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Reconstruction of Two- and Three-Dimensional Images from Synthetic-Collimator Data D. W. Wilson*, H. H. Barrett, and E. W. Clarkson

Abstract—A novel SPECT collimation method, termed the synthetic collimator, is proposed. The synthetic collimator employs a multiple-pinhole aperture and a high-resolution detector. The problem of multiplexing, normally associated with multiple pinholes, is reduced by obtaining projections at a number of pinhole-detector distances. Projections with little multiplexing are collected at small pinhole-detector distances and high-resolution projections are collected at greater pinhole-detector distances. These projections are then reconstructed using the ML–EM algorithm. It is demonstrated through computer simulations that the synthetic collimator has superior resolution properties to a high-resolution parallel-beam (HRPB) collimator and a specially built ultra-high-resolution parallel-beam (UHRPB) collimator designed for our 0.38-mm pixel CdZnTe detectors. It is also shown that reconstructing images in three dimensions is superior to reconstructing them in two dimensions. The advantages of a high-resolution synthetic collimator over the parallel-hole collimators are apparently reduced in the presence of statistical noise. However, a high-sensitivity synthetic collimator was designed which again shows superior properties to the parallel-hole collimators. Finally, it is demonstrated that, for the cases studied, high-resolution detectors are necessary for the proper functionality of the synthetic collimator. Index Terms—High-resolution collimators, high-sensitivity collimators, pinhole collimators, SPECT.

I. INTRODUCTION

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HEN DESIGNING a system for imaging single-photon emitters, a major consideration is the device the system will use to form an image of incoming photons. A number of image-formation mechanisms have been proposed, but by far the most common is a collimator. A collimator employs a highattenuating material such as lead to limit the photon pathways between the source and the detector. In doing so, it reduces the possible emission locations of a detected photon and increases the photons spatial information. The collimator most prevalently used in the clinic is the parallel-hole collimator, although fanbeam and conebeam collimators are also utilized [1]–[4]. All of these collimators apply an arrangement of long thin lead tubes to allow only photons arriving at the collimator face from a given angle to pass to the detector, with the angle constituting the difference between these

Manuscript received August 31, 1999; revised January 19, 2000. This work was supported by the National Cancer Institute under Grants CA 52643 and CA 75288. The Associate Editor responsible for coordinating the review of this paper and recommending its publication was F. J. Beekman. Asterisk indicates corresponding author. *D. W. Wilson, H. H. Barrett, and E. W. Clarkson are with the University of Arizona, Department of Radiology, Tucson, AZ 85724 USA. Publisher Item Identifier S 0278-0062(00)05301-5.

collimator types. The parallel-hole collimator has no focus point and ideally allows only photons with paths perpendicular to the face of the collimator to strike the detector. The fanbeam and conebeam collimators have, respectively, two-dimensional (2-D) and three-dimensional (3-D) focus points. An alternative to these types of collimators is the pinhole collimator [5]. Pinhole collimators operate on the same principle as a pinhole camera, with the pinhole acting as an aperture through which a photon must pass in order to be seen by the detector. This produces a conebeam image on the detector, with the focus point at the pinhole. One major advantage of the pinhole collimator is the ease of production. While the parallel-hole collimators require special manufacturing techniques and facilities only available in a few locations, the pinhole collimator is easily produced by anyone with access to lead and a drill press. This is particularly important for our group, as we must design collimators for the high-spatial- and high-energy-resolution solid-state detectors being developed in our laboratory [6]. A pinhole can quickly be constructed for high resolution [7], [8] or with increased angular sampling [9]. However, pinhole collimators also possess disadvantageous properties. These include low sensitivity [10] and the fact that they produce conebeam images. The goal of our research is to overcome these disadvantages using a novel pinhole-collimator design. In this paper we examine this design, termed the synthetic collimator, and present simulated comparisons that show advantages for the synthetic collimator over parallel-hole collimators when the high-resolution detectors are employed. Analytic computer simulations were used to explore the properties of the various collimator types. While Monte Carlo methods can more accurately recreate the effects of scatter in the collimator and patient [11], we felt that the excellent energy resolution of the solid-state detectors, along with the possibility of using high-attenuating inserts for pinholes, reduced the importance of modeling these effects. Using the analytic simulation methods we demonstrate that the synthetic collimator can be configured to produce superior 2-D images compared to parallel-hole collimators. We also show that further advantages may exist by reconstructing 3-D rather than 2-D images from the synthetic-collimator projections. The latter is a surprising result, as data from only one collimator-object angle are collected. At first glance, one would expect that this limited angular sampling would prohibit 3-D object estimates. However, each of the pinholes provides sampling from a different angle and, as we shall show, it is this increased angular sampling that allows us to reconstruct the tomographic images.

