726 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 51, NO. 3, JUNE 2004 Speed Up of an Analytical Algorithm for Nonuniform Attenuation Correction by Using PC Video/Graphics Card Architecture Junhai Wen, Member, IEEE, Zigang Wang, Member, IEEE, Bin Li, Tianfang Li, Member, IEEE, and Zhengrong Liang, Member, IEEE Abstract—A major task in quantitative SPECT (single photon emission computed tomography) reconstruction is compensation for object-specific attenuation, which is usually nonuniform. Mathematically this task is expressed as the inversion of the attenuated Radon transform. Novikov had derived an explicit inversion formula for the attenuated Radon transform for parallel-beam collimation geometry. In our previous work, we ex- tended his work to variable focusing fan-beam (VFF) collimators. A ray-driven analytical inversion formula for VFF reconstruction with nonuniform attenuation was derived. The drawback of ray-driven methods is that they are time consuming. In this work, we proposed a fast implementation method, which includes algorithm optimization and acceleration by the texture-mapping architecture of PC graphics/video card. We further investigated the noise properties and associated artifacts of the analytical inversion formula. The artifacts were remarkably reduced when more projections were sampled to mitigate the problem of wide bandwidth of the discrete Hilbert transform. The reconstruction from noisy data demonstrated the accuracy and robustness of the presented ray-driven analytical inversion formula with dramatic speed acceleration by the PC graphics/video card. Index Terms—Analytical reconstruction, nonuniform attenua- tion, PC video/graphics card, ray driven, speed up. I. INTRODUCTION I N single-photon emission computed tomography (SPECT) imaging, because of photoelectric absorption and Compton scatter, the gamma photons of the intravenously injected radiotracer are attenuated inside the body before arriving at the detector. Quantitative reconstruction of the radiotracer uptake concentration at any location inside the body requires accurate compensation for the body’s object-specific attenuation, which is usually nonuniform. Mathematically, the compensation and reconstruction task can be formulated as the inversion of the atten- uated Radon transform [1]. A great research interest in analytical inversion of the attenuated Radon transform for quantitative SPECT image reconstruction with nonuniform attenuation and parallel-beam geometry has been seen in the last decade. Arbuzov Manuscript received November 12, 2003; revised April 7, 2004. This work was supported in part by the NIH National Heart, Lung, and Blood Institute under Grant HL54166. J. Wen, Z. Wang, and B. Li are with the Department of Radiology, State University of New York, Stony Brook, NY 11794 USA (e-mail: wenjh@mil.sunysb.edu). T. Li is with the Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794 USA. Z. Liang is with the Departments of Radiology and Computer Science, State University of New York, Stony Brook, NY 11794 USA. Digital Object Identifier 10.1109/TNS.2004.829788 et al. [2] presented an inversion of the attenuated Radon trans- form, but their result was not cast in the well-established filtered backprojection (FBP) form [3], [4], which was later derived by Novikov [5]. In Novikov’s work, an explicit inversion formula was derived in the FBP form for the attenuated Radon transform with parallel-beam geometry. This formula was implemented and good reconstruction results were obtained [6]. Another version of the explicit inversion formula was later reported by Natterer [7], also for parallel-beam geometry. For many clinical applications, however, fan-beam and variable focusing fan-beam (VFF) collimation geometries are preferred. Fan-beam collimator improves count density and spatial resolution, as compared to parallel-beam collimator, for imaging small objects such as animals and human head and breasts. An analytical inversion of the attenuated Radon trans- form for fan-beam geometry was recently reported, with some assumptions and approximations [8]. For cardiac studies, how- ever, the fan-beam geometry encounters the truncation problem due to its limited acceptance angle across the field-of-view (FOV), which can cause artifacts in the reconstructed SPECT images. Variable focusing fan-beam collimator overcomes this truncation problem, while preserving the improved count density and spatial resolution [9]–[11]. In our previous work, we derived an approximate inversion formula for the attenuated Radon transform for both fan-beam and VFF collimator geometries and obtained very good reconstruc- tion results [12], [13]. An exact ray-driven analytical formula for fan-beam reconstruction with nonuniform attenuation was obtained in [14] and for VFF reconstruction with nonuniform attenuation in [15]. But the drawback of ray-driven methods is that they are time consuming. In this work, we propose a fast implementation method, which includes algorithm optimization and acceleration by the texture-mapping architecture of PC graphics/video card. We further investigated the noise properties and associated artifacts of the analytical inversion formula. II. THE RAY-DRIVEN METHOD FOR INVERSION OF THE NONUNIFORMLY ATTENUATED RADON TRANSFORM WITH VFF COLLIMATORS In SPECT imaging, the measured projection data at angle can be expressed as (1) 0018-9499/04$20.00 © 2004 IEEE