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.
KeywordsNoisy Data Continuous Linear Functional Reproduce Kernel Space Inverse Radon Spline Form
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