c154-FDK-TS-SCT(3)

小波分析论文

386IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 3, MARCH 2009where(7) and (8) , We list the following two properties of which will be used in the derivation of the FBP reconstruction formula: (9) andFig. 2. C , C , and C are direction curves on the unit sphere centered at the point x. These curves indicate the directions from which x are illuminated. A and A are the start and end positions of C , respectively, j 2 f1; 2; 3g. The arrows are from the start to the end positions. The dashed curves are on the back surface of the sphere, which would be invisible if the sphere is opaque. The dotted lines are through the center of the sphere O . (a) A 3-D view, and (b) the top view of the unit sphere., [12], [13]. We denote the intersecand respection points as and tively. For brevity, we use the notations . Then, the inter-helix PI-line for and can be represented as . The intersection points and are mapped onto the unit and respectively, for , as sphere for as and are the start point and end point of shown in Fig. 2. respectively. Next, we dene the inter-helix PI-intervals for . In fact, the inter-helix PI-arcs (refer to Fig. 7) can be determined with (3) and the corresponding inter-helix PI-interfor as vals. Denote the inter-helix PI-interval of , . For simplicity, we omit the variable in the functions to use , , directly instead of , and , respectively. We introduce the Hilbert transform operator for , and . It will become clear later that the index determines the inter-helix PI line along which the ltrawith tion is done. We dene for any given unit vector . Let (4) , depending on Then, we dene the Hilbert transform over the projection space , the mapping as follows:(10) Before stating the FBP reconstruction formula, we deneTheorem 2: [FBP reconstruction formula] Let parameterized by a function dard helix in and . Then, the inversion formula(11) be a stan(3), ,(12) holds for all malization condition , provided that the mapping ( ) satises the nor-Ij(13) The normalization condition (13) holds for(14) (5) Next, the backprojection operator is dened as The index determines the inter-helix PI line along which the ltration is done. Note that in (14), occurs only at the position or for the source . In that case, , which should be ruled out by the condition . Therefore, .(6)Authorized licensed use limited to: IEEE Xplore. Downloaded on March 9, 2009 at 03:06 from IEEE Xplore. Restrictions apply.