Isotropic and anisotropic filtering norm-minimization: A generalization of the TV and TGV minimizations using NESTA

Abstract

Compressive sensing (CS) allows for the reconstruction of sparse signals based on measurements acquired at sub-Nyquist sampling rates. Amongst the common CS methods, total variation (TV) minimization is a common approach used to reconstruct images that are approximately piece-wise constant. The discrete gradient operation in TV can be described as a filtering operation using the horizontal and vertical finite differences filters. In this paper, we generalize the TV minimization procedure for any set of digital filters. The proposed method allows one to reconstruct signals that are sparse when filtered with some set of filters, other than the finite difference operators. Our implementation is based on a fast and accurate first-order optimization algorithm which is called NESTA, in a reference to the Nesterov’s algorithm. We incorporate isotropic and anisotropic filtering combinations to the original TV minimization method implemented in the original NESTA algorithm. We also propose 3 forms of the second order total generalized variation (TGV) when using first and second order filters. In order to evaluate the method, we perform a systematic set of experiments using synthetic and real magnetic resonance images, with several sets of filters and cost functions and under different undersampling factors and noise levels. A statistical analysis of the results shows that the best configurations of our method provide a significantly better image quality when compared to the TV and TGV for MRI reconstruction.