CLINICALSCIENCES

Functional images are usually created in an attemptto portray someaspectof organ behavior as a singleimage. Many such functional images have been described for a variety of organs. These images permit easysubjective assessment of function, and provide numericdata describing function on a regional basis. Despite theirquantitative nature, functional images are still frequentlyevaluated subjectively—by observation of the image.This paper describes an objective method for assessingfunctional images. It can be used to produce a singlenumber that may be a descriptor of overall organ function. The method is based on producing a distributionfunction of pixels (i.e., a histogram of the number ofpixels possessing a certain value plotted against thatvalue). The distribution function (DF) is produced in aunique way that overcomes the three principal objectionsassociated with such analyses: (a) The method is insensitive to selection of the precise organ borders; (b) itcompensates for random fluctuations due to the smallnumber of pixels in each “bin―(i.e., range of functionalvalue) of the DF, and the method is also independent ofthis bin width; and (c) most importantly, it allows spatial

Received Aug. 10, 198 1; revision accepted Oct. 27, 1981.

For reprints contact: Stephen L. Bacharach, National Institutes ofHealth, Building I0, Room IB38, Bethesda, MD 20205.

information to be included in the DF. That is, the methodcan be used to emphasize regional dysfunction as opposed to global dysfunction.

In order to illustrate the application of the techniqueand its advantages, an example of its use is given. Theexample chosen is the analysis of Fourier “phase―mapsof the left ventricle ( I —7)produced from gated equilibriumcardiacstudies.

METHODS

Calculational.To applythe proposedmethodforcalculating the distribution function of values within afunctional image, two images are required—the functional image itself and an error image. The latter consistsof an image in which each pixel contains an estimate ofthe error of the corresponding point in the functionalimage. Usually the error image will be calculated byapplying standard error propagation relations. This isquite straightforward (albeit tedious) for ejectionfraction maps, phase maps, stroke-count maps, etc. Ifother (not counting-rate-limited) sources of error areknown, they also should be included in the errorimage.

Given the functional map and its associated errorimage, a region of interest (ROl) is drawn around the

Volume 23, Number 4 285

DIAGNOS11C NUCLEAR MEDICINE

A Methodfor ObjectiveEvaluationof FunctionalImages

StephenL.Bacharach,MichaelV. seen, Robert0. Bonow,CornelisN.deGraaf,andGeraldS.Johnston

National Institutes of Health, Bethesda, Maryland and the Institute of Nuclear Medicine, Utrecht, The Netherlands

A methodIs descrIbedforcharacterIzIngfunctIonalmapsby meansof a frequencydIstrIbutIonfunction(DF)ofpIxels.Themethodmakesuseofa mapofthestandarddevIationofthefunctionalparameter.Byuseofthismapthemethod(a)becomeslesssensftlvetoImprecIsedefInitIonoftheorganborders,(b) compensatesfor randomfluctuatIonsIn thefunctionalmap,and(mostImportantly)(c)permitsspatIalInformationtobeIncludedIntheDF ThisspatialInformationisincludedby performIngspatialclusterweighting—aprocedurethat emphasizesreglonaldysfunctionoverglobaldysfunction.Themethodis Illustratedbyapplyingit to phasemapsoftheleftventricle.Inthisexample,It isshownthatanalysisoftheDFpermitsdetectionof regionalabnormalitiesof LVwallmotionandthatanimprovementindetectabilityisobtainedwithclusterweighting.

J Nuci Med 23: 285—290,1982

by on April 10, 2018. For personal use only. jnm.snmjournals.org Downloaded from

BACHARACH, GREEN, BONOw, DEGRAFF, AND JOHNSTON

organ of interest. The DF is then calculated on pixelswithin the ROl. To understand the method used herebest, consider first the conventional way of computingthe distribution function of pixels within the functionalimage. One would first examine the range of functionalvalues within the ROl ofthe map (they might range, forexample, from 1 unit to 100 units). This range wouldthen be divided into an arbitrary number of intervals or“bins―(for example, 50 bins oftwo units each). Then onewould simply tabulate the number of pixels (within theROl)possessingavalueofeither1or2 forthefirstbin,3 or4 forthesecondbin,andsoon.

