SPECT (single photon emission computed tomography)

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1 2628 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006 Detector Blurring and Detector Sensitivity Compensation for a Spinning Slat Collimator Gengsheng L. Zeng, Senior Member, IEEE Abstract This paper extends the well-known frequency-distance relation to a rotating-spinning parallel-slat collimator SPECT system, where the projection data are in the form of weighted planar integrals and the detection sensitivity function is inversely proportional to the distance between the point source and the detector. The three-dimensional (3D) Radon inversion formula is used in image reconstruction. The 3D backprojection is implemented as Marr s two-step two-dimensional (2D) backprojectors. The frequency-distance relation is applied in the second step of the 2D backprojection. Both the collimator blurring and distance dependent sensitivity effect are compensated for. Index Terms Analytical image reconstruction algorithm, frequency-distance relation, SPECT. I. INTRODUCTION SPECT (single photon emission computed tomography) cameras most commonly utilize parallel-hole collimators, which measure line integrals of the object. A complete projection data set is acquired when the detector rotates 180 around the object. In a parallel-hole collimator, the detection sensitivity is fixed, and does not vary with the distance from the detector. In SPECT, if a slat collimator is used, the planar integrals of the object can be measured. Due to the nature of planar integrals, the projection data are incomplete if the collimator does not spin by itself. Therefore, collimator spinning is required to guarantee a complete set of measurements. There are two ways to implement a spinning slat collimator. One approach is shown in Fig. 1, where a spinning parallel-slat collimator is attached to a conventional SPECT camera. Fig. 2. shows an alternative system, where the detector simultaneously spin with the collimator, and there is no relative motion between the collimator and the detector. The use of a slat collimator was first suggested by Keyes in 1975 [1]. In the early 1990 s Webb et al.. actually built a rotating slat system, similar to the one shown in Fig. 1 [2]. They mounted a rotating slat collimator on a conventional Anger SPECT camera. The configuration shown in Fig. 2 was adopted when we developed our CZT SPECT camera [3]. Since the CZT detectors are expensive, this configuration was designed to obtain a relatively large field of view in spite of having a relatively small detector. Compared with the configuration shown in Fig. 1, the main differences are the shape and size of the SPECT Manuscript received February 2, 2005; revised July 21, This work was supported by NIH Grants EB and EB The author is with the Department of Radiology, Utah Center for Advanced Imaging Research, University of Utah, Salt Lake City, UT USA ( larry@ucair.med.utah.edu). Digital Object Identifier /TNS Fig. 1. A rotating and spinning SPECT system with a slat collimator. Fig. 2. Another rotating and spinning SPECT system with a slat collimator and a rectangular detector. detector, since our only concerns were developing image reconstruction algorithms. As a result, the detection sensitivity functions are different for these two configurations. However, they provide the same spatial resolution if they have the same collimator slab length and slab gap as illustrated in Fig. 3. For both of the parallel-hole collimator and the parallel-slat collimator, the detection spatial resolution in terms of FWHM (full width at half maximum) is linearly proportional to the distance from the point source to the detector (see Fig. 3) [4]. One of the goals of this paper is to compensate for this distance dependent collimator blurring effect. In a parallel-hole collimator, the detection sensitivity is fixed, and does not vary with the distance from the detector. On the other hand, the detection sensitivity for the parallel-slat collimator is inversely proportional to the distance to the detector [5]. The second goal of this paper is to normalize this spatially varying sensitivity function via a filtering technique in the frequency domain /$ IEEE

