Analytical Correction Methods for Aperture and AttenuationEffects in SPECT

Akira Ishikawa and Koichi Ogawa

Graduate School of Engineering, Hosei University, Koganei, Tokyo, 184-8584 Japan

SUMMARY

This paper is concerned with the lowering of spatialresolution by γ ray attenuation and the collimator aperture,which are serious factors degrading the quantitative per-formance of SPECT (single-photon emission CT). Thelimit of analytical correction is investigated from the view-point of the aperture angle and noise. The methods consid-ered in this paper are the analytical attenuation correctionproposed by Kudo and Saito and the analytical aperturecorrection based on the stationary phase principle proposedby Lewitt and colleagues. Both of these methods are in-tended to correct projection data affected by attenuation andaperture blur in the Fourier space. Simulation results showthat the correction for the attenuation and the aperture iseffective if the single-sided aperture is up to 1°. It is alsoshown that the statistical noise should be carefully handledin applying the correction. © 2006 Wiley Periodicals, Inc.Electron Comm Jpn Pt 3, 89(7): 42–50, 2006; Publishedonline in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20253

Key words: SPECT; γ ray attenuation; collimatoraperture; analytical correction; Fourier space.

1. Introduction

SPECT (single-photon emission CT) visualizes thefunctional state of internal organs by means of γ rays from

a single-photon emission isotope such as 99mTc. A problemis that the γ rays interact with the atoms composing thehuman body (e.g., by the photoelectric effect, Comptonscattering, and coherent scattering). This lowers the γ raycount (also called the projection data) collected for imagereconstruction. Thus, the γ ray count does not reflect theoriginally existing radioisotope, and a low-frequency dis-tortion component is produced in the reconstructed image.This decay of γ rays is represented as the sum of the decayby the photoelectric effect and the apparent decay by Comp-ton scattering, etc. It is called attenuation of γ rays. Correc-tion for this distortion is indispensable if one is toquantitatively reconstruct SPECT images.

Furthermore, a single-photon emission isotope isused in SPECT, and the collimator must be used in dataacquisition in order to identify the direction of the incomingγ rays [1]. The basic structure of such a collimator is a leadplate several centimeters thick, penetrated by a large num-ber of holes of finite size. Then, the following problemarises.

It is not true that each hole of the collimator passesonly γ rays to the detector in a thin cylinder on the frontside, with the same diameter as the hole: it passes γ raysincident on the detector from a conical region with anaperture of several degrees, having the hole as the vertex.The angle of incidence is determined by the diameter anddepth of the hole. Because of this aperture angle, the spatialresolution is greatly degraded when the distance betweenthe collimator and the γ ray source is increased.

In SPECT, the projection data are acquired fromvarious directions around the human body and the image isreconstructed. Thus, shift-variant blur is produced in the

© 2006 Wiley Periodicals, Inc.

Electronics and Communications in Japan, Part 3, Vol. 89, No. 7, 2006Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J87-D-II, No. 1, January 2004, pp. 44–51

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reconstructed image, with different shapes between thecenter and the periphery. It is very important to correct thiseffect of the collimator aperture in order to improve thespatial resolution of SPECT images.

This study is concerned with a method for analyzingthe above attenuation and aperture problems. The limit ofcorrection and other aspects are investigated by computersimulation. We intend to show clearly the effectiveness ofthe correction method. Consequently, it is assumed thatthere is no scattering of γ rays and the target is a two-dimen-sional uniformly attenuating body.

2. Corrections for Attenuation andAperture

2.1. Correction for attenuation

The methods of attenuation correction can be broadlydivided into iterative methods and analytical methods. Theassumptions regarding the attenuation coefficient distribu-tion in the target object can also be divided into two types:a uniform distribution and a realistic nonuniform distribu-tion.

In the iterative method, the correction is performedby including an attenuation term for the projection data inthe statistical image reconstruction procedure, such maxi-mum-likelihood expectation maximization, or maximum apriori probability expectation maximization. The iterativemethod has the advantage that correction terms for variousphysical phenomena can be flexibly included in the recon-struction formula. On the other hand, iterative computationis required.

