- Email: [email protected]
Use of digital phantoms and a simulation of the scanning process to evaluate techniques of computer focusing of area scans
International
Journal
of Nuclear
Medicine
and Biology,
1973, Vol.
1, pp. 29-35.
Pergamon
Press.
Printed
in Northern
Ireland
Use of Digital Phantoms and a Simulation of the Scanning Process to Evaluate Techniques of Computer Focusing of Area Scans K. A. KRACKOW,
M . L. DUNCAN and R. J. GORTEN*
Department of Radiology (Nuclear Medicine Division) University Medical Center and Veterans Administration U.S.A. (Received 12 December 1971;
and Department of Medicine, Duke Hospital, Durham, North Carolina,
in revisedform 21 February 1972)
The usefulness of digital phantoms and computer simulation of the scanning process to evaluate computer methods for improving the presentation of scan data is demonstrated. Knowledge of the exact radioactivity distribution, i.e. the digital phantom, producing a given scan allows effective evaluation of computer techniques designed to improve that scan. As an example, the iterative focusing technique of Nagai et al. is analyzed in detail. Long computation time, previously considered to be the most significant limitation of this technique, is less limiting with ordinary line and column spacing factors. This analysis implicates instead other limiting The actual degree of focusing or deconvolution achieved, factors as being more significant. the sensitivity of the method to smoothing techniques, and the stated criterion for a satisfactorily focused result may represent more important limitations than computer time requirements. The criterion for satisfactory solution by a deconvolution technique is basic to all such methods. This work with digital phantoms and scan simulation has indicated possible inadequacies of relying on the similarity of images to infer the same degree of similarity between objects. Work is cited which may permit an investigator to know quantitatively the limitations of such an inference. This application of computer simulation of area scans from digital phantoms suggests the use of similar procedures first to analyze and then to aid in the modification and improvement of digital computer techniques of scan analysis. L’EMPLOI DES DE BALAYAGE
PHANTOMES DIGITAUX ET UNE SIMULATION DU PROCEDE POUR EVALUER LES METHODES DE MISE AU POINT PAR CALCULATEUR DES BALA YAGES DE SUPERFICIE On dtmontre l’utilitt des phantomes digitaux et de la simulation par calculateur du procede de balayage pour Cvaluer les methodes a calculateur pour ameliorer la presentation des don&es de balayage. La connaissance de la distribution exacte de la radio-activitt, c-a-d. le phantome digital, qui produit un balayage d onnt permet l’evaluation effective des m&odes a Comme exemple, on analyse en detail la calculateur destinees a amtliorer ce balayage. methode iterative de mise au point de Nagai et al. Un long temps de calcul, autrefois consider&e la limitation la plus importante de cette mtthode, donne moins de limitation avec les facteurs Cette analyse implique d’autres limitations a ordinaires d’espace de ligne et de colonne. Le degre actuel de mise au point ou de d&convolution sa place, comme &ant plus signifiantes. ache&, la sensibilite de la methode aux techniques de lissage, et le dit crittre pour un resultat bien mis a point peut presenter de plus graves limitations que les besoins du temps de calcul. Le critere pour une solution acceptable par une technique de deconvolution est a la base de toutesmtthodespareilles. Cet ouvrage avec les phantomes digitaux et la simulation du balayage
* Present address: 77550.
Division
of Nuclear
Medicine,
University 29
of Texas Medical
Branch,
Galveston,
Texas
30
K. A. Krakow,
M. L. Duncan and R. J. Gorten
a indique des insufhsances possibles de se fier a la similaritt d’images pour dtduire le mtme degre de similaritt entre les objets. On cite du travail qui laisserait savoir quantitativement a un rechercheur les limitations dune telle deduction. Cette application de la simulation par calculateur des balayages de superficie de phantbmes digitaux mtne I’idte a l’emploi de proctdes pareils dabord a analyser, ensuite a aider a modifier et a ameliorer les techniques a calculateur digital d’analyse a balayage. HPHMEHEHHE HM@POBbIX (DAHTOMOB H MOjJEJIJIBPOBAHBE HPOHECCA PA3BEPTHH AJIH OHEHKH METOAOB QOKYCIPOBKB PA3BEPTHM HO HOBEPXHOCTB HA BbIcIHCJIBTEJIbHOH MAIIIBHE nOKa3aHa nOJIe3HOCTb I@lpOBbIX @aHTOMOB H MOAeJIJIIlpOBaHIlR npOqeCCa pa3BepTKEi Ha KoMnbIoTepe,~T06bIO~eHElTbMeTO~bI~JIRy~y~~eHHRUpe~CT~B~eHEl~pe8y~bTaTOBpa8BepTKU. @aKToMa, namlqero 3namie TOYHOrO pacnpefienenna PaJJHOaKTEIBHOCTEI, T.e. I&IPOBO~O (PaKTHUeCKyIO pa3BepTKy, nO8BOJIEiT npUMeHliTb 8@$eKTMBHOe BbWWJIeHEle IIpH OJJeHKe MeTOAOB yJlyWIleHMR TOB pa8BepTKEl npM HOMOWU BblWiCJIWTe~bHOfi MaIlILIHbI. HanpnMep, HTepaqnoHKoro MeTona @OKYCHPOBKH Harall II up. &nrrenbnoe Aan no~po6~bl~ aHam BbIWfCJIPiTeJIbHOe BpeMff, 9TO paHbIIIe CYEITaIOT CaMbIM 8HElWJTeJIbHbIM OrpaHHYeHHeM 8TOrO MeTOna, OKa3bIBaeTCH OrpaHWIeHEleM B MeHbIIIei CTeneHEI B CJIyWe 06bIKHOBeHHbIX KO844iW BMecTe Toro ~ambdi aHam BBOAHT npyrne QHeHTOB 3anOJIHeHIIFI JIllHId II CTOJI~OB. OrpaHHWBaIOIIJHe YCJIOBHR KBK 6onee 3HaW%TeZbHbIe. npEl 8TOM B03MOPKHO OTMeTHTb $%lKTWIeCKH nOJlyqeHHaR CTeneHb +OKyCHpOBKM IUIH AeKOHBOJlEOlJMH, YyBCTBkiTeJIbHOCTb K MeTOAaM CrJlaWiBaHHH,EI.yKa3aHHdi KpHTepHti AJlII yAOBneTBOpHTeJIbH0 @OKyCHpOBaHHOrO co6ot 6onee 3HawTenbHbIe OrpaHwIeHHR, 9eM pe8yJIbTaTa BOBMOH(H0 IIpeACTaBJIHeT Tpe6OBaHHHB peMeHtI BbIWlCJIHTeJIbHOti MaJlIHHM. HpHTepd JQ'IR yAOBJIeTBOpHTeJIbHOr0 peIIIeHIUl HeKOTOpNM MeTOAOM AeKOHBOJIIOqEfI4 JIe?KllTBOCHOBeBCeXTaKHXMeTOAOB. Pa6oTaCqa@pOBarMa~aHTOMaMn&iMO~em~pOBaH~eM pa3BepTKIl nOKameT B08MOPKHbIe HeAOCTaTKH nOJIOH(HTbCH HanoAo6He H806pameK&,qTo6bI CAeJIaTb BJ~BOA 0 TOti CaMOi H(e CTelIeHK IIOAO6HH MemAy o6aeKTam. OTMeqeHa pa6oTa, KOTOpaHBO3MO?KHO nO3BOJIHT 3HaTb KOJlH9eCTBeHHble OrpaHWfeHHRTaKOrO BbIBOAa. 3T0 IlpHMeHeHHe KOMnIOTepH30BaHHOrO MOAeiWIKpOBaHHH pa8BepTKEi n0 nOBepXHOCTM OT Ql@pOBbIX @aHTOMOB rOBOpHT 0 nOJIe3HOCTII TaKUX MeTOAOB BO-nepBbIX aHaJII43MpOBaTb M nOTOM IIOMOraTb MOAH@KaIJHl¶ H yJIy=lIIIeHlWJ MeTOAHKIl aHaJIki3a pa8BepTKki npH nOMOI4H qll@pOBOti BbIWICJIBTeJIH.
