system morphometry Landmark registration constrained of high-genus surfaces applied tovestibular Computerized Medical Imaging and Graphics

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Computerized Medical Imaging and Graphics

jo u r n al ho me p ag e: w w w . e l s e v i e r . c o m / l o c a t e / c o m p m e d i m a g

Landmark constrained registration of high-genus surfaces applied to vestibular system morphometry

Chengfeng Wen

^a

, Defeng Wang

^b,g

, Lin Shi

^c,d

, Winnie C.W. Chu

^b

, Jack C.Y. Cheng

^e

, Lok Ming Lui

^f,∗

aDepartmentofComputerScience,StonyBrookUniversity,StonyBrook,NY11794,USA

bDepartmentofImagingandInterventionalRadiology,TheChineseUniversityofHongKong,Shatin,HongKong

cDepartmentofMedicineandTherapeutics,TheChineseUniversityofHongKong,Shatin,HongKong

dChowYukHoTechnologyCenterforInnovativeMedicine,TheChineseUniversityofHongKong,Shatin,HongKong

eDepartmentofOrthopaedicsandTraumatology,TheChineseUniversityofHongKong,Shatin,HongKong

fDepartmentofMathematics,TheChineseUniversityofHongKong,Shatin,HongKong

gShenzhenResearchInstitute,TheChineseUniversityofHongKong,Shenzhen,China

a r t i c l e i n f o

Articlehistory:

Received28November2014 Receivedinrevisedform17May2015 Accepted20May2015

Keywords:

Surfaceregistration Landmark High-genussurface Vestibularsystem Universalcoveringspace Beltramicoefficients Homotopicloops Minimalsurfaces

a b s t r a c t

Theanalysisofthevestibularsystem(VS)isanimportantresearchtopicinmedicalimageanalysis.VS isasensorystructureintheinnerearfortheperceptionofspatialorientation.Itisbelievedseveraldis- eases,suchastheAdolescentIdiopathicScoliosis(AIS),areduetotheimpairmentoftheVSfunction.The morphologyoftheVSisthusofgreatresearchsignificance.AmajorchallengeisthattheVSisagenus-3 surface.Thehigh-genustopologyoftheVSposesgreatchallengestofindaccuratepointwisecorrespon- dencesbetweenthesurfacesandwherebyperformaccurateshapeanalysis.Inthispaper,wepresenta methodtoobtainthelandmarkconstraineddiffeomorphicregistrationbetweentheVSsurfacesbased onthequasi-conformaltheory.GivenasetofcorrespondinglandmarksontheVSsurfaces,adiffeomor- phismbetweentheVSsurfacesthatmatchesthefeaturesconsistentlycanbeobtained.Thebasicideais toiterativelysearchforanadmissibleBeltramicoefficient,whichisassociatedtoourdesiredlandmark matchingregistration.Withtheobtainedsurfaceregistrations,vertex-wisemorphometricanalysiscan becarriedout.Twotypesofgeometricfeaturesareusedforshapecomparison.Oneisthecollectionof homotopicloopsoneachcanalsoftheVS,whichcanbeusedtomeasurethelocalthicknessofthecanals.

Fromthehomotopicloops,centerlinescanbeextracted.Byexaminingthedeviationsofthecenterlines fromthebestfitplanes,bendingsofthecanalscanbedetected.Thesecondgeometricfeatureisthe minimalsurfaceenclosedbythehomotopicloop.Fromtheminimalsurfacesofeachhomotopicloops, cross-sectionalareaofthecanalscanbeevaluated.Tostudythelocalshapedifferencemorecomprehen- sively,acompleteshapeindex,whichisdefinedusingtheBeltramicoefficientsandsurfacecurvatures, isused.Wetestproposedregistrationmethodon15VSofnormalcontrolsubjectsand12VSofpatients sufferingfromAIS.Experimentalresultsshowtheefficacyandaccuracyoftheproposedalgorithmto computetheVSsurfaceregistration.Shapeanalysishasalsobeencarriedoutusingtheproposedgeo- metricfeaturesandshapeindex,whichrevealsshapedifferencesintheposteriorcanalbetweennormal anddiseasedAISgroups.

©2015ElsevierLtd.Allrightsreserved.

1. Introduction

Surface-based morphometry has become one of the most ubiquitousshapeanalysistechniquesinmedicalimageanalysis.

Forexample,inhumanbrainmapping,themorphometryofbrain

∗Correspondingauthor.Tel.:+85239437975.

E-mailaddresses:[email protected](C.Wen),[email protected] (D.Wang),[email protected](L.Shi),[email protected](L.M.Lui).

cortical surfaces has been extensively studied to detect shape changes during disease progression or healthy aging. The hip- pocampalsurfacemorphometryhasalsobeenanactiveresearch fieldtostudytheAlzheimer’sdisease.

In order tocompare two anatomical surfaces, findingaccu- rate point-wise correspondences between them is of utmost importance.Such aprocessis calledsurface registration.Surface registrationsbetweensimply-connectedopenorclosedsurfaces havebeenextensivelystudiedandvariousalgorithmshavebeen developed. However, registration between high-genus surfaces http://dx.doi.org/10.1016/j.compmedimag.2015.05.006

0895-6111/©2015ElsevierLtd.Allrightsreserved.

Fig.1.(A)TheVSsurfaceofanormalsubject.(B)TheVSsurfaceofanAISsubject.

is comparatively less studied. The high-genus topology of the surfacesmakestheregistrationproceduremuchmorechallenging.

Onetypicalexampleofhigh-genusanatomicalstructuresiscalled thevestibularsystem(VS)(seeFig.1).TheVSisagenus-3structure situatedintheinnerear,whichisasensorystructureresponsible for detectinghead movements and sending postural signals to thebrain.ThemorphometryofVSplaysanimportantroleinthe analysis of various diseases such as the Adolescent Idiopathic Scoliosis(AIS) disease.The AISis a 3D spinal deformity which affects about4% schoolchildren worldwide. Theetiology of AIS isstill unclearbut believedtobea multi-factorial disease.One popularhypothesiswassuggestedtobethestructuralchangesin theVSthatinducethedisturbedbalanceperception,andfurther causethespinaldeformity.ThemorphometryoftheVSbecomes importanttounderstandthedisease.

Ourgoalinthispaperisdevelopaneffectivealgorithmtoobtain accurateregistrationsbetweenVSsurfacessothatsystematicshape analysiscanbeperformed.Inmedicalimaging, landmark-based surfaceregistrationmaysometimesbeadvantageous,sinceexpert knowledgeoffeaturecorrespondencescanbeincorporatedinto theregistrationmodel.Althoughvariouslandmark-basedregistra- tionmodelshavebeenrecentlyproposed,mostofthemfocuson computinglandmarkmatchingregistrationsforsimply-connected surfaces.Forhighergenussurfaces,theyareoftencutintoseveral simply-connectedpatches. Landmark-based registration is then obtained by registering every corresponding simply-connected patches.Thedrawbackisthatconsistentboundarycutsmustbe accuratelydelineated,whichisusuallydifficult.Inthiswork,we proposeaneffectivealgorithmtoobtainthelandmarkconstrained diffeomorphic(1-1andonto)registrationbetweentheVSsurfaces, whichisindependentoftheboundarycuts.Givenasetofcorre- spondinglandmarksontheVSsurfaces,adiffeomorphismbetween theVSsurfacesthatmatchesfeaturesconsistentlycanbeobtained.

