*Purpose*: Reconstruction of X-ray computed tomography data remains a mathematically challenging problem in medical imaging. Complementing the standard analytical reconstruction methods, sparse regularization is growing in importance, as it allows inclusion of prior knowledge. The paper presents a method for sparse regularization based on the curvelet frame for the application to iterative reconstruction in X-ray computed tomography. *Methods*: In this work we present an iterative reconstruction approach based on the alternating direction method of multipliers (ADMM) using curvelet sparse regularization. *Results*: Evaluation of the method is performed on a specifically crafted numerical phantom dataset to highlight the method's strengths. Additional evaluation is performed on two real datasets from commercial scanners with different noise characteristics, a clinical bone sample acquired in a micro-CT and a human abdomen scanned in a diagnostic CT. The results clearly illustrate that curvelet sparse regularization has characteristic strengths. In particular, it improves the restoration and resolution of highly directional, high contrast features with smooth contrast variations. We also compare this approach to the popular technique of total variation and to traditional filtered backprojection. *Conclusions*: We conclude that curvelet sparse regularization is able to improve reconstruction quality by reducing noise while preserving highly directional features.     «

*Purpose*: Reconstruction of X-ray computed tomography data remains a mathematically challenging problem in medical imaging. Complementing the standard analytical reconstruction methods, sparse regularization is growing in importance, as it allows inclusion of prior knowledge. The paper presents a method for sparse regularization based on the curvelet frame for the application to iterative reconstruction in X-ray computed tomography. *Methods*: In this work we present an iterative reconstruct...     »