VERWENDUNG VON DIGITALEN PHANTOMEN UND EINE NACHBILDUNG DES ABTASTVORGANGS ZUR BEWERTUNG VON RECHENMASCHINENVERFAHREN FUR DAS FOKUSSIEREN VON FLACHENABTASTUNGEN Die Brauchbarkeit von digitalen Phantomen und einer Rechenmaschinennachbildung des Abtastvorgangs zur Bewertung von Rechenmaschinenverfahren zur Verbesserung der Darbietung von Abtastdaten wird demonstriert. Die Kenntnis der genauen Radioaktivititsverteilung,, d.h. ein digitales Phantom, fiir eine gegebene Abtastung ermiiglicht eine wirksame Bewertung der zur Verbesserung dieser Abtastung ersonnenen Rechenmaschinenverfahren. Als Beispiel wird das iterative Fokussierungs-Verfahren von Nagai et al. ausfuhrlich analysiert. Lange Rechenzeiten, die friiher als die wichtigste Beschrankung dieses Verfahrens angesehen wurden, sind mit den tiblichen Zeilen-und Spalten-Abstandsfaktoren weniger als Grenzen anzusehen. Diese Analyse zieht andere Faktoren als ausschlagebender an. Der tatsachlich erzielte Fokussierungs- oder Dekonvolutionsgrad, die Empfindlichkeit des Verfahrens fur Glattungsmethoden und das angegebene Kriterium fur ein zufriedenstellend fokussiertes Ergebnis ktjnnen wichtigere Einschrankungen als die erforderliche Rechanmaschinenzeit darstellen. Das Kriterium fur eine zufriedenstellende Liisung durch em Dekonvolutionsverfahren ist fur alle diese Verfahren die Grundlage. Dieses Arbeiten mit digitalen Phantomen und Abtastnachbildung hat miigliche Unzulinglichkeiten aufgezeigt, falls man sich auf die Ahnlichkeit der Bilder zum Verleihen desselben Grads an Ahnlichkeit zwischen Objekten verlisst. Arbeiten werden angefiihrt, aus denen ein Forscher quantitativ die Beschrlinkungen einer solchen Folgerung ersehen kann. Die Verwendung einer Rechenmaschinennachbildung der Fllchenabtastungen von digitalen Phantomen regt zur Benutzung Ihnlicher Arbeitsweisen an, zunlchst zur Analyse und dann zur Hilfe beider Abanderung und Verbesserung von digitalen Rechenmaschinenverfahren zur Abtastanalyse.
31
Computer f0cu.rin.g of area scans
THE CLINICAL usefulness
of organ scans can be significantly improved by increasing the similarity between tracer distribution in the organ scanned and data presented for evaluation. The unavoidable discrepancies which exist with present scanning procedures are largely the result of collimator characteristics and the random nature ofradioactivity. Several methods involving digital computers have been developed for the purpose of reducing these discrepancies.(1-3) In order to appreciate fully the capabilities and limitations of these procedures, it is necessary to evaluate their effects objectively. Because such computer techniques operate on scan data in digital form, their evaluation, as shown here for the iterative method of NAGAI et al.(r), can easily be performed using digital phantoms and computer simulated scans.(4) The particular advantages of this type of evaluation derive from knowing exactly the which was used to “tracer distribution” produce the simulated scan.
DIGITAL
PHANTOMS SIMULATION
AND
SCAN
To understand the process of scan simulation, one can first consider a two dimensional array of numbers F = (&) where i = 1, . . . , h and w ,which is the digital phantom and ‘-1 .I--,***, represents the activity distribution in the object whose scan is to be simulated. A second array K = (k,) with i = -s, . . . , 0, . . . , +s and +t describes the collimator j= --t ,..., 0 )..., response characteristics for the scanning system being simulated. By aligning k,, with some fi, of F, the contribution of fi+,,,,+,, to counts observed over position (i,j) of the object matrix F may be obtained by multiplying fi+.+m ,j+n by
k,,.
= 2 i &+.+zl,,+ak,,, then the image p=-*p=--f array or simulated scan is the matrix G = (gij) i=l,..., hj=l,..., w, where gij is a number selected randomly from a collection of numbers with an essentially Poisson distribution about the mean mij. The array F is constructed arbitrarily to represent the distribution of radioactivity in the object whose scan is to be simulated. The collimator array K may be constructed as previously described(a) or in some other manner, e.g. scanning of a point or line source, which Ifmij
produces an array of values consistent with the definition of K. By calculating the mij from the elements of K and F, the effects of the collimator response pattern are simulated. An mij represents the average activity “seen” by the collifor mator when positioned above fij, allowing contributions from nearby elements of F. Selection of a random number as described@) simulates the stochastic nature of observations of radioactive events. Therefore, the array G is analogous to a scan from a source with radioactivity distribution as in F. It is the matrix G which is subjected to the computer method whose effects are to be analyzed.
