Wavelet methods for inverting the Radon transform with noisy data

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 Donoho (1995). 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, whose 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 parameters can be derived if one can estimate only two Besov-space parameters about an image f. Both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than Kolaczyk´s (1996) variant of Donoho´s method and the classical filtered backprojection method