小波分析论文

ZHAO et al.: A FILTERED BACKPROJECTION ALGORITHM FOR TRIPLE-SOURCE HELICAL CONE-BEAM CT385counterpart of the Tam–Danielsson window. Also, the role of the inter-helix PI-lines in triple-source helical CBCT is similar to that of the PI-lines in single source helical CBCT. In this paper, we present an exact FBP algorithm for triplesource helical CBCT. The algorithm also uses data from the three inter-helix PI-arcs associated with the inter-helix PI-lines and corresponding Zhao windows. In comparison with our previous triple-source BPF algorithm, this triple-source FBP algorithm is advantageous in terms of image reconstruction speed when the parallel implementation is in place. Therefore, this algorithm is technically and clinically very attractive. The paper is organized as follows. In Section II, the FBP formula is derived by the roadmap in [10]–[13], [48], [49] and the geometric relations specic to triple-source helical CBCT. Section III describes the simulation results. In Section IV, relevant issues are discussed and conclusions are drawn. II. FBP FORMULA FOR TRIPLE-SOURCE CBCT Let a trajectory be a piecewise differentiable curve in described by , , and an innitely differentiable real . is integrable function with a compact support and . The cone-beam transform the difference of sets along is dened by (1) is the direction of a ray emitted from the source , . Let be an object density function to be reconstructed. Assume that this function is smooth and vanishes outside the object cylinder whereFig. 1. Illustration of a triple-source helical cone-beam scanning. The three X-ray sources are rotated around the z -axis along the helices a (t), a (t), and a (t), respectively. The helices a (t), a (t), and a (t) are on a cylinder of radius R. An object to be reconstructed is conned within a cylinder of radius R , where radius R < R. Parameter h denotes the pitch of each helix. The inter-helix distance along the z -axis is h=3 between neighboring helices.(2) where is the radius of the object cylinder and the radius of the scanning cylinder. The scanning cylinder refers to the cylindrical volume conned by the helices dened by (3). , the triple-helix In the Cartesian coordinate system trajectories can be expressed as(3) where stands for the distance from each X-ray source to the rotation axis (i.e., the -axis), the pitch of each helix, and the rotation angle. Fig. 1 illustrates the triple-source helical CBCT conguration. Previously, we dened the inter-helix PI-lines and extended the traditional Tam–Danielsson window into the Zhao window in the case of triple helices [12], [13]. For convenience, here , we restate the Zhao window as follows: for each source the corresponding Zhao window is the region on the surface of the scanning cylinder bounded by the nearest helix turn of and the nearest helix turn of , [12], [13]. More specic denition of the Zhao Window is listed in Appendix II. Also, recall that an inter-helix PI-linefor and , , is the line that (1) intersects at one point and at the other point, (2) the absolute difference of the angular parameter values at [12], [13]. The dethe two intersection points is less than nition of inter-helix PI-line is given in Appendix I. In fact, the inter-helix PI-lines are a type of R-lines [7], [28]. We already proved the existence and uniqueness properties of the inter-helix PI-lines, which can be summarized as the following theorem [12], [13]: , there exists one and Theorem 1: Through any xed only one inter-helix PI-line for any pair of the helices from the three helices dened by (3). In the triple-helix case, there are three inter-helix PI-lines and three corresponding inter-helix PI-arcs for a xed whose end points are the intersection points of the inter-helix PI-lines with the corresponding helices. The three inter-helix PI-arcs represent the source trajectory segments from which the sources illuminate the point . Without loss of generality, we assume that the sources and the detectors are translated upwards and rotated counter-clockwise when being observed from the top. To describe the data completeness condition for a xed , we use the Orlov method, i.e., consider direction curves on the unit sphere [29]. As shown in Fig. 2, the direction curves , , on the unit sphere centered at correspond to the source scanning angular ranges for (refer to [12], [13] for details). It is a property of the inter-helix PI-line and that the intersection points represent for the beginning position of the source illuminating from and the end position of the other source illuminating fromAuthorized licensed use limited to: IEEE Xplore. Downloaded on March 9, 2009 at 03:06 from IEEE Xplore. Restrictions apply.