  Model-based methods are widely used for reconstruction in compressed sensing
(CS) magnetic resonance imaging (MRI), using regularizers to describe the
images of interest. The reconstruction process is equivalent to solving a
composite optimization problem. Accelerated proximal methods (APMs) are very
popular approaches for such problems. This paper proposes a complex
quasi-Newton proximal method (CQNPM) for the wavelet and total variation based
CS MRI reconstruction. Compared with APMs, CQNPM requires fewer iterations to
converge but needs to compute a more challenging proximal mapping called
weighted proximal mapping (WPM). To make CQNPM more practical, we propose
efficient methods to solve the related WPM. Numerical experiments on
reconstructing non-Cartesian MRI data demonstrate the effectiveness and
efficiency of CQNPM.
