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Energy-selective reconstructions in X-ray computerised tomography

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1976 Phys. Med. Biol. 21 733 (http://iopscience.iop.org/0031-9155/21/5/002) View the table of contents for this issue, or go to the journal homepage for more

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You may also be interested in: Principles of computer assisted tomography (CAT) in radiographic and radioisotopic imaging R A Brooks and G Di Chiro Physics of contrast mechanism and averaging effect of linear attenuation coefficients in a computerized transverse axial tomography (CTAT) transmission scanner C M Tsai and Z H Cho Correction for beam hardening in computed tomography G T Herman Optimal CT settings for bone evaluations A Muller, P Ruegsegger and P Seitz Beam hardening in X-ray reconstructive tomography R A Brooks and G Di Chiro K-edge imaging based on photon counting detectors E Roessl and R Proksa Photon-counting spectral computed tomography using silicon strip detectors Hans Bornefalk and Mats Danielsson

PHYS. MED. BIOL.,

1976,

VOL.

21,

NO. 5 ,

733-744.

@ 1976

Energy-selective Reconstructions in X-ray Computerized Tomography ROBERT E. ALVAREZ and ALBERT RTACOVSKI Department of Electrical Engineering, St'anford University, Stanford, CA 94305,

u.s.a.

Received 1 1 September 1975, i n final f o r m 17 February 1976 ABSTRACT.All X-ray computerized tomography systemsthat areavailable or proposed base their reconstructions on measurements that integrate over energy. X-ray tubes produce a broad spectrum of photon energies and a great deal of information can be derived by measuring changes in the transmitted spectrum. We show that for any material, complete energy spectral informationmay be summarized by afew constants which are independent of energy. A technique is presented which uses simple, lowresolution, energy spectrum measurements and conventional computerized tomography techniques to calculate these constants a t every point within a cross-section of an object. For comparable accuracy, patient dose is shown to be approximately the same as thatproduced by conventional systems. Possible uses of energy spectral information for diagnosis are presented.

1. Introduction

Thispaperpresentsthetheory of a technique for obtaining essentially complete energy dependent information in a computerized tomography system by making simple, low-resolution, energy spectrum measurements. The systems currently in use base their reconstructions on measurements of the intensity of an X-raybeam which essentially integrate overits energy spectrum. They attempt to reconst'ruct the linear attenuat'ion coefficient p at a single average energy B and thus ignore the information present in the behaviour of the complete attenuation coefficient function of energy p ( E ) . The theory for the extraction of complete energy dependent information is a method of representing the complete att'enuation presented in two steps. First, coefficient function a t every energy by a small number of constants which are independent of energy will be described. These constants specify the amounts of photoelectricand Compton scatteringinteractionscont'ributing tothe total linear attenuation coefficient. Sext, it will be shown that computerized t'omographytechniquesmay be used to calculate theseconstants a t every point within the cross-section of an object. Diagnostic information may then be derived directly from these const'ant's or they may be used to provide a reconstruction of t'he linear attenuation coefficient at any energy within the diagnostic range. Theutility of energy dependentinformationfordiagnosis is discussed. of informationmaybededuced.Thephotoelectric Essentiallytwokinds coefficient depends strongly on atomic number and thus provides an indication of the composition of the object'. The Compton scatteringcoefficient depends on

7 34

Robert E . Alvarex and Albert Nacovski

electrondensitywhich,in turn, is proportional to mass density for most materials.The use of thisinformation formedicalpurposescanbehighly significant. For example,a lesion of increased attenuation coefficient in existing systems can be due to either increased density or increased average atomicnumberdue to calcification. A system of thistypecandistinguish between these phenomena. However, the ultimate use of t'hese parameters will have to be determined by clinical experience. Preliminary result's are presented for the values of these coefficients for various body structures. These results indicate that energy dependent information should be very useful for diagnosis. The effects of noise and patient dose are considered.Generalexpressions are derived for the errors introduced by counting noise. These expressions are evaluated for a typical system andit is shown that thedose will be comparable to that produced by conventional systems. The effects of scatter and limited detector resolution are discussed. 2. Representation of attenuation coefficients

The attenuation coefficient as a function of energy p ( E ) may be represented by a small number of constants {ai, i = 1 , 2 ! , .,,n} by finding a set of basis functions {f,(E), i = 1 , 2 , . ..,n } such that p ( E ) can be expressed as a linear combination of these functions as given by

a,f,(E) + a2fAE)+ . + a,f,(E). (1) Various functions have been suggested for use as the basis set (Victoreen 1948, =

