IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 51, NO. 1, FEBRUARY 2004
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Evaluation of Maximum-Likelihood Position Estimation With Poisson and Gaussian Noise Models in a Small Gamma Camera Yong Hyun Chung, Yong Choi, Member, IEEE, Tae Yong Song, Jin Ho Jung, Gyuseong Cho, Member, IEEE, Yearn Seong Choe, Kyung-Han Lee, Sang Eun Kim, and Byung-Tae Kim
Abstract—It has been reported that maximum-likelihood position estimation (MLPE) algorithms offer advantages of improved spatial resolution and linearity over conventional Anger algorithm in gamma cameras. While the fluctuation of photon measurements is more accurately described by Poisson than Gaussian distribution model, the likelihood function of a scintillation event assumed to be Gaussian could be more easily implemented and might provide more consistent outcomes than Poisson-based MLPE. The purpose of this study is to evaluate the performances of the noise models, Poisson and Gaussian, in MLPE for the localization of photons in a small gamma camera (SGC) using NaI(Tl) plate and PSPMT. The SGC consisted of a single NaI(Tl) crystal, 10 cm in diameter and 6 mm thick, optically coupled to a PSPMT (Hamamatsu R3292-07). The PSPMT was read out using a resistive charge divider, which multiplexes 28(X) by 28(Y) cross wire anodes into four channels. Poisson and Gaussian-based MLPE methods have been implemented using experimentally measured detector response functions (DRF). The averaged intrinsic spatial resolutions were 3.14, 3.09, and 2.88 mm, the integral uniformities were 15.3%, 12.3%, and 11.4%, and the averaged linearities were 0.75, 0.33, and 0.22 mm for Anger logic, Poisson, and Gaussian-based MLPE, respectively. MLPEs considerably improved linearity and uniformity compared to Anger logic. Gaussian-based MLPE, which is easy to implement, allowed to obtain better linearity and uniformity performances than the Poisson-based MLPE. Index Terms—Position measurement, position sensitive PMT, small gamma camera.
I. INTRODUCTION
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MALL gamma cameras using a position-sensitive photomultiplier tube (PSPMT) have been developed for various applications especially for small field of view imaging in
Manuscript received January 28, 2003; revised September 19, 2003. This study was supported by Korea Institute of Science and Technology Evaluation and Planning (KISTEP) and by the Ministry of Science and Technology through their National Nuclear Technology Program and by Grant 02-PJ3-PG6EV06-0002 from the Korea Health 21 R&D Project, Ministry of Health and Welfare, Republic of Korea. Y. H. Chung is with the Department of Nuclear Medicine, Samsung Medical Center, Sungkyunkwan University School of Medicine, Seoul 135-710, Korea, and also with the Radiation Detection and Medical Imaging Laboratory, Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology (KAIST), Taejon 305-701, Korea (e-mail:
[email protected]). Y. Choi, T. Y. Song, J. H. Jung, Y. S. Choe, K.-H. Lee, S. E. Kim, and B.-T. Kim are with the Department of Nuclear Medicine, Samsung Medical Center, Sungkyunkwan University School of Medicine, Seoul 135-710, Korea (e-mail:
[email protected]). G. Cho is with the Radiation Detection and Medical Imaging Laboratory, Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea. Digital Object Identifier 10.1109/TNS.2003.823053
nuclear medicine including small animal and breast imaging [1]–[3]. Position estimation methods employed in these cameras should be different from those of general gamma camera because of the usage of PSPMT. The conventional position estimation method, Anger logic, works well in general gamma cameras using multiple PMTs with the careful gain calibration of individual PMTs, because Anger logic assumes a linear relationship between event locations and output signals. However, the nonlinear components of PSPMT response, such as nonuniform response of the photocathode and nonuniformity of the electric field within PSPMT, cannot be corrected because there is only a single gain control for the entire tube [4]. Therefore, more appropriate event positioning scheme, for example based on the statistical characterization of gamma ray detector, should be introduced for the development of small gamma camera using PSPMT. In this study, the maximum-likelihood position estimation (MLPE) was employed among alternative methods. It has been reported that MLPE algorithms offer advantages of improved spatial resolution and linearity over conventional Anger algorithm in gamma cameras [5]–[8]. Since the fluctuation of photon measurements is more accurately described by Poisson than Gaussian distribution model, most MLPE methods assumed the noise process follows Poisson distribution. However, if the event number of measurement is relatively large, the Gaussian model is widely applicable to many problems in counting statistics [9]. The likelihood function of a scintillation event assumed to be Gaussian model could be more easily implemented and might provide more consistent outcomes than Poisson-based MLPE. The purpose of this study is to evaluate the performances of noise models, Poisson and Gaussian, in MLPE for the localization of photons in a small gamma camera (SGC) using NaI(Tl) plate and PSPMT. II. MATERIALS AND METHODS Because of the stochastic nature of signal generation in a gamma camera, the probability of measuring a given set of given an event at output signals in the gamma camera will be described by the probability . The informadistribution as tion from this probability can be incorporated in a position estimation scheme. For a two-dimensional detector, the position likelihood function should be maximized with respect to the . MLPE was defined as the location of event position