Tosimplifytheregistrationproblem,theVSsurfacesarefirstly parameterizedconformallyintothehyperbolicdiskHinR²,which isendowedwiththehyperbolicmetric.Aniterativeschemeisthen usedtosearchforalandmark-matchingdiffeomorphism.Itisdone byiterativelysearchforanadmissibleBeltramicoefficient,which is associated to our desired landmark-aligned diffeomorphism.

Usingtheproposedmethod,accuratepoint-wisecorrespondences betweentheVSsurfacesthatmatchcorrespondingfeaturescanbe efficientlyobtainedwithin30s.

Once the surfaces have been registered, vertex-wise mor- phometric analysis can be carried out. Various corresponding geometricfeaturescanalsobeextractedontheVS,whichcanbe usedfortheshapecomparison.Inthispaper,weextracttwokinds ofgeometricfeaturesontheVSsurfaces.Oneisthehomotopic loopsoneachcanalsoftheVS.Asetofgeodesichomotopicloopsis firstlyextractedonthemeanVSsurface.Withtheregistration,cor- respondinghomotopicloopsoneveryVSsurfacescanbeextracted.

Thesehomotopicloopscanbeusedtostudythelocalgeometryand

thicknessateachpositionoftheVSsurface.Fromthehomotopic loops,thecenterlinescanbeextracted,whicharecurvespassing throughthecentroidsofhomotopicloops.Byexaminingthedevi- ationsofthecenterlinesfromthebestfitplane,bendingsofthe canalscanbedetected.Thesecondgeometricfeatureisthemini- malsurfacesenclosedbythehomotopicloops.Withtheseminimal surfaces,cross-sectionalareaateachpositionofthecanalscanbe evaluated.Tostudythelocalgeometricdifferenceindetails,acom- pleteshapeenergy,whichisdefinedusingtheBeltramicoefficients andsurfacecurvatures,isalsoapplied.

Wehaveevaluatedtheproposedregistrationalgorithmon15 VSofnormalcontrolsubjectsand12VSofAISpatients.Results demonstratetheefficiencyandaccuracyoftheproposedmethodto computetheVSsurfaceregistration.Shapeanalysishasalsobeen carriedoutusingtheproposedshapefeaturesandshapeenergy, whichrevealsshapedifferencesintheposteriorsemi-circularcanal betweenthenormalcontrolanddiseasedsubjects.

Inshort,thecontributionsofthisworkaretwo-folded.Firstly, weproposeamethodtocomputethelandmarkconstraineddif- feomorphicregistrationbetweenthehigh-genussurfaces,which isindependentoftheboundarycuts.Givenanytwocorrespond- ingsetsoflandmarks,wecanobtainasmoothandbijectivemap betweenthehigh-genussurfacesaligningthelandmarksconsis- tently.Secondly,weproposeseveralshapeanalysismodelstostudy thegeometricdifferencesbetweencorrespondingVSsurfaces.

2. Relatedworks

About4%ofteenagerssufferedfromAdolescentIdiopathicSco- liosis(AIS).DevelopedAISleadstophysicalpainsandlowerthe pulmonaryfunction.Vitalcapacityofthelungcouldbereduced duetodeficitrespiratory musclestrengthunderabnormalbone andmusclestructure[1].AISalsoaffectsthelivingqualityofthe patients,disabilityonworkhoursandactivitylevel.Perceptionof limitationduetothediseasemightincreasethechancefordepres- sion[2].RecentresearchessuggestthatAISmaybeacauseofother mentalillnessessuchaseatingdisorders[3].

TheetiopathologyofAISisstillunknownandthereareclin- icalandscientificstudiestoinvestigatethecausefromdifferent aspects.Poorposturalbalanceisoneoftheearliestcharacteristics recognizedandbeingstudiedinpatientswithAIS[3–5].Experi- mentsdemonstratedthatthedevelopmentofAISisrelatedtothe impairedvestibularsystem[6,7].Lambertetal.[8]demonstrateda possiblerelationshipbetweenthevestibularsystemandscoliosisin frogmodelsbyremovingthelabyrinthineendorgansattheirlarval stages.Characteristicssimilartoscoliosissuchasspinaldeforma- tionappearedaftertheirmetamorphosis.Therefore,themorpho- metricanalysisoftheVSisofutmostimportancetounderstandAIS.

Therehasbeenalotofworkonshape analysisforanatomi- calsurfaces.Inordertoperformshapeanalysiseffectively,surface registrationisnecessary.Surfaceregistrationhasbeenextensively studied.For genus0 closedsurface registration,conformalsur- faceregistration,whichminimizesangulardistortionandpreserves thelocalgeometrywell,hasbeenwidelyused[9–13].Hurdaland Stephenson[12]proposedtocomputetheconformalparameteri- zationsusingcirclepackingandappliedittoregistrationofhuman brains.Guetal.[10]andWangetal.[13]proposedtocompute theconformalparameterizationsofhumanbrainsurfacesforreg- istrationusingharmonicenergyminimizationandholomorphic1- forms.Quasi-conformalmapping,asageneralizationofconformal map,hasbeenappliedtoobtainsurfaceregistration[14–16].For example,Luietal.[15]proposedtocomputequasi-conformalregis- trationsbetweenhippocampalsurfacesbasedontheholomorphic Beltramiflowmethod,whichmatchesgeometricquantities(such ascurvatures)andminimizestheconformalitydistortion[14].

In many situations, shape information suchas curvature or intensityareimportant.Severalregistrationalgorithmshavebeen proposedtoincorporatetheseinformation intotheregistration models.Lytteltonetal.[17]computedsurfaceregistrationswith surfacecurvaturematching.Fischletal.[9]proposedanalgorithm forbrainregistrationthatbetteralignscorticalfoldingpatterns,by minimizingthemeansquareddifferencebetweentheconvexityof thesurfaceandtheaverageconvexityacrossasetofsubjects.Lord etal.[18]proposedtomatchsurfacesbyminimizingthedeviation oftheregistrationfromisometry.Yeoetal.[19]proposedthespher- icaldemonsmethod,which adoptedthediffeomorphic demons algorithm[20],todrivesurfacesintocorrespondencebasedonthe meancurvatureandaverageconvexity.

Feature landmarks,suchas sulcal landmarks onthe human brains,provideimportanthumanknowledge.Landmark-matching registrationalgorithmsareproposedtotaketheseinformations into account. Bookstein [21] proposed to obtain a landmark- matchingregistrationusingathin-platesplineregularization(or biharmonicregularization). Tosun et al.[22] proposedto com- bineiterativeclosestpointregistration,parametricrelaxationand inversestereographicprojectiontoaligncorticalsulciacrossbrain surfaces.Thesediffeomorphismsobtainedcanbettermatchland- markfeatures,althoughnotperfectly.Wangetal.[23]andLuietal.

[24]proposedtocomputetheoptimizedconformalparameteriza- tionsofbrainsurfacesbyminimizingacompoundedenergy.Lui etal.[25]proposedtheuseofvectorfieldstorepresentsurface mapsandreconstructthemthroughintegralflowequations.Addi- tionally,timedependentvectorfieldscanbeusedtorepresentthe setofallsurfacemaps.JoshiandMiller[26]proposedthegenera- tionoflargedeformationdiffeomorphismsforlandmarkmatching, wheretheregistrationsaregeneratedassolutionstothetransport equationoftimedependentvectorfields.