Grodstein 1967). The ultimate choice of basis functions is empirical. A set must be found which fits experimental data with errors smaller than those introduced by the proposedmeasurementsystem.Wehave been successful infittingexperimental data with functions of the form

where f K N ( E is ) the Klein-Nishina function

]

ln(l+2cu) +_ln(l+2cu)-

++

(1 301) (1 2a)2

(3)

and 01 = E/610.975 keV. This is a particularly conrenient basis set because the two functions have physical meaning. The function 1/E3approximates the gives the energy energy dependence of the photoelectric interaction andfKN(E) dependence of the total cross-section for Compton scattering. The dependence of a, and a2 on physical parameters is given below: a,

M

Kl-Zn, P A

a2 M K,-Z P

A

n

M

4 (5)

Energy Spectral Techniques in Computerized Tomography

735

where K, and K, are constants, p is mass density, A is atomic weight and Z is atomicnumber.The expressionsareonly approximateandindicatethe general behaviour. The usefulness of thesefunctionsinrepresentingattenuation coefficient functions is well known (Hale 1974). Recently, highly accurate measurements of the attenuation coefficients of body materials have become available (Rao and Gregg 1973, Phelps, Hoffman and Ter-Pogossian 1975). These data were approximated by the two functions using a least squares curve fitting routine withexcellentresults. Over the range of 30-200 keV the biggest difference between the calculated and experimental values was less than one per cent with the average error being a few t'enths of a per cent. 3. Incorporation of energy spectral measurements into computerized tomography I n computerized tomography we makemeasurements of line integrals of the function we are reconstructing. X-ray systems fundamentally measure line integrals of the linear attenuation coefficient : Sp(x,y ; E ) ds. This is equivalent to making measurements of line integrals of the coefficients a , and ad. That is, since 1 p ( x ,y; E ) = y) + a g ( x , Y ) f K N ( E ) (6)

then

where

.c

A , = a,@,y)dsand

A,

i

= a,(x, y) ds.

(8) Thereconstruction of a,(%,y) and a,(x,y) requires the measurement of line integrals A , and A, a t every point in the projections of the object. This means that two independent pieces of information must be available a t each point. These may be obtained by making intensity measurements with two different source spectra,

&(Al, A,)

=

J

T S,(E) exp [ - A,/E3- A P f K N ( E )d]E

(10)

where T is the total measurement time. I n these equations S, and S, may be photon number spectra with 11,I, total counts, or S , and S , may be energy spectra with I,, I, total energies. The case of photon number spectra will be considered with similar results obtainable for energy spectra. The spectra S, and S , may be formed in many ways. They could be monoenergetic radiation produced by two different isotope sources, They could be obtained by filtering a single source spectrum by alternatelypassing it through

736

Robert E. Alvarez and Albert Xacovski

two materials with transmissions g,(E) and g P ( E )where S , = S(E)g,(E) and X, = S(E)g,(E). I n a counting detector system one can do single level pulse height analysis where I, represents the counts from 0 to some threshold energy E , and I, the countsabove E,. This is a good model forsemiconductor detectors, but represents an idealization for lower resolution detectors. With these detectors the response function will be the convolution of the idealized rectangular response functionwith the detectorenergyresolution response. Since the energy dependent informationis relatively simple, all that is required is that the detector separate the photons into two distinct energy groups. Thus relatively low resolution detectors are sufficient t o extract the information. In any of these schemes, so long as the Jacobian

J

= det

2-ll/i3A1 211/8A2 2I,/2A1 i312i2A,

is nonzero one can solve the integral eqns (9) and (10) for A, and A,. Once the sets of line integrals {A,) and {A,} areknown for allprojections, the functions a,(x,y) and a,(%,y) may be reconstructed asin aconventional computerized tomography system. Scatter can be a serious practical consideration depending on the geometry of the scanning system. Because of the nonlinear nature of eqns (9) and (lo), it is difficult to predict the effect of adding a relatively constant scattering term. Conventional intensity-only systems must also operate in the presence of scatter. They use nonlinear processing by taking the logarithm of their measurements and excessive scatter can lead t o artifacts in the reconstruction (Stonestrom and Macovski 1975). The successful operation of these systems indicates that scatter may be reduced to negligible levels by the use of proper collimation. 4. Simulation of computerized tomography syst'emwith energy spectral measurements