ITERATIVE
TECHNIQUE
The iterative technique evaluated here is a method for determining the object matrix F or some approximation to it p, given the observed matrix G and the collimator matrix K. As described in the definition of mij above, disregarding the statistical nature of radioactivity, the elements of the scan matrix G are linear combinations of elements in F with coefficients from K. The iterative technique is actually a method of making successive approximations to the solution of a set of linear equations. Given G and K, the method produces a sequence of matrices {pn}, n = 1,2, . . . , which are intended to approximate the matrix Fwhen n is sufficiently large. Let G, = (g$)) be the matrix obtained from p,, by applying K just as the mi3 were obtained from F. As described elsewhere,(l) the iteration procedure is stopped, and r’, is accepted as a suitable approximation to F when the &) differ from the observed gij by less, on the average, than one standard deviation. The initial application of this method to the improvement of area scans suggests that the principal limitation of this computer focusing procedure is the requirement of inordinate amounts of computer time.(ls3)
ITERATIVE FOCUSING TECHNIQUE EVALUATED BY COMPUTER SIMULATION When application of the iterative focusing method was undertaken in this laboratory, it became readily apparent that computation cost would not be so limiting. This conclusion can be
K. A. Krackow,
32
M. L. Duncan and R. J. Gorten
DIGITAL 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
555555555 10 10 10 10 10 10
10 10
10 10
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 555555555
10 10
10 10 10 10 10 10 10 10
10 10
10 10
10 10 10 3 10 2 10 sl 1 10 10 10 10 10 10 10 10
10 10
io 4 3 2 10 10 10 io
10 10 10
10 10 io 10 20 10 20 10 Y 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 io 3 10 2 10 10 10 10 10 10 10 -~ 10 __ 10 10
5 10 10 10 10 10 IO 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
PHANTOM 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 IO 10 10 10 10 10 10 10 10 10 10 10 5
5 10 io 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
5 10 10 IO 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 IO 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 io 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 5 5 : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 z 5 5
Larger hot lesion in upper FIG. 1. Digital phantom with three “lesions” and otherwise uniform background. left region is designated Lesion 1, other hot lesion in upper right portion is Lesion 2, and the cold lesion in the lower portion is Lesion 3. appreciated by comparing the popular &in. line spacing and similarly sized column spacing to the l-mm spacing factors used in the work of NAGAI et al.(l) in which an 11 x 7 cm thyroid scan was focused. The computer time for the focusing process varies directly with the square of the number of data points per unit area in the data and the collimator function matrices. When data values are collected at $-in. line spacing and similar column spacing, an 11 x 7 cm area can be focused by the computer facility used in this study (IBM System 360/75) This in approximately l-5 set per iteration. computer time costs $0.15, which is not prohibitive. Thyroid scans averaging 5 x 7 in. can be focused in 4.3 set ($0.43) per iteration. Thus the reduction in computer time by using +-in. spacing brings scan processing costs to a reasonable level. The digital phantom shown in Fig. 1 was constructed to evaluate the ability of this iterative technique to produce a focused matrix more closely approximating the original distribution of radioactivity. A similar phantom with numbers three times as large was analyzed, with results similar to those discussed below.
The collimator response function used for the simulation and the focusing process is a matrix that is 7 rows by 5 columns. Therefore, the lesions in the phantom are separated enough that they do not interact when the collimator effects are superimposed. Figure 2 shows a simulated scan of this phantom after smoothing of statistical fluctuations, and Fig. 3 shows a focused version of this scan, When using this digital phantom-scan simulation method of evaluation, the manner in which the actual effects of a computer technique are compared to the desired effects should be chosen according to the nature of the technique being studied and its particular effects which are to be analyzed. The presentation of results here represents only one of many ways of making this comparison. The effect of the iterative procedure can be evaluated by visual inspection of Figs. l-3 and by consideration of Table 1. The table lists and compares averages of counts contained in corresponding portions of the three matrices. In addition the table presents ratios of “central average” to “border average” which indicate the degree of contrast between a lesion and the surrounding normal area. It is evident that this
33
Computerfocusing of area scans SIMULATED 322345 2124577755005677’38~7 335661 4 6 7 7 4
SCAN
AFTER
sH001H1NG
,55,55~“5”‘b55~~~~~~~~~~~~ 7
8
9
8
9
9
9
9 10 11 12 10 9 6 10 11 11 10 9 10 9 11 12 11 9 10 8 8101010 11 9 9 10 10 11
9
8
11
5 6 8 10 8 9 9 6 6 7 10 9 8 8 5 7 8 11 12 13 9 5 6 9 10 10 12 9 8 10 11 5 5 7 a 7 9 0 9 4 5 b 5 5 7 8 B 910 5 6 7 10 11 10 11 5 8 a 9 9 9 12 8 10 10 5 8 710 7 8 10 5 b 6 9 57776798 765b718 4 b 6 bb55665571544 4 5 6 0 44b4546
Fro. 2. Matrix
87 119 11 10 128 107
9a
9 9 12
9 e
8 10 11
11
11
9 9 8
11
8
7
110
9
8 8
7 9
8 9
7 6
9
7
9 10
8 9
8 10 12 13 10 10 9 8 10 10 12 10 10 10 9 910 9 910 8 9 9
B
7
8
9
10
9
8
910 8
11
8615*44
000797768
697777
11 11 14 15 12 12 11 10
8101010 B 9 .5 9 9 10 10 8 9 7 10 9 10 11 9 9 8 7 7 8 8979887991Q99 7 9 9 706677656
910
9 8
9
01010
8 9
8
6
11
910 810 810 11 91010 9 0 11 11 10 8 9 8 9 91112 9 9 9 9 10 9 10 10 10 81010 81010 8 7 9 810 910 0 9
9 9
76699990817 8 9 a
9 9 8 10 10 810 9
9
9 9 910
9
9
91010
910099076 9 9 9 118788755
9
0
6
5
5
4 5
7 B
6 6
6
5
3
6 7 6 7
6 5 6 6
4 5 5
8
6
5
a
5
5
6
55018b7675644
5445577056665b54 3
554555444444
obtained
5
6
6
55455544
after type I smoothing (see fOOtnOte, next page) of computer lated scan of digital phantom in Fig. I.
