VesselDetectionby Mean Shift BasedRay Propagation

Huseyin Tek Dorin Comaniciu JamesP. Williams

ImagingandVisualizationDepartmentSiemensCorporateResearch,Inc.

755CollegeRoadEast,Princeton,NJ 08540�tek,comanici,williams � @scr.siemens.com

Abstract

Arobustandefficientmethodfor thesegmentationofves-selcross-sectionsin contrastenhancedCT andMR imagesis presented.The primary innovationof the techniqueistheboundarypropagationby meanshift analysiscombinedwith a smoothnessconstraint. Consequently, therobustnessof themeanshift to noiseis enhancedby theuseof aprioriinformationonboundarysmoothness.Thisprocessingis in-tegratedinto ourcomputationallyefficientframeworkbasedon ray propagation. The new algorithm allows real timesegmentationof medicalstructuresfoundin multi-modalityimages(CT, MR) andvariousexamplesare shownto illus-trateits effectiveness.

1 Intr oduction

We focus on imagesproducedby contrast-enhancedmagneticresonanceangiography(CE-MRA) andcomputedtomographyangiography(CTA.) In the CE-MRA imagingprotocol,acontrastagent,usuallybasedontherare-earthel-ementGadolinium(Gd)(ahighly paramagneticsubstance),is injected into the bloodstream. In such images,bloodvesselsandorgansperfusedwith thecontrastagentappearsubstantiallybrighterthansurroundingtissues.In CTA, acontrastagentis injectedwhich increasestheradio-opacityof the blood making the vesselsappeardense. The goalof the majority of CTA/CE-MRA examinationsis diagno-sis andqualitative or quantitative assessmentof pathologyin the circulatorysystem. The mostcommonpathologiesareaneurysmandstenosiscausedby arterialplaques.Themodernclinical workflow for the readingof theseimagesincreasinglyinvolvesinteractive 3D visualizationmethods,suchasvolumerenderingfor quickly pinpointingtheloca-tion of thepathology. Oncethelocationof thepathologyisdetermined,quantitativemeasurementscanbemadeon theoriginal2D slicedataor, morecommonly, on2D multi pla-

nar reformat(MPR) imagesproducedat user-selectedpo-sitions and orientationsin the volume. In the quantifica-tion of stenosis,it is desirableto producea cross-sectionalarea/radiusprofile of a vesselso that one can comparepathologicalregionsto patent(healthy)regionsof thesamevessel.

A basicproblem in the segmentationof vesselsfrombackgroundis the accurateedgelocalization in the pres-enceof noise. Most CT andMR imageshave significantnoiselevels.Weemployonedimensionalmeanshift analy-sisalongasetof raysprojectingradially from auser-placedseedpointin theimage.Noisealongtheseraysis eliminatedwhile edgesarepreserved. Theenvelopeof theseraysrep-resentsan evolving contourwhich convergesto the vessellumenboundary. The algorithmis parsimonious,examin-ing only pixelsalongandadjoiningtheseraysandtheseedpoint.

It is importantthat the boundarydeterminedby the al-gorithm be consistentand invariant to medium to largescalelinearandnon-linearimageinhomogeneities.This isneededfor thealgorithmto beinsensitiveto MR distortions.Theuseof constantthresholdfactors(suchasHounsfield-basedthresholdsin CT) would limit theability of thealgo-rithm to adaptto varyingcontrastdosagesor varyingbeamenergiesin CT. Weseekto localizethemidpointof thestep-edgeastheboundaryof thevessel.

Thechoiceof risemidpointasthevessellumenbound-ary is similar to the choice made by full-width-half-maximum(FWHM) segmentationmethods[9]. It was,infact, the sensitivity of conventionalFWHM segmentationmethodsto noisewhich motivatedthedevelopmentof thismethod.

Our algorithmrequiresconstantparametersfor theevo-lution equationsandfor thewindow sizeusedfor themeanshift filter. To makethe methodadaptto varying imagequalitiesandmodalities,the evolution parametersarede-termineddynamicallyfrom thelocal statisticsof theimagecomputedfrom asmallneighborhoodimmediatelyadjacent

to theusersseedpoint. A limitation of our methodaspre-sentedis that the detectionscaleis determinedby userin-teraction.However, a fixedwindow sizefor themean-shiftfilter will serve for detectionover a broadrangeof vesselsizes.