2 ZENG: DETECTOR BLURRING AND DETECTOR SENSITIVITY COMPENSATION 2629 Fig. 4. In the 2D frequency domain, the negative slope corresponds to the signed distance in the spatial domain. Fig. 3. The collimator blurring effect worsens as the distance to the detector increases. phantom experiment are presented. Finally, summary remarks are given in Section V. Lodge et al. [5] investigated some advantages and disadvantages of using a slat collimator in SPECT. The advantages of using a slat collimator are mainly due to the advantages of planar integral measurements. The most attractive aspect of planar integral measurements (also known as 3D Radon transform) is the well-known local tomography property [6]. If a small region-of-interest (ROI) is always in the field of view of the detector, the ROI can be exactly reconstructed if planar integral data are measured. This small ROI may not be exactly reconstructed from line integral data if the data are truncated. Another advantage of using planar data measurements is that the reconstructed image has a better signal-to-noise ratio than the image reconstructed from line integral data, if the object is small. The slat collimator SPECT has some disadvantages. For example, it also requires spinning in addition to the conventional rotation; the slat collimator data contain more scattered photons than parallel-hole collimator data; and the image reconstruction algorithms are in general more sophisticated for planar data than for line integral data. In SPECT, the distance dependent collimator blurring effect is usually compensated for by using an iterative algorithm [7] [13]. The frequency-distance method, which is an analytical method to compensate for collimator blurring, was pioneered by Edholm et al. [14] and Lewitt et al. [15]. The frequency-distance method has been shown effective, and many improvements and applications have been made [16] [20]. This paper extends the frequency-distance principle to parallel-slat collimation imaging geometry, and uses the principle to compensate for the detector blurring and varying sensitivity function. This paper is organized as follows. We review the theory of the frequency-distance principle and the image reconstruction algorithm for the 3D Radon transform in Section II. The proposed collimator blurring correction technique of applying the frequency-distance principle to the 3D Radon reconstruction is also described. In Section III and IV, computer simulations and II. METHODS A. Theory Review The frequency-distance relation [14], [15] is used as the main tool for collimator blurring and detection sensitivity compensation in this paper. This section briefly reviews the frequency-distance relation as follows. In 2D tomography, the Radon transform (also known as the sinogram) of an object is denoted as, where is the detector rotation angle and is the projection location on the 1D detector. Let the 2D Fourier transform of the sinogram be In fact, the Fourier transform with respect to is a Fourier series expansion because is periodic with period. In this 2D Fourier domain, the frequency components along a line passing through the origin and with a slope - are mostly contributed by the activities of the object at distance : where is a signed distance that can be positive or negative as shown in Fig. 4. The zero distance corresponds to the distance from the axis of rotation to the detector. Positive distances are in the region labeled far, and negative distances are in the region labeled near. Equation (2) is referred to as the frequency-distance relation. This relationship has been successfully applied for compensation of the depth-dependent parallel-hole collimator blurring [14]. (1) (2)

3 2630 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006 If the detector s point response function at distance is and its Fourier transform with respect to is the transfer function, the ideal transfer function that compensates for the point response function would be. In the 2D Fourier domain of the sinogram, the point response compensation could be implemented as (3) When, let because the tomographic filters such as the ramp-filter or the second-order derivative filter have a DC frequency component of zero value. Note that when (3) is actually implemented, regularization must be used for noise control and stability considerations. B. Application to Planar Integral Data For a rotating and spinning SPECT system mounted with a parallel-slat collimator, the projection data are in the form of planar integrals of the object. Thus the frequency-distance relation for 2D line-integrals cannot be directly applied here. However, if the projection data are approximated as unweighted planar integrals, the 3D Radon inversion formula can be used to reconstruct the image [6]. The 3D Radon inversion algorithm consists of a second-order derivative procedure and a 3D backprojection: where is the planar-integral data at detector orientation is the unit sphere, and is the reconstructed 3D image. Notice that for 2D line-integral data, the usual filtered backprojection (FBP) algorithm is a ramp filtering followed by backprojection. The ramp-filtering can be decomposed into a derivative operation and a Hilbert transform. In general, for -dimensional Radon inversion, the image can be reconstructed by performing an th order derivative, a Hilbert transform, and a backprojection if is even [6], [21]. When is odd, the image can be reconstructed by performing an th order