9 11 11 10 11
9
86 108 129 10 12 11 10 109 109 11 10 11 10
9 10 11 12 11 11 9 10 11 12 10 10 10 9 a 9 9 9 10 9 3 8 8 9 10 9
8 8 8 9
8 8 8 8
8 8 5 8
9 9 9 9:
simu-
10 9 lo 9 10 10 lo 10 10 10 10 10 10 10 10 10
9 9 9 9
8 7 7 7
FIG. 3. Output from iterative focusing program(') applied to scan of Fig. 2. This matrix was produced by the sixth iteration, after which no further consistent decrease in the criterion value was observed. contrast is improved somewhat by the iterative method. However, the data suggest that the focused scan differs significantly from the original distribution in the digital phantom. Whereas the ratio of lesion to surrounding area is 1 to 5 for the cold spot, the same ratio for the focused 3
result is greater than 1 to 2. Similarly for the larger hot lesion the focusing technique improved the ratio from 3.4 to 4-0, but 4.0 falls very short of the 10.4 present in the ideal answer and represents only small improvement when compared to the ratio for the smoothed, unfocused matrix.
34
K. A. Krackow, M. L. DutlGan and R. J. Gorten TABLEI Average of counts in lesion
Average of counts in central portion of lesion
Average of counts in lesion border
45.4 40.0 2.3
104.0 78.3 -
10-o 10.0 10-o
10.4 7.8 0.2
35.3 27.9 6.8
55.8 34.7
16-5 16.6 8.4
3.4 2-l O-8*
38.0 30.3 6.1
62-O 36.3
15.4 17.1 8.3
4.0 2.1 o-7*
Ratio
central average border average
Digital phantom Lesion 1 Lesion 2 Lesion 3 Smoothed scan Lesion 1 Lesion 2 Lesion 3 Focused scan Lesion 1 Lesion 2 Lesion 3
* This number is the ratio of average counts in lesion to average counts in border.
In order to eliminate meaningless, oscillating results, the originators of the iterative method suggest smoothing the matrix p2 before proceeding to the (k + I)th iteration. Present analysis indicates however that the selection of the smoothing technique has very important bearing on the output of the iterative procedure. Two different smoothing techniques designated here I and II* were compared. Use of type I smoothing frequently produced matrices flh with negative numbers. When the greater degree of smoothing characteristic of II was employed, this problem of negative numbers did not arise. A different and unexpected result related to smoothing occurred when scan data were subjected to the iterative technique. Regardless of whether type I or type II smoothing was employed, the smoothed scan satisfied the criterion mentioned above for being a suitably * In type I smoothing a value in the scan matrix is compared to the mean of its eight nearest neighbors. If the value does not differ from this mean by more than one standard deviation (s.d.), it is left unchanged. If it differs by more than one s.d., the value is replaced by the mean of its neighbors. In type II smoothing the same comparison is made, and the same change is made if the two numbers differ by more than one s.d. If their difference is less than one s.d., the value is replaced by 4 x (original value + mean of the neighbors).
focused result. Because one seeks a better focused image than the smoothed data itself, this finding suggests that re-evaluation of this specific criterion is in order. Furthermore, examination of Figs. l-3 and of Table 1 demonstrates that a matrix, e.g. the smoothed scan, meeting this specific criterion may be much “farther” from the ideal answer than the criterion would suggest. These findings raise some basic questions re arding the decision as to whether a matrix j , a computed estimate of an unknown object matrix F, can be taken as a good approximation to F. In order to discuss some of these points concisely, it is necessary to introduce some mathematical notation. If A is a matrix we may use IAl to denote a matrix norm, i.e. a measure of the magnitudeof the matrix A. IfA = P - Q, where P and Q are matrices, then 1P - Q 1 = IA 1relates the magnitude of the difference between the matrices P and Q. For F a matrix representing the distribution of activity in an object, let T be the operation of scanning such that G, the scan matrix, is G = T(F). The criterion established for the iterative procedure(l) is such that F, is accepted as a suitable approximation to F if IG, - Cl is sufficiently small, where G, = T(FJ and G = T(F). As others”) have stated, when 1G, - GI approaches zero, so does IFn - F(. However, until IG,, - GI = 0,
35
Computerfocusing of area scans
i.e. until G, = G, it is desirable to know what the magnitude of IG, - GI implies about the size of I#, - FI, for it is this latter expression, lie, F1, which is of most significance. It relates how close the computed object is to the actual object. Obviously the problem in knowing lfl, - FI is that F is still undetermined. It has been mentioned that the iterative technique is actually a method of solving a system of linear equations. For certain systems of linear equations it is well known(5) that two matrices corresponding to the scan image, may be extremely close together while the matrices from which they are derived, i.e. their corresponding “object matrices”, are very far apart. Although the scan focusing problem is not so extreme as the example cited,c5) appreciation of this problem is mandatory. In fact, specific knowledge of the quantitative implication of a given IG, - GI for the corresponding lfln - F( is desired, and for this purpose the reader is referred to a discussion A complete treatment of the by hNCZOSc6). problem of establishing the suitability of a
focused result, i.e. that fi is a good estimate of F, is not intended here. However, the point should be clear from this analysis by computer simulation that there are significant inadequacies in relying on the similarity of images to infer the same degree of similarity between objects. Acknowledgements-The authors gratefully acknowledge the support given this project by the WamerLambert Research Institute, Morris Plains, New fe;;eand by the Veterans Administration Research
REFERENCES 1. NACAI T., IINUMA T. A. and KODA S. J. nucl. Med.
9,507 (1969). 2. BROWN D. W. and KIRCH D. L. J. nucl. Med.
3.
11,
304 (1970). NACAI T., FUKUDA N. and IINUMA
Med.
T. A. J. nucl.
10,209 (1969).
4. KRACKOW K. A. and GORTEN R. J. J. nucl. Med. 11,523 (1970). 5. J.,ANCZOSC. Applied AnalyJis, p. 149. Prentice-Hall, Englewood Cliffs, N. J. (1956). 6. Ibid, p. 167.