The new method described here differs from thisconventional technique. No “bin―interval range ischosen. Instead, the value of each pixel in the functionalimage is considered to represent the mean of a normaldistribution function of values. The width o of this distribution function is given by the corresponding point inthe error image. Each pixel in the functional image, then,has no definite value, but rather a range of possible values with the most probable being the measured value andwith less-probable values occurring further from themeasured value. That is, the single pixel itself can bethought of as possessing a Gaussian distribution functionof values. If P@,@(I)is the probability that this single pixelat (x,y) possesses a value I, then we assume:

P@,@(I)= ,_@- exp[—(I—Im)2/2c@,y],(1)V

where ‘mequals the measured value of the pixel and@is the estimated error in that measuredvalue.

The distribution function of values in the whole ROlis then just the sum of all the single-pixel distributionfunctions:

DF(I)= @:all piXelS

where DF(I) is the “number―of pixels possessing afunctional value I. The word “number―is in quotes because the number of pixels will be nonintegral. Each pixelcontributes some fraction of “itself―to every possiblefunctional value, as given by Eq. (1). The overall distribution function DF(I) is simply a sum of Gaussians.Pixels with low errors contribute strongly to their corresponding measured functional value (a peakedGaussian), whereas pixels with large errors contributeweakly to both the measured value and a range of nearbyvalues (a broad Gaussian). The DF described by Eqs. (1)and (2) will henceforth be referred to as the error-corrected DF.

In order to emphasize regional abnormalities, thedistribution function can be “cluster-weighted.―Thatis, if several pixels all possessingsimilar values aregrouped together spatially in a small region of the organ,these pixels can be counted more heavily in the distri

LOW WEIGHT HIGH WEIGHT

I@C'\

:@\ • I—'%\

\\@ it‘ \\•\ \‘@• I

RV'\@@157― RV\@ LV

. =PIXELWITHCERTAINPHASEVALVEFIG.1. Illustratingconcept of clusterwei@iting.Allpixelswithsomearbitrarybut equalphasevalueare shownas dots.Left: pixelspossessingthis phaseare spreadoutoverLV, resultingin a lowwei@t factor for thisvalueof phasein thedistributionfunction.Ri@t:pixelspossessingthis valueof phaseare “clustered―together,resulting in a high weight factor for this phase value.

bution function than if these same pixels were spread outover the whole organ. Figure 1 illustrates this concept.To implement the spatial-clustering algorithm, the distribution function is weighted by the reciprocal of thespatial variance of the pixels. That is, for each value ofthe functional parameter I, all pixels possessing thatvalue are examined, and their spatial variance computed.In Eq. (1), every pixel possesses some “amount―of everypossible value of the functional parameter. Thus incomputing the spatial variance of pixels possessing valueI,everypixelwithintheROlmustbeused,andincomputing this variance the “number―of pixels is derivedfrom Eq. (1):

SV(I) = @: P@,@(I)* [(x —xo)2+ (y —yo)i,all x,ywithin ROl

(3)where SV(I) is the spatial variance of pixels possessing

(2) avaluel.The distribution function of Eq. (3) can then be rec

omputed as:

DF'(I) = DF(I)/SV(I), (4)

where DF'(I) is the “cluster-weighted―distributionfunction.

