Slab-by-Slab Blurring Model for Geometric Point Response Correction and Attenuation Correction Using Iterative Reconstruction Algorithms

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1 2168 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL 45, NO. 4, AUGUST 1998 Slab-by-Slab Blurring Model for Geometric Point Response Correction and Attenuation Correction Using Iterative Reconstruction Algorithms Chuanyong Bai, Student Membel; IEEE ', Gengsheng L. Zeng, Membel; IEEE ', Grant T. Gullberg, Senior Membel; IEEE ', Frank DiFilippo? and Steven Mille? 'Department of Radiology, University of Utah, Salt Lake City, UT 2Picker International, Cleveland, OH. Abstract The distance-dependent geometric point response of a single photon emission computed tomography (SPECT) system and the attenuation effect of photons passing through the object are modeled in an iterative OS-EM reconstruction algorithm to improve both the resolution and quantitative accuracy of the reconstructed images. A specified number of neighboring vertical slices are grouped into a slab, and an efficient incremental slab-by-slab blurring model is introduced to accelerate the reconstruction. The advantage of the slab-byslab blurring model over the slice-by-slice model is that the computational time is reduced, while still maintaining the spatial resolution and quantitative accuracy of the reconstructed images. The application of this incremental slabby-slab blurring model with a slice-by-slice attenuation model to the image reconstruction of phantom, Monte Carlo simulated SPECT data, and patient data shows improved resolution and contrast over the images reconstructed without the corrections. The reconstruction is accelerated by a factor of about 1.4, and the projectionhackprojection operation is accelerated by a factor of about 5, using the slab-by-slab convolution implementation with 8 slices in a slab compared with the slice-by-slice convolution implementation. I. INTRODUCTION The geometric point response of a SPECT system is distance-dependent, and the width of the point response increases with increasing distance to the camera. This point response function of the system results in shape distortion and loss of resolution in the reconstructed images. The attenuation effect in SPECT results in quantitative inaccuracy. These two effects can be effectively and efficiently compensated by using a new blurring model proposed in this paper. The point response function of a SPECT system is fit to a 2D Gaussian function, of which the full-width at half maximum (FWHM) varies linearly with the distance from the point to the detector surface [l, 21. Efforts have been made to compensate for the system geometric point response and the attenuation effect based on modeling the geometric response and the attenuation effects and setting up a set of imaging equations: FX=P, where X is the source array, F is the The research work presented in this manuscript was partially supported by NIH Grant R01 HL39792 and Picker International. projector and P is the projection data. The purpose of this work is to correctly model the SPECT imaging system and solve for X. Due to the complexity of these equations, iterative reconstruction techniques are employed. Gullberg et a1 [3] proposed an iterative SPECT reconstruction using a ray-driven projector-backprojector which compensated for the attenuation effect. Formiconi et al [2] presented an iterative method that compensated for the 3D spatially varying point spread function in SPECT. Most current methods compensate for these two effects simultaneously. Tsui et a1 [4] incorporated a detector response based on attenuation and distance-dependent horizontal blurring within 2D transverse slices of the source volume. Zeng et a1 [5] incorporated a 3-D geometric point response and photon attenuation into a ray-driven projectorhackprojector and used an iterative EM algorithm for reconstruction. In recent years, many techniques have been used to reduce computational time in SPECT reconstruction. These techniques include: The iterative ML-EM reconstruction algorithm implemented as an ordered-subset version (OS-EM) [6, 71, the matrix rotation method [8, 91 incorporated with the OS-EM algorithm, the 3-pass shearing method with linear interpolation [ 101 used for rotation, and the layer-by-layer blurring [11,12,13] with small 2D blurring kernels which can be obtained by a least-squares method [13]. The application of these techniques significantly decreases the computation burden for SPECT image reconstruction [13]. Pan, et al [ 141 proposed a technique to compensate for the 3D point response in the frequency domain by using a raytracing technique without a rotator. In [ 141, the image array is divided into a set of planes parallel to the detector surface, which are used to calculate the distance-dependent blurring. Different number of planes can be used. Instead of using a ray-tracing method, in this paper we propose a new slab-by-slab blurring method to perform projection and backprojection. Each slab in thin technique contains a specified number of slices. Since slab-by-slab blurring, instead of slice-by-slice blurring, is implemented to model the geometric point response in a SPECT system, the number of convolutions for each projectionhackprojection operation is reduced, and as a result, the computational time is decreased /98$ IEEE