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a scintillation event, which was most likely to have produced a given set of output signals. Therefore, when the estimate of the probability distribution has been obtained, the map of position estimates can be determined. The MLPE schemes were two-stage mapping from the measured PSPMT signals to position estimates. The first stage was a mapping from the measured signal space to an event characterization space by machine training. In the second stage, a mapping from the event characterization space to the position estimates space is carried out. The MLPE solution was achieved by maximizing the likelihood function between the predetermined detector response function, which is a collection of the , and measured event signals. We have imlocations plemented two MLPE methods based on Poisson and Gaussian noise models. A. Poisson-Based MLPE Assuming the noise process in the detector response follows Poisson distribution, the likelihood is given by (1) where (
is the set of observing PSPMT outputs , and ) for a given event position and is the mean of the th output of PSPMT as a function can be obtained experimentally by of event position. moving a point source in the direction and observing the th output of PSPMT. Performing the maximization with respect to the log likelihood, the resulting approximation is (2) where
The MLPE solution is achieved by minimizing the absolute value of the middle expression in (2). B. Gaussian-Based MLPE Assuming the noise process in the detector response follows Gaussian distribution, the likelihood is given by (3) where is the set of observing PSPMT outputs , and ) for a given event position is ( is the standard deviation of the th output the mean and of PSPMT as a function of event position. Performing the maximization with respect to the log likelihood, the resulting approximation is
(4)
Fig. 1. Diagram of the experimental apparatus for training. SGC and training system consist of collimated source, motorized x–y stage, NaI(Tl) crystal, PSPMT, signal processing electronics, and data acquisition board.
The MLPE solution is achieved by minimizing the inside value of the parenthesis in (4). C. Experimental Setup for Training Detector response functions (DRFs) for a detector were determined by experimental training. Our small gamma camera consisted of a single NaI(Tl) crystal, 10 cm in diameter and 6 mm thick, optically coupled to a single PSPMT (Hamamatsu R3292-07) [10]. The PSPMT was read out using a resistive charge divider, which multiplexes 28(X) by 28(Y) cross wire an, and . Fig. 1 is odes into four position signals a diagram of the experimental apparatus employed. The analog signal processing electronics consisted of preamplifiers, NIM dual-spectroscopy amplifiers, and delay amplifiers. A summed signal drove a single channel discriminator for event triggering on the peak output of the shaping amplifiers. Digitization of the four PSPMT output signals was accomplished with a National Instruments NI-DAQ board. To characterize DRFs, data were acquired on 21 21 evenly spaced points on a 10 cm diameter crystal using a computer controlled X–Y stepper-motor stage and collimated Tc-99m source. Then DRFs were generated using Gaussian fitting and cubic spline interpolation [11]. After collecting all four digitized position signals of PSPMT for each position of the collimated source, Gaussian fitting was performed to extract and variance information from these the mean coarse training data. Then the cubic spline interpolation was applied to the coarse data of DRF samples of mean and variance to obtain the fine DRF with 0.4 mm search resolution. When an is generated, unknown set of position signals for all position is calculated using (2) and (4) with fine DRF and then the most likely event location , providing , is determined to the event position. maximum D. Performance Measurements The position estimation scheme is critical in the determination of the overall performance of a gamma camera. The performances of the small gamma camera obtained with MLPE were compared to those obtained with the Anger logic. The intrinsic spatial resolution, linearity, and flood field uniformity of the small gamma camera were measured based on the NEMA protocol for Anger logic and two MLPE methods [12]. The intrinsic spatial resolution and linearity of the small gamma camera were measured by imaging 21 equally spaced
CHUNG et al.: MLPE WITH POISSON AND GAUSSIAN NOISE MODELS IN A SMALL GAMMA CAMERA
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X+
Fig. 2. Acquired detector response function (DRF) of . (a) Experimentally generated DRF by moving a point source on a 5 mm step grid. (b) Fine DRF generated by cubic spline interpolation with 0.4 mm resolution.