Illustrative example. Fourier “phase―maps of thecardiac left ventricle (LV) have been chosen to illustratethe use of the error-corrected distribution function andto demonstrate the effects of cluster weighting. Clusterweighting should be most useful when applied to afunctional parameter possessing localized, regional abnormalities. Cardiac LV wall-motion abnormalities arefrequently regional in nature, especially in subjects withcoronary artery disease. There are many functional parameters that may be descriptors of wall motion. Thereis evidencethat Fourier phasemaps are influenced bythe presence or absence of regional wall-motion abnormalities. Their presence or absence was therefore chosen

286 THE JOURNAL OF NUCLEAR MEDICINE

by on April 10, 2018. For personal use only. jnm.snmjournals.org Downloaded from

.. a

,_ @@4'

CLINICAL SCIENCESDIAGNOSTIC NUCLEAR MEDICINE

as the functional parameter to illustrate the effects ofcluster weighting, and Fourier phase was chosen as aparameter that might be descriptive of wall motion.Phase maps of LV cardiac function were chosen notbecause they necessarily represent a “good―or “useful―functional parameter, but because they may be familiarto the reader. The phase maps were obtained from thefirst harmonic of the temporal Fourier expansion of thesingle-pixel time-activity curves (TAC), in the usual way(1—6).The temporal resolution of each TAC was 20msec. An error map was also calculated. For Fourierphase, the standard deviation of each phase value maybe computed as:

@x = *—*SQRT,y@@ (A/B)2 B

x @f(tsin2(2irt/T)@ @tcos2(2irt/T)A2 B2

—2* @tsin(2@rt/T cos(2irt/T) (5)

where A and B are the first harmonic coefficients of thesine and cosine, respectively, in the single-pixel TACFourier expansion, f(t) LV time-activity curve, andT = periodof thecardiaccycle.@ = sumoverall t.SQRTrepresentsthesquareroot.Thiscalculationtakesonly a small amount of additional computer processing(CPU)timewhenperformedduringcalculationof thephase values, and is easily implemented.

We desired to compare the cluster-weighted with thenoncluster-weighted distribution function. To this end,phase maps were produced from two groups of subjects:40 normal volunteers (NV) and 70 subjects having angiographically demonstrated coronary artery disease(CAD). The NVs (aged I8—66)each had a normal history, treadmill ECG, chest radiograph, and echocardiogram. Coronary arteriography was not performed onthe NVs. The 70 CAD subjects each had >50% stenosisin oneor moreof the three major vesselsasdeterminedby coronary arteriography. Both groups underwentECG-gated equilibrium blood-pool scintigraphy at rest.Three observers independently assessed each subject'scardiac LV wall motion from the cinematic display ofthe scintigraphic images. Each reader made a simplebinary “normal―or “abnormal―assessment. A CADsubject was considered to have “normal―or “abnormal―wall motion on the basis of agreement between two ormore of the observers. On this basis, 37 of the 70 CADsubjects had normal wall motion at rest.

Analysis of distributionfunction. The error-correcteddistribution function (cluster-weighted or not) can beanalyzed in several ways. Many well-known measures,such as variance, skew, kurtosis, etc., may be calculatedto give a single number that describes the distributionfunction. Which of these measures has the most physicalmeaning will depend to some extent on the application.

PHASE

FIG.2. Calculationof ‘@reflectedarea.―DOttedlinerepresentsareflectionof curveto leftof peak.

For the distribution function of phase values describedabove, the relatively simple measure of symmetry illustrated in Fig. 2 was chosen. For each subject, a “reflected―curve was created by reflecting the left portionof the distribution function about the maximum. Fromthis reflected curve a “reflectedarea―was calculated,defined as the difference in area between this reflectedcurve and the measured distribution function, expressedas a percentage of the whole area under the measuredcurve.

The “reflectedarea―parameter is not necessarily thebest or most sensitive measure of curve symmetry. Norisitevenknownthatcurvesymmetryisthemostsensitivedescriptor of the distribution function. Reflected areawas chosen simply to illustrate the application of clusterweighting using one of the many possible parameterscalculable from the distribution function.

RESULTS

Figure 3 shows a typical Fourier phase map (panel A)from the heart of a normal subject, and its associatederror map calculated from Eq. (5) (panel C). The imageswere acquired in a 400 modified left anterior obliqueview. Figure 4A shows the distribution function (after

AMPLITUDEMAP tilliull in rn@a@ @r

FIG. 3. A typical phase map (A), amplitude map (B), and map of onestandarddeviation in the phase(C).

