Mathematical Methods

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IEEE Transactions on Nuclear Science, Vol. NS-26, No. 2, April 1979

IMAGE RECONSTRUCTION WITH LIMITEDANGLE PROJECTION DATA

*Tamon Inouye

II. MATHEMATICAL THEORY

A new image reconstruction technique forcomputed tomography is described. Projectiondata obtained by a smaller angle rotationless than 180 degrees around the object areused to make the image. The main feature ofthe method is the estimation of missing re-gion in the Fourier transformed domain by ex-trapolation employing analytic continuity.Numerical simulations were carried out usingcomputer generated pattern data. The resultsshow strong effects of the content of noisycomponent on the reconstructed image. Themethod might be, however, practically appliedto some real fields for medical diagnosis.

I. INTRODUCTION

The technique of computed tomography isnow being extensively used for medical diag-nosis. Many types of CT scanners have beendeveloped. However, their image reconstruc-tion processings are carried out by applyinga simple mathematical principle, originallyconceived by Radonl). This image reconstruc-tion principle may be alternatively describedas follows:The one-dimensional Fourier transform of pro-jection data at angle e with spatially fixeddirection gives the component on a line withdirection angle 0 of two-dimensional Fouriertransform of the image distribution.

In order to obtain complete informationfor the image reconstruction, therefore, pro-jection data for the whole angle around theobject are needed. In an ideal case, wherethe mono-energetic X-ray beam is used to ob-tain the projection data, the symmetric char-acteristic of X-ray intensity decay in thematerial in both directions on the same linecan be assured, so that the needed projectionangle range is 0 < e < w. In actual cases,however, the scanning X-ray beam is not mono-energetic. For this reason, different dataare taken at the antipodal position of theobject. After obtaining such distribution,the effective projection data with symmetryare calculated. These data are actually usedfor processing, where the mathematical proce-dures for the image reconstruction are to beapplied to the whole data in the 0 < e < 7rrange.

In this paper, a discussion on the prob-lem of the necessity for a set of projectiondata obtained by a smaller angle rotationless than 180 degrees is proposed. This kindof approach for image reconstruction was firstsuggested by Ramachandran2) and a similartrial was reported by Kowalski3). This typeof image reconstruction seems to have verypractical applications, because a new type ofcomputed tomography is feasible, if a fan-beamscanning method is applicable.

* Toshiba Research & Development Center,Kawasaki 210, Japan

The image pattern is expressed by f(x,y)in the Cartesian coordinate. The Fouriertransformed image F(w,0) in the polar coordi-nate is given by

cf x00

F(wre) = fJ (x,y)exp[-iw(xcos 6_00 _00

+ ysin0)]dxdy . (1)

This function is analytic in the whole (w,0)plane. Therefore, even if a part of data forF(w,0) is missing, it can be estimated byusing the analytic characteristic of thefunction.

Since the object is a real valued dis-tribution, the real part of the Fourier trans-form F(w,0) is symmetric with respect toangle 0 as

ReF(w,0+Te) = ReF(w,0) . (2)

In the same way, the antisymmetry for theimaginary part of F(w,0) is stated as fol-lows:

ImF(w,0+Tr) = -ImF(w,0) . (3)

Accordingly, these Fourier transformed func-tions can be expanded into the Fourier seriesof projection angle e of even and odd orders,respectively.

ReF(w,0) = Z [a (w)cos2mem=0

+ bm(w)sin2me] ,

ImF(w,0) =

(4a)

00

[cn(w)cos(2n+l)6n=0

+ dn(w)sin(2n+l)e] . (4b)

These expansions of F(w,0) with respect to

parameter e converge everywhere in the finite

range of (w,0). Consequently, in the approx-imated expression, summations in Eqs.(4a) and

(4b) are carried out over limited numbers of

terms, as:

M

ReF(w,0) > [am(u)cos2mOm=0

+ bm(w)sin2me],

ImF(w,0)

(5a)

NEI [cn(w)cos(2n+l)en=0

+ dn(w)sin(2n+l)O] . (5b)

If the data for F(w,0) are given for a

certain region of as

e e , (6)

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ABSTRACT

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the coefficients am(w), bm(w), cn(w) and dn(w)in Eqs.(5a) and (5b) for the Fourier expan-sions can be determined. This is accomplishedby the following procedures.

Presuming that the data are given at thefollowing L points on radius w as

01 = Oa, 92, 03, .*.. I L0, 'O

then, it is possible to obtain the coeffi-cients am(w), bm(w), cn(w) and dn(w) by solv-ing

L IML | Z[am(w)cos2m6i + bm(w)sin2mei1

-ReF(w,6i)12 = min., (7a)

L

j=l

M> [cn(w)cos(2n+l)Oj + dn(W)sinn=O

obtained, the image reconstruction was easilycarried out by applying the usual algorithm.In this example, the filtered-back projectionmethod4) was applied to obtain the reconst-ructed image. This procedure is equivalentto calculating the following inverse Fouriertransform of F(w,0):

2Tr

f(x,y) = 1 f{f F(w,e)Iwlexp

[iw(xcosO+ysin) ]dw}d6 .

(9)

Some examples are shown in Figs. 1 to 4.Figure 1 is an original computer generatedpattern that consists of some superimposedcircular distributions.