Journal
of Nuclear
Medicine
and Biology,
1973, Vol.
1, pp. 29-35.
Pergamon
Press.
Printed
in Northern
Ireland
Use of Digital Phantoms and a Simulation of the Scanning Process to Evaluate Techniques of Computer Focusing of Area Scans K. A. KRACKOW,
M . L. DUNCAN and R. J. GORTEN*
Department of Radiology (Nuclear Medicine Division) University Medical Center and Veterans Administration U.S.A. (Received 12 December 1971;
and Department of Medicine, Duke Hospital, Durham, North Carolina,
in revisedform 21 February 1972)
The usefulness of digital phantoms and computer simulation of the scanning process to evaluate computer methods for improving the presentation of scan data is demonstrated. Knowledge of the exact radioactivity distribution, i.e. the digital phantom, producing a given scan allows effective evaluation of computer techniques designed to improve that scan. As an example, the iterative focusing technique of Nagai et al. is analyzed in detail. Long computation time, previously considered to be the most significant limitation of this technique, is less limiting with ordinary line and column spacing factors. This analysis implicates instead other limiting The actual degree of focusing or deconvolution achieved, factors as being more significant. the sensitivity of the method to smoothing techniques, and the stated criterion for a satisfactorily focused result may represent more important limitations than computer time requirements. The criterion for satisfactory solution by a deconvolution technique is basic to all such methods. This work with digital phantoms and scan simulation has indicated possible inadequacies of relying on the similarity of images to infer the same degree of similarity between objects. Work is cited which may permit an investigator to know quantitatively the limitations of such an inference. This application of computer simulation of area scans from digital phantoms suggests the use of similar procedures first to analyze and then to aid in the modification and improvement of digital computer techniques of scan analysis. L’EMPLOI DES DE BALAYAGE
PHANTOMES DIGITAUX ET UNE SIMULATION DU PROCEDE POUR EVALUER LES METHODES DE MISE AU POINT PAR CALCULATEUR DES BALA YAGES DE SUPERFICIE On dtmontre l’utilitt des phantomes digitaux et de la simulation par calculateur du procede de balayage pour Cvaluer les methodes a calculateur pour ameliorer la presentation des don&es de balayage. La connaissance de la distribution exacte de la radio-activitt, c-a-d. le phantome digital, qui produit un balayage d onnt permet l’evaluation effective des m&odes a Comme exemple, on analyse en detail la calculateur destinees a amtliorer ce balayage. methode iterative de mise au point de Nagai et al. Un long temps de calcul, autrefois consider&e la limitation la plus importante de cette mtthode, donne moins de limitation avec les facteurs Cette analyse implique d’autres limitations a ordinaires d’espace de ligne et de colonne. Le degre actuel de mise au point ou de d&convolution sa place, comme &ant plus signifiantes. ache&, la sensibilite de la methode aux techniques de lissage, et le dit crittre pour un resultat bien mis a point peut presenter de plus graves limitations que les besoins du temps de calcul. Le critere pour une solution acceptable par une technique de deconvolution est a la base de toutesmtthodespareilles. Cet ouvrage avec les phantomes digitaux et la simulation du balayage
* Present address: 77550.
Division
of Nuclear
Medicine,
University 29
of Texas Medical
Branch,
Galveston,
Texas
30
K. A. Krakow,
M. L. Duncan and R. J. Gorten
a indique des insufhsances possibles de se fier a la similaritt d’images pour dtduire le mtme degre de similaritt entre les objets. On cite du travail qui laisserait savoir quantitativement a un rechercheur les limitations dune telle deduction. Cette application de la simulation par calculateur des balayages de superficie de phantbmes digitaux mtne I’idte a l’emploi de proctdes pareils dabord a analyser, ensuite a aider a modifier et a ameliorer les techniques a calculateur digital d’analyse a balayage. HPHMEHEHHE HM@POBbIX (DAHTOMOB H MOjJEJIJIBPOBAHBE HPOHECCA PA3BEPTHH AJIH OHEHKH METOAOB QOKYCIPOBKB PA3BEPTHM HO HOBEPXHOCTB HA BbIcIHCJIBTEJIbHOH MAIIIBHE nOKa3aHa nOJIe3HOCTb I@lpOBbIX @aHTOMOB H MOAeJIJIIlpOBaHIlR npOqeCCa pa3BepTKEi Ha KoMnbIoTepe,~T06bIO~eHElTbMeTO~bI~JIRy~y~~eHHRUpe~CT~B~eHEl~pe8y~bTaTOBpa8BepTKU. @aKToMa, namlqero 3namie TOYHOrO pacnpefienenna PaJJHOaKTEIBHOCTEI, T.e. I&IPOBO~O (PaKTHUeCKyIO pa3BepTKy, nO8BOJIEiT npUMeHliTb 8@$eKTMBHOe BbWWJIeHEle IIpH OJJeHKe MeTOAOB yJlyWIleHMR TOB pa8BepTKEl npM HOMOWU BblWiCJIWTe~bHOfi MaIlILIHbI. HanpnMep, HTepaqnoHKoro MeTona @OKYCHPOBKH Harall II up. &nrrenbnoe Aan no~po6~bl~ aHam BbIWfCJIPiTeJIbHOe BpeMff, 9TO paHbIIIe CYEITaIOT CaMbIM 8HElWJTeJIbHbIM OrpaHHYeHHeM 8TOrO MeTOna, OKa3bIBaeTCH OrpaHWIeHEleM B MeHbIIIei CTeneHEI B CJIyWe 06bIKHOBeHHbIX KO844iW BMecTe Toro ~ambdi aHam BBOAHT npyrne QHeHTOB 3anOJIHeHIIFI JIllHId II CTOJI~OB. OrpaHHWBaIOIIJHe YCJIOBHR KBK 6onee 3HaW%TeZbHbIe. npEl 8TOM B03MOPKHO OTMeTHTb $%lKTWIeCKH nOJlyqeHHaR CTeneHb +OKyCHpOBKM IUIH AeKOHBOJlEOlJMH, YyBCTBkiTeJIbHOCTb K MeTOAaM CrJlaWiBaHHH,EI.yKa3aHHdi KpHTepHti AJlII yAOBneTBOpHTeJIbH0 @OKyCHpOBaHHOrO co6ot 6onee 3HawTenbHbIe OrpaHwIeHHR, 9eM pe8yJIbTaTa BOBMOH(H0 IIpeACTaBJIHeT Tpe6OBaHHHB peMeHtI BbIWlCJIHTeJIbHOti MaJlIHHM. HpHTepd JQ'IR yAOBJIeTBOpHTeJIbHOr0 peIIIeHIUl HeKOTOpNM MeTOAOM AeKOHBOJIIOqEfI4 JIe?KllTBOCHOBeBCeXTaKHXMeTOAOB. Pa6oTaCqa@pOBarMa~aHTOMaMn&iMO~em~pOBaH~eM pa3BepTKIl nOKameT B08MOPKHbIe HeAOCTaTKH nOJIOH(HTbCH HanoAo6He H806pameK&,qTo6bI CAeJIaTb BJ~BOA 0 TOti CaMOi H(e CTelIeHK IIOAO6HH MemAy o6aeKTam. OTMeqeHa pa6oTa, KOTOpaHBO3MO?KHO nO3BOJIHT 3HaTb KOJlH9eCTBeHHble OrpaHWfeHHRTaKOrO BbIBOAa. 3T0 IlpHMeHeHHe KOMnIOTepH30BaHHOrO MOAeiWIKpOBaHHH pa8BepTKEi n0 nOBepXHOCTM OT Ql@pOBbIX @aHTOMOB rOBOpHT 0 nOJIe3HOCTII TaKUX MeTOAOB BO-nepBbIX aHaJII43MpOBaTb M nOTOM IIOMOraTb MOAH@KaIJHl¶ H yJIy=lIIIeHlWJ MeTOAHKIl aHaJIki3a pa8BepTKki npH nOMOI4H qll@pOBOti BbIWICJIBTeJIH.
