To increase the detection efficiency and improve the spatial resolution, a coded-aperture imaging methodis applied to nuclear medicine. The aperture consists of nine pinholes arranged in a square grid. Threekinds of coding are sequentially used to record the same number of projections including parallax and over-lap. The overlapped images are partially separated, and good tomograms of a ring phantom and a humanmyocardium are reconstructed using a modified backprojection algorithm with variable damping factor.
1. Introduction
In gamma-ray imaging systems for medical diagnosis,it is desirable to improve the image quality of distrib-uted gamma-ray sources in a human body. As is wellknown, there are two ways to make the image qualityhigh; one is to increase the total counts of detectedphotons. For this purpose we should extend theimaging time or use more radioisotopes. But neitherof these methods is favorable. Alternatively, the de-tection efficiency should be improved to diminish thestatistical deviations. The other way is to improve thesystem resolution. This can be realized either by usinga high-resolution collimator, which unfortunately haslow sensitivity, or by enlarging the projection images;this is difficult for tomographic systems because manyimages must be projected in different directions.
Coded-aperture imaging uses a specially designedaperture instead of conventional collimators to increasethe detection efficiency and enlarge the projection im-ages. Such apertures, however, might cause someoverlap of the images. So a certain postprocessing willbe necessary. Up to now many studies about variousapertures and decoding methods have been reported.For example, Fresnel zone plates,1 random pinhole
Kenji Ishimatsu is with Hitachi Medical Corporation, KashiwaWorks, 2-1 Shintoyohuta, Kashiwa 277, Japan; T. Matumoto and T.Iinuma are with National Institute of Radiological Sciences, 4-9-1Anagawa, Chiba 260, Japan; the other authors are with Tokyo Insti-tute of Technology, Imaging Science & Engineering Laboratory, 4259Nagatsuta, Midori-ku, Yokohama 227, Japan.
apertures,2 Fourier apertures, 3 time-modulated aper-tures,4 and a pair of coherent coded apertures5-7 havebeen reported and a certain possibility of practical useis shown by them. Among these methods, a multipin-hole system is expected to reconstruct good tomogramsas shown by the seven-pinhole system,8 but this systemdoes not tolerate any overlap between the projectionimages.
With nine pinholes arranged in a square grid, asshown in this paper, a coded-aperture imaging system,which is designed for myocardial diagnosis, can givetomograms of high quality. Three kinds of coding aresequentially used to achieve partial separation of ov-erlapped images, so each projection image can be en-larged to improve the spatial resolution. Good tomo-grams of a phantom and a human myocardium are ob-tained by using a modified backprojection and iterationmethod.
11. Principle
A. General Coding TheoryA schematic diagram of system operation is shown in
Fig. 1, where an object is projected through an array ofpinholes onto a position-sensitive detector. When theaperture consists of n pinholes, it makes n projectionimages denoted by oi (i = 1 n), and this is expressedmathematically by
nR = Ej oi-s
i=1(1)
where si indicates the sign of the ith pinhole; if si = 1,the ith pinhole is open and if si = 0, it is closed.Equation (1) can also be represented in matrix formas
R = 0 t,
where 0, a set of n different images, is defined as
Fig. 1. Schematic diagram of the system operation.
0 = (01,2, * * * , .n),
Detector
Aperture
Object
(3)
and At is the transpose of a matrix I, which is a set ofsigns given by
= (,S 2, * * * . Sn)
Fig. 2. Overlapping of the projection images. 01 and 02 are over-lapped at a point K.
