Universi of Utah Medical Center ... along t km e ray from the center of the pro'ection bin. .... Med., vol. 28, p. 566, 1987. C. E. Floyd, R. J. Jaszczak, K. L. Greer, et.
IEEE Transactions on Nuclear Science, Vol. 35, No. 1, February 1988
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IMPLEMENTATION O F SIMULTANEOUS AWENUATIPN AND DETECTOR RESPONSE CORRECTION IN SPECT Benjamin M.W. Tsui, Hong-Bin Hu, p v i d R. Gilland, and Grant T. Gullberg Department of Radiology and Curriculum in Biomedical Engineering The University of North Carolina at Chapel Hill CB# 7575,152 MacNider Hall Chapel Hill, NC 27514 'Department of Radiology Universi of Utah Medical Center 50 gorth Medical Drive Salt Lake City, U T 84132
Abstract Simultaneous correction of attenuation and detector response was implemented in Single Photon Emission Computed Tomography (SPECT) image reconstruction. We have developed a ray-driven projector-backprojector which exactly models attenuation in the reconstructed image slice and the spatially variant detector response. The projector-backporojector was used in the iterative maximum likelihood algorithm t o simultaneously correct for nonuniform attenuation and detector response. A computer generated heart-lung phantom was used in simulation studies t o compare the simultaneous correction method with the intrinsic attenuation correction method with a smoothing filter, the intrinsic attenuation correction method with a deconvolution filter, and a modified Chang attenuation correction method which uses nonuniform attenuation distribution. The results demonstrate that the simultaneous correction method using the iterative maximum likelihood algorithm provides more accurate quantitation and superior image quality compared t o the other correction techniques tested. Introduction The major factors which degrade the quality
of SPECT images a r e phaton attenuation, geometric detector response, and scatter radiation. Most attenuation correction methods assume the attenuation coefficient is constant throughout the reconstructed image slice. These methods can b e categorized into the intrinsic correction technique [ 11-[3], algorithms which preprocess the projection data [4,5], and algorithms which postprocess the reconstructed image [6]. Methods which apply nonuniform attenuation correction a r e mainly iterative techniques [7-lo]. Correctin for detector response in SPECT has typically invofved linear filtering or iterative reconstruction methods. Deconvolution filters have been used to compensate for the spatial resolution degradation in SPECT images [ 11-15]. However, since the collimator response function varies with distance from the collimator face, the deconvolution technique does not provide an exact solution t o the problem. The variation in detector response has previously been incorporated in the projector-backprojector of iterative reconstruction algorithms [16] and in a generalized reconstruction algorithm which includes compensation of attenuation, detector *This work was supported Service Grant No. R 0 1 CA 39463.
by
Public Health
response and scatter [17]. However, the later requires a Monte Carlo simulation which is computationally intensive. This paper implemented a correction method which accurately models the attenuation distribution in the reconstructed image slice and the spatially variant geometric detector response. T h e method requires a priori knowled e of the attenuation coefficient distribution w%ch can b e obtained using a transmission C T stud [18,19]. The geometric detector response as a L n c t i o n of distance from the collimator face is stored in a look-up table t o increase processing speed. The attenuation distribution and detector response a r e incorporated in a ray-driven projectorbackprojector [ l o ] used in iterative reconstruction algorithms for simultaneous correction for the two image degrading factors. The correction method was evaluated using a simulated heart-lung phantom having a nonuniform attenuation distribution. The results were compared with conventional correction methods with noise-free and noisy data.