Fig. 4. Averaged spatial resolutions of center nine points in the CFOV in the x- and y -directions for Anger logic, Poisson, and Gaussian-based MLPE.
Fig. 3. Collimated source images obtained at 1.5 cm equally spaced 21 points in the CFOV with a small gamma camera using Anger logic (left), Poisson (center), and Gaussian (right)-based MLPE.
points using the X–Y stepper-motor stage and a 1 mm diameter lead-collimated point source of Tc-99 m by scanning over the center field of view (CFOV) of the small gamma camera with 1.5 cm spacing. The profiles were fitted to a Gaussian function to determine the peak location and the full width at half maximum (FWHM) values were computed to estimate the intrinsic spatial resolution. To measure the differential linearity expressed as a standard deviation of the line spread function peak separation, the best-fit line for the peak location of each point source at the same - or -axis was computed and the standard deviation of the peak separation between each point and the best-fit line in center field of view were calculated. The flood field image was acquired using the detector masked to useful FOV (UFOV) with a lead mask and the point source located at least five UFOV diameters from the center of the uncollimated detector. The acquired image was smoothed by conas volution with 3 3 filter function specified in the NEMA protocols. The integral uniformity was calculated by identifying the maximum and minimum values within the CFOV as follows: (5)
III. RESULTS After collecting the data of 21 21 evenly spaced points, DRFs were generated using Gaussian fitting and cubic spline interpolation. Fig. 2 shows the experimentally generated coarse signals generated by cubic spline DRF and the fine DRF of interpolation. The collimated source images obtained at equally spaced 21 locations in CFOV using Anger logic, Poisson, and Gaussianbased MLPE are illustrated in Fig. 3. The photon counts of each image were identical. While the inward shifts, caused by mislocating the scintillations close to the edges of the crystal toward the center, were observed in the image obtained by Anger logic,
Fig. 5. Linearity in the x- and y -directions for Anger logic, Poisson, and Gaussian-based MLPE.
MLPEs provided more linear images. Fig. 4 summarizes the averaged spatial resolutions of the center nine points in Fig. 3 and they are 3.10, 2.89, and 2.87 mm in the -direction and 3.18, 3.29, and 2.89 mm in the -direction for Anger logic, Poisson, and Gaussian-based MLPE, respectively. For the 12 points at the edge of Fig. 3, the averaged spatial resolutions are 3.19, 3.23, and 3.20 mm in the -direction and 3.29, 3.63, and 3.35 mm in the -direction for Anger logic, Poisson, and Gaussian-based MLPE, respectively. MLPEs, especially for the Gaussian-based MLPE, allowed to obtain a better performance in linearity as shown in Fig. 5. Linearities in CFOV are 0.57, 0.30, and 0.23 mm in the -direction and 0.92, 0.36, and 0.21 mm in the -direction for Anger logic, Poisson, and Gaussian-based MLPE, respectively. The flood field uniformity was also measured and Fig. 6 demonstrates the flood field images obtained by the three methods with same photon counts. The integral uniformities in CFOV are shown in Fig. 7 and are 15.3%, 12.3%, and 11.4% for Anger logic, Poisson, and Gaussian-based MLPE, respectively. IV. DISCUSSION AND CONCLUSION Gaussian-based MLPE was more robust to statistical noise than Poisson-based MLPE because the likelihood function of Gaussian model is derived from the mean and the deviation of the measurements while the likelihood function of Poisson model is derived from the derivative of the mean. Therefore,
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Fig. 6. Flood field images of the flood source obtained with a small gamma camera using Anger logic (left), Poisson (center), and Gaussian (right)-based MLPE.
Fig. 7. Integral uniformity of images in the CFOV obtained with Anger logic, Poisson, and Gaussian-based MLPE.
