S rect distortions in single photon emission computed

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1 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 41, NO. 6, DECEMBER A Rotating and Warping Projector/Backprojector for Fan-Beam and Cone-Beam Iterative Algorithm G. L. Zeng, Y.-L. Hsieh, and G. T. Gullberg Abstract-A rotating-and-warping projector / backprojector is proposed for iterative algorithms used to reconstruct fanbeam and cone-beam single photon emission computed tomography (SPECT) data. The development of a new projector/ backprojector for implementing attenuation, geometric point response, and scatter models is motivated by the need to reduce the computation time yet to preserve the fidelity of the corrected reconstruction. At each projection angle, the projector/ backprojector first rotates the image volume so that the pixelized cube remains parallel to the detector, and then warps the image volume so that the fan-beam and cone-beam rays are converted into parallel rays. In our implementation, these two steps are combined so that the interpolation of voxel values are performed only once. The projection operation is achieved by a simple weighted summation, and the backprojection operation is achieved by copying weighted projection array values to the image volume. An advantage of this projector/backprojector is that the system point response function can be deconvolved via the Fast Fourier Transform using the shift-invariant property of the point response when the voxel-to-detector distance is constant. The fan-beam and cone-beam rotating-and-warping projector/backprojector is applied to SPECT data showing improved resolution. I. INTRODUCTION EVERAL papers have reported on methods to cor- S rect distortions in single photon emission computed tomography (SPECT) images caused by the spatially varying collimator geometric point response and scatter, in addition to the commonly recognized effects caused by attenuation [ 11-[ 151. Usually, these distortions are corrected by using an iterative reconstruction algorithm [ 161 which requires one projection and one backprojection operation per iteration. The attenuation, geometric response, and scatter are modeled in the projector/backprojector. It has been found that the 3-D implementation of collimator response and scatter is computationally very expensive [6]. Originally, we implemented a ray driven projector/ Manuscript received November 1993; revised May This work was supported in part by the National Institutes of Health under grant number R01 HL 39792, by the Whitaker Foundation, and by Picker International. The authors are with the Department of Radiology University of Utah, Salt Lake City, UT This paper was originally presented at the 1993 Nuclear Science Symposium and Medical Imaging Conference held on October 30-November 6, 1993 in San Francisco, CA. IEEE Log Number /94$ IEEE backprojector [171 which modeled both attenuation and the 3-D geometric point response [61. This proved to be computationally very intensive. For parallel geometry, the computational speed was improved by developing a projector/backprojector pair that rotates the image volume for each projection angle so that the projection operation can be achieved by a simple weighted summation, and the backprojection operation can be achieved by copying weighted projection array values to the image volume [ll]. The group at Washington University in St. Louis also implemented a rotating projector/backprojector for parallel geometry [SI, [ 151. For converging geometries, they suggested, but did not implement, using a warping of the image to convert the convergent rays into parallel rays [IS]. In this paper, an additional warping step is added to the rotating projector/backprojector which we developed in [ll]. The image volume is warped so that fan-beam and cone-beam rays are converted into parallel rays. A fast rotating-and-warping projector/backprojector is implemented for modeling both attenuation and geometric point response. This paper describes the implementation of the projector/backprojector algorithm and shows improved resolution when applied to SPECT data. 11. ROTATING-AND-WARPING PROJECTOR / BACKPROJECTOR A. Projection and Backprojection First, we consider the simple operations of projection and backprojection without modeling attenuation or collimator response. The unique features of the new projector/backprojector are that it rotates the image volume at each projection angle so that the front face of the voxelized cube is always parallel to the detection plane as shown in Fig. l(a>, and it warps the volume so that the fan-beam and cone-beam rays become parallel inside the warped volume as shown in Fig. l(b). In the fan-beam case, warping is performed horizontally. In the cone-beam case, warping is performed in both horizontal and vertical directions. The projections are computed by summing up the image volume layer-by-layer from front to rear. Here, a layer is defined as a one voxel wide slice in the image volume, parallel to the detection plane, as shown in Fig. 2. All the voxels on a layer have the same distance to the detector.

