Vector-valued Image Restoration with Applications to Magnetic Resonance Velocity Imaging

The analysis of blood flow patterns and the interaction between salient topological flow features and cardiovascular structure plays an important role in the study of cardiovascular function. Flow velocity images acquired by Magnetic Resonance (MR) velocity imaging are generally subject to noise that are intrinsic to system hardware setup and those specific to patient movement in relation to imaging sequence designs. To improve the accuracy of the quantitative analysis of the evolution of topological flow features, it is essential to restore the original flow fields so that the associated critical points can be more reliably detected. In this study, we propose a total variation based variational method for the restoration of flow vector fields. The method is formulated as a constrained optimisation problem by minimizing the total variation energy of the normalized velocity field subject to a constraint that depends on the noise level. The effectiveness of this restoration method greatly depends on the choice of the regularization parameter in the formulation of the optimisation problem. A new computational algorithm based on the First Order Lagrangian method is proposed, which determines the optimal value of the regularization parameter while solving the minimisation problem. The proposed method has been validated with both simulated flow data and MR velocity maps acquired from patients with sequential MR examination following myocardial infarction.