Most of theexisting algorithms for surface registration and shapeanalysiscanonlydealwithsimply-connectedopenorclosed surfaces.Analyzinghigh-genussurfacesis generallychallenging because of theircomplicated topologies. Recently, some works havebeencarriedouttoregisterandanalyzehigh-genussurfaces.

Zeng et al. [27] proposedto measure the geodesic spectra on VSsurfaces using the Ricciflow. Length and thickness of each canalweremeasuredandthedifferencebetweentwogroupsof subjectswasshownstatisticallysignificant.Partitionapproaches havealsobeenproposedtoanalyzeand registerVSsurfacesby partitioningthemintosimpleopensurfacepatches[28].However, partition approaches greatly rely on theconsistency of surface cuts,whichareusuallydifficulttoobtain.Geometricregistration ofhigh-genus surfacesthat matches thesurfacecurvatures has beenproposedin[29].Surfaceregistration isobtainedbymin- imizing thecurvature mismatching energy, Since theenergy is non-convex,theobtainedregistration canpossibly bethelocal

minimizer.Thus,thisalgorithmcangivegoodregistrationresults onlyiftwosurfaceshavesimilarcurvaturedistributions.

3. Background

Inthissection,wedescribesomebasicmathematicaltheories closelyrelatedtoourwork.

3.1. Uniformizationofhigh-genussurfaces

Every Riemannsurface of genusg>1 is conformallyequiva- lenttoaRiemannianmetric,calledtheuniformizationmetricorthe hyperbolicmetric,whoseinducedGaussiancurvatureisequalto−1.

Thehigh-genussurfaceScanbeconformallyembeddedintothe PoincarédiskH⊂R².ThePoincarédiskisaunitdiskequippedwith thehyperbolicmetric:

ds²= dx²+dy²

(1−x²−y²)² (1)

In fact, Scan besliced openalong thecanonical homotopic basis {a₁,b₁,...,a_g,b_g}of thefundamental domain(S,p)ata point p∈S, for whichany two loopsofthe basis intersectonly at p (see Fig. 2(A)). Slicing along the basis, we get a simply- connected open surface S^cut. The boundary of S^cut is given by a₁b₁a⁻₁¹b⁻₁¹···agbga⁻_g¹b⁻_g¹.S^cutcanthenbeconformallyparameter- ized1-1andontoadomain

D_i⊂H,whichiscalledafundamental polygon(seeFig.2(B)).Denotetheparameterizationby

^:Scut→

D_i.Theedgesof

D_isatisfytheperiodicboundaryconditions.More specifically,thereexistMöbiustransformations{ϕ1,1,···,ϕg,g} (calledtheFuchsiangroupgenerators)suchthat:

ϕ_i(^(a

i))=

^(a−_i^1);i(

^(bi))=^(b

⁻_i^{1) (2)}

Bygluinginfinitelymanycopiesof

D_itogetheralongitsbound- aries,wegetthePoincarédiskH.

^{isextendedtoasurjectivemap}

:H→S,whichiscalledthecoveringmap,satisfying

⁻¹(S)=

i∈I

D_i, (3)

where

D_iand

D_jintersectsonlyattheedgesofthefundamental polygon(seeFig.2(C)).

3.2. Basicquasi-conformaltheories

Let ₁ and ₂ be 2D domains. Every diffeomorphism f:₁→₂ is associated to a unique Beltrami coefficient (BC), which measures the conformality distortion of the f:=u+iv.

Fig.2. (A)Avestibularsystem(VS)surface.(B)Itsfundamentalpolygoninthehyperbolicdisk.(C)TheuniversalcoveringspaceoftheVSsurface.

Here,weconsiderfasacomplex-valuedfunctionon₁.TheBC, :1→Ccanbecomputedbythefollowingequation:

(x,y)=

∂f

∂x+i∂f

∂y

/

∂f

∂x−i∂f

∂y

. (4)

GivenasmoothBC:C→Cwith∞<1.Thereisalways a diffeomorphism of Cthat satisfiestheEq. (4) [30].However, suppose1 and2 arearbitrarydomainsandextraconstraints (suchaslandmarkconstraints)areenforced.Then,anarbitraryBC :₁→Cmaynotbeassociatedtoadiffeomorphismf:₁→₂ subjecttotheextraconstraints.Inthiscase,aBCiscalledadmissi- bleifitisassociatedtoaquasi-conformalmapsubjecttothegiven constraints.

Inthiswork,weuseBCstocontrolthebijectivityofthemap- pings[31].

4. Constraineddiffeomorphicregistrationforhigh-genus surfaces

Inthissection,wedescribetheproposedlandmarkconstrained surfaceregistrationalgorithmforhigh-genussurfacesindetails.

4.1. Motivation

SupposeS₁andS₂aretwogenus-3VSsurfaces.Let{p_i∈S₁}ⁿi=1

and{q_i∈S2}ⁿi=1bethecorrespondingfeaturelandmarksonS1and S₂ respectively. Note that the feature landmarkscan either be landmarkpointsorcurves.Ourobjectiveistolookforasmooth diffeomorphismf:S1→S2thatmatchesthecorrespondingfeature landmarksconsistently.Thatis,f(p_i)=q_ifori=1,2,...,n.Although varioussurface registrationalgorithms have beenrecentlypro- posed, the topology of the high-genus surfaces poses a great challengetowardthisgoal.

In[29],avariationalmodelhasbeenproposedtoobtaingeomet- ricregistrationbetweenhigh-genussurfacesthatmatchessurface curvatures.Thebasicideaistoconformallyembedeachhigh-genus surfaceintothePoincarédiskHinthe2DplaneusingtheRicciflow method.RegistrationisthencarriedoutonHthroughminimiz- inganenergyfunctionalinvolvingacurvaturemismatchingterm.

Thisalgorithm givesgoodsurfaceregistrationresultsifthetwo surfacesaregeometricallysimilar.Theregistrationresultmaybe inaccurateifthetwosurfaceshaveverydifferentcurvaturedis- tributions.Forexample,theVSsurfacesofanormalsubjectand adiseasedpatienthavequitedifferentsurfacecurvatures.Curva- turemaynotbeavalidinformationtoguidethesurfacematching.

Applyingthealgorithmin[29]onthesesurfacesmaybesusceptible tomis-registration.Inthissituation,landmark-basedregistration maybemore advantageous, since expert knowledgeof feature correspondences can be incorporated into the model. Various landmark-basedsurfaceregistrationshavebeenrecentlyproposed.

However,mostofthemfocusoncomputinglandmarkconstrained registrationsforsimply-connectedsurfaces.Thismotivatesusto lookforaneffectivealgorithmtoobtainadiffeomorphismbetween high-genussurfacesthatalignsfeaturesconsistently.