2 IEEE TKANSACTIONS ON NUCLEAR SCIENCE, VOL 41, NO 6, DECEMBER 1994 \\ View 2 or Fig. 1. (a) At each rotation angle, the image volume is rotated so that the front face of the cube is always parallel to the detection plane (viewed along the axis of rotation). (b) The image volume is warped so that all diverging rays become parallel inside the new volume (viewed along the axis of rotation). Step 1 ' rotating step 2 (tan-beam): warping!, a layer Step 2 (cone-beam): warping ' a layer Fig. 2. A layer is ii plane parallel to the detection plane. All the voxels on a layer have the same distance to the detector. In the warping step, [.he layers are expanded. Only the central area of the expanded layer is used in projection. The backprojection is achieved by copying the values in the projection bins to each corresponding voxel in the layer from front to rear. In this way, we are able to avoid the heavy burden of calculating the geometric weighting factors corresponding to voxel contributions to each detector bin. When the converging beaim volume is warped into a pseudoparallel beam volume, a Jacobian factor of l/cos 8 is introduced for each converging ray, where 8 is the angle between the ray and the pseudoparallel ray as shown in Fig. 3. Therefore, after the warping, the line integral for the pseudoparal.le1 ray is the same as the original line integral with a converging ray. We also notice that warping changes the number of the voxels on each layer and the ratio of these two numbers is the factor. In the backprojector, all the rays are scaled by l/cos 8 first and all the layers are scaled down in the warping step with detector Fig. 3. Cone-beam case. In the back projector, all the layers arc scaled down from the size of the detector.

3 ~ ZENG et a[.: ROTATlNG AND WARPING PROJECTOR / BACKPROJECTOK 2809 a scaling factor of (d/di2 as shown in Fig. 3, where D is the cone-beam collimator focal length and d is the distance from the focal point to the layer. In other words, the total count on the detector is P. After warping the total count in a layer is p with p = P(d/D)'. Before and after warping, the total count within the layer should remain unchanged. Therefore, the voxel values are scaled up by (D/d) for the fan-beam geometry and ( D/d2 for the cone-beam geometry. Both rotating and warping steps require interpolations between the voxels. However, in our implementation, these two steps are combined and only one interpolation is performed. B. Attenuation In order to model nonuniform attenuation, a map of the attenuation coefficient distribution volume is required. The volume is rotated and warped at each projection angle in the same way as described in Section II-A. The attenuated projection is evaluated by summing along rows of the attenuation weighted emission image from the front layer to the rear layer (note: not from rear to front). The attenuation weighting factor is calculated as follows: Simultaneously with the summation of the weighted emission data, the attenuation coefficient volume is summed layer-by-layer from front to rear. After calculating the sum up to the ith layer (note; one half of the attenuation coefficient in the ith layer is added to the cumulative sum) in the attenuation coefficient volume, the attenuation factor is determined by evaluating the exponential of the negative of the sum of attenuation coefficients. This factor is multiplied by the corresponding voxel value in the ith layer of the emission image volume to obtain the attenuation weighted emission data or that voxel. During backprojection, the attenuation factors are calculated in the same way, and these factors are used to multiply the corresponding projection data and adding the results to the appropriate layers of the image array. C. Geometric Point Response The spatially varying geometric point response function is shift-inuariant on a plane parallel to the detection plane [ll]. Taking advantage of the fact that all image layers are parallel to the detection plane, the geometric point response in a layer can be modeled as a 2-D convolution. Different layers have different 2-D convolvers, which can be analytically evaluated as the point response of a single collimator hole [18], [19]. An analytical formulation of the collimator response based on a circular collimator hole has shown to give good results [6], [lll. In our implementation, the 2D convolutions are realized in the frequency domain, taking advantage of the Fast Fourier Transform (FFT). The Fourier transforms of the 2-D convolvers are precalculated and stored for each depth. These transforms are given the name point response transfer functions. To compensate for the warping step, the width of the point response function is scaled up by D/d for each layer which is a distance d from the focal point as illustrated in Fig. 4. Fig. 4. Due to the warping operation, the point spread function becomes wider in the horizontal direction for fan-beam (viewed along the axis of rotation). For the cone-beam geometry, the point spread function becomes wider in both horizontal and vertical directions. D. Implementation Our projector/backprojector technique involves the following steps: Projector: 1) Rotate the image volume and the attenuation volume. 