Tomographic reconstruction with adaptive sparsifying transforms

A central problem in computed tomography (CT) imaging is to obtain useful, high-quality images from low-dose measurements. Methods that exploit the sparse representations of tomographic images have long been known to improve the quality of reconstructions from low-dose data. Recent work has shown that sparse representations learned directly from the data can outperform traditional, fixed representations, but are prohibitively expensive for practical use in CT. We propose a new method for tomographic reconstruction from low-dose data by combining the statistically weighted data fidelity term with an adaptive sparsifying transform regularizer. This regularizer can be fit to the data at lower cost than competing methods. Our algorithm alternates between reconstructing the image and learning the sparsifying transform. The Alternating Direction Method of Multipliers technique is used to provide an efficient solution to the statistically weighted minimization problem. Numerical experiments on data from clinical CT reconstructions indicate that adaptive sparsifying transform regularization outperforms synthesis sparsity methods at speeds rivaling total-variation regularization.