The analytical method of attenuation correction, onthe other hand, has the advantage that it is fast, since thecorrection is completed in a single process. After Belliniand colleagues [2] published a paper on the analyticalmethod of attenuation correction, various methods wereproposed [3–5]. Correction methods for a uniformly attenu-ating object are systematically integrated in the papers ofMetz and Pan [6] and Kudo and Saito [7]. This study isbased on the correction formula presented by Kudo andSaito.

The correction is performed by the following steps.For details, see Ref. 7.

(1) exp(µl(X, φ)) is multiplied by p(X, φ), andg(X, φ) is derived.

(2) The Fourier transform is applied to g(X, φ) andG(γ, φ) is obtained.

(3) Gn(γ) is derived from G(γ, φ).(4) The attenuation is corrected by the following

relation to obtain Fn(ρ):

where

(5) The inverse transform is applied to Fn(ρ), and thereconstructed image is obtained.

Table 1 summarizes the notations used in the attenu-ation correction.

2.2. Correction aperture

Methods of correcting the effect of the collimatorinclude a method based on a position-independent deblur-ring filter [8], correction by iterative approach [9–12], anda correction procedure in Fourier transform domain in-tended to achieve position-dependent correction [13–15].In this study, a method of performing attenuation correctionand aperture correction at the same time [15] is used.

The procedure for aperture correction used in thisstudy is as follows.

(2)

(3)

Table 1. Notations in attenuation correction

φ rotation angle of detector

X position of detector in the coordinate afterrotation by φ

p(X, φ) projection data affected by attenuation

l(X, φ) distance from X axis to attenuating object

g(X, φ) normalized projection data

G(γ, φ) Fourier transform of g(X, φ) with respect to X

γ spatial frequency with respect to X

Gn(γ) circular harmonic function of normalizedprojection data

Fn(ρ) polar coordinate representation of originalimage by Fourier series

ρ spatial frequency with respect to variable r ofpolar coordinate

µ attenuation coefficient

γ0 (ρ2 + µ2)12

(1)

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(1) The Fourier transform of the projection dataq(X, φ) with respect to X is taken. The result is expanded asa Fourier series in φ, and G(γ, n) is obtained.

(2) The aperture is corrected by the following rela-tion:

(3) The inverse transform of F(γ, n) is taken, and thecorrected image is obtained.

Table 2 shows the notation used in aperture correc-tion.

2.3. Correction for attenuation and aperture

When the attenuation correction and aperture correc-tion are performed by a single procedure, the aperturecorrection is applied first, followed by the attenuation cor-rection.

2.4. Reduction of effect of Poisson noise

In the simulation taking account of the effect of noise,in order to reduce excessive frequency emphasis due tonoise, the projection data in the frequency space are multi-plied by a Butterworth filter with the following response:

Here k is the order of the Butterworth filter and f0 is thecutoff frequency. In this study, the Butterworth filter is usedin order to reduce the noise emphasis produced in thecorrection process. Thus, it is considered as part of thecorrection process. Consequently, the Butterworth filter isnot applied to the result without correction.

2.5. Collimator model

In this study, the effect of the aperture is modeled asfollows. A collimator with an infinitesimally thin wall isassumed to be used in Fig. 1. The bin size, that is, the

sampling interval of the projection data at the detector, isset as 1. It is assumed that a pixel composing the objectcontributes to the projection data if the center of the pixelis located within the aperture angle of ±θ.

Based on this assumption, when the aperture angle is0°, that is, when the projection data are acquired by usingan ideal collimator, only the data in the region which is theintersection of a strip of width 1 and the image is consid-ered. The fractional contribution of the pixel to the projec-tion data is calculated as follows. The length subtended bythe aperture angle (2L tan θ in Fig. 1) is set as 4σ of thenormal distribution. The area of the normal distribution inthe pixel region under consideration is multiplied by thedata as a weight. Then, the results for the pixels aresummed.

3. Simulation

3.1. Simulation conditions

A Shepp phantom of 128 × 128 pixels [Fig. 2(a)] isused as the original image (γ ray source distribution). Thepixel size is set as 2.5 × 2.5 mm, and the actual size of thephantom is set as 32 × 32 cm. The Shepp phantom isessentially a numerical representation of the head structure.The effect of the aperture is enhanced when the crosssection of the target object is enlarged, that is, when theradius of detector rotation is increased. In this study, the

Table 2. Notations in aperture correction

q(X, φ) projection data affected by aperture

G(γ, n) Fourier series representation of g(X, φ)

H(γ, n) transfer function of collimator

R rotation radius of detector

F(γ, n) Fourier series representation of original image

(4)

Fig. 1. Aperture model.

