Wavelet Methods for Inverting the Radon Transform with Noisy Data
user correction - Legacy Corrections
Bradley J. Lucier
Because the Radon transform is a smoothing transform, any noise in the Radon data becomes magnified when the inverse Radon transform is applied. Among the methods used to deal with this problem is the Wavelet-Vaguelette Decomposition (WVD) coupled with Wavelet Shrinkage, as introduced by David L. Donoho. We extend several results of Donoho and others here. First, we introduce a new sufficient condition on wavelets to generate a WVD. For a general homogeneous operator, which class includes the Radon transform, we show that a variant of Donoho's method for solving inverse problems can be derived as the exact minimizer of a variational problem that uses a Besov norm as the smoothing functional. We give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter needed to recover an image from noise-corrupted data. We conduct tomographic reconstruction computations that support the hypothesis that near-optimal shrinkage pa...