelowwereweightedbythevoxel-wisetissueprobabilitiestoaccountforpartial-volumeeffectsandsegmentationambiguities.

3.2 Setup

Figure 2: Populationdensities,colour-codedbyimagingcentre

Wefirstlyfittedthenonparametricmixturemodelstothefullinteger-valuehistogramsoftherawimages(insidethebrainmasks),asdescribedinSection 2.1.WesettheDP’sconcentrationparameterto2anduseddata-drivenNormal–Gammapriorsforthecomponents.Asanad-hocpost-processingstep,weprunedtheleftovermixturecomponentswithweightssmallerthan.Intheabsenceofoneglobalreferencedistribution,weaffinelyalignedtheseDPGMMsandthecorrespondingdatatozeromeanandunitvariance(cf. Fig. 2,middle).Afterthisroughalignment,globalandcentre-wiseaveragedensitieswerecomputed.Thesewerethenconsideredashistogramstowhichwefittedglobalandcentre-wisereferenceDPGMMs.Fornormalisation,weconsidertwoscenarios.Thefirstistonormaliseeachcentre’sreferencedistributiontotheglobaltarget,thentoapplythissametransformationtoallsubjectsinthatcentre.Intheotherapproach,eachsubject’simageisindividuallynormalisedtotheglobaltargetdensity.Thesescenariosreflectdifferentpracticalapplicationswherethecentre-wisenormalisationaimstopreserveintra-centrevariation,whichmightbedesired.Ontheotherhand,theindividualnormalisationaimstomakeallscansassimilaraspossible.WecompareourtechniquetoNyúletal.’sprevalentquantile-based,piecewiselinearhistogrammatchingmethod[7],consideredstate-of-the-artforintensitynormalisationandreferredhereasNyul.Weacquiredthedefault11landmarks(histogramdecilesandupper/lowerpercentiles)fromtheaffine-aligneddataforallsubjects,thennormalisedeachsubjecttothissetofaveragelandmarks.

3.3 Results

HistogramFitness.
Figure 3: Affine
Figure 4: NDFlow
Figure 5: Nyul
Figure 6: HistogramsandQ–Qplotsofeachofthemethodsagainstthetargethistogram.Theshadingshowsthediscrepancybetweenthetransformed(black)andtargethistogram(lightred).Intherightmostplot,thelandmarksareindicatedbyverticallinesinthehistogramandticksintheQ–Qplot.

Figure 6illustratestheresultsofnormalisationbetweenthepairofimagesinFig. 1,whichhaveanotabledissimilarityintheCSFregionofthehistograms.WeobservethatbothourNDFlow-andNyul-transformedhistogramspresentsubstantiallylowermeanabsoluteandrootmeansquarederrors(MAEandRMSE)thantheaffine-alignedone,andourmethodperformedbestbyasmallmargin.Thisisconfirmedinanumberoftrialswithotherimages.AnoteworthyartefactofNyul areabruptjumpsproducedatthelandmarkvalues(e.g. Fig. 5),whichappearbecauseintervalareuniformlycompressedordilatedbydifferentfactors,andmaybedetrimentaltodownstreamhistogram-basedtasks(e.g. mutualinformationregistration).NDFlow causesnosuchdiscontinuitiesduetothesmoothnessofthemass-conservingflows.

TissueStatistics.
Method 1stQuartile Median 3rdQuartile
  WM Affine   0.900 0.040   1.024 0.045   1.126 0.055
\MethodName:Centre   0.898 0.040   1.020 0.040   1.121 0.043
\MethodName:Indiv.   0.890 0.029   1.014 0.018   1.120 0.016
Nyul   0.897 0.029   1.023 0.015   1.126 0.008
  GM Affine   0.296 0.142   0.025 0.117   0.344 0.080
\MethodName:Centre   0.297 0.139   0.025 0.114   0.344 0.076
\MethodName:Indiv.   0.312 0.094   0.027 0.065   0.351 0.058
Nyul   0.309 0.106   0.027 0.070   0.350 0.064
  CSF Affine   2.036 0.145   1.486 0.140   1.024 0.156
\MethodName:Centre   2.035 0.143   1.480 0.142   1.018 0.160
\MethodName:Indiv.   2.031 0.136   1.484 0.170   1.028 0.191
Nyul   2.025 0.111   1.474 0.178   1.029 0.207
  • Bold:,one-tailedBrown–Forsythetestforlowervariancethan‘Affine’

