A NURBS-BASED TECHNIQUE FOR THE SEGMENTATION OF MEDICAL IMAGES Ravinda G.N. Meegama and Jagath C. Rajapakse School of Computer Engineering, Nanyang Technological University Nanyang Avenue, Singapore 639798 e-mail: [email protected], [email protected]Abstract Extracting the human brain from magnetic resonance head scans is difficult because of its highly convoluted and nonuniform geometry. A technique based on Non-Uniform Rational B-Splines (NURBS) and energy minimising de- formable models to extract the brain surface accurately from MR head scans is presented. The weighting parameter that comes with the NURBS definition is explored to attract the surface into the regions showing high curvature. The weight at each control point is adjusted automatically ac- cording to the curvature properties of the evolving surface. This process facilitates a deformable surface with increased local flexibility that adapts to complex geometrical features of the brain. The results show that the proposed model is ca- pable of capturing the correct brain surface with a higher accuracy than the existing techniques. 1. Introduction Magnetic Resonance (MR) head scans indicate the skull, subarachnoid fluid and other anatomical structures that hamper the visibility of the brain surface. Therefore, in or- der to carry out an accurate morphometry in neurological studies, the brain has to be separated from these external structures. Such a segmentation routine is of immense as- sistance for various clinical studies because the surface pat- terns of the brain are considered to be natural pathways on the subarachnoid system that can be used to access patho- logical structures within the brain [10]. Several researchers have investigated the problem of modeling the brain surface using deformable models [1, 7, 14, 15, 17]. Our research, an extension to “Dynamic NURBS” [13], is motivated by the limitations found in the previous techniques that require manual initialisation, user interaction during deformation, poor local shape control and redundant points to capture the brain surface accurately. The approach presented in this paper emphasises on de- veloping a surface model to segment the brain accurately from MR head scans with minimum user intervention. The main reason for the selection of a NURBS surface for our algorithm is its inherent locally adaptable features [12]. In other words, a movement of a particular control point (or a change in the weight of a control point) alters the shape of the surface segment lying immediately close to that point. This is in sharp contrast to other deformable models based on irregular mesh configurations where such changes tend to propagate globally on the surface [1, 8, 9, 15]. The proposed method consists of two stages: First, we obtain the brain matter using an approximate segmenta- tion routine. Manual initialisation is avoided by taking the output image from this procedure to construct the initial NURBS surface that is positioned close to the actual bound- ary of the brain matter. The second stage involves deform- ing the above NURBS surface according to an energy min- imisation algorithm and also, modifying the weight associ- ated with each control point automatically depending on the curvature properties of the evolving surface. This weight modification forces parts of the surface near sulci (grooves on the brain surface) to attract towards the cavities with- out having to duplicate the control points near such regions. By using a NURBS-based deformable model, we have suc- ceeded in; (a) increasing the local flexibility of the surface by automatically adjusting the weights of the control points, (b) controlling the local shape with B-Spline basis functions and (c) automating surface initialisation. Experiments to determine the reliability of the proposed technique are car- ried out on MR head scans of 16 healthy subjects. 2. Initial Brain Segmentation The purpose of this stage is to obtain an initial surface lo- cated close to the actual brain surface. This surface may not necessarily be the true brain surface but an approximation that will be deformed to conform to the actual boundary of the brain matter. At the beginning, the voxels of the original MR head scans (Fig. 1(a)) are converted to have isotropic dimensions of mm to facilitate distance transfor- mations. The scans are then filtered using an edge preserv- ing 3D neighbourhood averaging routine [5]. The filtered
Transcript
A NURBS-BASED TECHNIQUE FOR THE SEGMENTATION OFMEDICAL IMAGES
Extractingthehumanbrain frommagneticresonanceheadscans is difficult becauseof its highly convoluted andnonuniformgeometry. A technique basedon Non-UniformRational B-Splines(NURBS)and energy minimising de-formable modelsto extract the brain surfaceaccuratelyfromMRheadscansis presented.Theweightingparameterthat comeswith theNURBSdefinitionis exploredto attractthe surfaceinto the regionsshowinghigh curvature. Theweightat each control point is adjustedautomaticallyac-cording to thecurvature propertiesof theevolvingsurface.Thisprocessfacilitatesa deformablesurfacewith increasedlocal flexibility thatadaptsto complex geometricalfeaturesof thebrain. Theresultsshowthattheproposedmodelis ca-pableof capturingthe correct brain surfacewith a higheraccuracythantheexistingtechniques.