A computer simulation was performed to show how energy spectral techniques may be used in a computerized tomography system. A test object, shown in fig l , was used. It, included a bone annulusfilled with uniform brain type t,issue except for a lesion at the centre consisting of fat type tissue. Experimentally measured attenuation coefficients were used for the bone (ICRU 1970) and the tissues (Phelps et al. 1975). The values of U , and a, for the brain and fat are: brain a,

=

4792 keV3 cm-l,

fat a, = 2868 keV3 cm-,,

cc2

= 0.1694 cm-l,

a2 = 0.1784 cm-l.

An ideal pulse height analysis measurement was assumed. With an experimentally measured 105 kV, X-ray tube spectrum (Epp andWeiss 1966) shown in fig. 2, the number of counts in the transmitted spectrum below and above 59 keV were calculated a t 160 points across the projection. This was the raw data used in the reconstruction scheme. Scatter is neglected. At each of the 160 points in the projection, the integral eqns (9) and (10) had to be solved. An analytic solution is not possible because the exact analytic

adius

Energy Spectral Techniques in Computerized Tomography

737

Fattype tissue Intensityonlyreconstruction

Braln

7 5cm

Photoelectrlc part

0

1

(cm1

2

3

L

5

6

7

08

1

2

3

L

5

6

7

8

Radtus

Fig. 1. Results of computer simulation: ( a ) experimental corhguration, ( b ) conventional reconstruction of the logarithm of the transmitted intensity, (c) reconstruction of the photoelectric part al(z, y), ( d ) reconstruction of the Compton scattering part a&, y).

forms of the spectra S, and S, are notknown. Instead, thefollowing technique was used to solve the equations. A general form of eqns (9) and (10) with undetermined coefficients was introduced. These coefficients were determined experimentally by measuring the transmission through materials with known procedure. values of A , and A , and then using a least squares curve fitting This method could be used in a practical system.

Energy IkeV)

Fig. 2. Incident spectnun assumed in computer simulation (from Epp and Weiss 1966). 27

738

Robert E. Alvarez and Albert Macovski

A suitable general form with undetermined coefficients may be derived by noting that thetransmission integrals define smoothly varying functions. Thus, for the ranges of values of A , and A , of interest, a power series form may be used lnI, = b , + b l A l + b , A , + b 3 A 1 2 + b 4 A 2 2 + b 5 A l A 2 + b 6 A 1 3 + b , A 2 3(12) and lnl, = c , + c , A , + ~ , A , + ~ ~ A , ~ + c ~ A ~ ~ + c ~ A ~ A (13) ~+c~A~~+c, Once the sets ofcoefficients {b,} and {ci} are determined, eqns (12) and (13) become two simultaneous cubic equations which may be solved numerically using a generalization of the Newton-Raphson method (McCracken and Dorn 1964). The accuracy of eqns (12) and (13) may be demonstrated as follows : The coefficients {bi} and {ci} are determined by calculating I, and I, for various values of A , and A,. The integrals Il and I, are then calculated for arbitrary known values of A , and A , and eqns (12) and (13) are solved for the predicted values A , and A , . A comparison of the actual and calculated values is shown in table 1 . Table 1. Accuracy of solution of integral equations

75 000 75 000 155 000 155 000

1.5

2.7 1.5 039 155 2.7

74 963 75 029 154 959

1.5001 2,6999 1.5001 2.8999

Three reconstructions were carried out. As shown in fig. 1, these are: (1) a conventional reconstructionusing the logarithm of the total numberof photons transmitted; (2) a reconstruction of a,, the photoelectric part and (3) a2, the Compton scattering part. The density of the ‘lesion’ at the centre of the skull was chosen so that at theaverage energy of the transmitted spectrum,63 keV, its attenuation coefficient was equal to that of the surrounding tissue. Thus, in the intensity-only reconstruction it is nearly invisible. Since its attenuation coefficient a t other energies is different from that of the surrounding tissue,it is quite visible inthe Compton and photoelectricreconstruction. A simple convolutionreconstructionalgorithm (Bracewell and Riddle 1967) was used which accounts for the relatively large artifa,ctscaused by discontinuities in the object. As shown intable 1, the line integralsthemselves areaccurate to within a few tenths of a per cent. Also evident in the intensity reconstruction is a large ‘cupping’ in the soft tissue region near the skull. This is a spectral-shift artifact (Macovski, Alvarez, Chan and Stonestrom 1975) caused by a hardening of the beam near the edges of the skull where there are larger amounts of bone in the path. The hardening causes a shift in average energy which depends on the amount of material in the path andis the physical basis for the nonlinear nature of eqns (12) and (13). Energy spectral analysis provides a general and accurate technique for correcting this important artifact. Intensity-only systems must use water vessels

