Controlled Wavelet Domain Sparsity in X-ray Tomography

Tomographic reconstruction is an ill-posed inverse problem that calls for regularization. One possibility is to require sparsity of the unknown in an orthonormal wavelet basis. This in turn can be achieved by variational regularization where the penalty term is the sum of absolute values of wavelet coefficients. Daubechies, Defrise and De Mol (Comm. Pure Appl. Math. 57) showed that the minimizer of the variational regularization functional can be computed iteratively using a soft thresholding operation. Choosing the soft threshold parameter $\mu>0$ is analogous to the notoriously difficult problem of picking the optimal regularization parameter in Tikhonov regularization. Here a novel automatic method is introduced for choosing $\mu$, based on a control algorithm driving the sparsity of the reconstruction to an {\it a priori} known ratio of nonzero versus zero wavelet coefficients in the unknown function.