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II. BACKGROUND AND METHODS A. Collimators for Planar and Tomographic Imaging Clinical applications call for both planar and tomographic single-photon nuclear-medicine images. The planar image expected by the physician is a simple projection of the 3-D radiopharmaceutical distribution, while the tomographic image expected is a representation of the full 3-D distribution of the radiotracer. In both cases the image parameters to be estimated are (1) is the continuous distribution of radiotracer within where are the basis functions upon the imaged object and the are a which the continuous distribution is projected. The finite set, as necessitated by computational constraints, and cannot fully describe a general continuous object. Common are examples of (2) otherwise for 2-D planar imaging and (3) otherwise for 3-D tomographic imaging. These functions are shown (compressed to two dimensions and one dimension) in Fig. 1(a) and (b). The original goal of this study was to develop a collimation method for use with planar imaging [using (1) and (2)]. The type of (2) of collimator that would seem to directly estimate the is a parallel-hole collimator [as shown in Fig. 1(a)] and a parallel-hole collimator is frequently employed for planar imaging. Such a collimator, however, is not ideal for two reasons. First, , ) in (2) estimating is pointless if the area ( is too large to be clinically useful. In order to limit the size of this region, small collimator-bore diameters are required [12], [5]. Second, a parallel-hole collimator can exactly estimate the ’s of (2) only if it has infinitely long bores. Otherwise, the resolution [the width of the column in Fig. 1(a)] will decrease with distance from the collimator face. Unfortunately, a collimator with long thin bores will allow only a very small percentage of the photons to pass. Thus, a tradeoff exists between collimator resolution and collimator sensitivity. For a pinhole collimator, the pinhole diameter determines the resolution and, unlike parallel-hole collimators, the resolution of a pinhole collimator can be easily improved by decreasing the diameter of the pinhole and/or by moving the detector farther from the pinhole [5]. Unfortunately, as in the case of parallel-hole collimators, the resolution-sensitivity tradeoff still exists since the pinhole size affects resolution and sensitivity in opposite manners. Additionally, pinhole collimators do not directly measure the functions of (2), but rather measure cone-beam integrals. Some method of reconstruction is necessary if a true planar image is desired

Fig. 1. Integral regions for (a) (2) and (b) (3), shown with a parallel-hole collimator.

and reconstruction generally requires data from more than one angular view. B. Collimators for Low- and High-Resolution Detectors All commercial single-photon imaging systems in clinical use today employ sodium-iodide [NaI(Tl)] scintillators and arrays of photomultipliers to detect photons. These detectors typically have a spatial resolution of 2–3 mm. Therefore, due to the resolution/sensitivity tradeoff, it is counterproductive to use a collimator with significantly better resolution, and commercial clinical collimators have resolution within this order of magnitude. While collimators with this resolution are fine for scintillation detectors, our group has been developing high-resolution solid-state cadmium zinc telluride (CdZnTe) detectors [6]. These detectors have pixel sizes on the order of hundreds of microns and subpixel resolution has been demonstrated [13]. Clearly, with such a detector it is counterproductive to use a collimator designed for NaI detectors, so typical commercial collimators were proscribed. The two collimator possibilities that were considered for the CdZnTe detectors were an ultra-high-resolution parallel-beam (UHRPB) collimator and a high-resolution pinhole collimator. In collaboration with Tecomet Inc., we have demonstrated the possibility of imaging with a collimator having the same 380 m pitch as our current generation of detectors, where the pitch