1. Introduction
MagneticResonance(MR) headscansindicatethe skull,subarachnoidfluid and other anatomicalstructuresthathamperthevisibility of thebrainsurface.Therefore,in or-der to carry out an accuratemorphometryin neurologicalstudies,the brain hasto be separatedfrom theseexternalstructures.Sucha segmentationroutineis of immenseas-sistancefor variousclinical studiesbecausethesurfacepat-ternsof thebrainareconsideredto benaturalpathwaysonthe subarachnoidsystemthat canbe usedto accesspatho-logicalstructureswithin thebrain[10].
Several researchershave investigatedthe problem ofmodeling the brain surface using deformablemodels[1,7, 14, 15, 17]. Our research,an extensionto “DynamicNURBS” [13], is motivatedby the limitations found in theprevioustechniquesthat requiremanualinitialisation,userinteractionduringdeformation,poorlocalshapecontrolandredundantpoints to capturethe brain surface accurately.The approachpresentedin this paperemphasiseson de-velopinga surfacemodel to segmentthe brain accuratelyfrom MR headscanswith minimumuserintervention.The
main reasonfor the selectionof a NURBS surfacefor ouralgorithmis its inherentlocally adaptablefeatures[12]. Inotherwords,a movementof a particularcontrolpoint (or achangein theweightof a controlpoint) alterstheshapeofthe surfacesegmentlying immediatelycloseto that point.This is in sharpcontrastto otherdeformablemodelsbasedon irregularmeshconfigurationswheresuchchangestendto propagategloballyon thesurface[1, 8, 9, 15].
The proposedmethodconsistsof two stages:First, weobtain the brain matter using an approximatesegmenta-tion routine. Manualinitialisation is avoidedby taking theoutput image from this procedureto constructthe initialNURBSsurfacethatis positionedcloseto theactualbound-ary of thebrainmatter. Thesecondstageinvolvesdeform-ing theaboveNURBSsurfaceaccordingto anenergy min-imisationalgorithmandalso,modifying theweightassoci-atedwith eachcontrolpointautomaticallydependingonthecurvaturepropertiesof the evolving surface. This weightmodificationforcespartsof thesurfacenearsulci (grooveson the brain surface)to attract towardsthe cavities with-outhaving to duplicatethecontrolpointsnearsuchregions.By usinga NURBS-baseddeformablemodel,we havesuc-ceededin; (a) increasingthe local flexibility of thesurfaceby automaticallyadjustingtheweightsof thecontrolpoints,(b) controllingthelocalshapewith B-Splinebasisfunctionsand (c) automatingsurfaceinitialisation. Experimentstodeterminethereliability of theproposedtechniquearecar-ried outon MR headscansof 16 healthysubjects.
2. Initial Brain Segmentation
The purposeof this stageis to obtainan initial surfacelo-catedcloseto theactualbrainsurface.Thissurfacemaynotnecessarilybe the true brain surfacebut an approximationthatwill bedeformedto conformto theactualboundaryofthebrainmatter. At thebeginning,thevoxelsof theoriginalMR headscans(Fig. 1(a)) areconvertedto have isotropicdimensionsof
���������mm� to facilitatedistancetransfor-
mations.Thescansarethenfilteredusinganedgepreserv-ing 3D neighbourhoodaveragingroutine[5]. The filtered
imageis thenclusteredinto fiveclasses;cerebro-spinalfluid(CSF), gray matter (GM), white matter (WM), skull andbackground(Fig. 1(b)). Next, a binary maskcontainingGM andWM is generated(Fig. 1(c)). To openthenarrowpathsthatlinks thebrainwith otherexternalstructures,bor-derpointsof brainmatterareextracted(Fig. 1(d))anda3Ddistancetransformation[2] is appliedto yield all the con-nectedregionslocatedoutside2 pixelsfrom theseboundarypointsresultingin a binaryimage(Fig. 1(e)).Thisdistanceis chosenbasedon the percentageof CSFfound betweentheskull andthebrain.