2 11. METHODS A projector-backprojector pair is developed to perform projection and backprojection computations for an iterative reconstruction algorithm which performs geometric point response correction (GPRC) and attenuation correction by modeling the spatial1.y varying geometric response of the collimator as well as the attenuation effect of photons as the photons travel through the object tissues. In this paper, the iterative reconstructiori algorithm was the iterative orderedsubset expectation maximization (OS-EM) algorithm. This algorithm was used to reconstruct SPECT projection data obtained from both parallel-beam and fan-beam collimators. To reconstruct the image from parallel projection data, the image volume and the attenuation map at each projection view were rotated so that the face of the voxelized image volume and the voxelized attenuation volume were parallel to the detector surface (Fig. 1). Therefore, we were able to use the shift-invariance property of the geometric point response function in a plane parallel to the detector surface [13]. Small 2D distance-dependent blurring kernels were computed only once as a function of distance from the slab to the detector surface for a particular SPECT collimator specification. For fan-beam collimators, the geometric point response function is not shift invariant in a plane parallel to the detector. However, under some reasonable assumptions [ 151, the function can be approxiniated as shift invariant. Warping does not change the property of shift-invariance of a point response function. Under these assumptions, both the image volume and attenuation map were rotated and warped [16]. Then the same reconstruction techniques used to reconstruct parallel-beam data were applied. A small cross-shaped 2D distance-dependent convolution kernel, obtained by a least-squares method [13, 171, was used 2169 to model the geometric response blurring. The kernel had five elements in a shape of a cross, with b, a, b horizontally and c, a, c vertically positioned, and a at the center. The convolution was implemented slab-by-slab instead of slice-byslice. A slice was defined as a vertical layer in the image may which was parallel to the detector surface. A slab was a group of a specified number of slices (Fig. 1). The number of slices in a slab was referred to as the slab size. For instance, when two slices were grouped into a slab, the slab size was two. The number of slices in a single slab could be adjusted. Since the quantitative accuracy of the reconstructed images is highly sensitive to the attenuation effect, the attenuation effect was modeled slice-by-slice. Activities in the slices contained in a slab were summed up for each projection bin to form the activity of the corresponding slab bin. This was done taking into account the effect of attenuation. The slab as a whole was then blurred slab-by-slab to model the geometric point response of the system. At the same time, the attenuation effect between the slabs was considered. A. Mathematical Expressions The details of implementing the projector using the slabby-slab blurring model are described here. For the results presented in this paper, the backprojector was implemented modeling only the geometric point response. At each projection angle, the voxelized image volume rotates with the detector, so the image face is parallel to the detector surface. In forming the projection from back to front, a collection of slices making-up one slab are summed for each projection bin, taking into account the attenuation effect [see (3)]. The distance-dependent 2D kernel is applied to the slabsum and the result is added to the result for the next slab, which is closer to the detector [see (5)]. Attenuation Activity Ixl one slab AI one slice XI a1 I Detector I Figure 1. Illustration (top view) of slabs, slices and pixels. A slice is a plane of pixels. A slab is a group of specified number of slices. Attenuation correction is performed slice-by-slice, and blurring is performed between slabs.