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is defined as the distance between pixels for the detector and distance between individual bores for the collimator. With this collimator/detector combination we were able to produce excellent high-resolution images [14]. However, there were three problems associated with the collimator. First, the sensitivity of the collimator was only 5 10 , about half of what would be expected from a high-resolution parallel-beam (HRPB) collimator used for NaI detectors. Second, in order to achieve sensitivity even this high, it was necessary to use a very short (7-mm) bore length. Thus, the resolution of the collimator dropped off quickly with distance from the collimator face. Finally, this collimator required special manufacturing techniques that did not lend it to the quick imaging-system-design changes that we frequently require in our laboratory. While a high-resolution pinhole collimator is much easier to construct, in order to get the resolution required for the CdZnTe detectors, a pinhole of diameter on the order of 0.5 mm is required, and a collimator with a single 0.5 mm diameter pinhole has a sensitivity of about 5 10 over a reasonable field of view. Since this is unacceptable for most imaging tasks, a method was needed to increase the sensitivity of a pinhole collimator. One means of achieving greater sensitivity is to use multiple pinholes rather than a single pinhole, and this is the approach we explore in this study. C. The Synthetic Collimator A major problem introduced with multiple-pinhole collimators is multiplexing [15]. Multiplexing occurs when projections through different pinholes overlap on the detector. A photon hitting a given detector pixel will have reduced information, as it could have come through any of two or more pinholes. The multiplexing problem increases if the detector is moved farther from the pinhole aperture in an attempt to improve the resolution of the imaging system. In this paper we demonstrate that the problems of multiplexing, sensitivity, and cone-beam projections can be overcome with the synthetic collimator. The physical configuration of the synthetic collimator is quite simple. It consists of a multiple-pinhole aperture and a high-resolution detector [Fig. 2(a)]. The central idea of the synthetic collimator is to collect multiple-pinhole projection data using a number of pinhole-detector distances, as shown in Fig. 2(b). Since the amount of projection overlap varies as a function of this distance, the data contains information on how the projections are multiplexed and on how to remove the multiplexing effects. Ideally, low-multiplexed data from small pinhole-detector distances and high-resolution data from large pinhole-detector distances would be collected. The synthetic-collimator concept is then to combine these data in a manner that leads to a nonmultiplexed high-resolution projection. D. Methods of Simulation and Reconstruction In order to properly analyze an imaging system, it is necessary to use a phantom of sufficient complexity to challenge that system [16]. For this study, we chose a textural phantom, with texture similar to what has been observed for both digital mammograms and mammoscintigrams. The phantom filled

Fig. 2. (a) A 3-D view of the synthetic collimator. (b) A schematic view of images collected at two different pinhole-detector distances. Note that at the smallest pinhole-detector distance no multiplexing occurs but the magnification on the detector is small. At the greatest pinhole-detector distance there is greater magnification but also significant multiplexing.

a volumetric area of 10 cm 10 cm 5 cm. Nine slices of the phantom, separated by a 2-mm distance, are shown in Fig. 3(a). A Gaussian lesion with a full-width at half maximum (FWHM) of 4 mm and a peak-to-background ratio of seven was then placed at the phantom’s center. The same nine slices, with lesion, are given in Fig. 3(b). Throughout this study, a 10 cm 10cm detector with a pixel size of 0.33 mm was assumed. The projection data, , were generated using a numerical-integration technique. This technique involved sampling the object on a 0.2 mm 0.2 mm 0.2 mm grid, calculating the system response to that sample point, and multiplying the response by the value of the object at the point. The system response function was generated on a 1-mm cubic grid, with the response of a voxel assumed to be the average response of the 125 sample points contained within the voxel. The effect of depth of interaction within the detector was not modeled since we have developed a method of estimating, and therefore compensating for, this effect in the CdZnTe devices [8]. Attenuation was included in both the projection operation and the model used for the reconstruction. Scatter was not included, but we would expect the scatter effects to be rather small due to the

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a similar manner so that the single resulting had elements representing the probability that a photon emitted in voxel would be detected in pixel of the total data vector . For this study the data sets were ordered from smallest to largest was the data collected pinhole-detector distances, so that the data collected at the at the smallest separation and largest separation. The ML–EM algorithm was then run with the combined and as inputs. The stopping point was ten ML–EM iterations. This is an unusually low number of iterations for the algorithm [20], [21], but produced images of better apparent quality than at higher iteration stopping points. The reason that this collimator configuration required fewer iterations than is generally considered optimal for more traditional emission-tomographic configurations is unknown and will be the focus of future study employing more sophisticated methods of image-quality analysis. The physical configuration for the 2-D and 3-D synthetic collimators was the same and both the 2-D and 3-D images were initially reconstructed in 3-D on the 1-mm cubic voxelized grid. For the 2-D synthetic collimator, the reconstructed 3-D image was summed along the tubes shown in Fig. 1(a). This summation constituted the only difference between the 3-D and 2-D collimators. III. RESULTS AND DISCUSSION A. High-Resolution Imaging with the Synthetic Collimator