Otherthanthebackgroundregion, thelargestconnectedcomponentin thedistance-transformedbinary imageis thebrain matter. A 3D binary closing [6] is applied to thiscomponentandthis morphedimage(Fig. 1(f)) is maskedwith thefilteredMR sourceto obtainanapproximatebrainmatter(Fig. 1(g)) that will be usedto constructthe initialNURBSsurface.
2.1. Initial NURBS surface
Theborderpointsof theextractedbrain,asshown in Fig. 2,form a continuousstreamof voxels aroundthe brain mat-ter. Fromthis representation,we samplea setof coordinatepositionsby rotatingan axisaboutthe centreof gravity of
theboundaryin eachslice(the � and � directionsareonthex-y planeand in the � direction, respectively). The inter-sectingpointsof thisaxisandtheboundarygivethecontrolpointsof theNURBSsurface.Theangleof rotationof theaxisateachsampledeterminesthenumberof controlpointsneeded.
2.2. Energy Minimising SurfaceThe brain surfaceis a two-parametervector-valuedmath-ematicalfunction ��� ���������� �������� ��� such that anarbitrarypointon thesurfaceis parametricallyexpressedas��� �!��"$# , �%�&�'��#)(*� ������ . Theinternalenergy +�,.-0/ at a point��� �!��"$# on thebrainsurface � is definedas
13254�6'798:7<;>=@?BACA!DFEG H!I@JLK3MMMM9N 8O7<;P=@?BAN ; MMMM Q:R ISKTJUMMMM9N 8O7<;P=@?VAN ? MMMM QRXW JCJUMMMM N Q 8O7<;P=@?BAN$Y�N$Z MMMM Q R�W Q K�MMMM N Q 8O7<;>=[?BAN$Y Q MMMM Q R\W K Q MMMM N Q 8O7<;P=@?]AN$Z Q MMMM QT^
(1)
where _�,a`�(b� [3]. Theexternalenergy +dcSe�/ at a point��� �!��"$# on thebrainsurface � is definedas
» ¢ � ³ jL¥OnT¦Pn � p�q ² i¢ � ³�½ ¢ � ¾ jL¥zp ½ ³'� ¿ j%¦�p£�µ!¶· À ¸!¹ º µ:¶·Á ¸!¹ ² iÀ � Á ½ À � ¾ j%¥!p ½ Á � ¿ jL¦$parethe rationalbasisfunctionsin which ½ÃÂ%Ä Å givestheÆ iLÇB-Splinebasisfunctionof degree È |ÊÉ ´
suchthat
½Ã � ¹Ëj9Ì�p�qÎÍ � n ifÌÃ|Ï� Ì Â n@Ì Â ´ ¶ p��n
otherwise
and
½  � Å jlÌ�p3q jlÌ)rÐÌ Â p ½  � Å µ!¶ j9Ì�pÌ Â ´ Å rÐÌ Â � j9Ì Â ´ Å ´ ¶ rÐÌ�p ½  ´ ¶ � Å µ!¶ j9Ì�pÌ Â ´ Å ´ ¶¯rÐÌ Â ´ ¶where
Ì Â arepositive real numbersreferredto asknots[12].
