# Controlled Wavelet Domain Sparsity in X-ray Tomography

## TL;DR

This paper introduces an automatic method for selecting the regularization parameter in wavelet-based X-ray tomography, improving the sparsity control of the reconstructed image through a novel control algorithm.

## Contribution

A new automatic parameter selection method for wavelet sparsity regularization in X-ray tomography using a control algorithm.

## Key findings

- Effective control of wavelet coefficient sparsity.
- Improved reconstruction quality with automatic parameter tuning.
- Demonstrated robustness across different datasets.

## Abstract

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.