Fig. 3. Slices from the phantom used for this study. (a) Without lesion. (b) With a Gaussian lesion with FWHM of 4 mm and signal-to-background ratio of 7.

small object size and the good energy resolution of the CdZnTe detectors (5% at 140 KeV [8]). Reconstruction of the synthetic-collimator data was performed using the maximum likelihood–expectation maximization (ML–EM) algorithm [18], [19]. The algorithm requires both the data, , collected by the imaging system and an estimate, , of the system’s response. The synthetic-collimator data collected at each pinhole-detector distance can be viewed as a separate data set and we shall refer to the th set (of total sets at pinhole-detector distances) as . Similarly, there are system-response functions, with the th referred to . For reconstruction, the were combined to form a as total data vector, . This combination was carried out by taking elements of , concatenating the elements of , the elements of and so on until a data vector of then the elements was formed. The were concatenated in

The first experiment involved assessing the benefits, in terms of resolution, of the synthetic collimator over an HRPB collimator and the UHRPB Tecomet collimator. The HRPB collimator was simulated with a bore diameter of 1.4 mm and a bore length of 34.0 mm and the UHRPB Tecomet collimator was simulated with a bore diameter of 0.26 mm and a bore length of 7 mm. In both cases it was assumed that there was no septal shadow or septal penetration. The synthetic collimator was simulated with 25 pinholes of diameter 0.5 mm. No septal penetration was included in the model and we would expect very little for 140 KeV photons since inserts of highly attenuating gold can easily be installed. To reduce possible artifacts that might arise from sampling the object space repeatedly at the same frequencies, the pinholes were randomly positioned in a 2 mm 2 mm area about the center of a regular 5 5 grid, with grid points separated by 2 cm. The projection set consisted of data collected from pinhole-detector distances of 6, 10, 18, and 30 mm. Fig. 4(a) shows the noise-free projection of the phantom with lesion through the HRPB collimator and Fig. 4(b) gives the noise-free projection through the Tecomet collimator. Fig. 4(c) is the noise-free 2-D synthetic collimator image (the 3-D image summed in the along the axis, with defined as the direction perpendicular to the collimator face). Since the lesion is clearly visible in each of these images, we compared profiles through the lesions. Fig. 5 gives these profiles for the HRPB-, UHRPB-, and synthetic-collimator images. It appears that the synthetic collimator better resolves the lesion, both in terms of contrast and spatial resolution. We note that the synthetic collimator reconstructs on a 1-mm grid while the parallel-hole images are collected by a detector with a 0.33-mm pixel size. In

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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 19, NO. 5, MAY 2000

Noise-free planar images using (a) a high-resolution parallel-hole collimator, (b) the Tecomet collimator, and (c) the 2-D synthetic collimator.

Fig. 5. Profiles through the images of Fig. 4. The irregularities in the synthetic-collimator profile result from the expansion of the synthetic-collimator image.

order to compare the images, the synthetic-collimator reconstruction was expanded without smoothing. The irregularities caused by this expansion are seen in the synthetic-collimator profile (Fig. 5). Fig. 6 shows nine slices of the 3-D reconstruction of the synthetic-collimator data, and Fig. 7 compares profiles through the lesion in the 2-D and 3-D synthetic-collimator reconstructions. It is apparent that although the synthetic collimator collects data from only one projection direction, sufficient angular sampling is achieved to reconstruct tomographic images, even near the edges of the field of view. The improved contrast resolution of tomographic images over planar images is obvious and no artifacts from the reconstruction process are seen. We concluded that, in this noise-free case, the 3-D synthetic collimator produces images superior to the 2-D synthetic collimator and, therefore, superior to the parallel-hole collimators. B. Importance of Multiple Pinholes and Multiple Pinhole-Detector Distances One obvious benefit of multiple-pinhole imaging over singlepinhole imaging is increased sensitivity. A somewhat less obvious benefit is increased angular sampling. It was pointed out in

Fig. 6. Nine slices from the 3-D synthetic-collimator reconstruction of the noise-free data.