Fig. 2. Original image (a) and regions of interest (b).

(5)

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effects of attenuation and aperture are enhanced by increas-ing the phantom size so that the correction performance iseasily investigated.

The attenuation coefficient for γ rays (ray decayfactor) is assumed to be uniform within the phantom and tobe 0.15 cm–1. This is the attenuation coefficient of water for140-keV γ rays. The number of projections is set as 256 inthe range of 0 to 360° (uniform angular interval). Theprojection data along a line are composed of 128 samplingpoints. Considering the inherent resolution of widely usedscintillation detectors, the normal distribution function ofFWHM 3.2 mm is applied as convolution to the projectiondata.

The projection data, as affected by the aperture, areconstructed as follows. The radius R of the detector rotationis set as 16 and 20 cm. The aperture angle θ is set as 1° and2°. As regards the Poisson noise, the case without noise andthe cases of acquisition counts of 8 M and 1.6 M areconsidered. The attenuation correction and the aperturecorrection are applied to these projection data. In order toinvestigate the relation between the number of projectiondirections and the correction accuracy, the correction isperformed for data with 256, 128, and 64 projection direc-tions.

When the Fourier transform in the X direction isapplied to the projection data in the correction, 64 samplepoints with a value of 0 are added to both sides of theprojection data. When Gn(γ) is constructed from Fn(ρ) inthe attenuation correction, Lagrange interpolation is ap-plied, and the frequency is determined by combining the 10points preceding and 10 points succeeding the samplepoint.

3.2. Evaluation of corrected image

The corrected image is evaluated by three methods.The first is evaluation in terms of the mean-square error(MSE) between the original image and the target image.The second is evaluation in terms of the mean and standarddeviation in the two regions of interest [ROI1 and ROI2 inFig. 2(b)]. The third is visual evaluation of the imagequality. The number of pixels in the ROI is set as 100.

4. Results and Discussion

First, the attenuation correction is applied to theprojection data affected only by the attenuation. The aper-ture correction is applied to the projection data affected onlyby the aperture. Then, the attenuation and aperture correc-tions are applied to the projection data affected by both theattenuation and the aperture and the results are discussed.

4.1. Effectiveness of attenuation correctionalone

Figure 3 shows the result when the attenuation cor-rection is applied to the projection data (without noise, 8 Mand 1.6 M count) affected only by attenuation. The left sideis the case without attenuation correction. The SPECTvalue is low in the reconstructed image. If the display rangeis set the same as in the image after attenuation correction,it is difficult to observe the details. Consequently, the dis-play range for the SPECT value is set as 0 to 60 in the figure.

The right side is the image after attenuation correc-tion. The display range for this case is 0 to 200. Thecorrection used in this case [7] takes account of noise, andis considered as the best among the correction methodsusing the analytical approach. We actually performed cor-rection using the methods of Refs. 2–7, and the abovemethod is used here because it gave the best results. We seethat the proposed method reduces density distortion with-out excessively emphasizing the noise component, even forprojection data containing noise.

4.2. Effectiveness of aperture correction alone

Figure 4 shows the result when aperture correction isapplied to projection data affected only by the aperture. Wesee from the result without noise that the spatial resolution

Fig. 3. Attenuation correction to the attenuatedprojection data.

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is improved when the aperture angle is 1°, but when theaperture angle is 2°, ringing similar in shape to the outercontour of the Shepp phantom is produced inside the phan-tom.

When noise is present, it is amplified in the correctionprocess. Consequently, the Butterworth filter shown in Eq.

(5) is used to cut off the high-frequency component of theprojection data beforehand. The parameters of the Butter-worth filter were set as k = 5 and f0 = 0.273 for an apertureangle of 1° and k = 5 and f0 = 0.156 for an aperture angleof 2°. These parameter values were set by performingcorrections using various values and evaluating the results

Fig. 4. Aperture correction to the projection data with aperture effect.