A*B

PHASEMAP

Volume 23, Number 4 287

REFLECTEDAREA

@,‘ ;@

by on April 10, 2018. For personal use only. jnm.snmjournals.org Downloaded from

A@w

DFCJi't@_

B

CLUSTERv-wm. wEIGHTEDDF

A

ERRORCORRECTED

t'@ C

BACHARACH, GREEN, BONOW, DEGRAFF, AND JOHNSTON

REFLECTED AREAROC

t 1.0 •.

@ .2(.1)@@@@ I I

1.0.8 .6 .4 .2 0

—SPECIFICITY

FIG.6. SensitIvityversusspecificity(asdefinedintext)usingreflected area as quantitativedescriptor of distribution function.

in which SV(I) of Eq. (4) is replaced with its square root,the spatial standard deviation.

The error-corrected DFs of phase, with and withoutcluster weighting, were used to try to detect visuallydetermined wall-motion abnormalities in the group of40 NVs and 70 CAD subjects. The previously described“reflectedarea―parameter was used as the descriptorof the distribution function. The ability of this parameterto detect wall-motion defects was evaluated using receiver operating characteristics (ROC) curve analysis.Figure 6 shows the ROC curves obtained with andwithout cluster weighting. Sensitivity, as used in Fig. 6,is defined as true positivesdivided by total number ofsubjects with wall-motion abnormalities (as describedpreviously). Specificity, as used in Fig. 6, is defined asnumber of false positives among the normal volunteersdivided by the total number of NVs. It is again emphasized that the goal was not to determine sensitivityspecificity relationships carefully from phase maps, butonly to illustrate the effects of cluster weighting.

DISCUSSION

Frequency functions have been used previously inattempts to evaluate functional images (1—3,5,6).Thereare several objections associated with such analysesobjections that are partially overcome by the use of Eqs.(1) through(3). Figure3 illustratesthefirstof theseobjections and how it is overcome. Panel A of this figureshows the phase map from the cardiac chambers. Valuesof phase are calculated everywhere in the image. Outsidethe cardiac structures, the values of phase fluctuatenearly randomly over the entire range of 3600. Ifan ROlisdrawn slightly too large, the resultingraw distributionfunction will include pixels from these nonmeaningfulregions. Such pixels will distort any measurements made(e.g., variance, skew, etc.) on the raw DF. One mightthink that the amplitude image (Fig. 3B) could be usedto overcome this problem. By choosing only those pixelsin the phase map that have large amplitudes, one indeed

ERRORCORRECTEDDF

ERRORCORRECTED+

CLUSTERWEIGHTED

1.OrI ii'

.8 I-,@.. .@•t>-I->

I-C,)zwU)

/\

1.0.8 .6 .4 .2 0

B+CDF

PHASEDISTRIBUTiONFUNCTIONS

FIG. 4. Frequencydistribution fUnCtiOns(DF)of phase(I.e., histograms of phasevalues) in normal subject. Vertical axis is numberof pixelswith equalvaluesof phase.Horizontalaxis is valueofphase.A: unmanipulatedDFof phase,from LV. B: DFof phaseafterclusterwei@'itlngwIthspatialvariance.C:OFofphaseafterapplyingEq. (2). D: B and C overlapped showing that in this normal subjectcluster weighting narrows the DF. Narrower curve is from PanelB.

applying an LV RO! to Fig. 3A) as it is usually calculated. Figure 4C shows the same distribution functionafter applying Eq. (2) and using the errors determinedfrom the error map of Fig. 3B. Applying the clusterweighting operation to this normal volunteer's distribution function, we obtain Fig. 4B.

