A New Approach to the Numerical Evaluation of the Inverse Radon Transform with Discrete, Noisy Data
The inner (singular) integral in the inverse Radon transform for parallel beam computerized tomography devices can be integrated analytically if the Radon transform considered as a function of the ray position along the detector, is a cubic polynomial spline. Furthermore by using some spline identities, large terms that cancel can be eliminated analytically and the calculation of the resulting expression for the inner integral done in a numerically stable fashion. We suggest using smoothing splines to smooth each set of projection data and by so doing obtain the Radon transform in the above spline form. The resulting analytic expression for the inner integral in the inverse transform is then readily evaluated, and the outer (periodic) integral is replaced by a sum. The work involved to obtain the inverse transform appears to be within the capability of existing computing equipment for typical large data sets. In this regularized transform method the regularization is controlled by the smoothing parameter in the splines. The regularization is directed against data errors and not to prevent unstable numerical operations. Strip integral as well as line integral data can be handled by this method. The method is shown to be closely related to the Tihonov form of regularization.
KeywordsAttenuation Convolution Radon Lution Reso
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- 2.T. Chang and G.T. Herman (1978). Filter selection for the fan beam convolution reconstruction algorithm. TR No. MIPG 6, Dept. of Computer Science, SUNY Buffalo.Google Scholar
- 4.J. Fleisher (1979) Spline smoothing routines reference manual for the 1110. Academic Computing Center, The University of Wisconsin- Madison.Google Scholar
- 5.G. Golub, M. Heath and G. Wahba (1979). Generalized cross-validation as a method for choosing a good ridge parameter Technometrics 21, 2, 215–223.Google Scholar
- 7.P.H. Merz (1979). Spline smoothing by generalized cross-validation, a technique for data smoothing. Chevron Research Corp., Richmond, CA.Google Scholar
- 9.F. Natterer (1979). A Sobolev space analysis of picture reconstruction, to appear, SIAM J. Applied Math.Google Scholar
- 10.F. Natterer (1980). Efficient implementation of ‘optimal’ algorithms in computerized tomonraphy, manuscript.Google Scholar
- 13.F. Utreras (1979). Cross validation techniques for smoothing spline functions in one or two dimensions. In “Smoothing Techniques for Curve Estimation”, T. Gasser and M. Rosenblatt, eds. Lecture notes in Mathematics, No. 757, Springer-Verlag, Berlin.Google Scholar
- 15.G. Wahba (1979a). Spline interpolation and smoothing on the sphere. University of Wisconsin-Madison Statistics Dept. TR No. 584.Google Scholar
- 16.G. Wahba (1979b). Ill posed problems: Numerical and statistical methods for mildly, moderately,and severely ill posed problems with noisy data. University of Wisconsin-Madison Statistics Dept. TR No. 595.Google Scholar
- 17.G. Wahba and J. Wendelberger (1979). Some new mathematical methods for variational objective analysis using splines and cross-validation. University of Wisconsin-Madison Statistics Dept. TR No, 578.Google Scholar