2.4. Deformable NURBS SurfaceThe brain image in Fig. 1(g) doesnot exhibit the true3D patternsof the brain surfacebecauseit containslowintensity CSF that fills sulci cavities. In the secondstage of our technique, the approximateNURBS sur-face obtained previously is deformed so that it wrapsitself onto the cavities and the hills on the brain sur-face. The NURBS surface given in Eq. (4) can alsobe expressedin matrix form as
û î û è%é&ê'ë>êTìSí�î./. ï�ñãLå æ ò ã è%é!í ò æ è%ë�í î óÐèLézíS÷øó�ù3èSú$í0 Proceedingfur-ther, thesecondpartial integrationsarederivedas:
â ç�çãLå æ è%é&ê'ë>êTìSí�îðï�ñã%å æ ò æ è%ë�í!è û ó ã è%é!í�12�436587Ëó ô è ö>í�1Êíû � and
â¼þ[þã%å æ]èLéOêTëPê@ìSí�î ï�ñãLå æ ò ã è%é!í:9 û<; óÏÿ �� èSú$í=�>3�5:? ; óÏÿ� èSúËíA@û �where 1 î�÷øó�ù3èSú$í and
; î�óÐè ö>íC÷ . Further,
â ç�þãLå æ è%é&ê'ë>êTìSí�î ï�ñã%å æ û ó ã è%é!íC÷øóCB� è úËí=�D3 � � è úËíE5 ? ó ô èSö>íA1û �where óCBæ èLë$íÃî ò æ è%ë�íSó�ù � èLë$í8F ò æ è%ë�íSó�ùXè%ë�í , û ç î� û �E�àé and
û þ î2� û �E�àë . The above partial integrationsareusedto assesG 7 , G ? , G 77 , G ?�? , G 7 ? from G .
2.5. Weight AdjustmentWhenthedeformableNURBSsurfaceevolvesthroughtheenergy minimising states,somesegmentsmay not be sit-uatedcloseto the actualbrain matter. In sucha case,theonly wayto movethesurfacetowardstheseregions,withouttranslatingthecontrolpoints,is by increasingtheweightoftherelevantcontrolpoints[11]. Moreover, thisweightmod-ificationmustnot effect partsof thesurfacelocatedoutsidesulci cavities.
As shown in Fig. 3, let s be a point at the bottomofa sulci towardswhich the NURBS surface H needsto beattractedby adjustingthe weight of the control point
[ ñ ã%å æ(notethat s is notacontrolpoint). If H is at s at time ìtF �,
theNURBSform of theevolving surfacebecomes
u>u ï ñ B �ã%å æ ò ã è%é!í ò æ èLë$í [ ñ ãLå æu>u ï ñ B �ã%å æ ò ã è%é!í ò æ è%ë�í � upu ï ñãLå æ ò ã è%é!í ò æ è%ë�í [ ñ ã%å æu>u ï ñãLå æ ò ã è%é!í ò æ è%ë�íî s � H è ö!ê�úPê_vSí(7)
Note that during weight adjustments,the position of acontrol point remainsunchanged;i.e.
[ ñ B �ãLå æ î [ ñ ãLå æ . If theweightchangeis denotedby w ïdñãLå æ î ï ñ B �ã%å æ � ï�ñãLå æ , Eq. (7)canbewrittenas
û X ./. ïdñãLå æ�ò ã è%é!í ò æ èLë$í [ ñ ãLå æ F ./. w ïdñãLå æ�ò ã è%é!í ò æ è%ë�í [ ñ ã%å æ \� ./. ï�ñã%å æ ò ã è%é!í ò æ è%ë�í [ ñ ã%å æ X û F ./. w ï�ñãLå æ ò ã è%é!í ò æ èLë$í \î û X û F ./. w ï�ñã%å æ�ò ã èLézí ò æ èLë$í \ è s � H èSö!ê�ú�ê_v'í í(8)
Multiplying Eq. (8) by� � û yields
( ÿ �x ã�y �, ÿ �xæ�y � w ï ñã%å æ ò ã èLézí ò æ è%ë�í X [ qô å � � s \
î è s � H èLéOêTëPê@ìSí í û èLéOêTëPê@ìSí (9)
Now, Eq. (9) gives the weight changesw ï�ñãLå æ to expandthedeformableNURBSsurface H towardsthe sulci cavitybecauseit causesa perspective functionaltranslationof thepointson theeffectedNURBSsegmenttoward
[aqô å � .Due to the continuousnatureof the NURBS surface,
an analyticalsolutionfor the curvatureof the surfacenearthe control point
[:qô å � is computedby taking the distancebetween
[aqô å � and the averagevector z è [aqô å � í of all theconnectedneighbouringcontrol pointsof
[ qô å � , denotedby{ è [aqô å � í , which approximatesthesurfacecurvatureat[aqô å � .
