小波分析论文

ZHAO et al.: A FILTERED BACKPROJECTION ALGORITHM FOR TRIPLE-SOURCE HELICAL CONE-BEAM CT389Fig. 5. Representative reconstruction results with the Shepp–Logan phantom in the Cartesian coordinate system. (a)–(c) The reconstructed slices of the 3-D Shepp–Logan phantom at x ,y : , respectively. These results are obtained by interpolating the images in the inter-helix PI-line , and z based coordinate system to Cartesian coordinates. The display window is [0.99, 1.05]. Typical proles are shown along the lines specied by (d) x and z , (e) y and x : , and (f) z : and x .= 0 cm= 0 cm = 0 cm = 0 cm = 03 2 cm= 03 2 cm = 03 2 cm= 0 cm= 0 cmStep 1) Differentiate each projection with respect to variable , keeping the directions of involved X-rays the same for each differentiation operation. , perform the Hilbert transStep 2) For each form of the differentiated projection data along the direction of the cone-beam projection of the on the corresponding detector plane. Step 3) Backproject the ltered data at on the inter-helix PI-line to obtain . Step 4) Transform the reconstructed image from the inter-helix PI-line based coordinate system into the Cartesian coordinate system. The linear interpolation technique is used. In fact, one does not necessarily need to backproject the ltered data onto the inter-helix PI lines. One can directly backproject onto an image matrix and needs no rebinning. This is one of the main advantages of FBP algorithms over BPF algorithms. Backprojecting the ltered data onto the inter-helix PI lines is one of possible ways, which is similar to backprojecting the ltered data onto the PI lines in [5]. In the simulation, three rectangular planar detectors were used. Fig. 3 shows a rectangular planar detector corresponding . The detector was so arranged that the line to the source through the source and the detector center intersected the -axis and detector perpendicularly. The parameters used in the simulation are listed in Table I. The ltered data are backprojected onto 512 inter-helix PI-lines for a xed endpoint. There were 512 xed endpoints for one reconstructed inter-helix PI-linesurface, and 1536 reconstructed inter-helix PI-line surfaces in total. Both the 3-D Shepp–Logan phantom [37] and Defrise disk phantom [38] were used. These phantoms were respectively located at the center of the global coordinate system - - within the same spherical support. Fig. 4 presents the reconstructed images of the 3-D Shepp–Logan phantom in the inter-helix PI-line coordinates system. Fig. 5 gives the images reformatted into the Cartesian coordinate system. Fig. 6 shows the results for the Defrise phantom in the Cartesian coordinate system. All of the reconstructed images are in excellent agreement with the ideal phantom slices. IV. DISCUSSIONS AND CONCLUSION Although the symmetrical arrangement of triple sources is preferred, it is not a necessary condition for deriving the triplesource FBP formula (12). For a more general conguration(21) and , the where triple-source FBP formula (12) still holds exactly. We can also extend the triple-source FBP formula (12) to sources, perform exact reconstruction in the case of . Let -source helical loci be(22)Authorized licensed use limited to: IEEE Xplore. Downloaded on March 9, 2009 at 03:06 from IEEE Xplore. Restrictions apply.