Thereis an extensive body of work on the segmenta-tion of vesselsandothercurvilinearstructures(suchasair-ways)from CTA andMRA images. Ray propagationhasbeenusedin a numberof prior relatedworks. Perhapsthemostcloselyrelatedexampleis thework of Wink etal. [24]who useimagegradientto control ray termination. Theyuseconventionalsmoothingtechniquesto dealwith imagenoise. Thedrawbackof suchsmoothingis that it canshiftor entirelyeliminatelow-contrastboundaries.

Ourapproachis alsorelatedto otherdeformablemodels:snakes[11, 2, 3, 16, 25], balloons[7], levels-sets[14, 20,22, 12, 20], region-competition[26], skeletally-coupledde-formablemodels[18]. Thepositionof this work in thede-formablemodeltaxonomywill beelaboratedin Section2.

In this paperwe presentthe simplestusagecase: Theuserspecifiesthe vesselto besegmentedby placinga sin-gle seedinside it. A boundarycontouris then automati-cally generatedvia the propagationof rays from the seedpoint. The propagationis guidedby imageforcesdefinedthrough meanshift analysisand smoothnessconstraints.The gradient-ascentmeanshift localizesedgesaccuratelyin thepresenceof noiseandprovidesagoodcomputationalperformance,being basedon local operators. The incor-porationof a smoothnessconstraintinto our modelallowsboundaryfindingin thepresenceof eccentricitieslike calci-fications(CT)or smallbranchingvessels.Also, theadditionof smoothnessconstraintsto ouralgorithmfurtherimprovestherobustnessto isolatednoise.Althoughthealgorithmisnot optimizedyet, it is very fastfor detectingvesselsin or-thogonalviews, under ����� secondson a PenthiumIII , 500Mhz PC.

This algorithm is also suitablefor applicationswheremultiple contours are to be segmented along a pre-determinedvesselaxis. In our integratedsystemwe im-plementtheaxisextractionmethoddescribedin [4] to pro-vide orthogonalslicepositions.Validationis vital to makethis or any methodusefulin a clinical setting.In this paperexamplesareshown with phantomsof known groundtruthanddatafrom clinical practice. Completevalidationwillrequirea formal multi-siteclinical testwhich is plannedtobegin within 6 months.

2 Active Contours

In thissectionwebriefly review the“active contour”lit-eratureasis pertinentto thispaper[11, 16, 7, 14,20, 22,4,10, 24]. Seealso [26, 18] for relatedactive contourmeth-odsfor imagesegmentation.

2.1 Snakes

Active contours,or snakes[11, 3, 25], aredeformablemodels based on energy minimization of controlled-continuitysplines.Whenthey areplacedneartheboundaryof objectsthey will lock onto salientimagefeaturesunderthe guidanceof internalandexternal forces. Formally, let��� ������������������� ���

be the coordinatesof a point on thesnake,where

is the lengthparameter. The energy func-

tionalof a snakeis definedas

� ������� �"!#%$ �'&)(+* �,��� ���+- �'&).'/�0�1 ���2����3�4- �65378( ���2����3�:9<;=4�(1)

where�'&)(4*

representsthe internalenergy of thesplinedueto bending,

�'&).'/�0�1representsimageforces,and

�65378(are

theexternalconstraintforces.First, theinternalenergy,�'&)(+* � > !@? �BA8�� � ? C -D> C ? �EA�A��� � ? C � (2)

imposesregularity on the curve, and,> ! and

> C corre-spondsto elasticityandrigidity, respectively. Second,theimageforcesareresponsiblefor pushingthesnaketowardssalientimagefeatures.Thelocalbehavior of asnakecanbestudiedby consideringtheEuler-Lagrangeequation,FHG ��> ! � A � A -I��> C � A�A � A�A �KJL�,�M�N��� � � , � A � � � , ���O � and