Fig. 7. Profiles through the lesion for the 2-D and 3-D synthetic-collimator images. For this comparison, the images were rescaled to the same total counts.

the previous section that this configuration of the synthetic collimator collects data from only one projection direction, but for a

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Fig. 8. The (a) 2-D and (b) nine slices from the 3-D synthetic-collimator reconstructions for single-pinhole data.

Fig. 9. The (a) 2-D and (b) 3-D synthetic-collimator reconstructions with data taken only at the 30-mm pinhole-detector distance.

multiple-pinhole collimator each pinhole furnishes information from a different projection angle. These multiple pinholes still produce limited-angular data, since for a given projection angle at most 180 of cone-beam sampling is possible. However, as was demonstrated in the previous section, sufficient information is provided by the multiple pinholes to allow reconstruction of a complex object. In order to demonstrate the importance of the multiple pinholes we simulated a system with a single 0.5-mm pinhole and generated the noise-free data for the phantom of Fig. 2(b). With the exception of number of pinholes, system parameters were the same as in the previous experiment, and data were collected from 6, 10, 18, and 30 mm. Fig. 8(a) shows the 2-D syntheticcollimator reconstruction of these data and Fig. 8(b) gives nine slices from the 3-D synthetic-collimator reconstruction. The reconstructed images are seen to give a very poor estimate of the original radiotracer distribution. We concluded that, as would be expected, one pinhole does not provide sufficient angular sampling for reconstructing the phantom. We also studied the importance of multiple pinhole-detector distances for the synthetic collimator. The hypothesis was that reconstructing data from only one pinhole-detector distance would lead to a tradeoff between multiplexing and resolution in the reconstructed synthetic-collimator images. In order to test

this hypothesis, we obtained noise-free data from only the 30and 6-mm pinhole-detector distances. Fig. 9(a) and (b) shows the 2-D and 3-D reconstructions of the noise-free data (25 pinholes) from 30 mm. The lesion is clearly resolved, but the effects of multiplexing are seen in the form of image artifacts. While this does not affect the ability to perform the task of determining the presence of the lesion, it can be inferred that the multiplexing would degrade the ability to perform more complex tasks where the lesion’s shape and position were not known a priori. Fig. 10(a) and (b) give the 2-D and 3-D reconstructions from only the 6 mm data. Here the images appear to provide a good estimates of the phantom’s radiotracer distribution, but an apparent reduced contrast and spatial resolution of the lesion is seen when compared to the multiple pinhole-detector-distance reconstructions of Fig. 6. A more quantitative analysis is required to determine what types of tasks would be negatively affected by this reduced resolution, but the goal for our syntheticcollimator design was to maximize spatial resolution and the multiple pinhole-detector distances produced images of better apparent resolution. We concluded from these studies that multiple pinhole-detector distances were requisite for maximizing the spatial resolution of the synthetic collimator without introducing multiplexing artifacts.

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Fig. 11. Sensitivity, as a function of collimator-source distance, for the Tecomet, high-resolution, and 25-pinhole synthetic collimators.

Fig. 10. The (a) 2-D and (b) 3-D synthetic-collimator reconstructions with data taken only at 6-mm pinhole-detector distance.

C. Effects of Noise One concern for any imaging system that requires a reconstruction step is how differences between the true projection process and the projection model used by the reconstruction algorithm will affect image quality. As mentioned above, the modeling inconsistency of numerically integrated projection data (the system response sampled on a 0.2 mm 0.2 mm 0.2 mm grid) and voxelized reconstruction model (the system response averaged over a 1- mm cubic area) has already been introduced. Another important inconsistency between real and expected data results from statistical noise resulting from the limited number of photons collected in a detector pixel. In order to properly compare the noise properties of the synthetic collimator with those of the parallel-hole collimators it was necessary to calculate their sensitivities. In this study we defined sensitivity at a point as the fraction of photons from that point that would be allowed to pass through the collimator and average sensitivity over the field of view as the fraction of photons from a nonattenuating and uniformly emitting source distributed throughout the field that would be allowed to pass. While parallel-hole collimators have, to a good approximation, sensitivity independent of distance from the collimator face, the sensitivity of pinhole collimators is strongly dependent upon source distance. Fig. 11 gives a plot of sensitivity of the UHRPB