Table 1: Tissuestatisticsafternormalisation(meanstd. dev.,)

InTable 1wereporttheWM,GMandCSFintensitystatisticsfordifferentnormalisations.Firstly,weseethatthecentre-wisenormalisationhadasmallbutsignificanteffectontheoveralldistributionstatistics.Moreimportantly,thevariancesofthestatisticsafterindividualNDFlow andNyul transformationsweretypicallysimilar,andbothwerealmostalwayssubstantiallysmallerthanthevarianceafteronlyaffinealignment,withtheexceptionofCSF.Itisknownthattheamountofintra-cranialfluidcanvarysubstantiallyduetofactorssuchasageandsomeneurodegenerativeconditions,andthisreflectsonthedistributionsofintensitiesinbrainMRIscans,whichisevidentinFig. 2.Asaresult,normalisingallsubjectstoa‘mean’distributionfailstoidentifyaconsistentreferencerangeforCSFintensities.Afundamentallimitationofanyhistogrammatchingschemeisthatitisunclearhowtoproceedwhenthedistributionsaregenuinelydifferent.Intensitydistributionscanbestronglyaffectedbyanatomicaldifferences;forexample,wecanobservelargevariationsintheamountsoffluidandfatinbrainorwhole-bodyscans,whichmayheavilyskewtheoveralldistributions(moderateexampleinFig. 6).Theunderlyingassumptionofthesemethods(includingours)isthatthedistributionsaresimilarenoughuptoanaffinerescalingandamildnonlineardeformationofthevalues,thushandlinghistogramsoftrulydifferentshapesremainsanopenchallenge.Forimageswithdifferentfieldsofview,itmaybebeneficialtoperformimageregistrationbeforeapplyingintensitynormalisation.

CentreClassification.

Toevaluatetheeffectivenessofintensitynormalisationfordataharmonisation,weconductedacentrediscriminationexperimentwithrandomforestclassifierstrainedonthefullimages.Wereportthepooledtestresultsfromtwo-foldcrossvalidation(detailedresultsinAppendix 0.C).Relativetoaffinenormalisation,centre-wiseandindividualNDFlow andNyul showedaslightdropinoverallclassificationaccuracy(94.1%vs. 92.7%,93.6%,92.9%,resp.).Ontheotherhand,theuncertainty,asmeasuredbytheentropyofthepredictions,wassignificantlyhigher(paired-test,all).Nonlinearintensitynormalisationthereforeseemstosuccessfullyremovesomeofthebiasingfactorswhicharediscriminativeoftheoriginoftheimages.

4 Conclusion

Inthispaper,wehaveintroducedanovelmethodforMRIintensitynormalisation,callednonparametricdensityflows(NDFlow).ItisbasedonfittingandmatchingDirichletprocessGaussianmixturedensities,byminimisingtheirdivergence,andonmass-conservingflows,whichensurethattheempiricalintensitydistributionagreeswiththematcheddensitymodel.Wedemonstratedthatournormalisationapproachmakestissueintensitystatisticssignificantlymoreconsistentacrosssubjectsthanasimpleaffinealignment,andcomparesfavourablytothestate-of-the-artmethodofNyúletal. [7].WehaveadditionallyverifiedthatNDFlow isabletoaccuratelymatchhistogramswithoutintroducingspuriousartefactsproducedbythecompetingmethod.Finally,wearguedthatbothnormalisationtechniquescanreducesomediscriminativescannerbiases,inasteptowardeffectivedataharmonisation.Byemployingnonparametricmixturemodels,weareabletorepresentarbitraryhistogramshapeswithanynumberofmodes.Inaddition,ourformulationhastheflexibilitytomatchonlypartofthedistributions,byfreezingtheparametersofsomemixturecomponents.Thismaybeusefulforignoringlesion-relatedmodes(e.g. multiplesclerosishyperintensities),ifthecorrespondingcomponentscanbeidentified(e.g.,viaanomalydetection).Evaluatingthisapproachanditsrobustnessagainstlesionloadisacompellingdirectionforfurtherresearch.