(4)
If the recording step is implemented m times, m dif-ferent records including many overlapped images willbe obtained. Denoting the jth record as rj (j = 1 - m)obtained in the jth recording step, we have
(5)
where 'I 'F-1 becomes a unit matrix multiplied by thegain constant 2:
where tj = (s1j.,s2j,. . . , snj) expresses the sign of eachpinhole set in the ith recording step. The whole set ofm different records R = (r,r 2,.. , rn) is expressedby
R = O T, (6)
where 'I, a coding matrix, is defined in an abbreviatedform as
IV = [si](1 < i _ n~l<J_ m). (7)
Sij denotes the sign of the ith pinhole in the jth re-cording step. In postprocessing to solve Eq. (7) for 0,the inverse matrix of N! is multiplied:
R- T= 04T-1=K-O, (8)where K is an integer that means a gain in detection ef-ficiency, and T-1 is the inverse matrix of 'I. Hence, ifan inverse of the coding matrix exists, we can separatethe overlapped images and improve the detection effi-ciency. For this condition it is well known from thematrix theory that the rank of the coding matrix shouldbe equal to n and also that n corresponding to thenumber of pinholes cannot be larger than m denotingthat of recording steps. General solutions for T andT- are not discussed in detail in this paper. Here wepresent only an example of three pinholes used in thesystem; the coding matrix for three pinholes is
0 1 1
t= 1 0 1
1 1 0
and the inverse matrix is
4-1= ( 1 1 1-1 1 ,
1 -1
(9)
(11)
B. Practical Codes
To reconstruct better tomograms of a human myo-cardium the aperture consists of nine pinholes arrangedin a square grid. As there is much overlap in projectionimages, the codes explained in the previous sectionshould be used to separate the overlapped images fromeach other as if they are recorded by the respectivepinholes individually. In a practical system, however,any one of the projection images seldom overlaps theothers except for the neighbors. Then, it is not neces-sary to make every image independent of the others.This means that we can use the coding for less than ninepinholes.
It is already known that the number of codings for npinholes must be greater than or equal to n times thenumber of recording steps, and this leads to two im-portant points that should be noted; one is about thenumber of recording steps. Since the change of aper-ture should cause some time loss, it is more favorable toreduce the number of recording steps. The other pointis about the propagation of statistical deviations, whichis briefly explained below; codings with the least numberof recording steps and the maximum detection effi-ciency for n pinholes are given by
it, (O diagonal elements,
t1 nondiagonal elements,
and the corresponding matrix for decoding is
(12)
1 (2 - n diagnonal elements, (13)\1 nondiagonal elements.
Now, suppose that 1 and 02 are partially overlappedas shown in Fig. 2, the number of detected photons at
(10) K in the ith recording step ri and its statistical deviationare given by
rj = c1 + c 2 ±\/ (sj = 0, others = 1) (3 -j i n),
01 0 03 0 02 03(14) ri = 04 5 0° r2 = 04 0 °6
08 09 07 08 0
where c is the mean count of o at K to be detectedduring a single recording step, and the last term in Eqs.(14) following the + sign represents the standard de-viation of the mean counts. In the decoding step, forexample, the total count of 01 at K, Ik, is calculated bythe following operation using Eqs. (8) and (13):
The first term of Eq. (16) represents a signal amplifiedby (n - 1), and the second term represent the statisticaldeviation which is increased by some overlap and anunfavorable subtraction implemented in the decodingstep. This subtraction, producing the term (n - 2)2C2,is likely to be dominant in the noise term for a large n,so it will be better to make the number of independentpinholes, not the total number of pinholes, as small aspossible.
Then, nine pinholes are divided into three indepen-dent groups, in which there may be little overlap; imagesidentified by these groups g are given by
g = (01,05,09),
92 = (02,06,07), (17)
g3 = (03,04,08),
and this is realized by using the coding for three pin-holes as shown in the previous section. According tothe practical arrangement of the projection images,these codings are expressed in another form
tj = (SS2,s3, ... S9)
S1
= S4
S7
S2 S3\
S5 S6 .S5 39
(18)
Therefore, Eq. (10) is represented by
2o 0 0
d3 = r3 + ri - = 0 205 0 J0 0 209
are implemented by a computer to separate almost alloverlapped images, which means that most of the im-ages are separated but there may still remain a littleoverlap between one image and its neighbors in a diag-onal direction. Decoded images, thus obtained are tobe processed, and tomograms are reconstructed by themodified backprojection algorithm.
