Fast Hyperparameter Calibration of Sparsity Enforcing Penalties in Total Generalised Variation Penalised Reconstruction Methods for XCT Using a Planted Virtual Reference Image

The reconstruction problem in X-ray computed tomography (XCT) is notoriously difficult in the case where only a small number of measurements are made. Based on the recently discovered Compressed Sensing paradigm, many methods have been proposed in order to address the reconstruction problem by leveraging inherent sparsity of the object’s decompositions in various appropriate bases or dictionaries. In practice, reconstruction is usually achieved by incorporating weighted sparsity enforcing penalisation functionals into the least-squares objective of the associated optimisation problem. One such penalisation functional is the Total Variation (TV) norm, which has been successfully employed since the early days of Compressed Sensing. Total Generalised Variation (TGV) is a recent improvement of this approach. One of the main advantages of such penalisation based approaches is that the resulting optimisation problem is convex and as such, cannot be affected by the possible existence of spurious solutions. Using the TGV penalisation nevertheless comes with the drawback of having to tune the two hyperparameters governing the TGV semi-norms. In this short note, we provide a simple and efficient recipe for fast hyperparameters tuning, based on the simple idea of virtually planting a mock image into the model. The proposed trick potentially applies to all linear inverse problems under the assumption that relevant prior information is available about the sought for solution, whilst being very different from the Bayesian method