niques to identify hibernating myocardium (2) and to monitor heart function duringthe treatmentof cancer patientswith cardiotoxic cytostatica (3). Equilibriumradionucideangiocardiography is also used for risk assessment in patients with coronary heart disease (1,4), for assessment ofthrombolytic therapy in patients with acute myocardialinfarction (5,6) and for the assessment of other therapeuticinterventions (5,7).

Correct and reproducible measurements of LV function,in terms of parameterssuch as the global and regionalejection fraction,requirean accurateandreproduciblealgorithmto delineate the left ventricle (8,9). Delineation can be accomplishedmanuallyor automaticallyusing an edge detection algorithm (EDA). The manual algorithm usually suffersfrom low reproducibility, whereas automatic algorithms suffer from low precision due to nonuniformbackground,lowsignal-to-noise ratios (SNR) and the complicated structure ofthe heart,which may result in an imagewith the rightyentide and left atrium overlapping the left ventricle.

The most commonly used EDA is based on the assumption that the LV border ventricle coincides with the zerocrossing ofthe second-order derivative in a radial search withits origin in the left ventricle (10). To limit the search, themethod is often combined with thresholding (11). The EDAbased on combined thresholding and the second-order derivative is referredto here as the thresholding/derivativeEDA.A linearcombinationof first-and second-orderderivativeshas also been suggested (12). Also, Reiber et al. used analgorithm based on a minimal cost function (13).

The aimof this studywas to develop an EDA which is capable of handling the problems of low SNR and overlappingstructures in the image. We have used the concept of closedline, integral optimization. The regions of interest (ROIs)generated by our algorithm were compared with the ROIsgenerated by a thresholding/derivative EDA and with manwilly-definedROIs. The comparison was performed on bothsoftware phantom images and images from patient studies.

MATERIALSAND METhOD

Basic Principle of the AlgorithmThealgorithmis basedon thehypothesisthattheborderof the

LV ROlis definedas theclosedcurvewhichgivesthemaximum

Automaticevaluationof leftventricular(LV)functionusing equilibriumradionudide angiocardiographyrequires an edge detection algorithmto correct and reprOdUciblydelineate the leftvanbide. Available algorithms, usually based on differentiation of aradial profile,generally suffer from low precision due to lowsignal-to-noiseratios and overIapç@ngstructures, for example,the leftatrium.Methods: An edge detection algorithmwas de@ieIopedbased on the assumption that the LVborder can bedefined as the madmum, normalized,closed-line integralof adosed curve in a vector fieldderhied by image differentiation.Itis furtherassumed thatthe dosed curvecan be describedbyaFourier expansion with a limited number of harmonics. Regionsof interest(ROls)generatedby this algorithmwere comparedwithROIsgenerated by an algorithmbased on a combinationofthreshokiingand second-order derivatives.Results: This algorithmdelineates the leftventiide and gives resufts more doselyrelatedto ROIsgeneratedmanuallythanthe algorithmcombiningthresholdingand the second-order derivative.Our algorithmcan also handle the problemof overlappingstructures, as damonstrated in phantom simulations.Conclusion: The concept ofa mwdmum, normaliZeddosed-line integral will improve thedelineation ofthe LVin an equilibrium radionudkie angiocardiographystudy.The problemofoverlappingstructures is overcomebythisalgorithmbecause ittakes intoconsiderationglobaledgeinformation.

KeyWords: edge detectionalgorithms;leftventride;equilibriumradionudide angiocardiography

J NucI Med 1995; 36:1014—1018

quilibrium radionucide angiocardiography is an important clinical and research tool for the evaluation of leftventricular (LV) heart function and has become the goldstandardin prognostic studies in patientswith LV dysfunction (1). It has been used in studies in which the valuesfrom global and regional LV functions are used to assessthevalue of differentmetabolic andperfusionimagingtech

ReceivedJul 18, 1994;revisionacce@*edNov.10, 1994.For correspondenceor reprintscontnot MlkaeIEkman,MSc Departmentof

RadiationPhysics,SahlgrenSkaUniversityHospital,S-41345 GOteborg,Sweden.