3 2170 Fig. 1 shows the relationship between the slab and the slices. Each slab is made up of 4 slices. The pixel activities (array x) are attenuated by the attenuation matrix a. In the implementation of the algorithm, pixel activities in each slice of a slab are attenuated, summed up, then stored as the pixel activities for that slab (matrix x>. The slab is then convolved with the convolution kernel H. The following gives the mathematical expressions for the method (for slab size equal to 4): Slice-by-slice blurring can be expressed as: p = (((Xl*hl +X2)*h2+X3)*h3+X4)*h4+..., (1) where, p is the projection, x, is the nth slice, h, is the blurring kernel for the nth slice, and "*" is the 2D convolution operator. Slab-by-slab blurring can be expressed as: where X, is the nth slab, H, is the blurring kernel for the nth slab, and ''* " is the 2D convolution operator. When the attenuation effect is considered, slab activity X, is computed in the following way: and the slab attenuation A, is A, = aril x an2 x an3 x an4, (4) where xni (for i=l, 2,3,4) is the array of activities of the ith slice in the nth slab, ani is the array of attenuation factors of the ith slice in the nth slab, and "x" is point-by-point multiplication. As a whole, the projection using the slab-by-slab blurring model is expressed as: x A,*H, + X4). (5) B. Projection Data and Reconstruction Three sets of projection data were studied. 1) Projection data of the Jaszczak resolution phantom were acquired with a Picker PRISM 3000XP three-detector SPECT system, using 140 kev high resolution parallel-beam collimators with 120 projection views over 360". The cameras rotated on a circular orbit with a radius of rotation of cm. A dose of 3.44 mci of Tc-99m was injected before scanning. For each projection view, data were stored in a 128 x 128 array. The reconstruction voxel size was x x cm3. 2) A set of Monte Carlo simulated projection data of the MCAT phantom with parallel-beam collimation was generated using the PHG software [IS]. Photon energy was set at 140 kev, with a 10 percent energy window. A total number of 28 M events were simulated and 19.7 M were detected. The projection data were 120 projection views of 64 x 64. The radius of rotation was 22 cm for a circular camera orbit. For this data set, the reconstruction voxel size was x x cm3. 3) Patient data were acquired with a Picker PRISM 3000XP three-detector SPECT system using cardio fan-beam collimators. The focal length was 65 cm. Cameras rotated on a non-circular orbit. A dose of 6 mci of Tc-99m was injected into the patient before scanning. The acquisition obtained 120 projections over 360" from 3 detectors at 40 stops of 1 minute per stop. The total scanning time was 40 minutes. The projection data were stored in a 64 x 64 x 120 short-integer array. The reconstruction voxel size was x x cm3. Images were reconstructed with both geometric point response correction and attenuation correction for a varying number of slices per slab. The number of subsets used in the OS-EM algorithm was 30 for 120 projection views. All reconstructions were performed on a SUN Enterprise 3000 workstation. C. Image Comparison Two methods were used to compare the images reconstructed using different slab sizes. The first method involved calculating the normalized rootmean-square-differences (NRMS). This gives the measurement of global difference between two images, Two reconstructed images are scaled to have the same image mean before calculating NRMS. Mathematically, NFWS is given as: where A, and B, are the intensity values of the corresponding voxels of the two images to be compared, N is the total number of voxels, and (A) is the image mean. The second method involved comparing the resolution and contrast of images, using profiles through reconstructed images RESULTS AND DISCUSSIONS Fig. 2 shows the reconstructions for the same transverse slice of the Jaszczak resolution phantom. In Fig. 2(a) to Fig. 2(d), the images were reconstructed with both geometric point response correction and attenuation correction, using the slabby-slab blurring model with slab sizes 1, 8, 32, and 64, respectively. Table 1 lists the NRMS between the reconstructed images and the image reconstructed with slab size 1, the reconstruction time, and the projectionhackprojection time per transverse slice, per iteration. Figs. 3(b) to 3(e) show the reconstructions for the same transverse slice of the MCAT phantom; 3(a) is that of the MCAT phantom. The number of slices per slab used in Figs. 3(b) to 3(e) are I, 8, 16, and 64, respectively. Table 1 lists the NRMS difference between the images constructed with slab size 1 and those with different slab sizes. Line profiles were drawn from pixels (1 1, 64) to (64, 12). Fig. 4 shows the reconstructions for one of the transverse