Acknowledgements.

ThisprojectwassupportedbyCAPES,Brazil(BEX1500/2015-05),andbytheEuropeanResearchCouncilundertheEU’sHorizon2020programme(grantagreementNo757173,projectMIRA,ERC-2017-STG).

References

  • [1] Bergeest,J.P.,Jäger,F.:AComparisonofFiveMethodsforSignalIntensityStandardizationinMRI.In:BildverarbeitungfürdieMedizin2008,pp.36–40.Springer(2008)
  • [2] Blei,D.M.,Jordan,M.I.:VariationalInferenceforDirichletProcessMixtures.BayesianAnalysis1(1),121–144(2006)
  • [3] Ferguson,T.S.:BayesianDensityEstimationbyMixturesofNormalDistributions.RecentAdvancesinStatistics24(1983),287–302(1983)
  • [4] Hasanbelliu,E.,Giraldo,L.S.,Principe,J.C.:Arobustpointmatchingalgorithmfornon-rigidregistrationusingtheCauchy-Schwarzdivergence.In:2011IEEEInternationalWorkshoponMachineLearningforSignalProcessing.IEEE(2011)
  • [5] Hellier,P.:ConsistentintensitycorrectionofMRimages.In:Proceedingsofthe2003InternationalConferenceonImageProcessing(ICIP2003).IEEE(2003)
  • [6] Jian,B.,Vemuri,B.C.:RobustPointSetRegistrationUsingGaussianMixtureModels.IEEETransactionsonPatternAnalysisandMachineIntelligence33(8),1633–1645(2011)
  • [7] Nyúl,L.G.,Udupa,J.K.,Zhang,X.:NewvariantsofamethodofMRIscalestandardization.IEEETransactionsonMedicalImaging19(2),143–150(2000)
  • [8] Roy,A.S.,Gopinath,A.,Rangarajan,A.:DeformableDensityMatchingfor3DNon-rigidRegistrationofShapes.In:MedicalImageComputingandComputer-AssistedIntervention–MICCAI2007.pp.942–949.Springer(2007)
  • [9] Shah,M.,Xiao,Y.,Subbanna,N.,Francis,S.,Arnold,D.L.,Collins,D.L.,Arbel,T.:EvaluatingintensitynormalizationonMRIsofhumanbrainwithmultiplesclerosis.MedicalImageAnalysis15(2),267–282(2011)
  • [10] Villani,C.:OptimalTransport:OldandNew,GrundlehrendermathematischenWissenschaften,vol.338.SpringerBerlinHeidelberg(2009)

Appendix 0.A DivergenceGradients

Itcanbeshownthatthederivativeofthedivergencebetweendensitiesandwithrespecttosomeparameterofisgivenby

(5)

LetanddenotetwoGaussianmixtures,withcomponentsand.Giventhatthederivativesofw.r.t. itscomponentparametersareand,thegradientsofthedivergencecanbewrittenas

(6)
(7)

whereand.Tomakesuretheprecisionsremainnon-negativethroughouttheoptimisation,wecansimplyreparametrisethemas,with.