III. Modified Backprojection MethodAfter separating the projection images from each
other, tomograms are reconstructed using a modifiedadditive SIRT algorithm. First, referring to the prin-ciple of the simple backprojection method in brief, somemodification, which is useful in a practical system, isexplained. Now supposing that an object consists ofN layers, where in practice N is determined by theobject size and the depth resolution, the recorded imageR can be expressed as
NR = E Li *Pi
i=1(22)
where Li is the ith layer of the object and P is the pointspread function expressing an ideal image of a pointsource located within the ith layer. Consequently, Pidiffers from the others only by a slight change of mag-nification in the coordinates. To get Lk (k = 1 - N)from R, image R is first correlated with Pk. The cor-relation means simple backprojection. Then we obtainthe initial estimate of the kth layer:
L = R * Pk,
(19)
where the suffix indicates the order of recording steps;the first recording step is performed with No. 2, No. 6,and No. 7 pinholes closed and others open. The secondis done with No. 1, No. 5, and No. 9 closed, and so on.Thus, three kinds of record are obtained as follows:
(23)
where * denotes the correlation operation and the su-perscript indicates the initial. Then, substituting Eq.(22) into Eq. (23), we have
NL = Z Li * Pi * Pk.
i=1(24)
As the projection images do not overlap each other, thePk autocorrelation must be a delta function within the
where av is a constant called the damping factor to avoidthe divergence of the image data by adjusting the totalcounts of reconstructed images equal to that of projec-tion images.
In a practical system, however, each viewing field ofpinholes may be differently restricted; some points arenot recorded in all projection images, and this easilyproduces serious errors in reconstruction. Therefore,this system uses a variable damping factor, which isbriefly explained by taking a simple case as illustratedin Fig. 3. In this case 03 is recorded in every projectionimage, while 02 is not recorded in r3. Furthermore, 01is recorded only in r1 . With a variable damping factor,the images are reconstructed as
01 = I/al,
02 = (2 + I5)/a2,
Fig. 3. Illustration of object points and their projections. In practicerecords of each pinhole are differently restricted, i.e., 03 is recordedin all projections but 02 is not in r3. Moreover, 01 is recorded only
in r l.
limited reconstruction area. So, Eq. (24) can be writtenas
NL = Lk + 'Li * Pi * Pk. (25)
The second term of Eq. (25) expresses undesirableout-of-focus images of other layers to degrade the imageof the kth layer, so we want to make them as small aspossible. To accomplish this, we use an iterativemethod. Representing the second term by the symbolof ghost g, Eq. (25) is rewritten as
L = Lk + g (26)
The initial estimates L are projected by computer toform a virtual record:
NVO= L * Pi. (27)
A u=i
As the subtracted image,NDO=V0-R= g*Pi,
i= I
(32)
03= ( + I4 + 6)/a3.
Inspection of Eq. (32) shows that Oi (i = 1 - 3) can beadequately reconstructed when ai are given by
a1 = ,
a 2 = 2,
a 3 = 3/,
(33)
where 3 is also determined by the same relationshipbetween the reconstructed images and the records. Inother words, : denotes how many points in the recon-structed layers are projected to a certain point in therecord, and this is also diagrammed in Fig. 4.
Thus, by adopting a variable damping factor, tomo-grams can always be reconstructed well without diver-gence even when the areas recorded by each pinhole arerestricted differently. And it should be noted that thismodification results in expanding the image area of eachlayer, because those objects which are only recorded ina small number of projected images can be recon-structed.
(28)
is the projection image of the ghosts go}, we can getthem by performing the same operation as expressedby Eq. (24). Namely, correlation operation is carriedout:
NG= D * Pk = go + E'g? * Pi * Pk
i=1
=g +gk- (29)
The final step of one iteration cycle is image subtrac-tion,
L =LO-G =Lk -g, (30)
so, if g' is less than g, La will become approximatelyequal to Lk. This iteration method expressed by Eqs.(27)-(30) can be summarized in one equation as
ReconstructedRegion
Fig. 4. Illustration of reconstructed objects and their projections.The reconstructed region is determined by the common view of anumber of pinholes, not all of them. With the variable dampingfactor Ii(i = 1 3) are calculated as 11 = (04 + 05 + 06)/3, 2 = (02+ 03)/2, and I3 = 01. Then they are compared with real records.
Fig. 5. Design of a collimator. The diameter of the pinholes is 7 mm;they are arranged in a quare grid at 50-mm intervals.