(2n+1)eJ] - ImF(W,0.)12 = min. (7b)

Solutions of these equations are uniquely ob-tained, when the number of angular samplingpoints L exceeds the number of expansion coef-ficients 2M+1 for Eq.(7a) and 2(N+l) for Eq.(7b), respectively. In actual cases, in or-der to obtain the smooth connection betweenthe given and estimated regions, the follow-ing conditions are added when solving Eq.(7a)and (7b):

ReF(w,0i) =

ImF(w,Oj) =

MZ (am(w)cos2m6im= 0

+ bm(w)sin2m6i) , i=a,r

(8a)NZ fcn(w)cos(2n+l)Ojn= 0

Fig. 1. Computer generated pattern forsimulation calculation. Redius ofobject: 512 units

+ dn(w)sin(2n+l)Gj], j=ac,IO

(8b)

After obtaining coefficients am(w), bm(w), Cn(w) and dn(w) by such procedures, the data forF(w,6) in the missing region of 0 is easilycalculated by substituting the required valueof 6 into Eqs.(5a) and (5b). Once this pro-cess to estimate the missing range of Fouriertransformed function has been set up, theimage reconstruction can be carried out by thetwo-dimensional inverse Fourier transform.

III. NUMERICAL SIMULATIONS

In order to confirm the effectiveness ofthe method, numerical simulations were madeusing a computer. First, a computer generatedsimple pattern was used for the purpose.Using such an artificial pattern, projectiondata were calculated at each sampled projectionangle 8i in the allowed region of 6a <e _< .

Then, the one-dimensional Fourier transformof projection data was calculated to obtain apart of the two-dimensional Fourier transformF(w,0) in the region of 0 Oe0 Further,the missing part of F(w,6) was estimated bythe methods described in the preceding section.

Once the whole projection data had been

In the first simulation, three quarters of thewhole projection angle data between 0 and 3ir/4of 0 were taken into the calculation. Theestimation of missing portion of data for the

0 t/4r, 1/2r 3/4r r

Fig. 2. Comparison between real and esti-mated Fourier transforms of objectdistribution shown in Fig. 1. X =4 5/512, andRe /512 rad/unit length.

:real distribution----- :estimated distribution

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x102

U-

-5-

0 1/4r 1/2, 3/4r r

9

Fig. 3. Comparison between real and esti-mated Fourier transforms of objectdistribution shown in Fig. 1. w =16Xr/512 rad/unit length.

Fig. 4. Reconstructed image of the objectshown in Fig. 1, using 3/4 pro-jection data and estimation.

angle 3Xr/4 <0 <Tr were made by taking theFourier series terms in Eqs.(7a) and (7b) upto M=9, and N=9. In the simulation, the dataobtained by estimation can be compared withthe true distribution. Figure 2 and 3 showthe comparison between the thus obtainedguessed distribution and the real one atcertain points of frequency w. Figure 4 showsthe thus obtained reconstructed image usingthe limited projection data at 0 Ke .<3XT/4.For the same object using different amountsof projection angle data, results of imagereconstruction are shown in Fig. 5 and 6.

The same technique was applied to theprocessing of real CT scanner data. The rawdata were obtained by a TCT-60A TOSHIBA bodyscanner. Since this is a fan-beam scanningmachine, the projection data for the simula-tion calculation was obtained after the fan-to-parallel-beam transformation. Of thesedata the component corresponding to the region0 e-6 K3,a/4 was chosen and the missing region

data were estimated. The reconstructed imageof this real object was obtained as shown inFig. 7. Parameters for this processing werethe same as in the case shown in Fig. 4.

Fig. 5. Reconstructed image, using 7/8 pro-jection data and estimation.

Fig. 6. Reconstructed image, using 5/8 pro-jection data and estimation.

Fig. 7. Reconstructed image of a real ob-ject using 3/4 projection data andestimation.

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The calculation was made by using aTOSBAC-5600 computer. Average processing forthe estimation of missing region data took250 seconds.

IV. DISCUSSION

This method to obtain a reconstructedimage from limited angle projection data isbased on the analytical characteristic of theFourier transformed function of the 6bject.The technique for the estimation of missingregion is actually carried out by continuingthe function. This extrapolation is, there-fore, strongly affected, when the analyticalcharacteristic is disturbed by the fluctuat-ing component. Accordingly, the effective-ness of this method is restricted mainly bythe content of noisy component in the data.The results in the previous section seem toexplain the relationship between the imagequality and the fluctuating component involvedin the data.

The accuracy of the guess concerning themissing region is also affected by the methodused for the estimation. In this example,the analytical characteristic with respect tothe angular parameter 0 in the Fourier trans-formed domain is used. The missing regiondata are more effectively estimated, if thesame characteristic with respect to the radialparameter w in the Fourier transformed domainis jointly taken into the calculation. Thisis accomplished by applying the Fourier-Bessel expansion to the Fourier transformedfunction in Eqs. (4a) and (4b).

This technique seems to be very practi-cally applied, if the data acquisition is madeby using fan-beam scanning. In this case,the information concerning different projec-tion angle is simultaneously obtained simplyby the linear movement of X-ray source anddetector array. Therefore, a quite differenttype of CT scanner might be established usingthis image reconstruction technique.

ACKNOWLEDGEMENT

Thanks are due to Mr. H. Mizutani ofToshiba Research and Development Center forhis help in making numerical simulations.

REFERENCES

1. J. Radon, Berichte uber die Verhandlungender koniglichen sachsischen Gesellschaftder Wissenschaften zu Leipzig, Vol.69,p.262, 1917.

2. G. N. Ramachandran, Reconstruction ofSubstance from Shadow, Proc. Indian Acad.Sci., Vol.73A, p.14, 1971.

3. G. Kowalski, Fast 3-D Scanning SystemsUsing a Limited Tilting Angle, AppliedOptics, Vol.16, No.6, p.1686, June 1977.

4. L. A. Shepp and B. F. Logan, The FourierReconstruction of a Head Section, IEEETrans. Nucl. Sci., Vol.NS-21, No.3, P.21June 1974.

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