2) Warp the image volume and the attenuation volume. 3) Compute the 2-D FFT of each layer in the image volume. 4) Multiply each Fourier transformed layer by a precalculated point response transfer function according to the depth of the layer. 5) Compute the 2-D inverse FFT. 6) Sum up all the layers using weighting factors calculated from the attenuation volume. Bac kprojector: 1) Compute the 2-D FFT of the projection plane, and make n copies where n is the number of layers. 2) Multiply each of these n copies by the geometric point response transfer function corresponding to the depth of the layer. 3) Compute the inverse FFT of these n copies. 4) Add each of these n copies to its corresponding layer with the attenuation weighting factors calculated in the projector. 5) Unwarp the image volume back to normal. 6) Rotate the image volume back to 0" position and update the image METHODS The iterative emission EM-ML algorithm [16] was used for image reconstruction. The algorithm is given as where Xi is the ith voxel in the image volume, < is the jth projection bin, and Fi, is the contribution of the ith voxel Xi to the jth projection bin <. In (11, the summation over k is the projection operation and the summa-

4 tions over j and 1 are the backprojection operations. The summation over 1 is a hackprojector which back-projects a constant one. The algorithm with rota1:ing-and-warping projector/ backprojector was coded in :Fortran for distributed computing. Eight IBM RS computers that communicated with each other over a local-area network were used to reconstruct the image simultaneously, utilizing the PVM (parallel virtual machine) software. A. Fun-Ream Cardiac I'utient Study A PRISM 3000 three-detector SPECT system (Ohio Imaging of Picker International, Bedford Heights, OH) was used to perform the siniultaneous transmission and emission data acquisition [20]. Three fan-beam collimators were used. each having a focal length of 65 cm. A line sourcc filled with Tc-99m was mounted on the opposite site of detector # 1. During data acquisition, each detector rotated continuously over 360". Projection data were digitized into x 64 arrays. The projection bin size was cm and the image voxel size was also cm. The reconstructed image from the transmission data was used as the attenuation map to correct for the nonuniform attenuation in the emission image. The image was reconstructed in a 64 X 64 X 64 array. Each reconstruction was obtained after 100 iterations of the EM-ML algorithm. The depth-dependent point spread function and the attenuation map were precalculated and stored on disc. However, the rotation indices and interpolation weighting factors werc not stored but were calculated when needed. B. Cotic.-Beum Hofiaii Brain Phuntom Study A 3-D Hoffman brain phantom (Data Spectrum Inc., Hillsborough, NC) was used in this study. The diameter of the phantom was 18 cm and the length was 15 cm. The phantom was filled with 20 mci of Tc-99m. A PRISM 2000 two-detector SPECT system (Ohio Imaging of Picker International, Bedford Heights, OH) was used to acquire the data. Two cone-beam collimators were used, each with a focal length of 65 cm. During data acquisition, each detector rotated 180" with 60 stops. I'rojection data were acquired in 64 X 64 arrays. The projection bin size was cm and the image voxcl sizc was also cm. No transmission studies were performed. The attenuation map was assumed to be constant within the phantom. The reconstruction method was the same as that in the fan-beam study, except that, in the fan-beam code, warping was performed horizontally, while in the cone-beam code, warping was performed both horizontally and vertically. IV. Ri:sui.