Figure 5 presents the same kind ofdata as in Fig. 4 butfrom a CAD subject with a wall-motion abnormality.Panel 5A is the error-corrected distribution function,while panel 5C represents the cluster-weighted DF.Panel SB represents a milder form of cluster weighting

PHASEDISTRIBUTIONFUNCTIONS

FIG.5. DistributIonfunctions(DF)ofphaseintypicalCADsubjectwith wall-motion defect. lop: error-corrected DF. Left: clusterweighted with spatial standarddeviation. Right: cluster weightedwithspatialvariance.

S-CLUSTERWTD. V-CLUSTERWTD.

288 THE JOURNAL OF NUCLEAR MEDICINE

by on April 10, 2018. For personal use only. jnm.snmjournals.org Downloaded from

CLINICAL SCIENCESDIAGNOSTIC NUCLEAR MEDICINE

avoids including nonmeaningful phase values accidentally. By so doing, however, one also excludes akineticand poorly moving regions of the LV—thevery regionsone is attempting to detect. A large Fourier amplitude,while usually a sufficient condition for a meaningfulphase value, is not a necessary condition. Poorly movingregions of the LV may have quite low amplitude; yet ifthey possess high count densities, these regions maypossess statistically reliable values of phase. It is thestatistical reliability, then, that must be used as the critenon for inclusion of a pixel within the distributionfunction. Figure 3C illustrates a typical error map,showing the statistical reliability of phase at each pixel.The errors are, in general, low inside the cardiac structures and high outside of them. Equations (1) and (2)essentially allow creation of a “weighted―distributionfunction (the “error-weighted―DF) in which each pixelisweightedwith its reliability. Even if the ROI isdrawnso as accidentally to include noncardiac (time-independent) structures, such structures will not contributeheavily to the error-corrected DF, since they have lowreliability (a high error value).

Figures 4A and 4C illustrate the application of Eqs.(1) and (2) to the caseof a phasemapof a normal LV.Figure 4A shows the raw distribution function. Thehorizontal-axis resolution (i.e., the bin width) is onedegree. The phase distribution function in a normal LVis typically a narrow, symmetric, approximatelyGaussian bell-shaped curve. Although this general behavior can be appreciated from Fig. 4A, note the largefluctuations of pixels within each bin and the many binsin which the DF iszero,due to the sharphorizontal-axisresolution (i.e., small bin width) chosen. Note also thatthere are a few pixels in the DF considerably to the rightof the main peak. These pixels probably result from aslightly oversized LV region of interest. They mighteasily be interpreted as an abnormality. In reality, thesepixels have high errors associated with them. When Eqs.(1) and(2)areapplied,Fig.4Cresults.Theextraneouspoints are no longer noticeable owing to their high error.Figure 4C is obviously much more amenable to analysessuch as the “reflectedarea―calculation illustrated in Fig.2. Note that for some calculations Eq. (2) is not necessary. For example, if the variance or skew of the distribution function were desired, they could be computedin the usual way from the raw DF, but weighting eachpoint inversely with the square of the standard deviationas determined from the error map. In general, however,applying Eq. (2) to produce the error-corrected DF asshown in Fig. 4B is useful.

Amajordisadvantageofusingadistributionfunctionof pixels to characterize an image is that all spatial information is lost. Cluster weighting (illustrated in Figs.4Band4DandinFig.5)attemptstoputregional,spatialinformation back into the distribution function. Figure4B shows the cluster-weighted DF from a normal vol

unteer. It was produced by applying cluster weightingto the error-corrected DF of Fig. 4C. Note that for thisnormal subject, cluster weighting does not appreciablyalter the shape of the DF. The DFs with and withoutcluster weighting are shown in Fig. 4D. In this casecluster weighting actually narrows the DF. In general,the distribution functions of phase in normal subjects donot change greatly with cluster weighting. In CADsubjects with wall-motion defects, on the other hand,cluster weighting frequently produces dramatic effects.Figure 5 demonstrates this. Panel A of this figure illustrates the error-corrected DF. The distribution is slightlyskewed to high phase values (right-hand side), but anasymmetry of this magnitude is occasionally also seenin normal individuals.After clusterweighting [Eq. (4)J,this asymmetry is markedly emphasized (Fig. SC), indicating that a regional defect caused it. Figure 5B issimilar to 5C but uses a weaker cluster weight. Thisweaker weighting may be useful with very noisy data.