Energy Spectral Techniques

in Computerized Tomography

739

(McCullough, Baker, Houser and Reese 1974) and computational techniques to attempt tocorrect the artifact. 5. Noise and patient dose considerations

As in other X-ray systems, the accuracy in the measurement of A , and A , is limited by patient dose considerations. The calculation of errors is complicated because two equations must be solved simultaneously to determine the line integrals. Assuming Poisson counting noise is the limitingfactor, the procedureforcalculating thestandard deviation of the measurementsis shown in the Appendix. The results for the standard deviations of A , and A , are given by

and

I n these expressions Il and I, are the numberof counts in measurements1 and 2 as discussed in section 3 and a In I, a In Il m - aA, "

The physical interpretation of theseequationsisquiteinteresting.The denominator of both expressions is essentially a determinant which, by analogy with sets of linear equations, determines the conditioning of eqns (9) and (10). A small determinant implies an ill-conditioned system of equations and thusa A , and A,. Thenumerators of the expressions largeerrorindetermining become smaller as the number of counts increases. It is apparent that an ill-conditioned system requires more photons to give the same accuracy as a better-conditionedsystem.The conditioning is controlled bythe choice of filter functions g,(E) and g,(E). We are studying the problem of the choice of optimum filter functions. The calculation of dose is a complicated problem. To gain a rough estimate of the dose, a configuration of atypicalheadscannermaybeassumed. Assuming acountingdetector using pulse heightanalysis, the errors are determinedmainlybythreshold energy and bodythickness.Thespectrum shown in fig. 2 was used. The total number of incident photons was 5 x 107. The average energy of the incident spectrum is approximately 50 keV. Thus, assuming all photons are stopped within the body, the energy deposited will be approximately 4 erg. Assuming the energy is uniformly distributed within t'he irradiated volume (a cylinder 18 cm in diameter and 1.3 cm thick) and 160 positions at 180 angles are measured, the dose is 3-5 rad. This is comparable to the dose in a conventional scan.

740

Robert E . Alvarez Albert Macovski and

The errors due to counting noise are shown in fig. 3. They are plotted for various threshold energies and body thicknesses. Note that the error in the 0 081

to countlng noise

Errors due

0 06t Oh

0o q

Error In cornpion port

t

O "'i:hBss{ 0

10

20

;;L 30 LO

50 60 7080

90 100

'

0 08,

photoelectric In Errorpart 18, '3,

~~

0

10

20

30

LO

~~

50 60 70 80 90 100

Threshold energy

ikeV1

Fig. 3. Computercalculations of the standard deviations of the estimates of the line integrals A , and A , as a function of threshold energy. The results are calculated for soft tissue thicknesses of 12, 18 and 24 cm.

photoelectric line integral is substantially higher than theerror of the Compton line integral. This is to be expected since the photoelectric integral is mainly determined by low energy photons which are more highly attenuated by the body. 6. Utilization of energy spectral information for diagnosis I n currently used systems of computerized tomography, the resultantimage, except for artifacts, represents the average attenuation coefficient. The diagnostic capabilities of a one-dimensional representation of this type are limited. For example,a lesion exhibiting an increased average attenuation coeecient can be a result of either calcification or increased density. Fig. 4 shows the type of data generated by the proposed system. This represents in vitro samples (Phelps et al. 1975). It is clear that a conventional presentation, representing a projection along an axis nearly parallel to the horizontal axis, will fail to delineate many of the structures. We are beginning an experimental programme to obtain in vivo data. Delineation of calcifications is an example of a significant diagnostic parameter which, given adequate quantification,will allowcomputerized tomography toextendits usefulness into new areas. Much of breast cancer diagnosis depends on mammogramswithvery high spatialresolution to observe the