� A ��O �given,

(3)

whereJL�����

capturesthe image and external constraintforces.Notethat theenergy surface

�is typically not con-

vex andcanhave several local minima.Therefore,to reachthe solutionclosestto the initialized snake,the associateddynamicproblemis solved insteadof the static problem.When the solution

��QPR�stabilizes,a solution to the static

problemis achieved.FTSNUS * G ��> ! � A � A -K��> C � A�A � A�A �IJL�,�M�V�W�V P V�X+Y -[ZN\^] W ; X=_ �a`N\ W ; V P V \ W (4)

Snakesperformwell whenthey areplacedcloseto thedesiredshapes.However, a numberof fundamentaldiffi-cultiesremain;in particular, snakesheavily rely onaproperinitializationcloseto theboundary, multiple initializations,oneperobjectof interest.

To overcomesomeof the initialization difficultieswithsnakes,Cohen and Cohen [7] introduceda deformablemodel basedon the snakesidea. This modelsresemblesa “balloon” which is inflatedby anadditionalforcewhichpushestheactive contourto objectboundaries,evenwhenit is initialized far from theinitial boundary. However, bal-loonsstill have someproblems:First, like snakes,balloonscannothandletopologicalchanges, i.e., merging andsplit-ting. It shouldbe notedthat McInerney andTerzopoulospresenteda topologicallyadaptablesnakesin [15].

2.2 Level SetEvolution

Oneof thetraditionalproblemsof snakesandballoonsisthatthey cannoteasilycapturetopologicalchanges.There-fore, for imageswith multipleobjects,thesnakeor balloonmethodsrequireextensive userinteraction. This problemcanbe resolved by the useof the level setevolution, pro-posedby OsherandSethianfor flamepropagation[17], in-troducedto computervision for shaperepresentation[13],andfirst appliedto active contoursin [6, 14].

The level set approachconsidera curve�

as the zerolevel setof a surface,b �������+�c� � . Caselleset al. [6] pro-posedthat the zerolevel setof the function b , d �"eIf C�gb �)P ���B�2� �ih , evolve in thenormaldirectionaccordingtoj bj P �lkM�������+� ? m b ? ��; V�n � m b? m b ? ��- n ��� (5)

wherekM���o���+�p� !!3q�rtsvuow�x3y�z�{ , n is a positiverealconstant,|~}��'�is theconvolution of theimage

�with theGaussian|~}

, and, b # is the initial datawhich is a smoothedversionof thefunction

O GL���, where

���is thecharacteristicfunc-

tion of aset � containingtheobjectof interestin theimage.The gradientof the surfacem b is the normal to the levelset�, �� , andthe term

; V�n � sv�� sv� � � is its curvature � . Sev-eralinterestingrelevantapproaches,e.g., [12, 21] havebeenproposedafter the original work of Caselleset al. [6], andMalladi et al. [14].

Unlike snakes,thisactivecontourmodelis intrinsic,sta-ble, i.e., thePDEsatisfiesthemaximumprinciple,andcanhandletopologicalchangessuchas merging andsplittingwithoutany computationaldifficulty. A key disadvantageofthelevel setmethodis theirhighcomputationalcomplexity,due to the additionalembeddingdimensional,even whenthe computationis restrictedto a narrow bandaroundthecurve. To overcomethecomputationalcomplexity, narrowbandlevel setevolutionshave proposed[14, 1, 23]. How-ever, thesemethodsarestill not fast enoughfor real-timeimagesegmentation.Thus,in this paper, weproposeto useray propagationfor fast imagesegmentation,which is de-scribednext.

3 Ray Propagation

Let the front be representedby a 2D curve�2��4�8PR���������4�8PR�N�R����4�8PR���

where�

and�

are the Cartesiancoordi-nates,

is thelengthparameter, and

Pis time.Theevolution

is thengovernedby [13]F SNU rt��� * zS * � `^�������+� �����+� � ���l� # ���� (6)

where� # ����'��������4� � ��������+� � ��� is theinitial curve,and ��

is theunit normalvectorand`<�����R�4�

is thespeedof a rayatpoint

�������+�.