Gaussian-based MLPE provided considerably better resolution, linearity, and uniformity performances than the Poisson-based MLPE. The PSPMT used in this study has slightly better performances in the -direction than the -direction. Therefore, the linearity of Anger logic demonstrated a large difference between the - and the -direction, but the MLPEs provide uniform linearities in the - and -direction. While the standard deviations of the peaks of Gaussian fitted profiles from the best-fit line are worse at the edges of the FOV than at the center in Anger logic due to the falloff of detector response at the edge of the FOV, MLPEs have a good linearity performance by correcting the shrinkage and distortions occurred due to nonlinearity of the PSPMT output signals. MLPEs also provided better spatial uniformity, and experimental results yielded that the Poisson-based % and % improvement and GaussianMLPE allowed % and % improvement over Anger based MLPE did logic in terms of linearity and uniformity, respectively. Gaussian-based MLPE provided better and more consistent resolution in the - and -direction than Poisson-based MLPE. For more accurate position estimation, the DRFs of all the anodes and the fine sampling, especially for the edge of FOV due to the abrupt change of detector response at the edge of PSPMT, are required. In this paper, since the DRFs were generated using coarse sampling with 5 mm resolution and using the , and , which four weighted position signals, were multiplexed from 28(X) by 28(Y) cross wire anodes, there
IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 51, NO. 1, FEBRUARY 2004
is no meaningful improvement in the spatial resolution of the MLPEs compared to Anger logic. In the statistics-based positioning (SBP) method by Jung et al. [11], the DRFs were acquired from all the anode signals and the averaged spatial resolutions obtained by SBP were 44% better than those by Anger logic. Dead times of Poisson and Gaussian-based MLPE were 110 s and 75 s, respectively. The Gaussian-based MLPE allowed about 30% improvement in execution speed over the Poisson-based MLPE. The dead times of MLPEs were worse than that of Anger logic, 18 s due to the time consumable data processing procedures. The improvement of the dead time is under investigation and could be achieved by using faster CPUs and by reducing the number of search points per event. In conclusion, Poisson and Gaussian-based MLPE were successfully implemented to the small gamma camera using the machine training method. MLPE methods improved the linearity and uniformity compared to Anger logic but there is no meaningful improvement in the spatial resolution. The results of this study indicate that Gaussian-based MLPE, which is convenient to implement, has better performances than the Poissonbased MLPE. REFERENCES [1] N. J. Yasillo, R. N. Beck, and M. Cooper, “Design consideration for a single tube gamma camera,” IEEE Trans. Nucl. Sci., vol. 37, pp. 609–613, Apr. 1990. [2] D. Steinbach, S. Cherry, N. Doshi, A. Goode, B. Kross, S. Majewski, A. G. Weisenberger, M. Williams, and R. Wojcik, “A small scintimammography detector based on a 5 PSPMT and a crystal scintillator array,” in Proc. Conf. Rec. 1997 IEEE NSS/MIC, Albuquerque, NM, 1997, pp. 1255–1259. [3] R. Wojcik, S. Majewski, B. Kross, D. Steinbach, and A. G. Weisenberger, “High spatial resolution gamma imaging detector based on a 5 diameter R3292 Hamamatsu PSPMT,” IEEE Trans. Nucl. Sci., vol. 45, pp. 487–491, June 1998. [4] K. L. Matthews II, “Development and Application of a Small Gamma Camera,” Ph.D. dissertation, Univ. Chicago, Chicago, IL, 1997. [5] R. M. Gray and A. Macovski, “Maximum a posteriori estimation of position in scintillation cameras,” IEEE Trans. Nucl. Sci., vol. NS-23, pp. 849–852, Feb. 1976. [6] N. H. Clinthorne, W. L. Rogers, L. Shao, and K. F. Koral, “A hybrid of maximum likelihood position computer for scintillation cameras,” IEEE Trans. Nucl. Sci., vol. 34, pp. 97–101, Feb. 1997. [7] J. Joung, R. S. Miyaoka, S. Kohlmyer, and T. K. Lewellen, “Implementation of ML based positioning algorithms for scintillation cameras,” IEEE Trans. Nucl. Sci., vol. 47, pp. 1104–1111, June 2000. [8] J. Joung, R. S. Miyaoka, S. G. Kohlmyer, and T. K. Lewellen, “Investigation of bias-free positioning estimators for the scintillation cameras,” IEEE Trans. Nucl. Sci., vol. 48, pp. 715–719, June 2001. [9] G. F. Knoll, Radiation Detection and Measurement, 2nd ed. New York: Wiley, 1989. [10] J. H. Kim et al., “Development of a miniature scintillation camera using NaI(Tl) scintillator and PSPMT for scintimammography,” Phys. Med. Biol., vol. 45, no. 11, pp. 3481–3488, 2000. [11] J. Joung, R. S. Miyaoka, and T. K. Lewellen, “cMICE: A high resolution animal PET using continuous LSO with a statistics based positioning scheme,” in Proc. Conf. Rec. 2001 IEEE NSS/MIC, vol. 2, San Diego, CA, Nov. 2001, pp. 1137–1141. [12] Performance Measurements of Scintillation Cameras. Washington, DC: Nat. Elect. Manuf. Assoc., 1994, NEMA Std. Publ. NU1.