741

minute calcifications in many of the malignant lesions. Computerized tomography has relatively poor spatial resolution so that the degree of calcification will have to be averaged over larger volume elements. It is therefore important that an accuratemeasurement be made of the density and degree of 5300- Photoelectric component 5200-

:m 51004

50001

L700Comptonscattering component

L

6 0 0 d 9, 0.150 0 155 0 160 0165 0

170

0.175

0 180

02

Fig, 4. Two-dimensionalplot of the information available from energy spectral analysis. The values of the coefficients U, and u2 are calculated from measurements of the attenuation coefficients of body materials at 16 energies in the diagnostic region. Each point represents the (U,,u z )values for a given body material. 1. Clotted blood 6. Meningioma 2. Clotted blood 7. Meningioma 3. Subdural haematoma 8. Medullablastoma 4. Water 9. Astrocytoma 5. Neuroma 10. Human grey matter 11. Human white matter

calcification since these parameters, rather than the spatial distribution, must determine the diagnosis.

A. Macovski would like to acknowledge the support of NIH grant GM 2167 and GM 20984. APPENDIX Analysis of errors in estimation of line integrals Themeasured data are the number of counts I, and 12. For almost any measurement scheme in which the photons aremeasured by different detectors or with pulse height analysis, these counts are independent Poisson random variables with parameters h, and X, given by

S

h,(& 4 ) = T SIP) exp I: - A P - A2fm(E)I

I

h2(.4,,A,) = T S2(Wexp - A1/E3-A2&@)] d E where T is the counting time.

(AI) (A2)

Robert E. Alvarez and Albert Macovski

742

We wish to estimate A , and A , given I, and I, and also to derive expressions for the variance of these estimates. A maximum likelihood estimation procedure may be used (Van Trees 1968). This involves maximizing the probability that I, and I, are measured given that A, and A , assume some values. The probability is

W,,I, I A,>A,) = XlllexpI,!( -X,)

exp ( - X,) I,! Since the logarithmisamonotonically increasing functiononemay maximize the logarithm of the probability function

L

X21s

= lnPT(Il,12~Al,Az) = I,lnh,-X,+I,lnX,-h,-ln(I,!

also

12!). (A4)

This function will be a maximum when

Eqns (A5) and (A6) are a set of homogeneous linearequations quantities [(I,/A,)- l ] and [(12/h2) - l]. The only unique solution is

for the

% 4 - l = 0; & 12 -l = 0

which occurs when the determinant of the coefficients is nonzero as given by

(Note thisis equivalent to saying the Jacobianof the transformationpreviously defined by eqns (9) and (10) is nonzero.) The maximumlikelihood estimation thus involves solving the setof equations :

I1- l = 0 =-I, h,

= X, = T

exp [ - Al/E3- A2fm(E)] dE

-I2 1 = 0 I, = h, = T b , ( E )exp [ - AJE3- A,f,,(E)]

dE.

(A9)

(A10)

X2

The variance of this estimate may be calculated by using the Cramer-Rao bound (Van Trees 1968, p. 79). This may be done since all maximum likelihood estimators have variances which asymptotically approach this bound for large sample size. I n order to reconstruct accurately a large number of counts must be accumulated so the estimator given by eqns (A9) and (A10) is always used in the asymptotic region. The Cramer-Rao bound for the variances of the estimates in this multivariable system is given by


in Computerized Tomogrqhy

743

where J i j are elements of a matrix defined by

The symbol E, denotes expected value. Using the Poisson nature of I, and I2 one can show after some algebra that

so E,[I,] = X, and E,[12] = X,

Dehing

the expressions (A13), (A14)and (A15) become J2, = h, m,:

and J12

+ X, m222

= J21 = ~ , m 1 1 ~ , 2 + ~ 2 m 2 1 ~ 2 2 .

After some algebra one can show that the variances of A , and A , are: 2

- (m1221X2) + (m2z2/X1) - (m11m22 - m12m21)2

Assuming h, z Il and X, z I, and using the power series forms of eqns ( A l ) and (A2) we obtain 4 ( A 1 , A 2 )= e x p ( - b o - b , A , - b 2 A 2 - b , A 1 2 - b 4 A 2 2 - b 6 A l A 2 - b 6 A ~ - b 7 A 2 3 ) (A22)

and h2(A,,A,) = ~ X ~ ( - ~ ~ - ~ , A , - ~ ~ A ~ - C , A , ~ - C ~ A ~ ~ - C ~ A ~ A ~ The variances may be evaluated from these expressions.