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visually so as to determine when the effect of the aperturewas best corrected while the effect of the noise was notmuch emphasized. Specifically, the values were set so thatthe maximum variation in the flat part of the phantomremained within about 30% of the true value.

When the aperture angle was 2°, the blur was greaterthan at an aperture angle of 1°, also necessitating strongerhigh-frequency component emphasis in aperture correc-tion. But as a result the noise was also emphasized. For thisreason, the cutoff frequency of the Butterworth filter was

Fig. 5. Attenuation and aperture correction to the attenuated projection data with aperture effect.

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set lower. Thus, the image after correction had lower spatialresolution than at an aperture angle of 1°.

In this study, the effect of the noise was compared attwo noise levels: 8 M count and 1.6 M count. The cutofffrequency of the Butterworth filter was set the same for thetwo cases. The reason is that the frequency band containingstronger noise does not change when the noise power isvaried. Naturally, when the Butterworth filter is not used,the image reconstructed from the 1.6 M count projectiondata is more degraded than the image reconstructed fromthe 8 M count projection data. In any case, the imagecorrected without using the Butterworth filter was muchworse, and was far from the level suitable for visual evalu-ation. Although the reduction of noise and the improvementof spatial resolution are contradictory requirements, it isevident from Fig. 4 that the image can be improved byadequately setting the cutoff frequency.

4.3. Effectiveness of attenuation and aperturecorrections

Figure 5 shows the result when attenuation and aper-ture corrections are applied to projection data affected byattenuation and the aperture. The display range is the sameas in Fig. 3, namely, 0 to 60 before correction and 0 to 200after correction. The corrected image without noise is al-most the same as the image in Fig. 4. That is, ringing is alsoobserved in the reconstructed image for a projection angleof 2° in the case without noise.

The ringing is of nearly the same magnitude as in thecorrected image for the projection data affected only by theaperture. This implies that the effects of the attenuationcorrection and of the aperture correction can be consideredseparately. It is evident that the image quality of the finallycorrected image is greatly affected by the characteristics ofthe aperture correction. Thus, the effect of attenuation iswell corrected.

Figure 6 shows the density profile of the vertical crosssection at the center of the reconstructed image for the casewithout noise, with an aperture angle of 1° and a rotationradius of 20 cm. It is quantitatively evident from the figurethat the density distortion is almost exactly corrected by theattenuation correction, and that the spatial resolution isimproved by the aperture correction. For projection datacontaining noise, noise elimination was performed by usinga Butterworth filter. The same parameters as in Section 4.2were used. In this case too, almost the same results as inFig. 4 were obtained.

4.4. Number of projection data

Figures 7 to 10 indicate the effect of the number ofprojection data on the correction accuracy. Figures 7 and 8

show the basic statistics for two regions of interest (ROI1and ROI2), for an aperture angle of 1° and a rotation radiusof 16 and 20 cm. Figure 9 shows the MSE and Fig. 10 showsthe density profile at the center of the reconstructed image.

Comparing the means and standard deviations in thecases of 256 and 128 projections, it is evident that thecorrection accuracy differs little. However, the image qual-ity is greatly degraded in the case of 64 projections. Thisindicates that the correction is limited when the number ofprojection data is smaller. When the number of projectionsis decreased, only coarse angular sampling is achieved,preventing improvement of the correction accuracy.

By using the SPECT system with two or three detec-tors, it is generally possible to acquire 120 to 180 projectiondata at a clinical site. Since a satisfactory image is obtainedwith 128 projections, it appears that the proposed methodwill be useful in clinical applications.

Fig. 6. Longitudinal profiles of reconstructed images atthe center (θ = 1°, R = 20 cm). A: attenuation corr. (–),aperture corr. (–), B: attenuation corr. (–), aperture corr.

(+), C: attenuation corr. (+), aperture corr. (–), D:attenuation corr. (+), aperture corr. (+), E: original image.

Fig. 7. Mean and standard deviation in ROI1 (θ = 1°, R = 16, 20 cm).