Tecomet, HRPB, and 25-pinhole synthetic collimators as a function of source-collimator distance. It is seen that, for 25 pinholes, the synthetic collimator has advantages only for small 10 cm collimator-source distances. Within the ten 10 cm 5 cm field of the phantom, the average sensitivity of the 10 , the average sensitivity of HRPB collimator was 1.1 the UHRPB Tecomet collimator was 5.1 10 , and the average sensitivity of the 25-pinhole synthetic collimator was 7.8 10 . Poisson noise was added to the HRPB collimator data at a level appropriate for 100 000 total projection counts. Using the sensitivity numbers above, the count levels were adjusted to 46 400 for the Tecomet collimator and 70 900 for the synthetic collimator. Fig. 12(a) and (b) gives the noisy projections for the HRPB and Tecomet collimators. Fig. 12(c) and (d) gives 2-D and 3-D reconstructions or the noisy synthetic-collimator projection data. It appears that in the presence of statistical noise the advantages of the synthetic collimator over the parallel-hole collimators have decreased and we cannot conclude without more careful quantitative analysis involving a number of options, including low-pass filters, that the synthetic collimator offers any advantages at this particular count level. D. High-Sensitivity Synthetic Collimation It was shown in Section III-A that the 25-pinhole synthetic collimator has resolution advantages over parallel-hole collimators, but the results of the previous section cast doubt on whether these advantages hold up for low- or medium-count imaging tasks. It should be noted that while in the preceding study we did not assume an inordinately low number of counts, the radiotracer was distributed in a complex manner throughout the entire field of view. Thus, both the reconstruction and projection images were more noise-limited than would be expected for a task that involved estimating distributions in features with concentrated radiopharmaceutical such as would be the case for a bone scan. Nevertheless, since very low-count tasks are included in some of the small-animal imaging our group proposes to carry out using a synthetic collimator, it was determined that

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Fig. 12. Noisy images from (a) the high-resolution parallel-hole collimator and (b) the Tecomet collimator. Noisy reconstructions from (c) the 25-pinhole 2-D synthetic collimator and (d) the 25-pinhole 3-D synthetic collimator.

the performance of the collimator should be improved in such situations. For this reason, we sought to exploit the flexibility inherent in the synthetic-collimator design parameters to improve its performance for the imaging task and noise level studied in the previous section. In order to improve the sensitivity of the synthetic collimator, we increased the number of pinholes from 25 to 400. The 400 pinholes were randomly positioned in 0.5 mm region on a regular 20 20 grid a 0.5 mm with the grid points separated by 5 mm. This gave it a sensitivity of 1.3 10 over our field of view—an order of magnitude better than the HRPB collimator. This sensitivity corresponded to 1 180 000 total projection counts for the same imaging time used in the study of Section III-C.

Due to the high degree of multiplexing for the 400-pinhole collimator, we reduced the pinhole-detector distances to 2, 3.5, 5, and 8 mm. Fig. 13(a) and (b), show the 2-D and 3-D synthetic-collimator reconstructions of the noisy projection data. The 400-pinhole synthetic collimator appears to perform well compared to the parallel-hole collimators [Fig. 12(a) and (b)], particularly in the 3-D case. Due to the smaller pinhole-detector distances a reduced spatial resolution is seen compared to the noise-free 25-pinhole synthetic collimator. However it was still comparable to the resolution of the noise-free parallel-hole images, while the sensitivity, better by an order of magnitude, provides obvious advantages for the synthetic collimator.

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Fig. 14. Nine slices from the 3-D synthetic-collimator reconstruction of noise-free data collected by a detector with 2 mm pixel size.

Fig. 13. Noisy (a) 2-D synthetic-collimator reconstruction and (b) nine slices from the 3-D reconstruction for the 400-pinhole synthetic collimator.