Appendix 0.B MassConservationProof

Afterperformingthedivergenceminimisationschemedescribedabove,wehaveaccesstothesequencesand(letusassumecontinuousfornow),where.Thisallowsustoevaluateand.Weseekamapunderwhichsamplesfromwillconformwith,i.e.

aimingatmakingthefinalapproximatelyagreewiththetargetdensity.Onekeyconceptinthefollowingdevelopmentsisthatofconservationofmass,acornerstoneoffluiddynamics.Takingaprobabilitydensityforthetypicalmechanicaldensity,theconservationof(probability)massprinciplestatesthattheprobabilityofapointbeinginafixedregionofspacechangesbythenetprobabilityinfluxthroughitsboundary.Alternatively,statedfromtheLagrangianperspective,theprobabilityofamovingregionremainsconstantasitsboundaryistransportedbyavelocityfield.Undersmoothnessassumptionsondensitiesandvelocities,bothpicturesareequivalenttothedifferentialconservationlaw.

Definition 1

Avelocityfieldissaidtoconservemassforanevolvingfamilyofdensitiesiffitsatisfiesthecontinuityequation:

(8)

Letdenotethetrajectoryofpoint,asitistransportedbytheflowfromtimeuntil,with.Itcanbeformulatedasthefollowingordinarydifferentialequation(ODE):

(9)

Ifa(locallyLipschitz-continuous)velocityfieldsatisfiesthecontinuityequationforanevolvingdensity,thentheinducedflowisuniquelydefinedand,i.e. thepushforwarddensitythroughcoincideswiththetarget[10, p. 15].333Herewehaveoverloadedthenotion(andnotation)ofpushforwardmeasuretothecorrespondingdensityfunction,denotingforsomePDFanddiffeomorphism.Therefore,iftheevolutionofisknown,weonlyhavetodetermineasuitable.RecallthattheevolvingdensitysatisfiestheJacobianequation:

(10)

wheretheJacobiandeterminant,,quantifiesthelocalcompression()andexpansion()ofthedensity.Itisoftenusefultocomputeitexplicitly,whichwecandobasedonEq. 9:

(11)

Theresultsshownherepertaintomultivariatedensitiesandflows,andcannaturallybespecialisedtotheone-dimensionalcasediscussedinthemainpaper.

Lemma 1

Thevelocityfielddefinedas

(12)

conservesmassforadensitybuiltviareparametrisationofafixeddensity.

Proof

Withdefinedasabove(wewilluseasimplifiednotationhereforclarity),wehaveitsdivergenceas

Let.Takingthetotaltimederivativeof,

wecanwrite

(13)

wherewehaveappliedJacobi’sformula,andusedthefactthat.Sincewehavetakentobediffeomorphic(hencesurjective),theresultinEq. 13mustalsoholdovertheentireimageof,i.e.

Proposition 1

Letdenotedensityfunctionsandvelocityfields,andsuchthatand.Iftheflowdeterminedbyeachconservesmassfortheevolutionoftherespective,thentheflowdeterminedbyconservesmassfortheevolutionof.

Proof

Letusassumethat,forsomechoiceofparametricdensityfamily,wehaveobtainedavelocityfieldthatsatisfiesthecontinuityequationfortheevolutionof,foreach:

(14)

Now,takingaconvexcombinationoftheabovewithweights,weobtain

wherewehavedefinedand.∎

Appendix 0.C CentreClassificationResults

Guy’s HH IOP Overall
Unnormalised 0.9906 0.9892 1.0000 0.9913
Affine 0.9687 0.9135 0.8904 0.9411
\MethodName:Centre 0.9687 0.9081 0.7945 0.9272
\MethodName:Indiv. 0.9749 0.9135 0.8219 0.9359
Nyul 0.9655 0.9189 0.7945 0.9289
Table 2: Centreclassificationaccuracy.ThecentralcolumnscorrespondtothethreeLondonimagingcentreswhereIXIdatawascollected:Guy’sHospital,HammersmithHospital(HH)andInstituteofPsychiatry(IOP).The‘Overall’columnshowstheclass-balancedaverageaccuracy.
Figure 7: Centrepredictionstatistics.Eachviolinplotindicatesminimum,maximumandmedian.Dashedhorizontallinesmarkmaximalpossiblevalues(forentropy).Confidencehereisthepredictedprobabilityofthechosenclass.
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