Theapproachthatwe considerin this paperis basedonexplicit front propagationvia normalvectors.Specifically,thecontouris sampledandtheevolution of eachsampleisfollowedin time by rewriting the Eikonalequation[19] invectorform, namely,�� � � * ��+��PR��� `^�����R�4� ���� � { � q � { �� * ��4��PR���I`^���o���+� � �� � { � q � {� � (7)

This evolution is the“Lagrangian”solutionsincethephys-ical coordinatesystemmoves with the propagatingwave-front. However, the applicationsof ray propagationforcurve evolution hasbeenlimited. Becausewhen the nor-mals to the wavefront collide (formation of shocks),thisapproachexhibits numericalinstabilitiesdueto anaccumu-lateddensityof samplepoints,thusrequiringspecialcare,suchasreparametrizationof thewavefront. Also, topolog-ical changesarenot handlednaturally, i.e., anexternalpro-cedureis required.

In this paper, we will show that ray propagationcanin-fact beusedfor segmentingcertainobjectsin medicalim-agesefficiently and robustly. Previously, ray propagationhas beenusedto implement full-width at half-maximum(FWHM) techniquefor quantificationof 2D airwaygeom-etry [9]. In FWHM technique,themaximumandminimumintensityvaluesalongthe raysarecomputedto determinethe “half-maximum” intensityvalue,which is the half in-tensityvaluebetweenmaximumandminimum. However,stablecomputationof maximumand minimum along theraysarequitedifficult dueto highsignalvariations.

In addition,raypropagationwasusedby Wink etal. [24]for fastsegmentationof vesselsanddetectionof their cen-terline. In this approach,Wink et. al. usedthe intensitygradientsto stopthepropagationof rays.However, thisap-proachfacesdifficultieswhenthevesselsboundariesarenotsharp,i.e., dueto partialvolumeeffects,andalsowhenves-selscontainisolatednoises,e.g., calcificationsin CT im-ages.

The main shortcomingsof theseapproachesstemfromthe computationof imagegradientswhich are not robustrelative to the imagenoise. In this paper, we proposetousemeanshift analysisfor detectingvesselsboundariesef-ficiently androbustly. ComaniciuandMeer[8] showedthatthe imagediscontinuitiesarerobustly revealedby a meanshift processwhich evolvesin boththeintensityandimagespace.

We will first describethe meanshift analysis;second,illustrateanapproachwherethemeanshift procedureis ap-plied to selectdiscontinuitiesin a onedimensionalsignal;and finally, presentthe meanshift-basedray propagationapproachfor segmentationof medicalstructures,e.g., ves-sels.

3.1 Mean Shift Analysis

Given the set d � & h &�� !������ ( of;-dimensionalpoints, the

meanshift vectorcomputedat location � is givenby [8]

��� � � ����� (&�� ! �o  � ��¡E�2¢� �� (&�� !   � �2¡E� ¢� � G � (8)

where   representsa kernelwith a monotonicallydecreas-ing profileand £ is thebandwidthof thekernel.

It can be shown that (8) representsan estimateof thenormalizeddensitygradientcomputedat location � , i.e.,the meanshift vector alwayspoints towardsthe directionof the maximumincreasein the density. As a result, thesuccessive computationof expression(8), followedby thetranslationof thekernel   by

��� � � � will defineapaththatconvergesto a local maximumof the underlyingdensity.This algorithmis calledthemeanshift procedure, a simpleandefficientstatisticaltechniquefor modedetection.

The meanshift procedurecan be appliedfor the datapointsin thejoint spatial-rangedomain[8], wherethespaceof the 2-dimensionallattice representsthe spatial domainandthe spaceof intensityvaluesconstitutesthe rangedo-main. In this approach,a datapoint definedin the jointspatial-rangedomainis assignedwith a point of conver-gencewhichrepresentsthelocalmodeof thedensityin thisspace,e.g., a 3-dimensionalspacefor gray level images.Onecandefinedisplacementvector in the spatialdomainasthespatialdifferencebetweenconvergencepointandtheoriginalpoint.