derivative and a backprojection (without the Hilbert transform). Using Marr s two-step backprojection technique, the 3D Radon backprojection can be decomposed into: first backprojecting the 1D planar-integral data into 2D planar views around the object, and slice-by-slice backprojecting the 2D views into a 3D image volume, as illustrated in Fig. 5 [22]. In both steps, a 2D imaging model with parallel line-integral data is assumed, and backprojection is along parallel lines in a 2D plane. Mathematically speaking, using two 2D backprojections is equivalent to performing a single 3D backprojection. The main advantage of using two 2D backprojections is its computation efficiency. The computation complexity of using two 2D backprojections is, while the computation complexity of a direct 3D backprojector is. (4) Fig. 5. Two-step backprojection. Step 1: Backproject to 2D pseudo-detectors. Step 2: Slice-by-slice backprojection. The dots on each vertical great circle are used to generate a backprojected 2D image. These 2D images are similar to the planar images acquired via a conventional SPECT system with a parallel-hole collimator. It is recognized that the second step of Marr s backprojection is the same as the parallel backprojector in 2D imaging, and it carries the distance information. The 2D frequency-distance relation is thus applied in the second step of Marr s backprojector. C. Algorithm Before image reconstruction, the transfer function of the rotating/spinning parallel-slat imaging system is determined as, which is the 1D Fourier transform of the system s point response function. The point response function models both the distance dependent collimator blurring and distance dependent detector sensitivity function. Here is the signed distance from the point source to the detector. An example of the point response function is shown in Fig. 6. The FWHM is a linear function of. The total area underneath the curve of for a given is inversely proportional to the unsigned distance from the point source to the detector (i.e., the sum of and the distance from the axis of rotation to the detector). The procedure to determine the distance-dependent blurring transfer function consists of the following steps: Step 1. According to the collimator design, the collimator s open angle (see Fig. 3) is determined. In the computer simulation,. In the phantom experiment,.

4 ZENG: DETECTOR BLURRING AND DETECTOR SENSITIVITY COMPENSATION 2631 Fig. 6. The point response function for computer simulation. The point blurring worsens as the distance D increases. Fig. 7. Computer simulation setup. Step 2. We assume that the system s point response function is a Gaussian function with a Standard Deviation, where is proportional to the distance times the tangent of half of the open angle. Step 3. The system blurring function is the Fourier transform of with respect to the variable. Since is a Gaussian with respect to is Gaussian with respect to.in, the Standard Deviation is. The image reconstruction procedure is as follows. First, calculate the second-order derivative of the projection measurements with respect to, obtaining. Second, is scaled by, because in the backprojection integral with defined as. Third, perform a 2D backprojection at each fixed SPECT view angle, obtaining a set of 2D images, where the -axis is the axis of the detector rotation. The backprojection is calculated as where is a raised-cosine window for regularization with being the natural frequency defined in, and being the order of the window function. In fact, this window function is a lowpass filter. When ; when (the Nyquist frequency),. The bandwidth of the lowpass filter decreases as increases. The choice of is mainly determined by the projection data statistics. Sixth, compute the 2D inverse Fourier transform with respect to and, obtaining Seventh, perform a 2D backprojection with each fixed (that is, perform slice-by-slice backprojection), resulting in the final reconstructed 3D image volume : (8) (9) (5) (10) Fourth, for every fixed, take the 2D Fourier transform with respect to and, obtaining with Fifth, use the frequency-distance relationship to compensate for the distance dependent collimator blurring and distance dependent sensitivity decay, obtaining (6) (7) III. COMPUTER SIMULATIONS The computer generated phantom was a large uniform sphere of density 1 and radius 50 units, that contained two hot small spheres and two cold small spheres (see Fig. 7). The hot spheres had a density 2 and the cold spheres had a density 0. One of the hot/cold spheres had a radius of 6 units, and one of the hot/cold spheres had a radius of 2 units. In order to verify an FBP reconstruction algorithm, we selected a phantom with a large constant region because an inappropriate filter in the FBP algorithm causes the flat region to curve up or curve down. We would like to verify whether the algorithm could reconstruct small details in the object, thus a small hot lesion and a small cold lesion were selected in the phantom. The linear dimension of a detector pixel

