IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 49, NO. 5, OCTOBER 2002
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A Channelized-Hotelling-Trace Collimator Design Method Based on Reconstruction Rather Than Projections Gengsheng L. Zeng, Member, IEEE, and Grant T. Gullberg, Senior Member, IEEE
Abstract—In this paper, a new single-photon emission-computed tomography (SPECT) collimator design technique is investigated. In this feasibility study, the collimator hole diameter is the only design variable changed. Other collimator parameters are fixed to a low-energy–high-resolution (LEHR) design. The design is based on a task of imaging a small hot lesion with a uniform background. A channelized hotelling trace is used to quantify the lesion detectability in a reconstructed image for a particular collimator parameter. The unique feature of this development is that the image quality is not evaluated for planar projection data, but rather is evaluated for the reconstructed tomographic image with an ordered-subset expectation-maximization reconstruction algorithm. Our results suggest that a collimator hole size that is larger than the LEHR collimator hole size is preferable for lesion detection. In SPECT, the optimal collimator design should be reconstruction algorithm dependent. Index Terms—Collimator, hotelling trace, emission-computed tomography (SPECT).
single-photon
I. INTRODUCTION
I
N NUCLEAR medicine imaging, collimator design is determined by parameters such as hole size, hole length, and septal thickness. The optimal design is achieved by maximizing the photon counts for the given desired spatial resolution and photon penetration [1]–[3]. The collimator’s spatial resolution is specified for different distances from the collimator face. Task-based collimator design techniques have also been investigated [4], [5]. In single-photon emission-computed tomography (SPECT), reconstruction algorithms that have the capacity to compensate for the distance-dependent collimator blurring have been developed [6]–[13]. It is of interest to investigate the optimal collimator parameters when compensation for the distance-dependent collimator blurring is performed by the image-reconstruction algorithm. In this paper, we study the feasibility of using reconstructed images to design a parallel-hole low-energy collimator. Our design criteria is based on small hot-lesion detectability, in which the hot lesion has a uniform background. Since the channelized hotelling observer has been shown to correlate very well with human observer studies for many signal-known-exactly detection tasks [14]–[17], the lesion detectability in this study is Manuscript received December 4, 2001; revised April 10, 2002. This work was supported in part by the National Institutes of Health under Grant R01HL50663. The authors are with the Medical Imaging Research Laboratory, University of Utah, Salt Lake City, UT 84108-1218 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TNS.2002.803775
chosen to be of the quantity equivalent to a channelized hotelling trace (CHT). II. METHODS First, a task was defined to establish parameters for the new collimator design: detecting small hot lesions in a uniform background. The radioactivity in the lesion and background had a ratio of 8 : 1. The lesion size was 1 cm in diameter. The collimator was designed to image gamma rays of 140 keV. A set of low-energy–high-resolution (LEHR) SPECT collimator paramcm, hole diameter cm, and eters (hole length cm) were used as the default values. septal thickness The detector orbit was a circle with a radius of 30 cm. In this study, we varied only one parameter—the hole diameter—for our proposed design method. The projection data were generated by a computer-simulated line-tracing projector that modeled the distance-dependent collimator point-spread function [6], but did not model the attenuation and scatter. In other words, our study concentrated only on the distance-dependent collimator blurring effects. The projection data were attenuation- and scatter-free. The projection array size was 64 64 and the pixel size was 0.712 cm. The total number of projection views was 120 over 360 . The projection data were randomized according to Poisson statistics and collimator sensitivity (1)
Sensitivity
where is the distance from the radioactive source to the is the Gaussian-hole-approximation detector and resolution in terms of the full-width-at-half-maximum. The following equation: Hole Area Hole Length
(2)
and (1) imply that if the hole diameter is increased by , then the photon count is increased by . Both (1) and (2) are from [3]. The images were reconstructed three-dimensionally with an ordered-subset expectation-maximization (OS-EM) algorithm [18] that modeled collimator point response function using the slice-by-slice blurring method [8]. This slice-by-slice blurring method of modeling the collimator point response function was described in [8]. Each subset consisted of four views, 90 apart
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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 49, NO. 5, OCTOBER 2002
from each other. The images were reconstructed with four iterations. The reconstructed image volume size was 64 64 64. The CHT is an effective computerized method that accurately replicates human-observer studies. It is a measure of class separability and is briefly described below. We considered a set of two-dimensional images, which were the three-dimensional OS-EM reconstructions in our case. There were 28 different potential positions at which a hot lesion could be placed. The lesions were in a uniform elliptical cylinder. For the elliptical cylinder, the longer semiaxis was 21.36 cm and the shorter semiaxis was 14.73 cm. One-half of the reconstructed images contained a lesion and the other half did not. We generated four different spatial-domain templates, which were four different bandpass filters. Each filter determined a channel. In the frequency domain, these four filters had passing bands of 1/64–2/64, 2/64 –4/64, 4/64–8/64, and 8/64–16/64 cycles per pixel, respectively. The spatial-domain templates were obtained by applying a two-dimensional Hankel transform [19] and were given by
Template
if if (3)