Methods Simultaneous Correction of Attenuation and Detector Response As shown in Figure 1, the detector response was modeled by a fan of multiple projection rays originating from each projection bin. The detector response as a function of distance from the collimator face was calculatedc alon the fan of projection rays and stored in a loo!-up table. The values of the look-up table was normalized such that
a r e the table values a t a distance Lk where along t e ray from the center of the pro'ection bin. T h e ray subtends an angle 8 from the central ray. Assuming symmetric3 detector response function, the table contains only values of the central ray and rays on one of its sides. The table is identical for each projection bin.
km
T o combine the effects of attenuation and detector response in the rojector, each projection bin is a sum o f weighted attenuated integrals along the fan of projection rays. The contribution of each image pixel t o the projection bin is the weighted attenuated integral across the pixel. For example, as shown in Figure 2, the value in projection bin pn is given by
0018-9499/88/0200-0778$01 .OO 0 1988 IEEE
119
(1) intrinsic
attenuation with a smoothing filter,
m ij
correction
method
[2,3]
(2) intrinsic attenuation correction with a deconvolution filter,
method
[2,3]
(3) attenuation correction using a modified Chang algorithm [6] which uses nonuniform attenuation distribution information, and (4) iterative maximum likelihood method with correction for both attenuation and detector response. where P . ' is the pixel value at (i,j) of the reconstru&ed image and p .. is the corresponding attenuation coefficient at the sam'd pixel.
LOOK- UP TABLE poJectlm
(detector response function)
,array
I
I
O
I
2
012
p0)ectial
To evaluate the corrective reconstruction algorithms, a simulated heart-lung phantom was used [22]. The attenuation coefficient and radioactivity distributions of the phantom are shown in Figure 3(a) and 3(b), respectively. The distributions were digitized in 64x64 matrices. The values of the attenuation coefficient range from .036/pixel in the lung to .25/pixel in the bone. Figure 3(c) and 3(d) show the two profiles of the radioactivity distribution in Figure 3(b). The profiles serve as a reference for evaluating the different reconstruction algorithms used in the simulation studies.
I
2
Simulated Phantom
~~
rays
Figure 1. The detector response is modeled by a fan of projection rays originating from each projection bin. projection ray
in k
bin (k+1)
//
reconstructed image pixel ( i , j )
Figure 2. The contribution of each image pixel to a projection bin is the weighted attenuated integral across the pixel.
In this example, three consecutive bins, Le. k, k+ 1 and k+2, in the look-up table tkm fall within the pixel ( i j ) resulting in three separate integrals across the pixel. The summation over all pixels ( i j ) depends on the projection bin n and the ray m in the projection fan. The implementation of the weighted attenuated integrals in equation (2) is similar to that described by Gullberg [ 101. The resultant projector-backprojector is used with iterative reconstruction algorithm for simultaneous correction of attenuation and detector response. Reconstruction Algorithms The corrective projector-backprojector was modeled in the iterative maximum likelihood algorithm for image reconstruction [20,21]. The following reconstruction methods were compared
I
A
B CC)
C
D
cd)
Figure 3. Heart-lung phantom used in the simulation studies. (a) attenuation coefficient distribution; (b) radioactivity distribution; (c) profile through (b) along the 4 5 O diagonal; (d) profile through (b) along the vertical direction in the middle of the simulated distribution. Emission projection data at various viewing angles were simulated from the radioactivity distribution shown in Figure 3(b). A look-up table storing the geometric res onse function of a low energy general purpose colEmator was generated based on the parameters of the collimator design and distance from the collimator face [23]. The radius of rotation was set at 22.5 cm, typical in cardiac SPECT imaging. The bin
id?
783
image quality when compared to other correction methods.
pp. 799-816, 1985.
The main disadvantage of the correction method using the iterative maximum likelihood algorithm is its slow convergence and long processing time. Presently, the calculating software is written in Fortran code and the processing time is about 17 min per iteration with a 64x64 image slice and 128 angles using a DEC micro VAX I1 computer. The calculating time can potentially be shortened substantially by using a fast maximum likelihood algorithm [24] and/or by implementin the software on an array processor or in special processing harcfware.
M. A. Ifing, R. B. Schwinger, P. W. Doherty, et al., Two-dimensional filtering of SPECT ima es using the Metz and Wiener filters," J. Nucf Med., vol. 25, pp. 1234-1240, 1984.
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