Wavelet methods for inverting the Radon transform with noisy data

Abstract

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.