Theapproximatesurfacecurvatureat[aqô å � , | ñ ã%å æ , is givenby| ñ ãLå æ î dd
in which}} { è [aqô å � í }} givesthenumberof controlpoints
in{ è [aqô å � í . If thesurfacecurvature| ñ ã%å æ exceedsathreshold
(say, the averagesurfacecurvature î �( *-, ./. | ñ ãLå æ ), theweight ï�ñã%å æ is updatedsuchthat
ï ñ B �ãLå æ î ï ñãLå æ�FD� | ñ ãLå æ�����ãLå æ $ | ñ ãLå æ &where �P��� controls the amount of attraction of
the surface towards the control point. This local shapecontrol of the surface,due to weight adjustments,comesfrom the B-Spline basis functions where ò�� å � è���íÎî kfor ������ � � ê�� � B � B � í . The knots for the two B-Splinebasis functions ò ãLå � è%é!í and ò æ'å � èLë$í are given by thevectors è%é � êTé � ê+��.ê'é ( B � íCù and è%ë � ê'ë � ê��.ê'ë , B � íCù , respec-tively. Eachknot pair èLé ã ê'ë æ í relatestheparametersé andë to a weight ï�ñãLå æ suchthat an adjustmentof ï�ñãLå æ affectsonly thesurfacesegmentH è%é&ê'ë>êTìSí , èLéOêTë$í��4� é ã ê'é ã B � B � í �� ë æ êTë æ B � B � í [12]. Finally, thebrainmatteris extractedaf-ter removing all the voxels situatedoutsidethe deformedNURBSsurface.
Figure4: MR imageslicesof thebrain: (top) original MRimages,(middle) NURBS surface initialised close to thebrain and (bottom) brain slicessegmentedwith NURBS-baseddeformablesurface.
Fig. 4 depictssegmentedresultsof an MR headscan.Although our algorithmis implementedin 3D, the imagesareshown as2D slicesfor enhancedclarity. Theaccuracyandtheshapedifferenceof theNURBS-basedandmanualsegmentationwasmeasuredusingtheaveragedistanceerrorbetweenthe two surfaces[16]. It is the averagedistancebetweeneachpointonthedeformedNURBSsurface � andtheclosestpoint on theactualbrainsurface� � . Theaveragedistanceerror, � , is givenby
Figure5: Performancemeasurementof theproposedtech-niqueon 3D MR imagesusingtheaveragedistanceerror.
where » is the surfaceareaof ¼ . It can be observedin Fig. 5, wherethe X-axis givesthe subjectnumber(i.e.the volunteer’s MR headscannumber), that the error isbelow 1 pixel in all the caseswhen the weights ½K¾¿³À Á areupdated;the lowest being 0.60 pixels and the highester-ror being0.71 pixels. Whenthe weightsarenot updated,the lowestandthe highesterrorsare1.11and1.23pixels,respectively. In other words, the brain, segmentedusingtheNURBSframework with weightadjustments,is situatedcloserto the true brain surfacethan the segmentationob-tainedwithout weightadjustmentsfor thesamenumberofiterations.Theseresultsmatchwith increasedperformanceoverthosereportedby otherinvestigators[4, 17]. Whentheweightsarenot updated,thedeformablemodelbehavesasa B-Splinesurfacethatdoesnot exhibit rationalpropertiesto give anextra degreeof freedomfor shapingthesurface.Theprocesstakes,on theaverage,10 minutesto peela 3Dvolumetricimagein a SunSparc60workstation.
4. ConclusionA new techniquebasedon NURBS to segmentthe brainfrom MR headscansis presented.ThedeformableNURBSsurface exploits the weighting parameterat eachcontrolpoint that is updatedautomaticallyto detectsharpcavitiesfoundon the brainsurface. Moreover, the modeldoesnotcontainaninitialisationproblembecausetheinitial NURBSsurfaceis formedautomaticallyusinganapproximatebrainsegmentationroutine. It guaranteesthat thesurfaceinitial-isation is closeto the actualbrain matterandthus,avoidsthe evolving surfacegettingstuck in a local minima. Thecurrentapproach,which integratesimageprocessing,com-puter graphicsand vision, providesclinicians an accuratetool to carry out surface-basedstructuralanalysisof thehumanbrainandothervisualisationtasksin neuroimagingstudies.
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