5 2632 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006 The reconstruction without compensation used the same reconstruction algorithm as described in Section III.C, except that (7) was replaced by (11) Fig. 8. The central (z =0) slice of the reconstructed image. (a) Reconstruction without compensations. (b) Reconstruction with detector blurring and sensitivity compensations. (c) Deblurring reconstruction without lowpass windowing. The images (d, e, and f) in the second row are reconstructed with the same methods as in the first row, except that noisy projection data are used. This equation means that the planar-integral projection data were only filtered by the tomographic filter, that is, the second-derivative operator; no deblurring filtering was applied. The image resolution and uniformity improvements made by applying the frequency-distance relation can be easily observed from the profiles in Fig. 9. Small lesion contrasts were evaluated and compared for reconstructions with and without the proposed deblurring method. The contrast is defined as (12) Fig. 9. Profiles drawn along the central vertical line in Figs. 8(a) and (b). (a) Without compensation. (b) With compensation. was defined as one unit. The distance from the detector to the axis of detector rotation was 95 units. The collimator open angle (see Fig. 3) was 8.6. The projection data were generated analytically and were weighted according to a sensitivity function of, where is the signed distance and is the unsigned distance from the axis of rotation to the detector. No noise was added to the data. The parameter in (8) was 2. The purpose of this window function (or lowpass filter) (8) was to regularize the noise and artifact amplification due to the second derivative tomographic filtering and deblurring filtering. There were 128 detectors on the 1D detector, 64 spin angles over 180, and 128 rotation angles over 360. The reconstructed image was stored in a array. The central slice of the reconstruction is displayed in Fig. 8(b). The same slice of a reconstruction without compensation is displayed in Fig. 8(a) for comparison. A profile (along the -axis) comparison is given in Fig. 9. Without deblurring, the contrast values for the small hot and cold lesions are 0.08 and 0.07, respectively. With deblurring, the contrasts for the small hot and cold lesions are improved to 0.15 and 0.16, respectively. The true contrasts for both are 1. The first row of Fig. 8 shows the reconstructed images using noiseless projection data. We would like to emphasize the necessity of using a lowpass window, for example, the one given in (8), because this deblurring method amplifies the noise or errors. The deblurring filter is a high-pass filter because the transfer function in (7) tends to zero at high frequencies. Thus the deblurring procedure is ill-conditioned. Even when the projection data are noiseless, computer round-off errors and high-frequency components in the projection data are amplified by the filter. Fig. 8(c) shows the deblurring reconstruction without using the lowpass window. The image is noisy and unrecognizable, even though the projection data are noiseless. The second row of Fig. 8 shows the same reconstructions as in the first row, except that the projection data were corrupted by Poisson noise. The total projection photon count in the projection data was. The central uniform cubical regions in the reconstructed images were used to measure the noise-to-signal ratio (NSR), which is defined as the Standard Deviation divided by the mean. The NSR for the regular Radon inversion algorithm was 6.26, and the NSR for the deblurring algorithm was IV. PHANTOM EXPERIMENT A line-source filled with 1 mci Tc-99m was used in a phantom experiment with a prototype spinning CZT gamma camera [23], [24]. The experimental setup is illustrated in Fig. 10. The line source was 6.5 cm away from the axis of camera rotation. The closest distance between the line source and the camera was 5 cm. The gamma camera had 192 detection pixels. The central 128 pixels of projection data were used in image reconstruction. The camera rotated around the line-source phantom 360 stopping at 120 SPECT views. At