4.2. Contributions

Similarto[29],wefirstsimplifytheregistrationproblembycon- formallyembeddingS1 andS2intoHinR².Aniterativescheme is then used to look for a landmark-matching diffeomorphism betweentheconformalparameterdomains.Thestrategyistoiter- ativelysearchforanadmissibleBeltramicoefficientassociatedto ourdesiredlandmark-aligneddiffeomorphism.Here,aBCiscalled admissibleifitisassociatedtoabijectivequasi-conformalmapsub- jecttothelandmarkconstraints.In[32],thesurfaceTeichüllermap

(T-Map)wascomputedthroughadjustingBCs.TheBCisnormalized ineachiterationtoanadjustedBCwithaconstantnorm.However, thisalgorithmdoesnotsuitourpurposefortworeasons.Firstly, itworksonlyonsimply-connectedopenorclosedsurfaces.Inour case,genus-3VSsurfacesareconsidered.Secondly,thealgorithm computessurfaceT-Map,whoseconformalitydistortionsareuni- formandspreadoverthewholesurface.Itisunnaturalinmedical applications.

Motivatedbytheaboveobservations,weproposealandmark constrainedregistrationalgorithmforhigh-genussurfaceswitha smoothandnaturaldistributionofconformalitydistortions.The proposedalgorithmcancomputeadiffeomorphismbetweentwo high-genussurfacesthatmatchesprescribedcorrespondingland- marksconsistently.Thecontributionsaretwo-folded:

1.First, wepropose a landmark-constrainedregistrationalgo- rithmforgeneralhigh-genussurfaces.Thisisanovelextension ofthecurvature-matchingregistrationalgorithmproposedin [29]. We stress that VS surfaces of different subjects have quitedifferentcurvaturedistributions,curvature-matchingreg- istration maybe susceptibleto mis-registration.Therefore, a landmark-basedregistrationforhigh-genus surfacesisneces- sary.

2.Second,weproposeamodifiediterativescheme,whichextends the scheme in [32], to search for an admissible BC under landmark constraints witha natural distribution for general high-genussurfaces.ThekeyideaistofindanadmissibleBC underlandmarkconstraintsbetweenthePoincarédisks.This admissibleBCgivesrisetoabijectiveandlandmark-matching registrationbetweenthePoincarédisks.Thedesiredconstrained registration betweenthe high-genussurfaces S₁ and S₂ can thenbeobtainedbythecompositionmapwiththeirconformal parameterizations.Theresultantregistrationhasasmoothand naturaldistributionofconformalitydistortions.

4.3. Detaileddiscussionoftheregistrationalgorithm

SupposeS₁^cutandS₂^cutarethefundamentaldomains,obtainedby slicingalongthehomotopicbases{a1,b1,a2,b2,a3,b3}and{c1,d1, c₂,d₂,c₃,d₃}ofS₁andS₂respectively.Let₁:S₁^cut→D

₁⊂Hand ₂:S^cut₂ →

D₂⊂HbetheconformalparameterizationofS^cut₁ and S^cut₂ respectively.WeapplytheRicciflowmethod[33]toobtain 1 and2inthiswork.TheideaofRicciflowistoconformally deformthesurfacemetricg=(gij(t))accordingtoitsinducedGaus- siancurvatureK(t)toitsuniformizationmetricthroughtheheat flowequationonthemetric:^dg_dt^ij(t)=−2(K(t)+1)g_ij(t).g(∞)isthe desireduniformizationmetric.Fromtheuniformizationmetric,the surfacecanbeconformallyembeddedontoafundamentalpolygon inH.

WiththeconformalparameterizationofS₁ andS₂,theregis- trationprocesscanbecarriedoutonH.Ourgoalistolookfora diffeomorphismg:H→Hsuchthat:

(1)Landmarkconstraints:g(1(pi))=2(qi)fori=1,2,...,n.

(2)Periodic conditions on boundary cuts: ϕ_i(g(ai))= g(a⁻_i¹)and_i(g(bi))=g(b⁻_i¹),

where:{a1,b1,a2,b2,a3,b3}areimagesofthehomotopic basisonS₁inH.{ϕ₁,₁,ϕ₂,₂,ϕ₃,₃}aretheFuchsiangroup generators.

(1)is thelandmarkconstraintswhich requirecorresponding featurelandmarkstobe alignedconsistently. For (2), notethat ourproposedregistrationprocessiscarriedonH,butnotonthe fundamentalpolygonsD

₁and

D₁,withoutenforcingthecorrespon- dencesofthehomotopicbasisofS^cut₁ andS₂^cut.Inotherwords,a_imay

notcorrespondtoc_iandb_imaynotcorrespondtod_ifor1≤i≤3.

Therefore,someperiodicconditionsmustbesatisfiedonH.More specifically,suppose⁻¹₁ (S₁)=

i∈I

D¹_i and⁻¹₁ (S₂)=

i∈ID

²_i,we requirethatgisassociatedtoamappingf:S₁→S₂suchthatf= 2◦g◦1|

^−1D1i

foranyi∈I.Inotherwords,werequirethatgcanbe liftedtoasurfacemapf:S1→S2.(2)ensurestheabovecondition tobesatisfied.

Weproceedtosolvetheaboveconstrainedproblemwithan iterativescheme,usingthequasi-conformaltheory.Sincegisadif- feomorphism,itsassociatedBC_g:H1→Cmusthavesupreme normstrictlylessthan1(||g||_∞<1).Infact,theJacobianJg is closelyrelatedtog:

Jg=

^∂_∂^g_z

^{2(1−|g|2) (5)}

Hence,Jg>0ifandonlyif||g||_∞<1.Byinversefunctiontheo- rem,gisadiffeomorphismifandonlyif||_g||_∞<1.

Therefore,toobtaing,ourstrategyistoiterativelysearchfor asequenceofBCs{_j}^∞_j=1associatedwithasequenceofmappings {g_j}^∞_j=1suchthat:_j→gwith||g||_∞<1andg_j→gasj→∞.More precisely,iterativeschemecanbeexplainedgeometricallyasfol- lows.

₀ −→ ... −→ _j −→ ... −→ _∞=g

g₀ −→ ... −→ g_j −→ ... −→ g_∞=g (6)

Fortheinitialmapg₀:H→H,wechoosetheharmonicmap satisfyingthelandmarkconstraintsandthecorrespondencesofthe boundarycutsgivenbythearc-lengthparameterization.Notethat withthelandmarkconstraints,g0isunlikelybijective.According tothequasi-conformaltheory,anon-bijectiveg₀isassociatedwith aBC₀withsupremenormgreaterthan1.Ourgoalistolookfor anadmissibleBC,,with||||∞<1,whichisassociatedtoaquasi- conformalmapbetweenH1 andH2satisfyingtheconstraints(1) and(2).Weproceedtoadjust₀iterativelytoobtainourdesired optimalBC.Moregenerally,supposegnisobtainedatthenthiter- ation,whoseBCisgiven by_n.Our proposedalgorithm canbe describedasachop-smooth-reconstructiterativeprocedure.We nowdescribeeachstepindetails.

4.3.1. Chop

Suppose gn is non-bijective. Then, ||n||>1. We proceed to projectn toaBCwithsupremenormstrictlylessthan1.This canbedonebychoppingdownthenormofn.Thechoppingis toeliminatetheoverlapsofgn,since|n|>1indicatestheoccur- renceofflips.Morespecifically,thechoppingoperationisdefined asfollows:

n= (1− )n

|n|, if|_n|≥1− n, if|n|<1−

(7)

where >0issmall.