-r-s The reconstructions were performed on eight CPUs of the IBM RS/ cluster using the PVM library. The reconstruction time for a fan-beam study was seconds, and seconds for a cone-beam study. Fig. 5 shows the reconstructions from the fan-beam cardiac patient study. Fig. 5 (left) shows the reconstruc- Fig. 5. Fan-beam reconstruction of a patient cardiac study. A noncircular orbit was used to acquire the data. The number of iterations of the EM-ML algorithm was 100. Left: neither effects of attenuation nor the spatially varying point response function was corrected. Right: attenuation and point re\ponse were corrected. Fig. 6. Cone-beam reconstruction of the physical 3-D Hoffman brain phantom. A circular orbit was used to acquirc the data. The number of itcrations of the EM-ML algorithm was 100. Top row: neither attenuation nor point response function was corrected. Bottom row: attenuation and point response function were corrected. tions with the EM ML algorithm which uses the rotating projector/backprojector [ll]. Attenuation and geometric point response were not corrected. Fig. 5 (right) shows the reconstructions with the EM-ML algorithm which uses the rotating-and-warping projector/backprojector. Both attenuation and geometric point response were corrected. Fig. 6 shows the reconstructions from the cone-beam Hoffman brain phantom study. The images on the top row are the EM-ML reconstructions without corrections for attenuation and geometric point response. The images on the bottom row are the EM-ML, reconstructions with corrections for attenuation and geometric point response. It is observed that the new algorithm is able to reduce the attenuation artifacts, increase the image resolution, and decrease the noise. V. CONCLUSIONS In this paper, wc investigated a new approach to performing projection and backprojection operations for fan-beam and cone-beam geometries by rotating and warping the image volume. The warping step converts the fan-beam and cone-beam projection geometries into parallel geometry. The rotation of the image volume makes it so the voxelized image cube is always parallel to the detector plane. Thus, for modeling attenuation. the projection operation reduces to a simple weighted summation. Also, the rotation step makes it easy to effectively implement depth-dependent geometric point response

5 ZENG et al.: ROTATING AND WARPING PROJECTOR / BACKPROJECTOR 2811 correction by using the shift-invariant property of the point response function at a ked distance from the collimator and deconvolving the point response using the FFT. The goal of this research is to develop a fast projector/ backprojector to reconstruct fan-beam and cone-beam projections, which corrects for attenuation and geometric point-response distortions. The ray-driven projector/ backprojector takes about 2 weeks to reconstruct converging beam projections using 100 iterations of the EM-ML algorithm on the IBM 3090 [6]. For the same reconstruction problem the rotating-and-warping projector/backprojector reduces the reconstruction time to a few hours running on 8 IBM RS computers. The computation time for the rotating-and-warping operation is almost the same as the rotating-only operation. Warping and rotating saves overall computation time by speeding up the projection and backprojection calculations, The major computation time is spent in reformation of the image using the rotating-and-warping transformation, which requires a 3-D interpolation of the closest eight neighboring voxels. In rotating and warping the image volume, voxel indexes and weighting factors are continually recalculated for each index of the reformatted image array. These calculations could be done more efficiently by storing factors that require repeated arithmetic operations (especially multiplication) and trigonometric function calculations. The interpolation required in the rotating-and-warping operation causes a smoothing of the image. However, it has previously been recognized that the correction for geometric point response distortions effectively increases resolution without noise amplification [6]. Therefore, it may be that the smoothing caused by interpolation is masked by the smoothing cause by the modeling of the geometric point response. The rotating-and-warping approach has significant potential for implementation on a massively parallel computer such as MasPar computers [15]. We are presently evaluating this using a MasPar 64 x 64 computer. VI. ACKNOWLEDGMENT A grant of computer time from the Utah Supercomputing Institute, which is funded by the State of Utah and the IBM Corporation, is gratefully acknowledged. The authors thank Dr. Stefan0 Foresti for his assistance in programing the cluster computers. The authors thank Paul Christian for acquiring the patient and phantom data. Also, the authors thank Biodynamics Research Unit, Mayo Foundation for use of the Analyze software package. REFERENCES [l] B. M. W. Tsui, H. B. Hu, D. R. Gilland, and G. T. Gullberg, Implementation of simultaneous attenuation and detector re- sponse correction in SPECT, IEEE Trans. Nucl. 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