1014 The Journalof NudearMedicine•Vol.36 •No. 6 •June 1995

Closed-Line Integral Optimization EdgeDetection Algorithm and Its Application inEquilibrium Radionuclide AngiocardiographyMikael Ekman, Milan Lomsky, Sven Olof Strömbladand Sten Carisson

Departmentc ofRadiation P1@ysics, Clinical P@solo@j@ and Cardiolo@, Sah@grenska Univers4y Hospita4GOtebo,g Sweden

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commonly encountered problems when trying to correctly delhieate the leftventricleusingan automaticEDA. &low followsabriefcharacterizationof thestudies:

Si: Fairseparationof the left ventricleand left atrium;theedgeof the leftventricletowards the left atriumis poorlydefined.Slightdilatationof the left ventriclewith severeanteroseptal hypokinesia and apical dyskinesia. Normalrightventricledimensionsand wall motion.

S2: Goodseparationofthe leftventricleandleftatriumaswellastheleftventricleandrightventricle.Moderatedilatationof the left ventriclewith an ellipticalshape, severe anteroseptal hypokinesiaand apical akinesia. Normal rightventricle dimensions and wall motion.

S3: Goodseparationof theleftventricleandleftatrium,modcrateoverlappingof the leftventricleandrightventricleapex otherwise good separation of the left ventricle andright ventricle. Normal left ventricle and right ventricledimensionsandwallmotion.

S4: Fair separation of the left ventricle and left atrium as wellas theleftventricleandrightventricle.Slightdilatationofthe left ventriclewith a sphericalshape and moderateanteroseptal and slight posterolateral hypokinesia. Normalrightventricledimensionandwallmotion.

S5: Goodseparationof theleftventricleandleftatriumaswellas theleftventricleandrightventricle.Normalleftventridc and right ventricle dimensions and wall motion.

Red blood cells were labeledwith 925 MBq @‘@‘Tcusing astandard in vivo/in vitro labelingprocedure. Imaghigwas performedwith an ElscintApex 415gammacamera system (Haifa,Israel) with the patient in the supine position. Images were recorded in the LAO projection,with a 20°caudal tilt, to give thebest separationofthe rightandleftventricles.Datawere collectedin a 64 x 64 matrix using 18—32frames per cardiac cycle and withanECGgatetoleranceof ±iO%.Extrasystolicandpostextrasystoliccycleswereexcluded.AcquisitionwasstoppedwhentheleftventricularROlcontainedapproximately150countsper pixel.Allstudieswerepreprocessedwithanine-pointsmoothingfilterpriorto edgedetection.

Evaluation of th AlgorithmPhantom Studies. ROIs generated by the thresholding deriva

the EDA(ElscintSP-i) andclosed-line,integralEDAwereevaluatedvisuallywithrespectto theirabilityto excludethe nonoverlapping part of the left atrium.

Patient Studies. In each patient study, two experienced observera, independentlyof each other, manually defined the left yentricle in nine frames representingdifferentphases of the cardiaccycle. Differencesbetween the two observerswere resolvedbyconsensus, in that a consensus RO! was created as a compromiseacceptable to both observers.

The45manualROIswerethencomparedwiththeROIsautomaticallygeneratedby the thresholding/derivativeEDA (ElscintSP-i) and the closed-line,integralEDA (using3, 4, 5 and 6harmonics).