Here,weremarkthatthechoppingofBCwasalsoappliedin[29]

tocontrolthebijectivityofthemapping.However,notethatthere isnolandmarkconstraintenforcedin[29].Hence,anadmissibleBC associatedtoabijectivemapcanbeeasilyobtainedbysolvingthe Beltramiequation.Inourcase,landmarkconstraintsareenforced andthechallengeistosearchforanadmissibleBCassociatedto abijectivemapsatisfyingthelandmarkconstraintsconsistently.

Hence,inthiswork,weneedtoiterativelysearchforanadmissible BCassociatedtoalandmark-matchingdiffeomorphism.

4.3.2. Smooth

AsmoothmappingisassociatedwithasmoothBC.Inorderto obtainasmoothregistration,weproceedtosmoothout

^n.This

canbedonebyaLaplacesmoothingon

n: n(vi)= 1

N_i

vj∈N_i

n(vj) (8)

where N_i is the one-ring neighborhood of vertex vi. Thissim- plesmoothingoperationallowsustofurtherreducethenormof BC(and thusreduce theconformalitydistortions). It alsohelps toobtaina smoothand naturallydistributedBC,and hencethe resultantregistrationhasasmoothandnaturaldistributionofcon- formalitydistortions.

4.3.3. Reconstruct

AftertheadjustedBCnisobtained,weneedtofindanassoci- atedmapgn+1:H→Hsatisfyingthelandmarkconstraints(1)and periodicconditions(2)ontheboundarycuts.Notethat_n may notbeadmissible.Inotherwords,subjecttothelandmarkcon- straints,theremaynotexistaquasi-conformalmapwhoseBCis exactlyequalto_n.Weproceedtolookforamapg_n+1whoseBC isasclosetonaspossible.

Asin[31],thereconstructionofthequasi-conformalmapfrom anadmissibleBC(withoutlandmarkconstraints)canbereducedto solvinganellipticPDE.Also,inourcase,thereconstructedmapg_n+1 fromHtoitselfcanbeliftedtoasurfacemapf_n+1:S₁→S₂.Hence, theperiodiccondition(2)ontheboundarycutsofthefundamental polygonmustbesatisfied.Following[29],theellipticPDEsubject totheperiodiccondition(2)canbereducedtothefollowingnon- linearequationinthediscretecase:

A˜z+Q(˜z)=b, (9)

where ˜z_i=g_n+1(vi)(vi∈H),A=(w_ij)isthematrixrepresentation discretizingthedifferentialoperator ∇·(D∇^{), and Q(i, j)is the} Mobiüstransformationthattransformsoutside vertex

^vjoffun- damentalpolygontoitsinsidecopyvjinthefundamentalpolygon, andiszeroelsewhere,multipliedbyw_ij.

Inourwork,landmarkconstraints(1)arealsoenforced.Inother words,thefollowingconstraintsshouldbeaddedtothenonlinear Eq.(9):

˜

z_j:=g_n+1(p_j)=q_j (10) for1≤j≤n,where{p_j}ⁿ_j=1and{q_j}ⁿ_j=1 arethecorrespondingland- markconstraints.

WeapplytheNewton’smethodtosolvethesimultaneousEqs.

(9)and(10).Wecallsuchprocessthelandmarkconstrainedhyper- bolicBeltramisolver,anddenoteitbyHBS_LM.

ThereconstructionofmapusingHBS_LMensurestheconstraints (1)and(2)arebothsatisfiedineachiteration.TheBCisiteratively choppeddowntoanoptimalBCwithsupremenormstrictlyless than1,whichisassociatedtoabijectiveandlandmark-matching quasi-conformalmap.

Inshort,motivatedby[29],wedevelopanefficientalgorithm tocomputeconstraineddiffeomorphicregistrationforhigh-genus surfaceswithasmooth andnaturaldistributionofconformality distortion.Thisproposediterativealgorithmisagradientdescent basedalgorithm.Itessentiallyminimizestheharmonicenergyof theBeltrami coefficientiterativelyoverthespace ofadmissible Beltramicoefficient.ThealgorithmstopswhenthechangeofBel- tramicoefficientsaresmallenough,whichconvergestoanoptimal solution.

Thealgorithmcanbesummarizedasfollows.

Algorithm1:

Input:universalcoveringspaceU₁andU₂,landmarkconstraints {p_i}ⁿ_i=1↔{q_i}ⁿ_i=1

Output:hyperbolicquasi-conformalmapf:U₁→U₂,s.t.f(p_i)=q_i, i=1,..,n

1.Computeinitialmapf₀andBeltramicoefficients₀=(f₀),n=0;

2.ChopandsmoothnbyEqs.(7)and(8)toobtain

n; 3.Reconstructf_n+1from

nviaHBS:f_n+1=HBS_LM(

n);

4.ComputeBeltramicoefficients_n+1=(f_n+1);

5.Ifn+1−n_∞< ,stoptheprocess,obtainthemapf=fn+1.Oth- erwisen=n+1,gotostep2;

5. Shapeanalysismodels

WiththeregistrationsoftheVSsurfaces,correspondinggeo- metricfeaturescanbeextractedtoanalyzetheshapedifferences.

Acompleteshapeindexcanalsobedefined,whichcanbeusedto performthelocalshapeanalysisoftheVSsurfaces.Inthissection, wewilldescribeourproposedshapeanalysismodelsindetails.

5.1. GeometricfeaturesontheVSsurface

Inthiswork,weproposetoextracttwogeometricfeaturesfor shapeanalysis.Theyare:(1)homotopicloops[34]and(2)minimal surfaces.Thesefeaturescanbeusedtounderstandthegeometric patternsoftheVSsurfaces.

Withtheregistration,wecancomputethemeanshapeofthe VSsurfaces.Denotethegenus-3VSmeansurfacebySmean.Ahomo- topicbasis based ata point p onthesurface can beextracted.

Bycuttingalong thehomotopic basis,Smean becomesa simply- connectedopensurface.Smeancanbeembeddedintoitsuniversal coveringusingRicciflow.Ontheuniversalcoveringspace,wecan easilyfindacanonicalhomotopicbasis{a₁,b₁,a₂,b₂,a₃,b₃},which intersectsonlyatthebasepointpandareallhyperbolicgeodesic.

Foreachpointqonthecurvea_i(i=1,2or3),wecanfindageodesic closedloopcq:[0,1]→S₁suchthatcq(0)=cq(1)=q.Thegeodesic loopcq(t)solvesthefollowingminimizationproblem:

c1(t)=argmin_(t)

1 0

g_(t)((t),(t))dt (11)

allclosedloop(t)satisfying(0)=(1)=q.Thecollectionofall loopscq(t)situatedatqonai(i=1,2or3)arecalledthehomotopic loops.Thesehomotopicloopsbelongstotheequivalenceclass[a_i] ofthehomotopicgroup.

Usingtheobtainedregistrationbetweenthemeansurfaceand anyVSsurface,correspondinghomotopicloopscanbedelineated oneachVSsurface.LetSbeanyVSsurface.Supposef:Smean→S istheregistrationbetweenSmeanandS.Thecorrespondinghomo- topicloopsonScanbeeasilyobtainedbyc^S_q:=f◦c_q:[0,1]→S.

Thesehomotopicloopscanbeusedtostudythelocalgeometryand thicknessateachpositionsoftheVSsurface.