6 ZENG: DETECTOR BLURRING AND DETECTOR SENSITIVITY COMPENSATION 2633 Fig. 11. Modified sinograms before the second backprojection (A) with collimator blurring and sensitivity corrections, and (B) without collimator blurring and sensitivity corrections. Fig. 10. Phantom experiment setup. each SPECT view, the camera spun continuously and the projection data were binned into 512 spin-angles. These 512 spin-angles were later rebinned into 128 spin-angles for image reconstruction. The image was reconstructed in a array. The detector was 53 mm wide, and the detector pixel size was 1.8 mm 53 mm. The distance between two adjacent slats was 1.8 mm. The image voxel size was 1.8 mm 1.8 mm 1.8 mm. The collimator slat length was 4 cm. The algorithm presented in Section II in this paper was used for image reconstruction. The main difference between our computer simulation and phantom experiment was that the data in the phantom experiment were noisy and regularization was required for the second order derivative operation in the reconstruction algorithm. The parameter in (8) was 4. An image was also reconstructed using the conventional Radon inversion formula (4) without the detector response correction; however, the same noise regularization window function (8) was used in this noisy data reconstruction. The second order derivative operation was implemented as finite difference (13) where is the original projection and is the approximation of the second order derivative of. After the second order derivative operation and the first backprojection (that is, the view-by-view 2D backprojection) of the two-step backprojection operation, modified sinograms were obtained for each slice. The modified sinograms are not line-integrals of a slice of the object, but look like sinograms in appearance. The modified sinogram of the line-source phantom study is shown in Fig. 11(B) for slice 84. The modified sinograms were processed with the proposed frequency-distance technique for collimator blurring and sensitivity corrections, and the processed modified sinogram is shown in Fig. 11(A) for slice 84. It is observed that in the unprocessed modified sinogram Fig. 11(B), the sinewave becomes wider and dimmer as the object is distant from the detector, while in the processed modified Fig. 12. Reconstructed line-source images (A) with collimator blurring and sensitivity corrections, and (B) without collimator blurring and sensitivity corrections. Top row: cross sectional views; Middle row: sagittal views; Bottom row: profiles drawn horizontally in the sagittal views. sinogram Fig. 11(A), the sinewave has almost the same width for far and near object locations, and the intensity is almost uniform. The images (see Fig. 12) were finally reconstructed after performing the second backprojection (that is, the slice-by-slice 2D backprojection) of the two-step backprojection operation. It is observed that the reconstructed line-source had a more circular cross-section (see the first row of Fig. 12) and better spatial resolution. The FWHM (Full Width at Half Maximum) value for the deblurred reconstruction of the line source was 21 mm, while the FWHM value for the reconstruction without the deblurring procedure was 28 mm. However, the resolution recovery filter caused some ripples in the reconstructed image; this is better seen in the profiles.

7 2634 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006 V. CONCLUSION The frequency-distance principle was originally developed for the 2D Radon transform. The principle was established on the 2D Fourier transform of the sinogram, and the frequency components in the 2D Fourier domain on a straight line with a slope - and passing through the origin are dominantly contributed by the activities at a signed distance from the detector. For a rotating and spinning SPECT system that uses a parallel-slat collimator, the projection data are weighted planar integrals of the object. This system has a similar collimator blurring effect to that in a parallel-hole collimator system. The detection sensitivity of this system is inversely proportional to the unsigned distance from the point source to the detector, while for a parallel-hole collimator system the sensitivity function does not vary with the distance. The 3D Radon inversion formula is used to reconstruct the image. The 3D backprojection is implemented as Marr s twostep 2D backprojectors [22]. The frequency-distance principle is applied in the second 2D backprojectors. The distance dependent detector blurring effect and distance dependent detector sensitivity decay are compensated for simultaneously. However, it is unknown whether the frequency-distance principle can be applied if the direct 3D backprojection method is adopted during image reconstruction. The main drawback of the proposed analytical deblurring method is its noise amplification. This is a concern when real projection data and noisy simulation data are used. The noise regularization method lowpass filtering reduces the noise, but degrades the overall image resolution as well. It is an important research topic to investigate methods that perform distance-dependent deblurring while maintain the overall spatial resolution. Iterative methods seem to be effective in reaching this goal. 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