Improving compressive sensing in MRI with separate magnitude and phase priors

Compressive sampling/compressed sensing (CS) has shown that it is possible to perfectly reconstruct non-bandlimited signals sampled well below the Nyquist rate. Magnetic Resonance Imaging (MRI) is one of the applications that has benefited from this theory. Sparsifying operators that are effective for real-valued images, such as finite difference and wavelet transform, also work well for complex-valued MRI when phase variations are small. As phase variations increase, even if the phase is smooth, the sparsifying ability of these operators for complex-valued images is reduced. If the phase is known, it is possible to remove it from the complex-valued image before applying the sparsifying operator. Another alternative is to use the sparsifying operator on the magnitude of the image, and use a different operator for the phase, i.e., one related to a smoothness enforcing prior. The proposed method separates the priors for the magnitude and for the phase, in order to improve the applicability of CS to MRI. An improved version of previous approaches, by ourselves and other authors, is proposed to reduce computational cost and enhance the quality of the reconstructed complex-valued MR images with smooth phase. The proposed method utilizes ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document} penalty for the transformed magnitude, and a modified ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _2$$\end{document} penalty for phase, together with a non-linear conjugated gradient optimization. Also, this paper provides an extensive set of experiments to understand the behavior of previous methods and the new approach.