Fromthehomotopicloops,thecenterlinescanbeextracted.Let {c^S_q}_q∈a

ibethecollectionofallhomotopicloopsononecanalofthe VSsurfaceS.Foreachhomotopicloop,wecancomputeitscentroid.

Byjoiningallthecentroids,wecanobtainacurvelyingintheinte- riorofonecanalofS.Thiscurveiscalledthecenterline.Usingthe centerline,bendingsofthecanalscanbeexamined.Eachcanalsof theVSsurfacecanroughlybefittedtoaplane.Andthethreeplanes areroughlyorthogonaltoeachothers.TostudyAIS,itiscommonly ofinteresttoexaminehoweachcanalsaredeviatedfromaplane.

Letc= c(t)beparametricequationofthecenterline.Wecanfinda bestfitplaneP:n·(x,y,z)=tothecenterlinebyminimizing:

E(n,,c)=

1 0

D(n,,c(t)) ²dt, (12) whereD(n,,c(t)) isthedistanceofthepointc(t)fromtheplane P:n·(x,y,z)=.ThedistanceD(n^∗,^∗,c(t)) ofthepointc(t) from

thebestfitplanecanbeusedtomeasurethedeviationofthecanal fromaplaneateachpointofthecenterline.

OncethehomotopicloopsoneachcanalsoftheVSisextracted, wecancomputetheminimalsurfacesenclosedbyeachhomotopic loops.Withtheminimalsurfaces,cross-sectionalareaateachposi- tionofthecanalscanbeevaluated.Minimalsurfacesaredefinedas surfaceswhicharecriticalpointsfortheareafunctional.Suppose thehomotopiclooplisprojectedorthogonallytoaconvexcurve enclosingadomainDinaplane.Theminimalsurfaceisagraphofa function:z=u(x,y),wherexandyarethecoordinatesontheplane (seeFig.3(A)).Supposethehomotopicloopisgivenbyl=l(x,y)for (x,y)∈∂D.Thefunctionu(x,y)oftheminimalsurfacesatisfiesthe followingEuler–Lagrangeequation:

∇·

∇^u 1+|∇u|²

=0, (13)

subjecttotheconstraintthat:u|_∂D(x,y)=l(x,y).Inthiswork,the minimalsurfaceiscomputedbysolvingEq.(13)usingthefinite elementmethod.Fig.3(B)showstheminimalsurfacesenclosedby thehomotopicloopsofastandard3-torus.

5.2. Completeshapeindexforlocalshapeanalysis

Itisoftennecessarytostudythelocalshapedifferencesbetween VSsurfaces.Forthispurpose,ashapeindexdefinedoneachvertex ofthesurfacemesh,whichquantifiesthelocalshapedifferenceat eachpositionofthesurface,isneeded.Inthiswork,weuseacom- pleteshapeindexbasedontheBeltramicoefficientsandthesurface curvatures.GiventwoVSsurfacesS1andS2.Supposef:S1→S2be theregistrationbetweenS₁andS₂,asobtainedinthelastsection.

Thecompleteshapeindexisdefinedasfollows.

E_shape(f)=˛||²+ˇ(H₁−H₂(f))²+(K₁−K₂(f))² (14) where is the Beltramicoefficients of f; H1,H2 are themean curvaturesonS₁ and S₂ respectively;and K₁, K₂ are theGaus- siancurvaturesonS₁ andS₂ respectively.Thus,E_shape:S₁→Ris anon-negativereal-valuedfunctiondefinedonS1.

Thefirsttermmeasurestheconformalitydistortionofthesur- faceregistration.Thesecondandthirdtermsmeasurethecurvature differences.ItturnsoutE_shapeisacompleteshapeindexmeasuring subtleshapedifferencesbetweentwoVSsurfaces.Moreprecisely, E_shape(f)=0ifand only iftwo shapesareidenticalup toarigid motion.Byadjustingtheparameters(i.e.,˛,ˇand),E_shapecan bemadeequivalenttootherexistingshapeindices.Forexample, whenˇ=0,E_shapeisequivalenttotheisometricshapeindex;when

˛=0,E_shapeisequivalenttothecurvatureindex; whenˇ==0, E_shapemeasurestheconformalitydistortion.Inourwork,weset˛, ˇandtobenonzerotomeasurecompleteshapechanges.

5.3. Vertex-wisestatisticalmorphometricanalysis

After the registration is obtained, a vertex-wise statistical morphometricanalysiscanbecarriedout.Thevertex-wisemor- phometric deviation can be investigated using a two-sample Hotelling’sT-squaredtest.Beforetheanalysis,theregisteredsur- facesarerigidlyalignedusingtheiterativeclosestpointalgorithm.

Thetestthenmeasurestheposition(coordinates)differenceoftwo correspondingpointsoftheVSsurfacesoftheAISgroupandthe normalcontrol(NC)group.Asignificantlevelof0.001willbeused inthiswork.

Fig.3. (A)Aspacecurve(left)andminimalsurfaceenclosedbythespacecurve(right).(B)(left)Thehomotopicloopsandcenterlinesofa3-torus,rightshowstheminimal surfacesenclosedbythehomotopicloops.

6. Experimentalresults

Totesttheefficacyoftheproposedalgorithms,experiments havebeencarriedouton27VSsurfacesfromthenormalandAIS groups.

SubjectandImagingProtocol:

Inthis research,27 left-sidevestibularsystemsampleswere studied.Thesampleswerefromtwogroupsofvolunteers,theAIS groupandtheNCgroup.TheAISgroupconsistsof12girlssuffer- ingfromright-thoracicAISwhilethereare15age-matchedhealthy girlsintheNCgroup.MRIscanningisperformedforobtainingT2 weightedimagesoftheinnerearsofthevolunteersusinga1.5TMR Scanner(Sonata,Siemens,Erlangen,Germany)withaquadrature headcoil.Theimagingprotocolisa3Dconstructiveinterference steadystate(CISS)sequenceand withparameters including,TR

=11.94ms, TE=5.97ms, flipangle =70^◦,FOV = 130mm,slice thickness s= 1mm, nogap,matrix =320×288 and number of excitation=1.Highquality imagesareobtainedwiththevoxel sizeof0.5×0.4×1.0mm³.Theresultantvoxelsizeisadjustedto 0.2×0.2×1.0mm³afteranon-siteimageinterpolation.

6.1. SurfaceregistrationoftheVS

OneachVSsurfaces,8featurelandmarksarelocated.Theland- marksareselectedasfollows:(i)4landmarksarechosentobethe saddlepointslocatedattheintersectionsofthethreecanals;(ii)3 pointsoneachcanalswhicharegeodesicmid-pointsoftheland- marksin(i);and(iii)1landmarkisselectedtobecenternearthe utricle.These8landmarksareconsistentamongstdifferentsub- jects.Fig.4showstheeightfeaturelandmarksononeVSsurface visualizedattwodifferentangles.

Fig.5(A) and(B)shows thecanonical fundamentalpolygons oftwoVSsurfacesofanormalandAISsubjectsrespectively.The canonicalcutsareshown oneach surfaces.By slicingalongthe cuts,wecanembedthesurfaceontoafundamentalpolygon.The fundamentalpolygonsofdifferentperiodscanbegluedtogetherto formtheuniversalcoveringspace.Notethatthecanonicalcutsare introducedtoobtaintheconformalembeddingofthesurfaceinto itsuniversalcoveringspaceonly.Duringtheregistrationprocess,

Fig.4.VisualizationoflandmarksonaVSsurfacefromtwoviewangles.

thecanonicalcutsonthesourcesurfaceareallowedtomovefreely onthetargetsurface,sincethewholeprocessiscarriedoutonthe universalcovering spaces. In otherwords,the correspondences betweencanonicalcutsarenotrequired.