where is the norm of the two-dimensional spatial-domain variis the upper cutoff frequency of the channel, is able, is the Bessel the lower cutoff frequency of the channel, and function of the first order. We calculated the inner product of each template and each at each potential image by placing the template center lesion position. Thus, for each potential lesion position, we obtained four inner product values because we had four templates. These four inner product values formed a feature vector . Next, we formed a 4 4 inter-class scatter matrix
where is the trace of the matrix, i.e., the sum of the main diagonal elements in the matrix. CHT is a scalar quantity and is an extended concept of the signal-to-noise ratio—with being being the noise. the signal and as a figure-of-merit to indicate the lesion deWe used tectability. There were five different collimator hole sizes (diameters) used in the CHT evaluations, i.e., 0.61, 1.22 (LEHR default value), 2.44, 3.66, and 4.88 mm. There were two different noise levels, each having four different noise realizations. Therefore, the total number of feature vectors was (7) where 28 number of potential lesion positions; 2 with or without lesion; 2 number of noise levels; 4 number of random noise realizations; 5 number of collimator hole sizes. The CHT value and its standard deviation were estimated by using a jackknife method [20]. This method was performed as without the data from follows. We calculated a CHT value without the data from lesion lesion location 1, a CHT value location 2, and so forth, consecutively, until the final calculation without the data from lesion location 28 was of CHT value made. The overall CHT value was then given by (8) where set
. A standard deviation was calculated from the as follows:
(9)
III. RESULTS (4) is the mean feature vector of the class of lesion where is the mean feature vector of the class of lesion present, is the grand mean feature absent, is the a priori probability of ocvector of all classes, is the a priori probability currence of a lesion, and of absence of a lesion. Here, the angle brackets indicate an ensemble average. We then formed a 4 4 intra-class scatter matrix
(5)
Fig. 1 shows a typical OS-EM reconstruction at two different noise levels, with and without a lesion, when the collimator hole size was set at the default value of 1.22 mm in diameter for an LEHR collimator. Fig. 2 shows the same images, but the hole size of 3.66 mm in diameter was three times larger. Even though the collimator blurring was modeled in the reconstruction algorithm, the resolution was unable to be completely recovered, as seen in Fig. 2. In general, the resolution recovery is limited by noise. The CHT results are listed in Table I for different collimator hole sizes. A hole size that was three times larger than the LEHR collimator hole size was chosen as best according to the CHT criteria. The favored hole-size according to our CHT criteria is larger than that of the LEHR collimator. This is consistent with the observation of other researchers [22]. IV. DISCUSSION
The CHT is defined as (6)
For a given task of imaging small hot lesions using SPECT, we have developed a collimator design technique that uses a
ZENG AND GULLBERG: CHT COLLIMATOR DESIGN METHOD
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Fig. 1. Reconstructed images with the collimator hole size of 1.22 mm in diameter. Projection data for (a) and (b) have twice the photon counts as in (c) and (d). A lesion is present in (a) and (c). No lesion is present in (b) and (d). The lesion location is identified by an arrow.
Fig. 2. Reconstructed images with the collimator hole size of 3.66 mm in diameter. Projection data for (a) and (b) have twice the photon counts as in (c) and (d). A lesion is present in (a) and (c). No lesion is present in (b) and (d). The lesion location is identified by an arrow. TABLE I RANKING OF DIFFERENT HOLE SIZES BY THE CHT
CHT method to select the optimum parameters. This task-based technique distinguishes itself from other collimator designs by considering reconstructed images and using an image reconstruction algorithm that models the collimator point-spread function. There are three main parameters (i.e., hole size, hole length, and septal thickness) to be considered when designing collimators. We only varied one parameter (i.e., hole diameter) and kept the other two parameters fixed to illustrate the feasibility of our technique. Of course, the other two parameters can also be varied. It is interesting to see the classic tradeoff between the resolution and sensitivity in a collimator design. This study shows that the CHT criteria resulted in a preference for a configuration with lower resolution and higher sensitivity than the default LEHR collimator’s configuration. It is understood that, due to noise, we must terminate the reconstruction iteration early and, therefore, the collimator blurring cannot be completely recovered. Based on a single study, however, we are unable to draw a general conclusion that a higher sensitivity/lower resolution collimator is preferable to the LEHR collimator configuration. For simplicity, we have fixed the number of iterations and used uniform background. A lesion is more difficult to detect in a lumpy background. In fact, the optimal iteration number depends on many factors such as noise level, background, and
image frequency spectrum [21]. In a more rigorous design task, an optimal iteration number should be determined and used in CHT studies. ACKNOWLEDGMENT The authors thank S. Webb, University of Utah, Salt Lake City, for editing this paper’s manuscript. The authors also thank Dr. B. Tsui and Dr. E. Frey, both of the University of North Carolina at Chapel Hill, and Dr. K. Gilland, University of Florida, Gainesville, for helpful discussions. REFERENCES [1] E. L. Keller, “Optimum dimensions of parallel-hole, multi-aperture collimators for gamma-ray cameras,” J. Nucl. Med., vol. 9, pp. 233–235, 1968. [2] R. N. Beck and D. L. Gunter, “Collimator design using ray-tracing techniques,” IEEE Trans. Nucl. Sci., vol. NS-32, pp. 865–869, 1985. [3] D. L. Gunter et al., “Collimator characteristics and design,” in Nuclear Medicine, R. E. Henkin et al., Eds. St. Louis, MO: Mosby, 1996, pp. 96–124. [4] S. C. Moore, D. J. de Vries, B. Nandram, M. F. Kijewski, and S. P. Mueller, “Collimator optimization for lesion detection incorporating prior information about lesion size,” Med. Phys., vol. 22, pp. 703–713, 1995. [5] T. A. White, H. H. Barrett, E. B. Cargill, R. D. Fiete, and M. Kerr, “The use of the hotelling trace to optimize collimator performance,” J. Nucl. Med., vol. 30, p. 892, 1989.
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