ItisclearfromFig.5 thatclusterweightingcanemphasize regional abnormalities. It is also possible to define the regional phase abnormality (the smaller peakin Fig. SC) and determine which pixels from the originalmap caused this abnormality. This may possibly beuseful in visually emphasizing the location of regionalabnormalities in the original image.

Note that without the error-corrected DFs, as described in Eq. 2, cluster weighting would be quite difficult to perform. When using the raw DF, there are so fewpixels in each bin that one often cannot compute thespatial variance. In the raw DF of Fig. 4A, for example,many bins have two or fewer pixels. A wider bin improvesthe situation, but at the expense of worsened resolutionon the phase axis. The scheme described by Eqs. ( 1) and(2) avoids these problems. The equations allow eachsingle pixel to possess a distribution of phase values [asin Eq. (1)]. Thus all pixels contribute to some extent toevery value of phase, making calculation of spatialvariance, and hence clustering, possible.

In order to investigate the effects of cluster weightingmore quantitatively, we studied distribution functionsof phase with and without weighting. The ability todiscriminate between normal and abnormal left-yentricular wall motion was used as the criterion of comparison. The group of normal volunteers, and subjectswith CAD described previously, allowed ROC curves tobe constructed with and without cluster weighting. TheROC curvesof Fig. 6 illustratethe sensitivity-specificityrelations obtained with and without cluster weighting.Cluster weighting is seen to result in a slight (but visuallyapparent) increase in sensitivity at high specificity values. The choice of reflected area as the parameter for thiscomparison is completely arbitrary. It was simply oneof several possible parameters, and was chosen for itssimplicity. It is certainly not (we have since discovered)the most sensitive or specific parameter for detection of

Volume 23, Number 4 289

by on April 10, 2018. For personal use only. jnm.snmjournals.org Downloaded from

BACHARACH, GREEN, BONOW, DEGRAFF, AND JOHNSTON

LVwall-motionabnormalities.It doesillustrate,however, the improvement that may be gained by clusterweighting. Other descriptors of the DF, or DFs offunctional parameters other than phase, may responddifferently to cluster weighting, depending on the extentto which such descriptors or parameters are regional innature.

There are several computational difficulties inimplementing Eqs. (1)—(4).First, the process is slowunless floating-point hardware is available. A minicomputer* with floating-point hardware typically required 30 sec to produce cluster-weighted DFs from aI 50-pixel ROI. Second,if single-precisionvariablesareused to perform the cluster weighting, considerationmust be given to round off errors. Such errors may occasionally cause the cluster-weighted DF to increasewithout bound at phase values far from the main peak.Use of double-precision variables and/or care inimplementing the algorithm will minimize this problem.Third, in Eqs. (1)—(3),integrals have been replaced bysums. Proper computer integration must be utilized.

The analysis embodied by Eqs. (1)—(3)can in principle be applied to any functional image for which anerror map has been created. In many cases the functionalparameter of interest is spatially uniform, approximately,over the organ of interest in normal individuals. This istrue for the phase maps considered here as well as formaps of LV time to minimum counts, EF maps, andmany others. This approximate spatial uniformity offunction over normal organs facilitates analysis of theDF, permittingsimpledescriptorssuch as “reflectedarea―to be used. There are, however, classes of functional maps for which normal function is not even approximately uniform over the organ of interest. In suchcases different methods of describing the DF may benecessary. Also, if large spatial inhomogeneities areexpected to be present even in normal maps, clusterweighting may be of less value.