Fig.6showstheuniversalcoveringspacesoftwoVSsurfacesof anormal(left)andanAISsubject(middle).Thepositionsofthecor- respondinglandmarksontheparameterdomainsareshown.Using theproposedregistrationalgorithm,wecomputetheconstrained diffeomorphismbetweentheparameterdomainsthatmatchescor- respondinglandmarks.InFig.6,therightshowstheregistration result,whichmatchesthelandmarksconsistently.Notethatthe boundary cutsonthesourceparameterdomainare allowedto movefreelyonthetargetparameterdomain,whilesatisfyingthe periodicconditions.

Through the composition map of the diffeomorphism with the conformal parameterizations, the registration between the VSsurfacescanbeobtained.Fig.7showsthesurfaceregistration result of the first pairof VSsurfaces (corresponding to Fig. 6) visualizedbythecolormap.Thecolormapontheleft(normal)VS surfaceismappedtotheright(AIS)surfacesusingtheobtained registration.Notethatcorrespondingregionsofthetwosurfaces areconsistentlymatched.Tobettervisualizetheaccuracyofthe registration,featureloopsareplottedonthesourcesurfaceand aremappedtothetargetsurfacesusingtheregistration.Notethat

Fig.5.(A)TheVSsurfaceanditscanonicalfundamentalpolygonsofanormalsubject,leftandright,respectively.(B)TheVSsurfaceanditscanonicalfundamentalpolygons ofaAISsubject,leftandright,respectively.Thecanonicalcutsareshownoneachsurfaces.

Fig.6.Registrationontheuniversalcoveringspaces.TheleftandmiddleshowtheuniversalcoveringspacesoftwoVSsurfacesandtheircorrespondinglandmarklocations.

Therightshowstheregistrationresult,whichmatchesthelandmarksconsistently.Notethatthecanonicalcutsonthesourcesurfaceareallowedtomovefreelyonthetarget surface.

Fig.7.SurfaceregistrationresultsbetweenanormalVSsurface(left)andanAISVS surface(right).

thefeatureloopsarematchedtocorrespondinglocationsonthe targetsurfaces.

Fig.8showsapairofVSsurfacesofnormalsubjects.Thesurface registrationresultsareshown. Again,theregistrationis visual- izedusingthecolormap.Thecolormaponthesourcesurfacesare mappedtothetargetsurfaces.Thefeatureloopsonthesource surfacesarealsomappedtothetargetsurfaces.Fromthecorre- spondencesofthecolormapandfeatureloops,weobservethat correspondingregionsoftheVSsurfacesareconsistentlymatched.

Itagaindemonstratestheaccuracyofourregistrationresults.

Using the proposedregistration algorithm, we computethe surfaceregistrations betweenallpairs of VSsurfaces.Withthe computedregistrations,wecomputethemeansurfaceofallVSsur- facesofthenormalgroup.Themeansurfaceiscomputedbytaking themeanoftheircorrespondingcoordinates.Themeansurfaceis showninFig.9(A).Similarly,wecomputethemeansurfaceofall VSsurfacesoftheAISgroup,whichisshowninFig.9(B).Notethat themeansurfacespreservethestructuresoftheVSsurfacewell.It meansthatourobtainedregistrationsareaccurate.Theregistration betweenthetwomeansurfacesisvisualizedbycolormap.Thecol- ormaponthemeanVSsurfaceofthenormalsubjectsismappedto theVSmeansurfaceoftheAISsubject.Notethatthecorresponding regionsarematchedconsistently.Thisdemonstratesourproposed

Fig.8. SurfaceregistrationresultsbetweentwonormalVSsurfaces.

Fig.9.(A)Meansurfaceofnormalsubjects.(B)MeansurfaceofAISsubjects.Regis- trationisvisualizedbythecolormap.(Forinterpretationofthereferencestocolor inthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

algorithm isable toobtainsurfaceregistrationbetweentheVS surfaceswithaccurateone-to-onepoint-wisecorrespondences.

Thecomputationaldetailsforthesurfaceregistrationbetween somepairsoftheVSsurfacesarerecordedinTable1.Wecompute thesurfaceregistrationsbetweenasourcesurface(VSsurfaceof thenormalsubjectN01)todifferenttargetVSsurfaces.Thecom- putationaltimesareshowninthesecondcolumn,whichareallless than30s.Itillustratesthatourproposedalgorithmisquiteefficient.

Table1

Computationaldetailsofthesurfaceregistrationalgorithm.

Subject Time(s) Overlaps Vertices Iterations

A01 13.16 0 7239 28

A02 18.35 0 7367 42

A03 27.34 0 6003 63

A04 28.30 0 7909 54

A05 17.25 0 7036 34

A06 10.89 0 6915 25

A07 27.30 0 7140 53

A08 13.96 0 7638 32

A09 29.02 0 7827 54

A10 27.16 0 6259 62

A11 17.48 0 6564 40

A12 13.79 0 6840 27

N02 19.27 0 7353 43

N03 23.36 0 7408 43

N04 13.51 0 6446 29

N05 8.38 0 7228 19

N06 15.16 0 6925 30

N07 13.18 0 6724 30

N08 7.95 0 7612 18

N09 21.95 0 7226 41

N10 9.17 0 8114 21

N11 16.76 0 5493 37

N12 13.45 0 6836 31

N13 8.90 0 7160 20

N14 22.46 0 8327 40

N15 21.70 0 6320 41

Fig.10.(A)Homotopicloopsandcenterlinesonthenormalmeansurface.(B)Homo- topicloopsandcenterlinesontheAISmeansurface.

Thesecondcolumnshowsthenumberofoverlappingfacesofthe obtainedregistrations.Notethatthenumberofoverlappingfaces are0inallcases.Itdemonstratesourobtainedsurfaceregistra- tionsareallbijective.Thethirdandforthcolumnshowthenumber ofverticesforeachtargetVSsurfacesandtheiterationsneeded fortheregistrationprocess.Dependingonthegeometryofthesur- faces,ouralgorithmgenerallyconvergesinabout50iterationson average.

Our registrationalgorithm is a gradient descentbased algo- rithm.AlgorithmstopswhenthechangeofBeltramicoefficients aresmallenough.Inallourcases,italwaysconverges,anoptimal solutionofthelocalminimizeroftheenergyobtained.

6.2. Extractionofgeometricfeatures

Withtheobtainedregistration,variousgeometricfeaturescan beextractedonVSsurfaces.Thesegeometricfeaturescanbeused fortheshapeanalysisandshapecomparisonoftheVSsurfaces.

WefirstcomputethehomotopicloopsofthenormalmeanVS surface,which areshown in Fig.10(A). Fromthesurfaceregis- tration,we canmap thehomotopicloopstothecorresponding locationsonotherVSsurfaces.Correspondinghomotopicloopson everyVSsurfacescanthenbeobtained.Fig.10(B)showsthecorre- spondinghomotopicloopsontheAISmeanVSsurface.Byjoining thecentroidsofthehomotopicloops,thethreecenterlinesofthe AISmeanVSsurfacecanbeextracted.

