Applications Of Numerical Approximation In CT Image Reconstruction

Numerical approximation theory includes mainly polynomial interpolation, orthogonal polynomial, numerical integral and spline function etc.. With the development of the computer technique, numerical approximation method is applied widely for science engineering computing. Especially, multivariate interpolation and multivariate orthogonal polynomial have widespread application in many fields, such as curves, surfaces building and approximation, surface figure design and finite element method[1]. However, during considerable long period, they aren't applied for CT image reconstruction.Since the first CT machine came out, CT is one of the research hotspots in home and oversea including hardware and software. In hardware, CT machines on low, mid, high or superhigh 7-ray, X-ray, positron ray have been developed, such as high-resolution spiral CT, super-speed positron ray CT in west developed country. In software, from the mathematical model, data collecting, image method, and the soft realizing, many application softwares have emerged combined with the detecting system, improving the density resolution and the detecting speed[2].The main research in this paper is the application of numerical approximation for 2D CT image reconstruction. Contributions lie in the following three aspects:1. Improvement and Analysis of Marr's AlgorithmAnalyzing the Marr's algorithm[3] in detail in this paper, we can draw a conclusion that the local large oscillation of the term rⁿQ_n,k(r²) leads to the distortion of the reconstructed image and its proof is given. To settle the problems we design a weight filter and introduce the FFT technique. As a result, a fast improved algorithm is got. The computer simulation experiments show its effectiveness.In Marr's algorithm, reconstruction formula for image function /(x,j/) is:M [(M-n)/2Jf(x,y)^Y2 Y^ (°2^n'fc+1 (5n+1.fclQn,o(i) = 1 for all n, all tThe Coefficient an,k>Pn,k satisfying the conditions:o < k < 0 1, k is large.Thus, the origin of the image degradation produced by direct implementation of Marr's algorithm is proved in theory. To facilitate improved image quality, we introduce a filter to modify the algorithm. Consequently, we multiply the corresponding term of formula (1) withsin ((im)/2M)sinc({nn)/2M) ={im)/2M 9At the same time, in order to improve the computing time and expand the algorithm's application, we introduce the equivalent transform for 7^ as follows:7n,fc = ^{Pn.k - ian,k)7=0 J=0 ï¿¡(e'**(I + !, J)).-1 N-l? lit I J\1=0 J=0 & (1) . (2) >n,k ' f n,.m n fim)(o)TTV—0 *> '7nlfc=ï¿¡^mr?= E 2,we choosevt , 2kn . 2kn.T , , nX — (cos-----,sin-----) , fc = 1,2, ? ? ,n.n nas the nodes of Hakopian interpolation on the disk. We introduce a linear functionalf f=f f{Xk + t{Xl-Xk))dt. J\xk,xl) Joand let\(k,l)= f fJ\Xk.X']l[xk,x<] be the projection of f{X).Liang Xuezhang has given the following theorems in [5]-[6]:Theorem A For any f(X) e C(R2),there exists a unique polynomial P{X) = Hn(f;X) € 7rn₂(R2), such thatI f-P = 0, 0 Jix^.x']Only give the Lagrange expression of Lk,i{X) when n is even: sin2^7rrm(t) (TI ^ Tm(t) 2Um.1(t) - -^hr I-k is oddmr{t — cos i—⁷t) { . t — cosr (y. sin'^TTC/n,-!^)/ (l-t2)C7m-i(t) )Lfc,i(X =-----JTT------PFT i 2T"ift) + 9, ^—i=lTT f I-k is evenm2(t - cos ^tt) ( m2(t-cos^tt) JTheorem B There exists an absolute constant A such that for any X eB,f(X) 6 C(D), and n > 2, we have\f{X) — Hn(f;X)\ < AEn-iif) n\ognIn order to get valid reconstruction image, the coefficient Ln^"K for Lk,i{X) is as follows:r /v\ o^-fc 2cos2 ^7rcos^tt l-k .Lkt,(X) = cos2 —-7T +--------^^----a— ? (t - cos------it)I sin ^77 n+ o((<-cos—tt)2) (9)Then, the modified algorithm(for n = 2m) is given in the following:? By numerical integral yielding projection data.? If \t - cos^p7r| < e, using modified formula (8) and (9) yielding Lk,i{X), if \t -cos—-n\ > e, using original formula yielding Lk,i{X). Where e is an arbitrary small positive number.? Summation.Further, based on the theory of the ridge polynomial approximating function[6], a fast algorithm based on the decomposition of Chebyshev-Fourier of Hakopian interpolation is given.By computing, we have the following Chebyshev-Fourier coefficient of Hn(f\ X):When i>n-1,^ = 0.Assuming n = 2m is even(The similar result can by attained if n is odd), then, n-2U tf-Y\ 4 ^,,/V^ 4-V^ \ ^n(/,X)=^^W|^l + ^2/Here!i — 1 1i — 1 f-7rsinw-T—jru^-iW / _<sub>...../ (10)2mm m—1 n7rsinw 2m 2m7r)+2/sin(—-----tt) ; = 1,2,- ? ,msin—7rsinw—7ruu,_i(i') / / (11) m mt = xcos j — n ) + ysin I — i \m ) \mOrtherwise, in(10)-(ll), the superscript k may be negative or greater than n, here, lety^k vkiinFor simplicity, let(1) A . 2t-l . 2i-l f . cHj = 5Sm^"7rSma;^<sup><sup>7rV-2' m' n² (2^(b) Composition of ^i(t,j) = J^ wc^ j-wu-i(t) and ^2(t,j) = J2 wc^jUu,_i(i).(c) Summation Hn(f;X) = 4r ^ {Ai(t,j) + A2(t, j)}.In step (a), the c^'- and c^^- can by directly calculated by discrete sine transform of2i — 1 r f j ? 2i f rIn step (b), we can also use the sine transform to speed up \\{tk, j) and \2{tk,j)-Here, tk = cos ^ fc = 1,2, ? ? , n — 1. Values at other points t, which are needed for the backprojection, are obtained by linear interpolation.3. On the Convergence of Hakopian Interpolation Polynomial...