VERWENDUNG VON DIGITALEN PHANTOMEN UND EINE NACHBILDUNG DES ABTASTVORGANGS ZUR BEWERTUNG VON RECHENMASCHINENVERFAHREN FUR DAS FOKUSSIEREN VON FLACHENABTASTUNGEN Die Brauchbarkeit von digitalen Phantomen und einer Rechenmaschinennachbildung des Abtastvorgangs zur Bewertung von Rechenmaschinenverfahren zur Verbesserung der Darbietung von Abtastdaten wird demonstriert. Die Kenntnis der genauen Radioaktivititsverteilung,, d.h. ein digitales Phantom, fiir eine gegebene Abtastung ermiiglicht eine wirksame Bewertung der zur Verbesserung dieser Abtastung ersonnenen Rechenmaschinenverfahren. Als Beispiel wird das iterative Fokussierungs-Verfahren von Nagai et al. ausfuhrlich analysiert. Lange Rechenzeiten, die friiher als die wichtigste Beschrankung dieses Verfahrens angesehen wurden, sind mit den tiblichen Zeilen-und Spalten-Abstandsfaktoren weniger als Grenzen anzusehen. Diese Analyse zieht andere Faktoren als ausschlagebender an. Der tatsachlich erzielte Fokussierungs- oder Dekonvolutionsgrad, die Empfindlichkeit des Verfahrens fur Glattungsmethoden und das angegebene Kriterium fur ein zufriedenstellend fokussiertes Ergebnis ktjnnen wichtigere Einschrankungen als die erforderliche Rechanmaschinenzeit darstellen. Das Kriterium fur eine zufriedenstellende Liisung durch em Dekonvolutionsverfahren ist fur alle diese Verfahren die Grundlage. Dieses Arbeiten mit digitalen Phantomen und Abtastnachbildung hat miigliche Unzulinglichkeiten aufgezeigt, falls man sich auf die Ahnlichkeit der Bilder zum Verleihen desselben Grads an Ahnlichkeit zwischen Objekten verlisst. Arbeiten werden angefiihrt, aus denen ein Forscher quantitativ die Beschrlinkungen einer solchen Folgerung ersehen kann. Die Verwendung einer Rechenmaschinennachbildung der Fllchenabtastungen von digitalen Phantomen regt zur Benutzung Ihnlicher Arbeitsweisen an, zunlchst zur Analyse und dann zur Hilfe beider Abanderung und Verbesserung von digitalen Rechenmaschinenverfahren zur Abtastanalyse.
31
Computer f0cu.rin.g of area scans
THE CLINICAL usefulness
of organ scans can be significantly improved by increasing the similarity between tracer distribution in the organ scanned and data presented for evaluation. The unavoidable discrepancies which exist with present scanning procedures are largely the result of collimator characteristics and the random nature ofradioactivity. Several methods involving digital computers have been developed for the purpose of reducing these discrepancies.(1-3) In order to appreciate fully the capabilities and limitations of these procedures, it is necessary to evaluate their effects objectively. Because such computer techniques operate on scan data in digital form, their evaluation, as shown here for the iterative method of NAGAI et al.(r), can easily be performed using digital phantoms and computer simulated scans.(4) The particular advantages of this type of evaluation derive from knowing exactly the which was used to “tracer distribution” produce the simulated scan.
DIGITAL
PHANTOMS SIMULATION
AND
SCAN
To understand the process of scan simulation, one can first consider a two dimensional array of numbers F = (&) where i = 1, . . . , h and w ,which is the digital phantom and ‘-1 .I--,***, represents the activity distribution in the object whose scan is to be simulated. A second array K = (k,) with i = -s, . . . , 0, . . . , +s and +t describes the collimator j= --t ,..., 0 )..., response characteristics for the scanning system being simulated. By aligning k,, with some fi, of F, the contribution of fi+,,,,+,, to counts observed over position (i,j) of the object matrix F may be obtained by multiplying fi+.+m ,j+n by
k,,.
= 2 i &+.+zl,,+ak,,, then the image p=-*p=--f array or simulated scan is the matrix G = (gij) i=l,..., hj=l,..., w, where gij is a number selected randomly from a collection of numbers with an essentially Poisson distribution about the mean mij. The array F is constructed arbitrarily to represent the distribution of radioactivity in the object whose scan is to be simulated. The collimator array K may be constructed as previously described(a) or in some other manner, e.g. scanning of a point or line source, which Ifmij
produces an array of values consistent with the definition of K. By calculating the mij from the elements of K and F, the effects of the collimator response pattern are simulated. An mij represents the average activity “seen” by the collifor mator when positioned above fij, allowing contributions from nearby elements of F. Selection of a random number as described@) simulates the stochastic nature of observations of radioactive events. Therefore, the array G is analogous to a scan from a source with radioactivity distribution as in F. It is the matrix G which is subjected to the computer method whose effects are to be analyzed.