The leftventriclewas dividedintofive sectorsusingthe geometric center of the manual ROl as the origin. The apical andoutflow sectors had an aperture of 60°,the anteroseptal and poeterolateral sectors had an aperture of 120°and the global sectorhadan apertureof 360°.The automaticandmanualROIswerecompared as follows: the average radius for each ROI was determined in every sector; the differencein the average radius be

FiGURE1. Thevectorsdefinedbydifferentiationoftheimage(left)are rotated through 90°,resuftingInvectors pointingalong theedge (right).

of the length-normalized,closed-lineintegralin a vector fieldderivedby differentiationof the image.It furtherassumesthat theclosedcurvecanbedescribedbyFourierexpansionwitha limitednumberof harmonics.In simpleterms,the normalized,closedlineintegralof anoutlinecanbe regardedas thesumof theslopesperpendicularto theoutlineat differentpointsalongtheoutline.Itis thennormalizedaccordingto thelengthof thecurveto preventwide outlines from being favored. The problem is then to find the

closedcurve whichgives the maximumvalue of the normalized,closed-lineintegral.

Descd@ of the AlgorithmA completemathematicaldescriptionof thisEDAalgorithmis

givenin theAppendix.After selectingan origininside the left ventricle, the imageis

differentiatedintheradialandtangentialdirectionsforeachpixel.Theradialandtangentialcomponentsdefinea vectordescribingthelocal changeincountdensity,both indirectionandmagnitude.All vectors in the image are rotated through 90°,resulting in avector fieldpointingalongthe (local)edges (Fig. 1). The closedobjectcurve is describedby Fourierexpansion.The simplestcurves (zero harmonics) will give a circle, while a more complexshaperequiresmoreharmonics.

Thus, the problemoffinding an outlineis reducedto findingthecurve, i.e., the coefficients in the Fourierexpansion thatoptimizethenormalized,closed-lineintegralforagivennumberofharmonics. We used a numerical mm/max algorithm, Simplex downhill(14).

Software PhantomThesoftwarephantomcontainedanellipsoidsimulatingtheleft

ventriclewitha 28-pixellongaxis anda 16-pixelshortaxis in a64 x 64 matrix. The left atrium was simulatedby a spherewith adiameter of 14 pixels. The simulated left ventricle, left atrium wasplaced in a homogeneousbackgroundof 100counts/pixel.Themaximum net counts were 100 per pixel for the simulated leftventricleand75perpixelforthesimulatedleftatrium.Thesimulatedimageswere convolutedwith the gammacamera'spointspreadfunctionand finallyGaussiannoise was added.Beforeanalysis,a common,nine-pointsmoothwas appliedto theimageto simulatea routineevaluationof a patientstudy.

Tenimageswithvaryingdistancesbetweenthesimulatedleftventricleandleftatriumweregenerated,startingwithaseparationof 2 pixels andendingwith an overlapof 3 pixels (see also Fig. 2).

PaUent Stud@eFive patient studies (S1—S5)were selectedfromroutineexam

inations.Thestudieswerecarefullychosento representthemost

1015New Edge DetectionftJgorfthm•Ekman at al.

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DWferenoe(pbc)Standard(plx)

H=5 H=6 TSDp(Wx)CUH=5 H=6 TSD H=3H=4CUSectionH=3 H=4

Thresholdlng/dertvativeEDAand closed-line,IntegralEDA(H = 6) failedin this regionki one study. Results are based on 36 images.The dosed-line integralEDAwas evaluated using 3, 4, 5, and 6 harmonIcs(H).Glob = globally,Sept = anteroseptal,Apic = apical, Let =

posteroleteral;Out = outflow;CU = dosed-line integral;TSD = thresholdkigderivative.The Wilcoxonsigned range test, p(Wx@,was applied todosed-line IntegralEDAH = 4 versus threshdding@derIvatIveEDA Data are based on 45 ROleunless othewise stated.

FiGURE 2. Phantom simulationof the left atrium and theleftventrideshowsframenumbersi (topleft),2(top,ight@,9(low left) and 10 (low right).FrameI showscorrectdelineation by both methods. Frames2,9and lOshowincorrectdelineationofthe leftventriclewiththe inclusion of the left atriumby the thresholdkigfdedvatlveEDAandcorrectdelineationbythe closed-line, Integral EDA.