(A231

R~STJM~ Reconstruction sQlectivesen Qnergie dans la tomographie h ordinateurs aux rayons X Tous les systbmes de tomographieaux rayons X, traitQe parordination, qui sontsoit disponibles ou propos6s, basent leurs reconstructions sur des relevQs s’intbgrant sur 1’6nergie. Les tubes B rayons X produisent un large Qventail d’Qnergies de photons e t l’on peut deriverbeaucoup d’informations en mesurant les changements dans le spectre transmis. Nous montrons que, pour n’importe quelle matibre, toutes les donnQesspectrales d’bnergie peuvent6trerQsumQes par


744


quelques constantes qui sont independantes de l’energie. On prirsente une technique utilisant des releves de spectresd’energie simples a basse resolution et des techniques de tomographie B ordination usuelles pour calculer ces constantes B tous les points de la section transversale d’un objet. Pour une precision comparable, on montre que la dose appliquee au malade est approximativement la m6me que celle produite par les systemes ordinaires. On presente des utilisationseventuelles de l’information spectrale d’energie pour la diagnose.

ZUSSAMMENFASSUNG Energieselektive Rekonstruktionen in komputerisierter Rontgentomographie Alle komputerisierten Rontgentomographiesysteme, die derzeit zur Verfugungstehen oder vorgeschlagen sind, basieren ihreRekonstruktionen auf uber Energie integrierten Ivlessungen. Rontgenrohren produzieren ein breites Spektrum von Photonenenergie, und aus Messung von Veranderungen in dem ubertragenen Spektrum liisst sich Information in betrachtlichem Masse ableiten. Wir zeigen auf, dass bei jedem Material die vollstandige Energiespektralinforation in einigen wenigen Konstanten ausgedruckt werden kann, die von Energie unabhangig sind. Diese Konstanten konnen fur jeden Punkt innerhalb des Querschnitts eines Objekts mit Hilfe einfacher Energiespektralmessungen mit niedriger duflosung und herkommlichen Techniken der komputerisierten Tomographie berechnet werden. Fur einen Vergleich der Genauigkeit wird gezeigt, dess die Patientendosis in etwa die gleiche ist wie die von herkommlichen Systemen aufgestellte. Die Energiespektralinformation konnte in der Diagnose eine mogliche Verwendung fbden.

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REFERENCES BRACEWELL, R. N., and RIDDLE,A. C., 1967, Astrophysical J . , 150, 427. EPP, E. R., and WEISS, H., 1966, P h y s . X e d . B i o l . , 11, 225. GRODSTEIN,G. W., 1957, U.S. National Bureau of Stds. Circ. 583. HALE,J., 1974, The Fundamentals of Radiological Science (Springfield, Illinois: Charles C. Thomas). ICRU, 1970, RadiationDosimetry: X R a y s GeneratedatPotentials of 5 to l 5 0 k l r , Report No. 17(ICRUPublications, P.O. Box 30165, Washington,DC 20014,

U.S.A.). MCCRACKEN, D. D., and DORN,W. S.,1964, Xumerical Methods and Fortran Programming (New York: J o h n Wiley) pp. 144-145. E. C., BAKER,H. L., HOUSER, 0. W., a n d REESE,D . F., 1974, Radiology, MCCULLOUGH, 111, 709. MACOVSKI,A,, ALVAREZ, R. E., CHAN,J. L.-H., and STONESTROM, J. P., Proc. Conf. on

Image Processing for 2 - 0 and 3-D Reconstructions from Projections, Stanford,C A , MB1-5.

PHELPS, M. E., HOFFMAN, J., E. and TER-POGOSSIAN, M. M., 1975, Radiology, 117, 573. RAO,P. S., a n d GREGG,E. C.,1974, Proc. A A P X Meeting, San Diego, 1973, Abstract, P h y s . M e d . Biol., 19, 231. STONESTROM, J. P., and MACOVSKI,A., 1975, Proc. Conf. on Image Processing f o r 2 - 0 and 3-D Reconstruction from Projections, Stanford, C A , WB3-1. VAN TREES,H. L., 1968, Detection, Estimation and 1Modulation Theory (New York, J o h n Wiley ) .

VICTOREEN, J. A., 1948, J. A p p l . P h y s . , 19, 855.