ITERATIVE
TECHNIQUE
The iterative technique evaluated here is a method for determining the object matrix F or some approximation to it p, given the observed matrix G and the collimator matrix K. As described in the definition of mij above, disregarding the statistical nature of radioactivity, the elements of the scan matrix G are linear combinations of elements in F with coefficients from K. The iterative technique is actually a method of making successive approximations to the solution of a set of linear equations. Given G and K, the method produces a sequence of matrices {pn}, n = 1,2, . . . , which are intended to approximate the matrix Fwhen n is sufficiently large. Let G, = (g$)) be the matrix obtained from p,, by applying K just as the mi3 were obtained from F. As described elsewhere,(l) the iteration procedure is stopped, and r’, is accepted as a suitable approximation to F when the &) differ from the observed gij by less, on the average, than one standard deviation. The initial application of this method to the improvement of area scans suggests that the principal limitation of this computer focusing procedure is the requirement of inordinate amounts of computer time.(ls3)
ITERATIVE FOCUSING TECHNIQUE EVALUATED BY COMPUTER SIMULATION When application of the iterative focusing method was undertaken in this laboratory, it became readily apparent that computation cost would not be so limiting. This conclusion can be
K. A. Krackow,
32
M. L. Duncan and R. J. Gorten
DIGITAL 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
555555555 10 10 10 10 10 10
10 10
10 10
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 555555555
10 10
10 10 10 10 10 10 10 10
10 10
10 10
10 10 10 3 10 2 10 sl 1 10 10 10 10 10 10 10 10
10 10
io 4 3 2 10 10 10 io
10 10 10
10 10 io 10 20 10 20 10 Y 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 io 3 10 2 10 10 10 10 10 10 10 -~ 10 __ 10 10
5 10 10 10 10 10 IO 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
PHANTOM 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 IO 10 10 10 10 10 10 10 10 10 10 10 5
5 10 io 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
5 10 10 IO 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10
10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 IO 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 10 10 io 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5
5 5 5 : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 z 5 5
Larger hot lesion in upper FIG. 1. Digital phantom with three “lesions” and otherwise uniform background. left region is designated Lesion 1, other hot lesion in upper right portion is Lesion 2, and the cold lesion in the lower portion is Lesion 3. appreciated by comparing the popular &in. line spacing and similarly sized column spacing to the l-mm spacing factors used in the work of NAGAI et al.(l) in which an 11 x 7 cm thyroid scan was focused. The computer time for the focusing process varies directly with the square of the number of data points per unit area in the data and the collimator function matrices. When data values are collected at $-in. line spacing and similar column spacing, an 11 x 7 cm area can be focused by the computer facility used in this study (IBM System 360/75) This in approximately l-5 set per iteration. computer time costs $0.15, which is not prohibitive. Thyroid scans averaging 5 x 7 in. can be focused in 4.3 set ($0.43) per iteration. Thus the reduction in computer time by using +-in. spacing brings scan processing costs to a reasonable level. The digital phantom shown in Fig. 1 was constructed to evaluate the ability of this iterative technique to produce a focused matrix more closely approximating the original distribution of radioactivity. A similar phantom with numbers three times as large was analyzed, with results similar to those discussed below.
The collimator response function used for the simulation and the focusing process is a matrix that is 7 rows by 5 columns. Therefore, the lesions in the phantom are separated enough that they do not interact when the collimator effects are superimposed. Figure 2 shows a simulated scan of this phantom after smoothing of statistical fluctuations, and Fig. 3 shows a focused version of this scan, When using this digital phantom-scan simulation method of evaluation, the manner in which the actual effects of a computer technique are compared to the desired effects should be chosen according to the nature of the technique being studied and its particular effects which are to be analyzed. The presentation of results here represents only one of many ways of making this comparison. The effect of the iterative procedure can be evaluated by visual inspection of Figs. l-3 and by consideration of Table 1. The table lists and compares averages of counts contained in corresponding portions of the three matrices. In addition the table presents ratios of “central average” to “border average” which indicate the degree of contrast between a lesion and the surrounding normal area. It is evident that this
33
Computerfocusing of area scans SIMULATED 322345 2124577755005677’38~7 335661 4 6 7 7 4
SCAN
AFTER
sH001H1NG
,55,55~“5”‘b55~~~~~~~~~~~~ 7
8
9
8
9
9
9
9 10 11 12 10 9 6 10 11 11 10 9 10 9 11 12 11 9 10 8 8101010 11 9 9 10 10 11
9
8
11
5 6 8 10 8 9 9 6 6 7 10 9 8 8 5 7 8 11 12 13 9 5 6 9 10 10 12 9 8 10 11 5 5 7 a 7 9 0 9 4 5 b 5 5 7 8 B 910 5 6 7 10 11 10 11 5 8 a 9 9 9 12 8 10 10 5 8 710 7 8 10 5 b 6 9 57776798 765b718 4 b 6 bb55665571544 4 5 6 0 44b4546
Fro. 2. Matrix
87 119 11 10 128 107
9a
9 9 12
9 e
8 10 11
11
11
9 9 8
11
8
7
110
9
8 8
7 9
8 9
7 6
9
7
9 10
8 9
8 10 12 13 10 10 9 8 10 10 12 10 10 10 9 910 9 910 8 9 9
B
7
8
9
10
9
8
910 8
11
8615*44
000797768
697777
11 11 14 15 12 12 11 10
8101010 B 9 .5 9 9 10 10 8 9 7 10 9 10 11 9 9 8 7 7 8 8979887991Q99 7 9 9 706677656
910
9 8
9
01010
8 9
8
6
11
910 810 810 11 91010 9 0 11 11 10 8 9 8 9 91112 9 9 9 9 10 9 10 10 10 81010 81010 8 7 9 810 910 0 9
9 9
76699990817 8 9 a
9 9 8 10 10 810 9
9
9 9 910
9
9
91010
910099076 9 9 9 118788755
9
0
6
5
5
4 5
7 B
6 6
6
5
3
6 7 6 7
6 5 6 6
4 5 5
8
6
5
a
5
5
6
55018b7675644
5445577056665b54 3
554555444444
obtained
5
6
6
55455544
after type I smoothing (see fOOtnOte, next page) of computer lated scan of digital phantom in Fig. I.