FiGURE3. Anearlysystolicimagefrom study3. Correctdelineationof the leftventriclebythe dosed-line, integral EDAIncorrectdelineationof the leftventricle by the thresholdinglderivative EDA with the Indueon of the apex of the rightventhde.

A fully automated EDA to delineate organs is of greatimportance in quantitative examinations using nuclearmedicine. It reduces both inter-and intraobservervariability. Several algorithms are available, ranging from thresholding to more advanced cost function analysis. The general drawbacksof simple algorithmsare their sensitivity tonoise and their inabilityto include cold spots at the edge ofthe organ or exclude overlapping structures.

The algorithm basedon the maximum, closed-lineintegral described in this article generates ROIs which aresignificantly better than those resulting from a commonlyused algorithmbased on the second-order derivative whencompared with ROIs manually defined by two experiencedobservers. Our algorithm has an inherent ability to handlethe problem of overlappingorgans,asshown in both softwarephantomexperimentsandin patientstudies.Onereason for this is the assumption that the left ventricle ROlcan be described by a Fourier expansion with a limitednumber of harmonics, thus giving a robustness to thesearch for the best closed curve. Another reason is that thealgorithm takes into account not only the information onthe local edge but also the informationof the whole ROl.The fact that the closed-line, integral EDA excludesconvex parts of the count density when generatingthe vectorfield, togetherwith the use of a limitednumberof harmon

tween thresholding/derivative EDA ROIs and the manual ROIswas determinedin every sector, as was the differencein theaverage radius between closed-line integral-EDA ROIs and themanualROIs;andtheaveragedifferenceandits standarddeviation were determinedfor each sector.

RESULTS

Phantom StudissThe threshuldinWderivativeEDA was successful in 2 of

10 images (images 1 and 3) and failed to exclude thenonoverlappingpart of the left atriumin 8 images (Fig. 2).The closed-line integral EDA, using four harmonics, wassuccessful in all images (Fig. 2).

Patient SbicflssThe differencesbetween manualand automaticROIs are

shown in Table 1. The closed-line integral EDA generatesan outline that is closer to the standardthan the thresholding/derivative EDA. The closed-line, integral EDA alsoresulted in a sightly lower standarddeviation of the differences than the thresholding/derivative EDA. The Wilcoxonsigned range test shows a statistically significant difference (p< O.O1)between the closed-line integral EDA and the thresholding/derivative EDA in all sectors. Examples of ROISfromtwo patient studies are shown in Figures 3, 4 and 5.

TABLE 1Radii Differences between EDAS and Manually Drawn ROls

Glob—0.23—0.11—0.10—0.08—0.410.470.490.480.460.49<0.01Sept—0.19—0.23—0.20—0.160.120.510.530.510.520.60<0.01ApIc@—0.160.200.330.42—1.270.860.700.690.640.91<0.01Lal—0.06—0.14—0.19—0.12—0.850.550610.570.610.65<0.01Out—0.62—0.07—0.06—0.020.531.301.121.171.181.12<0.01

1016 TheJournalof NuclearMedicine•Vol.36•No.6 •June1995

DISCUSSION

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FIGURE 4@ End-systolicImagefrom study3. Correct delin

@lonby both methods.

FIGURE 5. Latesystolicimage fromstudy1.Correctdelineatlonofthe leftventrdebytheclosed-line,IntegralEDA Thethreshodng/derlvatlveEDAexdudes the apex ofthe leftvantride and Includespart of theleftatrium.The upparborderofthethresholdlnglderlvaWeEDAROldiffersbyonly1 pIxelfromthe thasdic refarenc*ROlandthus lieswithinthe mask ROl.

ice in the Fourier expansion, makes its outlines wider andthus, closer to manually-producedROIs.