Fromthehomotopicloops,minimalsurfacescanbeextracted.

Fig.11(A)showstheminimalsurfacesofthenormalmeanVSsur- face.Fig.11(B)showstheminimalsurfacesoftheAISmeansurface.

6.3. Geometricpropertiesanalysis

Fromthegeometricfeatures,shapeanalysisoftheVSsurfaces canbecarriedoutforthepurposeofdiseaseanalysis.Inthiswork, wecomparetheshapedifferencebetweentheVSsurfacesofthe normalandAISgroups.

Fig.11.(A)Minimalsurfacesenclosedbythehomotopicloopsonnormalmean surface.(B)MinimalsurfacesenclosedbythehomotopicloopsonAISmeansurface.

We test theidea tocompare themean surfacesof the nor- malVSandtheAISVS.Fig.12(A)showstheperimetersofeach homotopicloopsof thenormalmean VSand theAISmean VS.

Loops1–25arethehomotopicloopsontheanteriorsemi-circular canal(SSC).Loops26–43arethehomotopicloopsonthehorizontal semi-circularcanal(LSC).Loops44–74arethehomotopicloopson theposteriorsemi-circularcanal(PSC).Thebluecurveshowsthe perimetersofthehomotopicloopsofthenormalmeanVS.Thered curveshowstheperimetersofthehomotopicloopsoftheAISmean VS.Note that thedifferences inperimeter betweenthenormal meanVSandtheAISmeanVSattheSSCandLSCareinsignificant.

However,thedifferenceinperimeterbetweenthenormalmeanVS andtheAISmeanVSisquiteobviousatthePSC.

Wehavealsocomputedthesurfaceareaofeachminimalsur- faces.Fig.12(B)showstheareaofeachminimalsurfacesenclosed bythehomotopicloopsofthenormalmeanVSandtheAISmean VS.Again,thebluecurveshowstheareaoftheminimalsurfaces ofthenormalmeanVS.Theredcurveshowstheareaofthemini- malsurfacesoftheAISmeanVS.Noticethatthedifferenceinarea betweenthenormalmeanVSandtheAISmeanVSisquiteobvious atthePSC.Itagaindemonstratetheshapedifferencebetweenthe normalandAISVSsurfacesatthePSC.

Usingthecenterlines,wecanalsomeasurethedeviationofeach canalsoftheVSfromthebestfitplane.Fig.13showsthedistances ofeachpointsonthecenterlinesfromthebestfitplanesofthe normalmeanVSsurface(left)andAISmeanVSsurface(right).The distancesarevisualizedbythecolormaps.Redcolorindicatesa largedeviationfromthebestfitplane.Asshowninthefigure,it wasfoundthatthePSCofAISmeanVSsurfaceismoredistorted anddeviatedfromthebestfitplane.

Foramorecomprehensivelocalshapeanalysis,weuseacom- pleteshapeindexbasedontheBeltramicoefficientsandcurvatures tocapturethelocalgeometricdifferencebetweentheVSsurfaces.

AfterthesurfaceregistrationbetweentwoVSsurfacesisobtained, thecompleteshapeindexthatgivesavalueforeachvertexcanbe

Fig.12.(A)PerimetersofeachhomotopicloopsofthenormalmeanVSandtheAISmeanVS.(B)Areaofeachminimalsurfacesenclosedbythehomotopicloopsofthe normalmeanVSandtheAISmeanVS.

Fig.13.(A)Meansurfaceofnormalsubjects.(B)MeansurfaceforAISsubjects.Dis- tancesofeachpointsonthecenterlinesfromthebestfitplanesofthenormalmean VSsurfaceandAISmeanVSsurface.Thedistanceisvisualizedbythecolormaps.

Redcolorindicatesalargedeviationfromthebestfitplane.(Forinterpretationof thereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversion ofthisarticle.)

Fig.14.VSsurfacesregistrationoftwonormalsubjects.Theshapeindexisshown ascolormap,plottedoneachVSsurfaces.(Forinterpretationofthereferencesto colorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

computed.Thevaluemeasuresthedegreeofgeometricdifference.

Fig.14showstheVSsurfacesoftwonormalsubjects.Theshape indexisshownascolormap,plottedoneachVSsurfaces.Theblue colormeansthegeometricdifferenceissmall.Theredregionsare thosewithhigherdegreeofgeometricdifference.Fig.15showsthe shapeindexofanotherpairofVSsurfacesoftwonormalsubjects.

Fig.16showstheshapeindexofapairofVSofthenormalandAIS subjects.Notethatthegeometricdifferencebetweenthetwosur- facesaremoreobviousthanthegeometricdifferencebetweenthe VSoftwonormalsubjects.Fig.17showstheshapeindexofanother pairof VSofthenormalandAISsubjects.Again,thegeometric differencebetweenthetwo surfacesaremoreobviousthanthe geometricdifferencebetweentheVSoftwonormalsubjects.These resultsdemonstratethatthecompleteshapeindexisaneffective measuretodetectlocalsurfacegeometricdifferences.

Wealsocomputethecompleteshapeindextodetectthegeo- metricdifferencebetweenthenormalmeanVSandtheAISmean

Fig.15.VSsurfacesregistrationoftwonormalsubjects.Theshapeindexisshown ascolormap.(Forinterpretationofthereferencestocolorinthisfigurelegend,the readerisreferredtothewebversionofthisarticle.)

Fig.16.VSsurfacesregistrationofanormalandanAISsubjects.Theshapeindex isshownascolormap.(Forinterpretationofthereferencestocolorinthisfigure legend,thereaderisreferredtothewebversionofthisarticle.)

Fig.17.VSsurfacesregistrationofanormalandanAISsubjects.Theshapeindex isshownascolormap.(Forinterpretationofthereferencestocolorinthisfigure legend,thereaderisreferredtothewebversionofthisarticle.)

VS.TheshapeindexisshowninFig.18.Notethatthelocalgeomet- ricdifferenceismoreobviousinthePSC.Thisagainsuggeststhat theshapedifferenceatthePSCbetweenthenormalandAISVSis moresignificant.

6.4. Morphometricanalysiswithvertex-wisecoordinates

Withtheobtainedsurfaceregistration,statisticalmorphomet- ricanalysisbasedonthecoordinates’deviationcanbeperformed.

Usingthetwo-sampleHotelling’s T-squaredtest,thep-valueat eachvertexonthesurfaceisfound.Itindicatestheprobabilityof theexistenceofstructuraldifferencesbetweentheVSoftheAIS groupandtheNCgroup.Ifthep-valueislessthanourpredefined significantlevelof0.001,apotentialmorphometricvariationcanbe concludedatthatvertex.OurresultsshowninFig.19illustratesthe locationofpossiblemorphometricdeviations.Itcanbeobserved thatthesignificantregionsliemainlyatthePSC.

Fig.18.ThefigureshowsthenormalmeanVS(left)andtheAISmeanVS(right).The shapeindexisshownascolormap,plottedoneachVSsurfaces.Thebluecolormeans thegeometricdifferenceissmall.Theredregionsarethosewithhigherdegreeof geometricdifference.(Forinterpretationofthereferencestocolorinthisfigure legend,thereaderisreferredtothewebversionofthisarticle.)