IV. Experimental Results
According to the practical coding theory explainedin the previous section, an aperture was made formyocardial diagnosis. As shown in Fig. 5, the apertureconsists of 3 X 3 pinholes arranged in a square grid.The diameter of the pinholes is 7 mm; they are placedat 50-mm intervals. On this aperture there is a slidingshutter moved by a step per motor to achieve threekinds of coding automatically. This collimator isequipped with a large-field gamma camera, HitachiRC1C-1635DL, controlled by an EDR4200 system, andthe detected event signals are sent to a minicomputer,HitaclO-II, to form digital projection images. Thenumber of pixels of one record is 128 X 128, while thatof the reconstructed image is 64 X 64. On a minicom-puter, Eclipse S-130, it takes -5 min in FORTRAN V toreconstruct nine layers with three iterations. Thiscomputing time includes the indispensable calculationof the variable damping factor that should be carriedout for every recorded image.
The upper row of Fig. 6 shows three kinds of recordof a human myocardium filled with 201 T1 viewing fromLA045 (left anterior oblique angle of 450). Theserecords are added to or subtracted from each other ac-cording to Eqs. (21) to give the decoded images in whichthere is a little overlap as shown in the middle row.Then, these decoded images are divided into nine pro-jection images of each pinhole, some of which are alsoshown in the lower row. Flood source data are used forsensitivity correction and determining the view field ofeach pinhole and the overlapped region that might stillremain in this system. In the reconstruction step thoseimage data within the overlapped region equally con-tribute to the neighboring images.
Figure 7 shows reconstructed tomograms of mountedring phantoms filled with 99mTc imitating a humanmyocardium. The total count of three records is-800,000 and the recording time is -3 min. Every ringphantom has a certain cold spot in its shell, and the coldspots are the regions of interest. Each phantom's sizeand figure as well as the mounting order are illustratedin Fig. 8. Tomograms of nine layers are reconstructedas 1-cm slices. The first layer is 10 cm away from theaperture and the last 18 cm. Inspection of this resulttells us that the depth resolution does not seem to be
Fig. 6. Recorded images and decoding process. The upper row,RC1-RC3, are three kinds of recorded image of a human myocardiumfilled with 20 1T1 viewing from LAO45. The middle raw, RD1-RD3,are decoded images of RC1-RC3. Thus, the overlapped images canbe partially separated. The lower row shows some of the separated
records of each pinhole number after sensitivity correction.
Fig. 7. Reconstructed tomograms of a three-layer phantom.
Images on the detector
l . e O b e I
9 9mTC
Fig. 8. Phantom experimental geometry. This phantom consistsof three layers filled with 99mTc.
Fig. 9. Reconstructed tomograms of a human myocardium viewedfrom anterior to posterior.
Fig. 10. Reconstructed tomograms of a human myocardium viewedfrom LAO45.
good, but all the cold spots, including their sizes andpositions, can certainly be recognized.
Figures 9 and 10 show tomograms of a human myo-cardium filled with 201T1 viewed from ANT (anteriorto posterior) and LA045, respectively. Reconstructiondepth is from 10 to 18 cm at 1-cm intervals in both cases.Recording time is 5 min and the number of detectedevents is -3,100,000. Thus, this system can reconstructgood tomograms of a human myocardium, which ismostly due to the overlap in records and the variabledamping factor.
V. Discussion and Conclusion
This paper shows a coded-aperture imaging systemand some good tomograms of a phantom and a humanheart. Although nine pinholes are used, necessary re-cording time can be properly reduced to three, which isfavorable for practical use. As the projection image ofeach pinhole is enlarged, reconstructed images shouldbe improved in quality. Furthermore, the variabledamping factor enables us to reconstruct even suchobjects as are recorded only in a few projections, whichresults in spreading the image area of each layer. Thisalso means that the operation to adjust the object to beat the center of the gamma camera lessens the totalimaging time.
As for the initial estimates, the simple backprojectionwith the variable damping factor can yield imagessimilar to those obtained by the impedance methodproposed in Ref. 8. Time cost for calculation is sub-stantially less by that method than by the impedancemethod, because the latter carries a reciprocation op-eration. Good results ensure that this system has theability to give good tomograms. So, it can be used inmyocardinal diagnosis with an expectation of producinga fine contribution.
The authors would like to thank Y. Tateno, H.Shishido, and S. Kaneko for their fruitful discussionsand considerable cooperation in the diagnosis experi-ment.
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