A similarapproachto edge detectionalgorithmshasbeen used by Reiber et al. (13). The main differenceis thatouralgorithmisbasedonclosed-line,integraloptimization,thus taking into account informationconcerning the direction of the (local) edge. Parametrizationin outliningthe leftventricle in equilibrium radionucide angiocardiographystudies using the Fourier expansion has been studied byHosoba et al. (9), who found six harmonics to be theoptimal choice in describing LV contour. Our results indicate that a lower numberof harmonics is sufficient.

The major drawback of our algorithm is its computationalcomplexity.Withthegammacameracomputersystem (Elscint SP-1) used in this study, the closed-line integral EDA takes about 30 sec per image. By using modernworkstations, however, this time can be reduced significandy. There is reason to believe that algorithms usingglobal edge information, as in the case with closed-line,integral evaluation, are less noise-sensitive than algorithmsusing local edge information.

Future work will include the reduction of the standarddeviation in the differences between ROIs generated byclosed-line, integral EDA and manually-definedreferenceROIs.The most crucialstep seems to be the numericalmm/max algorithm. Other mm/max algorithms will also beconsidered.

CONCLUSION

Our algorithmimproves delineation of the left ventriclein an equilibriumradionucide angiocardiographystudy because it applies both local as well as global edge information.

Polar Coordinate Systsm DsfinlUonThe imageis describedin polarcoordinatesI(r, @).This re

quires the definition ofan origin which is assumed to be anywhereinsidethe objectbeingoutlined.

Paramstrlzatlon of ths Clossd CurvsTheclosedcurveis definedbytheradiuswhichis a functionof

angleL(4). It is assumedthat there is i:i mappingbetweenangleandradius.Theparametrizationis performedby Fourierexpansion

@ Eq.A1

whereH is the numberof harmonicsused. Beforeperformingclosed-line integration, the curve derivative must be evaluated:

t(@)=;@-@t+L(@. Eq.A2

As can be seen, the curved derivativeis representedas a vectorandis interpretedasthemovementindicatedbythetangentof thecurve per unit step in@

Imags Dlffsrsntlatlon and creating ths Victor RaidThe imageis differentiatedin both the radialand tangential

directionsby applyingthe gradientoperator to the image.Thisformsa vector fieldpointingalongthe gradient,perpendiculartothe localedges.

äI(r4) 1 8I(r 4')VI(r,4')= ar@ ;: o4@ @- Eq. A3

To avoidtoo tighta delineation,allgradientsoriginatingfromtheconvexparts of the count densityare excluded:

/8@I(r4@) \ /821(r4,)VI(r,4,)=O if@@ <0.0) or@@ <0.0

Eq.A4

VI(r, 4,) = VI(r, 4,) otherwise.

To makethisvectorfieldpointalongthe edges in the counterclockwisedirection, the vectors are rotated using the rotationoperator R:

V1(r, 4,)@= VI(r, 4,)R, Eq. A5

where R=[01 1]

cMterla for Optimizing NOrmaIIZSdc@lossd-Un.Integrals

The normalized,closed-lineintegralis definedby the followingequation:

I.@bVI(r,4')@--@----d4,. Eq.A6JL(4.) IT(4')I

Theintegralis a scalarproductbetweenthevectorfieldgeneratedby thegradientoperator,andthederivativeof theclosedcurve.

New Edge Detection Aigorithm •Ekman at al. 1017

APPENI@X

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Thederivativeofthecurveisnormalizedtoachievearesultwhichis independentof curvelength.

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1018 The Journal of NuclearMedicine•Vol.36 •No. 6 •June 1995

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1995;36:1014-1018.J Nucl Med.   Mikael Ekman, Milan Lomsky, Sven Olof Strömblad and Sten Carlsson  Equilibrium Radionuclide AngiocardiographyClosed-Line Integral Optimization Edge Detection Algorithm and Its Application in

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