Finally, note that many descriptors of the DF do notrequire creation of the error-corrected DF [Eq. (2)J.Variance, skew, kurtosis, etc., may all be calculated directly, not from the raw DF but from the functional mapand its associated error map. The error map must be usedin order to weight properly the contribution of each pixelto the value of the parameter being calculated (variance,skew, etc.).

CONCLUSION

By use of Eqs. (1) and (2), we are able to producedistribution functions from functional images. The

method used to produce the distribution functions, byusing the error image, alleviates the usual statisticalproblems associated with raw DFs. In addition, a cluster-weighting scheme is proposed that should allow enhanced detection of regional abnormalities.

Application of the proposed method to phase mapsverifies that the cluster-weighting scheme has value inemphasizing regional LV wall-motion abnormalities. Wealso demonstrate that use of the error-corrected DFpermits an entire image to be characterized by one (ora few) descriptive parameters. For the case of phaseimages, the arbitrarily chosen parameter of “reflectedarea―gave a reasonably good sensitivity—specificityrelationship. By more careful selection of parameterscharacterizing the DF, or by combining several suchparameters, still better results may be hoped for. Although the method has been tested only on phase imagesof the LV, it is in principle applicable to nearly any organfrom which a functional map can be produced.

FOOTNOTE

aHewlettPackardHP-bOOFSeries.

REFERENCES

I . GEFFERSH, ADAMWE, BITTERF,etal: Dataprocessingandfunctionalimagingin radionuclideventriculography.Proceedingsofihe Fifth InternationalConferenceonInformationProcessingin Medical Imaging, Nashville, 1977(ORNL/BCTIC-2),pp322-332

2. BOSSuYTA, DECONINCKF, LEPOUDREM, Ctal:Theternporal Fourier transform applied to functional isotopic imaging.InformationProcessingin Medical Imaging.Paris,INSERM88, 1979,pp 397-408

3. ADAM WE, TARKOWSKAA, BITTER F, et al: Equilibrium(gated) radionuclide ventriculography. Cardiovasc Radio!2:161—173,1979

4. VERBAJW, BORNSTEINI, ALAZRAKI NP, Ctal: Onset andprogression of mechanical systole derived from gated radionuclide techniques and displayed in cine format. J Nuc! Med20:625-626, 1979 (abst)

5. LINKS JM, DOUGLASKH, WAGNER HN JR: Patternsofventricular emptying by Fourier analysis of gated blood-poolstudies. J Nuc! Med 21:978-982, 1980

6. BYROM E, PAVEL DG: Algorithm for quantification of regional wall motion abnormalities, using the phase image of theECG-synchronized cardiac study. J Nuc! Med 21 :47, 1980(abst)

7. BACHARACHSL,GREENMV, DEGRAAFCN, etal:Fourierphase distribution maps in the left ventricle: toward an understanding of what they mean. In Functional Mapping ofOrganSystems.ESSERPD, Ed. New York, Societyof NuclearMedicine, 1981, pp 139-148

290 THE JOURNAL OF NUCLEAR MEDICINE

by on April 10, 2018. For personal use only. jnm.snmjournals.org Downloaded from

1982;23:285-290.J Nucl Med.   Stephen L. Bacharach, Michael V. Green, Robert O. Bonow, Cornelis N. deGraaf and Gerald S. Johnston  A Method for Objective Evaluation of Functional Images

http://jnm.snmjournals.org/content/23/4/285This article and updated information are available at:

  http://jnm.snmjournals.org/site/subscriptions/online.xhtml

Information about subscriptions to JNM can be found at:  

http://jnm.snmjournals.org/site/misc/permission.xhtmlInformation about reproducing figures, tables, or other portions of this article can be found online at:

(Print ISSN: 0161-5505, Online ISSN: 2159-662X)1850 Samuel Morse Drive, Reston, VA 20190.SNMMI | Society of Nuclear Medicine and Molecular Imaging

is published monthly.The Journal of Nuclear Medicine

© Copyright 1982 SNMMI; all rights reserved.

by on April 10, 2018. For personal use only. jnm.snmjournals.org Downloaded from