小波分析论文

ZHAO et al.: A FILTERED BACKPROJECTION ALGORITHM FOR TRIPLE-SOURCE HELICAL CONE-BEAM CT387Proof: In the following proof, we always assume and . From [3], [48], [49], we have (15) Extending the right side of (14), we have(16) Note that (17) from (16) we have (18) Thus, the validity of (13) is equivalent to proveIj(19) Let us prove (19) for rst. Following from the contifor nuity of j and the change of variable each integrand, we have (20) shown at the bottom of the page, where we have used and since the givenlies on the inter-helix PI-segments and . Equation (20) leads to (19). Derivations analogous to (20) have also appeared in the literature [10], [11]. and . That Similarly, we can prove for (19) for completes the proof of Theorem 2. Although the above proof is based on Tuy’s formula, the same or similar results may be derived from other classic and recent ndings on exact conebeam reconstruction [3], [31]–[34], [46], [47]. Let us compare this FBP algorithm with our previous triplesource BPF algorithm [14], [15]. Both the algorithms are exact, based on the Zhao windows and the inter-helix PI-lines. It was reported that the FBP algorithm has some merits [8], [32], [33] in comparison with the BPF algorithm for single-source case. The triple-source FBP algorithm maintains these merits in comparison with its BPF counterpart. The most attractive feature of the triple-source FBP algorithm is its potential for practical applications when it is coupled with parallel computing techniques [35], [36]. With serial computing techniques, the triple-source FBP algorithm may be more computationally intensive than its BPF counterpart because the ltering direction on the detector plane depends not only on the inter-helix PI-line but also on the projection view while the ltering direction of the triple-source BPF algorithm just depends on the inter-helix PI-line. With precalculated interpolation weights and sufcient parallel processors, both the FBP algorithm and the BPF algorithm can be greatly accelerated for triple-source helical CBCT. As far as the computational efciency is concerned, FBP-type algorithms have an advantage over BPF algorithms in terms of data ow. Qualitatively speaking, in the FBP framework each projection frame can be ltered, backprojected and discarded immediately, while in the BPF framework the inverse Hilbert ltering can only take place after the backprojection is done, and furthermorejjjIj jjj jIjjIjjjIjj(20)Authorized licensed use limited to: IEEE Xplore. Downloaded on March 9, 2009 at 03:06 from IEEE Xplore. Restrictions apply.