Wheneachpixel in the imageis associatedwith thethenew range(intensity) information carriedby the point ofconvergence,thealgorithmproducesdiscontinuitypreserv-ing smoothing.Conceptually, this processis similar to theanisotropicdiffusion,nonlinearfiltering, or bilateralfilter-ing [5].

However, whenthespatialinformationcorrespondingtothe convergencepoint is also exploited, one can defineasegmentationprocessbasedon thedisplacementvectorsofeachpixel. Theconvergencepointssufficiently closein thisjoint domainaregatheredtogetherto form uniform regionsfor imagesegmentation[8].

In this paper, we will exploit the meanshift-generateddisplacementvectorsto guideactive contourmodels. Therobustnessof the meanshift is thus combinedwith apri-ori informationregardingthesmoothnessof theobjectcon-tours. This processingis integratedinto our computation-ally efficientframework basedonraypropagation.Thenewalgorithmallows real time segmentationof medicalstruc-tures.

3.2 Mean Shift Filtering Along a Vector

Let us first illustrate the meanshift procedureappliedto a 1-dimensionalintensityprofile which is obtainedfroma 2d graylevel image.Specifically, let d � & ��� & h &�� !���������� ¤ andd � x & �R� x & h &�� !�������� ¤ bethe2-dimensionaloriginalandfilteredN imagepoints in the spatial-rangedomain. In addition,the outputof the mean-shiftfilter includesa displacementvector d ; & h &�� !���������� ¤ which measuresthe spatialmovementof eachspatialpoint. In our algorithm,eachpoint in thisspatial-rangedomainis processedvia themeanshift oper-atoruntil convergence.Specifically, thealgorithmconsistsof 3 steps:

For eachV �¥O4� �)�)� � �

1. Initialize ¦ �§O and��� xN¨& �R� x�¨& ��; & ���©��� & ��� & � � �

2. Compute

� x ¨ q�!& � �Iª«�¬i­ � « 1i®=¯ °�±�² ¢ ®^° «�³ {{ w {° 1 ®=¯ ´�±R² ¢ ®+´ «�³ {{ w {´� ª«�¬i­ 1=®=¯ ° ± ² ¢ ®^° «R³ {{ w {° 1 ®=¯ ´ ± ² ¢ ®<´ «R³ {{ w {´� x ¨ q�!& � �Kª«�¬i­ y « 1 ®=¯ °�±R² ¢ ®^° «�³ {{ w {° 1 ®=¯�´ ²¢ ®+´ «�³ {{ w {´� ª«�¬i­ 1i®=¯ ° ± ² ¢ ®^° «�³ {{ w {° 1 ®=¯ ´ ± ² ¢ ®+´ «�³ {{ w {´(9)

until the displacementof spatialpoints,� &

aresmall,i.e., ? � x ¨ q�!& G � x ¨& ?4µ[¶

3. Assign; & �¥��� x ¨ q�!& G � & �

4. Assign��� x & ��� x & �v�§��� & ��� x& �

where · � and · y determinetheGaussianspatialandrangekernelssize,respectively. Observe that in the last stepoftheproceduretheoriginalspatiallocations,namely

� &’sare

assignedwith the smoothedintensityvalues. Figure1a il-lustratesanexamplewhere1-dimensionalintensitydataisobtainedfrom a sliceof a CT image.Figure1b andc illus-tratetheoriginal intensityprofileandthesmoothedintensityprofile, respectively. Observe thatthemeanshift proceduresmoothstheintensitydatawhile preservingandsharpeningitsdiscontinuities.Similarly, Figure1ddepictsthedisplace-mentvectorsalongthis1-dimensionalsignal.Ourboundarydetectionframework exploits the informationcontainedinthesedisplacementvectors.