4 2171 slices of a female patient heart. In Figs. 4(a) to 4(d), the images were reconstructed with both geometric point response correction and attenuation correction, using the slab-by-slab blurring model with different slab sizes. Comparison of the NRMS and reconstruction time are listed in Table 1. The profiles demonstrate the similarity of the spatial resolutions of the images reconstructed with different slab sizes. Line profiles were drawn from pixels (14, 63) to (55,8). One would think that noise level would increase when geometric point respclnse correction was applied compared with that without. However, we observed that noise level decreased when GPRC was applied. A similar observation has been made by others [9, 191. Reconstructions with larger slab sizes seem to reverse this effect slightly because of the decreased number of blurring convolutions. Another possible reason might be that thl: reconstructions have slightly different rates of convergence for different slab sizes. The NRMS betwe1:n the reconstructed images is small when slab sizes less than 16 are used. This indicates that the reconstructions with different slab sizes (less than 16) are of little difference. However, we used 4 iterations for all the reconstructions. For this number of iterations, as slab sizes become larger (greater than or equal to 16), the NRMS between the reconstructed images increases. The line profiles of the reconstructed cardiac images using the slab-by-slab blurring model with different slab sizes (less than 16) differ very little, which means the spatial resolution of the images is not degraded much. Thus, the images are expected to have the sarne diagnostic value. The slab-by-slab method is based on the assumption that the point response function within a slab is almost the same. When the collimator has very high resolution, the decrease in resolution with increasing source distance from the collimator is small. This makes the slab-by-slab blurring model more efficient and accurate. This is also the reason that reconstructions with slab sizes 1 and 8 give similar results. However, when the slab size increases, this approximation becomes poor, resulting in inaccurate reconstructions as indicated by the NRMS in Table 1 and shown by the images. Our experiments suggest not to use a slab size larger than 8, otherwise, the geometric response correction is not modeled adequately. By using this new slab-by-slab blurring model, when the slab size increases from 1 to 8, the projectionhackprojection operation is accelerated by a factor of about 5. However, the reconstruction is accelerated only by a factor of about 1.4. The reason for this is that the bilinear interpolation used in the rotation operation is slow [lo]. The reconstruction can be further accelerated when faster rotators are used, and the slabby-slab blurring model will demonstrate more advantage over the slice-by-slice blurring model. IV. CONCLUSION A slab-by-slab blurring model for geometric point response correction that uses a small 2D convolution kernel can be more efficient than the slice-by-slice model while maintaining approximately the same quantitative accuracy and spatial resolution when slab sizes between 1 and 8 are used. Images reconstructed by using the slab-by-slab model with different slab sizes are expected to have similar diagnostic value. The advantage of using slab-by-slab blurring model is that the reconstruction time can be reduced. Table 11. Comparison of the reconstructed images with different slab sizes for the Jaszczak resolution phantom, MCAT phantom, and patient studies. Data are listed as NRMS/timel/time2, where NRMS is the normalized root mean square difference between the image reconstructed with the specified slab size and that with slab size 1; time1 is the total reconstruction time and time2 is the time for projector and backprojector operation per transverse slice per iteration in CPU seconds. All images were reconstructed with 4 iterations

5 2172 ACKNOWLEDGMENTS The authors thank the Biodynamics Research Unit and the Mayo Foundation for use of their Analyze software. We also thank Sean Webb for editing the manuscript. REFERENCES [1] T. Hebert, P. Murphy, W. Moore, R. Dhekne, R. Wendt, and M. Blust, Experimentally determining a parametric model for the point source response of a gamma camera, IEEE Trans. Nucl. Sci., vol. 40, pp , [23 A. R. Formiconi, A. Pupi, and A. Passeri, Compensation of spatial system response in SPECT with conjugate gradient reconstruction technique, Phys. Med. Biol., vol. 34, pp , [3] G. T. Gullberg, R. Huesman, J. Malko, N. Pelc, and T. Budinger, An attenuated projector-backprojector for iterative SPECT reconstruction, Phys. Med. Biol., vol. 30, pp , [4] B. M. W. Tsui, H. Hu, D. Gilland, and G. T. Gullberg, Implementation of simultaneous attenuation and detector response correction in SPECT, IEEE Trans. Nuc. Sci., vol. 35, pp , [5] G. L. Zeng, G. T. Gullberg, B. M. W. Tsui, and J. A. Terry, Three-dimensional iterative reconstruction algorithms with attenuation and geometric point response correction, IEEE Trans. Nucl. Sci., vol. 38, pp , [6] H. M. Hudson and R.S. Larkin, Accelerated EM reconstruction using ordered subsets of projection data, IEEE Trans. Med. Imag., vol. 13, pp , [7] C. L. Byrne, Block-iterative methods for image reconstruction from projections, IEEE Trans. Image Processing, vol. 5, pp , [8] Z. Liang, R. J. Jaszczak, T. G. Turkington, D.R. Gilland, and R. E. Coleman, Simultaneous compensation for attenuation, scatter, and detector response of SPECT reconstruction in three dimensions, Phys. Med. Biol., vol. 37, pp , [9] C. Kamphuis, F. J. Beekman, and M. A. Viergever, Evaluation of OS-EM vs. ML-EM for 1D, 2D and fully 3D SPECT reconstruction, IEEE Trans. Nucl. Sci., vol. 43, pp , [IO] E. V. R. Di Bella, A. B. Barclay, R. L. gisner, and R. W. Schafer, A comparison of rotation-based methods for iterative reconstruction algorithm, IEEE Trans. Nucl. Sci., vol. 43, pp , [ 113 J. W. Wallis and T. R. Miller, Rapidly converging iterative reconstruction algorithms in single-photon emission computed tomography, J. Nucl. Med., vol. 34, pp ,1993. [12] A.W. McCarthy and M. I. Miller, Maximum likelihood SPECT in clinical computation times using meshconnected parallel computers, [I31 G. L. Zeng, G. T. Gullberg, C. Bai, P. E. Christian, E Trisjono, E. V. R. DiBella, J. W. Tanner, and H. T. Morgan, Iterative reconstruction of fluorine-18 SPECT using geometric point response correction, J. Nucl. Med., vol. 39, pp , T. S. Pan, D. S. Luo, and M. A. King, Design of an efficient 3-D projector and backprojector pair for SPECT, Conference Proceedings of the 1995 International Meeting on Fully Three-Dimensional Image Reconstruction, Aix-la-Bains, France, pp , B. M. W. Tsui and G. T. Gullberg, The geometric transfer function for cone and fan beam collimators, Phys. Med. Biol., vol. 35, pp , 1990, 1161 G. L. Zeng, Y.-L. Hsieh, and G. T. Gullberg, A rotating and warping projector/backprojector for fan-beam and cone-beam iterative algorithm, IEEE Trans. Nucl. Sci., VO~. 41, pp , [17] C. Bai, G. L. Zeng, and G. T. Gullberg, Evaluation of small 2D convolution kernel for slice-by-slice blurring model of ultra-high-energy collimation, [Abstract] J. Nucl. Med. vol. 38, p. 214P, [18] S. Vannoy, The Photon History Generator, University of Washington Medical Center, Release 1.14b, [19] F. J. Beekman, C. Kamphuis, and M. A. Viergever, Improved SPECT quantitation using fully threedimensional iterative spatially variant scatter response compensation, IEEE Trans. Med. Imag., vol. 15, pp ,1996.

6 2173 (4 (b> (c> (d) Figure 2. Reccmtructions of the Jaszczak resolution phantom with different slab sizes. Projection data were stored in a 128 x 128 x 120 array in short integer. Images were reconstructed with an iterative OS-EM algorithm with 4 iterations for (a) and (b), 2 for (c) and (d). Geometric point response correction and attenuation correction were applied. (a) 1 slab=l slice; (b) 1 slab=8 slices; (c) 1 slab=32 slices; (d) 1 slab=128 slices. Reconstruction time per slice, per iteration: (a)17.0 CPU seconds, (b) 10.0 CPU seconds, (c) 9.34 CPU seconds, and (a) 9.20 CPU seconds. (a). (b) (c> (4 (e) Figure 3. Reconstructions of Monte Carlo simulated MCAT phantom data with different slab sizes. Projection data: 64 X 64 X 120, Images were reconstructed with an iterative OS-EM algorithm, with 4 iterations for (b), (c), and (a), 2 iterations for (e). Both geometric point response and attenuation effects were modeled. (a) true activity of the slice of MCAT phantom; (b) 1 rlab=l slice; (c) 1 slab4 slices: (d) 1 slabz32 slices; (e) 1 slab-64 slices. Reconstruction time per transverse slice, per iteration: (b) 3.39 CPU seconds, (c) 2.72 CPU seconds, (d) 2.59 CPU seconds, and (e) 2.38 CPU seconds. The line profiles were drawn from pixel (1 1,64) to pixel (64, 12), i.e. along the white line in (a), and the same lines in (b), (c), (d) and (e). The dotted line profile in each diagram is the profile of (b), which serves as a gold standard for comparison. (4 (b) (c) (4 Figure 4. Reconstructions of patient data with different slab sizes. Projection data: 64 x 64 x 120. Images were reconstructed with an iterative OS-EM algorithm with 4 iterations for (a) and (b), 2 iterations for (c) and (a). Geometric point response and altenuation effects were modeled. (a) 1 slab=l slice; (b) 1 slab=8 slices; (c) 1 slabm16 slices; (d) 1 slab=64 slices; Reconstiuction time per slice per iteration: (a) 3.56 CPU seconds, (b) 2.53 CPU seconds, (c) 2.48 CPU seconds, and (d) 2.40 CPU seconds. The line profiles were drawn along the white line in (a), and the same lines in (b), (c) and (a). The dotted line profile in each figure is the profile of (a), which serves as a gold standard.