9 11 11 10 11
9
86 108 129 10 12 11 10 109 109 11 10 11 10
9 10 11 12 11 11 9 10 11 12 10 10 10 9 a 9 9 9 10 9 3 8 8 9 10 9
8 8 8 9
8 8 8 8
8 8 5 8
9 9 9 9:
simu-
10 9 lo 9 10 10 lo 10 10 10 10 10 10 10 10 10
9 9 9 9
8 7 7 7
FIG. 3. Output from iterative focusing program(') applied to scan of Fig. 2. This matrix was produced by the sixth iteration, after which no further consistent decrease in the criterion value was observed. contrast is improved somewhat by the iterative method. However, the data suggest that the focused scan differs significantly from the original distribution in the digital phantom. Whereas the ratio of lesion to surrounding area is 1 to 5 for the cold spot, the same ratio for the focused 3
result is greater than 1 to 2. Similarly for the larger hot lesion the focusing technique improved the ratio from 3.4 to 4-0, but 4.0 falls very short of the 10.4 present in the ideal answer and represents only small improvement when compared to the ratio for the smoothed, unfocused matrix.
34
K. A. Krackow, M. L. DutlGan and R. J. Gorten TABLEI Average of counts in lesion
Average of counts in central portion of lesion
Average of counts in lesion border
45.4 40.0 2.3
104.0 78.3 -
10-o 10.0 10-o
10.4 7.8 0.2
35.3 27.9 6.8
55.8 34.7
16-5 16.6 8.4
3.4 2-l O-8*
38.0 30.3 6.1
62-O 36.3
15.4 17.1 8.3
4.0 2.1 o-7*
Ratio
central average border average
Digital phantom Lesion 1 Lesion 2 Lesion 3 Smoothed scan Lesion 1 Lesion 2 Lesion 3 Focused scan Lesion 1 Lesion 2 Lesion 3
* This number is the ratio of average counts in lesion to average counts in border.
In order to eliminate meaningless, oscillating results, the originators of the iterative method suggest smoothing the matrix p2 before proceeding to the (k + I)th iteration. Present analysis indicates however that the selection of the smoothing technique has very important bearing on the output of the iterative procedure. Two different smoothing techniques designated here I and II* were compared. Use of type I smoothing frequently produced matrices flh with negative numbers. When the greater degree of smoothing characteristic of II was employed, this problem of negative numbers did not arise. A different and unexpected result related to smoothing occurred when scan data were subjected to the iterative technique. Regardless of whether type I or type II smoothing was employed, the smoothed scan satisfied the criterion mentioned above for being a suitably * In type I smoothing a value in the scan matrix is compared to the mean of its eight nearest neighbors. If the value does not differ from this mean by more than one standard deviation (s.d.), it is left unchanged. If it differs by more than one s.d., the value is replaced by the mean of its neighbors. In type II smoothing the same comparison is made, and the same change is made if the two numbers differ by more than one s.d. If their difference is less than one s.d., the value is replaced by 4 x (original value + mean of the neighbors).
focused result. Because one seeks a better focused image than the smoothed data itself, this finding suggests that re-evaluation of this specific criterion is in order. Furthermore, examination of Figs. l-3 and of Table 1 demonstrates that a matrix, e.g. the smoothed scan, meeting this specific criterion may be much “farther” from the ideal answer than the criterion would suggest. These findings raise some basic questions re arding the decision as to whether a matrix j , a computed estimate of an unknown object matrix F, can be taken as a good approximation to F. In order to discuss some of these points concisely, it is necessary to introduce some mathematical notation. If A is a matrix we may use IAl to denote a matrix norm, i.e. a measure of the magnitudeof the matrix A. IfA = P - Q, where P and Q are matrices, then 1P - Q 1 = IA 1relates the magnitude of the difference between the matrices P and Q. For F a matrix representing the distribution of activity in an object, let T be the operation of scanning such that G, the scan matrix, is G = T(F). The criterion established for the iterative procedure(l) is such that F, is accepted as a suitable approximation to F if IG, - Cl is sufficiently small, where G, = T(FJ and G = T(F). As others”) have stated, when 1G, - GI approaches zero, so does IFn - F(. However, until IG,, - GI = 0,
35
Computerfocusing of area scans
i.e. until G, = G, it is desirable to know what the magnitude of IG, - GI implies about the size of I#, - FI, for it is this latter expression, lie, F1, which is of most significance. It relates how close the computed object is to the actual object. Obviously the problem in knowing lfl, - FI is that F is still undetermined. It has been mentioned that the iterative technique is actually a method of solving a system of linear equations. For certain systems of linear equations it is well known(5) that two matrices corresponding to the scan image, may be extremely close together while the matrices from which they are derived, i.e. their corresponding “object matrices”, are very far apart. Although the scan focusing problem is not so extreme as the example cited,c5) appreciation of this problem is mandatory. In fact, specific knowledge of the quantitative implication of a given IG, - GI for the corresponding lfln - F( is desired, and for this purpose the reader is referred to a discussion A complete treatment of the by hNCZOSc6). problem of establishing the suitability of a
focused result, i.e. that fi is a good estimate of F, is not intended here. However, the point should be clear from this analysis by computer simulation that there are significant inadequacies in relying on the similarity of images to infer the same degree of similarity between objects. Acknowledgements-The authors gratefully acknowledge the support given this project by the WamerLambert Research Institute, Morris Plains, New fe;;eand by the Veterans Administration Research
REFERENCES 1. NACAI T., IINUMA T. A. and KODA S. J. nucl. Med.
9,507 (1969). 2. BROWN D. W. and KIRCH D. L. J. nucl. Med.
3.
11,
304 (1970). NACAI T., FUKUDA N. and IINUMA
Med.
T. A. J. nucl.
10,209 (1969).
4. KRACKOW K. A. and GORTEN R. J. J. nucl. Med. 11,523 (1970). 5. J.,ANCZOSC. Applied AnalyJis, p. 149. Prentice-Hall, Englewood Cliffs, N. J. (1956). 6. Ibid, p. 167.