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5. Conclusions

This paper has considered attenuation and the colli-mator aperture as factors that degrade the quantitative qual-ity of SPECT images. Analytical methods of correction forthese two factors are combined, and the effectiveness of theproposed method in clinical application is investigated. Itis shown that an improvement in spatial resolution can beexpected if the single-sided aperture angle is up to about 1°.It is also clear that the effect of noise should be carefullyhandled in the analytical corrections for attenuation andaperture, and that satisfactory correction is achieved whenas few as 128 projections are used.

REFERENCES

1. Moore SC, Kouris K, Cullum I. Collimator design forsingle photon emission tomography. Eur J Nucl Med1992;19:138–150.

2. Bellini S, Piacentini M, Cafforio C, Rocca F. Com-pensation of tissue absorption in emission tomogra-phy. IEEE Trans Acoust Speech Signal Process1979;27:213–218.

3. Tretiak OJ, Metz CE. The exponential Radon trans-form. SIAM J Appl Math 1980;39:341–354.

4. Tanaka E. Quantitative image reconstruction withweighted backprojection for single photon emissioncomputed tomography. J Comput Assist Tomogr1983;7:692–700.

5. Inouye T, Kose K, Hasegawa A. Image reconstructionalgorithm for single photon emission computed to-mography with uniform attenuation. Phys Med Biol1989;39:299–304.

6. Metz CE, Pan X. A unified analysis of exact methodsof inverting the 2D exponential Radon transform withimplications for noise control in SPECT. IEEE TransMed Imag 1995;14:643–658.

7. Kudo H, Saito T. A systematic organization of ana-lytical image reconstruction in SPECT and the noisepropagation characteristics. Trans IEICE 1996;J79-D-II:977–988.

8. King MA, Glick SJ, Penney BC, Schwinger RB.Interactive visual optimization of SPECT prerecon-struction filtering. J Nucl Med 1987;28:1192–1198.

9. Katsu H, Ogawa K. Iterative correction of collimatoraperture in SPECT. Trans IEICE 1993;J76-D-II:199–205.

10. Ogawa K, Katsu H. Iterative correction method forshift-variant blurring caused by collimator aperturein SPECT. Ann Nucl Med 1996;10:33–40.

11. Tsui BMW, Hu HB, Gilland DR, Gullberg GT. Im-plementation of simultaneous attenuation and detec-

Fig. 10. Longitudinal profiles of reconstructed imagesat the center (θ = 1°, R = 20 cm). Numbers of

projections: 256, 128, and 64.

Fig. 8. Mean and standard deviation in ROI2 (θ = 1°, R = 16, 20 cm).

Fig. 9. Mean square error versus numbers ofprojections.

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tor response correction in SPECT. IEEE Trans NuclSci 1988;35:778–783.

12. Gifford HC, King MA, Well RG, Hawkins WG,Narayanan MV, Pretorius PH. LROC analysis ofdetector-response compensation in SPECT. IEEETrans Med Imag 2000;19:463–473.

13. Ogawa K, Paek S, Nakajima M, Yuta S, Kubo A,Hashimoto S. Correction of collimator aperture usingshift-variant deconvolution filter in gamma camera

CT. SPIE Proc Med Imag II: Image Processing1988;914:699–706.

14. Lewitt RM, Edholm PR, Xia W. Fourier method forcorrection of depth-dependent collimator blurring.SPIE Proc Med Image III: Image Processing1989;1092:232–243.

15. Xia W, Lewitt RM, Edholm PR. Fourier correctionfor spatially variant collimator blurring in SPECT.IEEE Trans Med Imag 1995;14:100–115.

AUTHORS (from left to right)

Akira Ishikawa received a B.S. degree from the Department of Electronic Information, Hosei University, in 2003,completed the M.E. program in 2005, and joined Canon Company. His student research focused on medical image processing.

Koichi Ogawa (member) received a B.S. degree from the Department of Electrical Engineering, Keio University, in 1980,completed the M.E. program in 1982, and became a research associate there. He moved to the Faculty of Engineering, HoseiUniversity, in 1991, and has been a professor since 1998. His research interests are image processing and radiation measurement.He received a Paper Award from the Medical Image Engineering Society in 1989 and an award from the Japan Nuclear MedicineSociety in 1991. He holds a D.Eng. degree, and is a member of IEEE, SNM, the Japan Radiologic Medicine Society, and theJapan Nuclear Medicine Society.

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