E. Importance of High-Resolution Detectors for Synthetic Collimation A question that arises is whether the synthetic collimator could serve to improve clinical imaging systems employing lower resolution detectors. One disadvantage of our current CdZnTe technology is that it is difficult to produce detectors of sufficient size for the large field-of-view imaging commonly performed in nuclear-medicine clinics. If the synthetic collimator could provide advantages over parallel-hole collimators for lower-resolution detectors, it would have important implications both for our research systems that employ modular NaI cameras and for clinical imaging systems using large NaI crystals. In order to answer this question, we increased the simulated detector pixel size from 0.33 to 2 mm, similar to the resolution of an NaI detector. The 400-pinhole synthetic collimator with the same system parameters as in Section III-D was used to generate the projection data. The pinhole-detector distances were again 2, 3.5, 5, and 8 mm. Fig. 14 shows nine slices from the 3-D synthetic-collimator reconstruction of the noise-free data. It is clearly seen that the lower-resolution detector greatly reduces the resolution of the reconstructed images. There is almost no apparent difference between the slices taken at different distances in the direction

perpendicular to the face of the detector, and the visibility of the lesion appears greatly diminished when compared to Fig. 10(b). We concluded that the synthetic collimator requires high-resolution detectors, such as our CdZnTe cameras, for this configuration and this imaging task. However, it is possible that for some configuration and some imaging tasks the synthetic collimator will provide advantages over a parallel-hole collimator when NaI(Tl) detectors are used. This is an area we shall explore in future studies.

IV. CONCLUSIONS AND FUTURE DIRECTIONS We have introduced the concept of the synthetic collimator, studied a number of its properties, and compared its performance with the performance of parallel-hole collimators. The synthetic collimator consists of a physical aspect: a multiplepinhole aperture and a high-resolution detector; a motion aspect: the ability to change pinhole-detector distances; and an algorithmic aspect: a method for estimating the radiotracer distribution given the collected data set. We have demonstrated that synthetic collimators can be designed with superior spatial resolution or superior noise properties compared to parallel-hole collimators. Our studies have also established that it is possible to tomographically reconstruct data collected from only one projection direction if sufficient angular sampling is provided by multiple pinholes. The synthetic collimator thus offers the possibility of tomographic imaging in situations where rotation of the detector is difficult or impossible. These 3-D reconstructions are shown to produce images with much better contrast between lesion and background than is provided by either parallel-hole-collimated images or 2-D synthetic-collimated images. The reconstructions appeared to give good estimates of the true distribution up to the edge of the field of view which leads to optimism that the method will work for larger fields. However

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the full size of the field over which the synthetic-collimator data could be tomographically reconstructed was not determined in this study. We have shown some limitations of synthetic collimation. These include poor performance when few photons are collected or when a lower resolution detector is used. We note, however, that we have only explored a small portion of the parameter space, and one advantage of the synthetic collimator is the large number of parameters available that can be used to optimize imaging-system performance. Configurations possibly exist that would greatly reduce the problems we encountered. The exploration of the full parameter space of the synthetic collimator is a logical next step for this study. In order to carry out this exploration some additional tools are required. The first is a quantitative means of assessing image quality. In order to optimize a system some definition of optimality is required, and the qualitative methods used in the present study lead to uncertainty in ascertaining the performance of the imaging systems. A number of quantitative methods for measuring image quality have been proposed using human- and ideal-observer models [22]–[24]. These tools will be applied to the optimization of the synthetic collimator. Also necessary for the full exploration of the parameter space are methods for determining optimal solutions for computationally large models once the definition of optimality has been established. One reason more synthetic-collimator parameters were not studied was the computational burden required to simulate the system, even for the relatively small field of view we employed. An obvious way to increase our ability to study imaging systems is to bring more powerful computers on line. Another is to develop algorithms for more systematically exploring the parameter space. We are currently studying the use of genetic algorithms for providing more powerful means to optimize imaging systems, and we shall apply these algorithms to the synthetic-collimator problem. Finally, we shall seek new designs that further increase the capabilities of the synthetic collimator. Two such designs that we lacked the computation resources to explore are a rotating synthetic collimator and a multi-head synthetic collimator. These would improve the angular sampling, and presumably further increase resolution. One means of increasing the sensitivity would be to use more than one detector for a single aperture. The system modeled in this study employed detector motion to collect data from multiple pinhole-detector distances. However, since a detector allows a percentage of photons to pass through unperturbed, with that percentage dependent upon the thickness of the detector material, multiple pinhole-detector distances could also be achieved by the use of multiple detectors stacked at the desired pinhole-detector distances. Not only would this reduce the collimator complexity by removing the need for a motion system, but it would improve system sensitivity since the number of detectors could be increased to a point where virtually all photons transmitted through the collimator would be detected. An obvious problem with such a system is interdetector scatter. This effect has not been explored, but will be built into our

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