3.3 Mean Shift-BasedRay Propagation

Semi-automaticsegmentationproceduresarevery well-acceptedin medical image applicationsbecauseof theirfast executiontimesandtheir stability. Infact, active con-tourshavebeenextensively usedin medicalimagesegmen-tation. In this paper, we advocateray propagationfrom a

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Figure 1. (a)A 1-dimensionalintensityprofile is obtainedalongtheredline from a CT imagecontainingtheaorta.Theleft-endpointof theredline is thebeginningof dataandright-endpoint is theendof data.(b) Original intensityprofile. (c) Intensityprofileaftermeanshift filtering, with ¸4¹�º"» and ¸=¼'º¾½�¿ ¿ . (d) Thedisplacementvectorsof eachspatialpoint andtheir first derivatives(e).

singlepoint for vesselsegmentation. Specifically, we as-sumethat the vesselsareorthogonalto the viewing planeand their boundariesarevery similar to circular/ellipticalobjects. Thus,ray propagationis very well suitedfor thisproblemfor the following reasons.First, ray propagationis very fast. Second,no topologicalchangesarenecessaryandnoshocksform duringthepropagation,sinceraysfroma singlesourcepoint do not collide with eachother. Re-call that level setshave beenthechoiceof curve evolutionproblems[13] due to formation of shocksand topologi-cal changesthat may happenduring the evolution. How-ever, basedon our experiments,level setbasedsegmenta-tion techniques,e.g., [14, 21] arestill slow for real timeimagesegmentation.

Let us now presentthe meanshift basedray propaga-tion: Theapproachis basedonanexplicit front propagationvia normal vectors,rays. Specifically, the evolving con-touris sampledandtheevolutionof eachsampleis followedin time by rewriting theevolution equationin vectorform,namely,

�� � � * ��4��PR���KÀÁ���o���+� � �� � { � q � {�� * ��4�8PR�2� ÀÁ�����R�4� � �� � { � q � {� � (10)

whereÀa���o���+�

is a speedfunctiondefinedasÀa���o���+�Á�  à ���o���+�O ��� - ? m ;E���o���+� ? C -ÅÄ � ���o���+� (11)

where;E�����R�4�

is thedisplacementfunctioncomputedby themeanshift procedure,� ���o���+� is the discretecurvature,

ÂandÄ

areconstants,and à �����R�4� is givenby

à �����R�4�a� FHG V k W ��;B���o���+��� V à ? m ;E�����R�4� ?iÆ[Ç ·EÈO É Y É(12)

Ideally, rays should propagatefreely towards the objectboundarieswhenthey areaway from themandthey shouldslow down in the vicinity of theseobject boundaries. Ifthey crossover theboundariesthey shouldcomebackto theboundary. Observe thatall theserequirementsaresatisfiedby thechoiceof speedfunction. Themeanshift-generateddisplacementvectorshave high gradientmagnitude(Fig-ure1e)anddiverge(Figure1d)at theaortaboundary. Notethatthehigh gradientmagnituderesultsin low propagationspeed,while thedivergencepropertydeterminesthedirec-tion of the propagation,i.e., outwardinside the aortaandinwardoutsidetheaorta.

Often intensity valuesinside vesselsare not smoothlychanging.In fact, it is possiblethat theremay be isolated

θ

ri rrrl

P

Figure 2. Thecurvatureof a ray Ê�Ë at P is approximatedby theangleÌ at thatvertex.

noisewhich thencreatessevereproblemsfor the propaga-tion of rays. In addition,theobjectboundariesmaybedis-tortedat isolatedlocationsdue to interactionwith nearbystructures. Becauseof theseirregularities inside vesselsandon its boundaries,we believe that theremustbe somesmoothnessconstrainson the evolving contour via rays.Thus,we add � �����R�4� to thespeedfunctionof rays,whichforcesthe front to be smoothduring the propagation.Wecurrentlyuse � ���o���+���H�3O G �+ÍÎ � C � asthe curvaturevalueof a ray at point

�����R�4�where Ï is the anglebetweentwo

contoursegments,Figure2. TheratioÂ2Ð^Ä

controlsthede-greeof desiredsmoothness.

Our speedfunctioncontainsthe · È parameter. This sta-tistical parameteris the variationof the magnitudeof dis-placementfunctionin aregionandis learnedfrom thedata.Specifically, first raysareinitially propagatedvia constantspeedin a small region (circular region). Second,first or-der statistics,namely, mean, Ñ andstandarddeviation, ·EÈof gradientdisplacementfunction arecomputedfrom thissample.It is assumedthat locationsof thesmall gradientsin adisplacementfunction(lessthan Ç ·EÈ ) shouldnotbepartof any objectboundary.

4 Results

Figure3 illustratesthemeanshift-basedraypropagationfor a CT (top) anda MR image(bottom). Specifically, wedepict the displacementvectorsobtainedfrom meanshiftfiltering in middlecolumnin orderto show thestrengthofthemeanshift filtering. In this figure,blackcolor indicatesavectorpointingtowardsthecenterandsimilarly whitecol-orsarethevectorspointingoutwardfrom thecenterof rays.Observe the divergenceof the displacementvectorsin thevicinity of aortaboundaries.This divergenceof vectorsareintegratedvia rays,which thenleadsto thesegmentationofaorta.

Wehavetestedthestabilityof ouralgorithmonavarietyof CT and MR imagesand as well as on a CT phantomdata,Figure4. Furthervalidationstudieswill be donebytheexperts.

Figure 5 illustrates the need for shapepriors, e.g.,smoothnessconstraintsin segmentationon an MR imageandaCT image.Shapepriorsarenecessaryfor tworeasons:� V �

Figure 5a depictsa casewhere renal arteriesbranchfrom the aorta. Thesegmentationof aortafor quantitativemeasurementsvia ray propagationwithoutany smoothnessconstraintwould resultin largeerrorsdueto renalarteries,Figure 5b. The addition of strongsmoothnessconstraintresultsin bettersegmentationof aortafor quantitativemea-surements,Figure5c.Thisexampleillustratesthatouralgo-rithm is capableof incorporatingsimpleshapepriors.

� V�V �Thesmoothnessconstraintis oftenneededfor the stabilityof a segmentationprocess.Figure5aillustratesa structurein a CT imagewhich hasdiffusedboundariesanda circu-lar darkregion in it. Observe that theray passingover thecirculardarkregionis stoppeddueto thehighdisplacementvector(or high intensitygradient),Figure5b. In addition,oneray did notstopat thecorrectboundarydueto theverylow gradient.Figure5c illustratesthat theseisolatederrorsin thesegmentationprocesscanbecorrectedby theadditionof smoothnessconstraints.

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[5] D. Barash.Bilateralfiltering andanisotropicdiffusion: To-wardsa unified viewpoint. TechnicalReport HPL-2000-18(R.1),Hewlett-Packard,2000.

[6] V. Caselles,F. Catte,T. Coll, and F. Dibos. A geometricmodel for active contoursin imageprocessing. TechnicalReportNo 9210,CEREMADE,1992.

[7] L. D. Cohen. Note on active contourmodelsandballoons.CVGIP: ImageUnderstanding, 53(2):211–218,1991.

[8] D. ComaniciuandP. Meer. Meanshift analysisandappli-cations.In IEEE InternationalConferenceon ComputerVi-sion, pages1197–1203,1999.

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(a) (b) (c)

(a) (b) (c)

Figure 3. This figureillustratesthemeanshift-basedray propagationon a CT (top) andMR imageaorta(bottom).Both imagescontainaorta.(Right)Original image(Middle) Theunit displacementvectorsof meanshift procedurearecomputedvia propagatingunit speedrays,i.e., speedfunctionis unity. Thewhite indicatesavectorpointingoutwardfrom thecenterandblackcolor indicatesavectorpointingtowardsthecenter. Observethatourspeedfunctionin Equation(12)includesthenegativesignof thedisplacementvectors.Thusinitially raysarepushedtowardsto thevesselsboundariesandif they crossover themthey comebackdueto signchangein thespeedfunction.(Right)Thesegmentationof vesselvia meanshift-basedraypropagation.

(a) (b) (c) (d)

Figure 4. Thisexampleillustratesthestability of our resultson a CT phantomdatadepictingcoronoryarteries.Theboundariesaredetectedby asingleclick insidethevessels.

(a) (b) (c)

(d) (e) (f)

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