Sparse MRI Reconstruction via Multiscale L0-Continuation

"Compressed Sensing" and related L1-minimization methods for reconstructing sparse magnetic resonance images (MRI) acquired at sub-Nyquist rates have shown great potential for dramatically reducing exam duration. Nonetheless, the non-triviality of numerical implementation and computational intensity of these reconstruction algorithms has thus far precluded their widespread use in clinical practice. In this work, we propose a novel MRI reconstruction framework based on homotopy continuation of the L0 semi-norm using redescending M-estimator functions. Following analysis of the continuation scheme, the sparsity measure is extended to multiscale form and a simple numerical solver that can achieve accurate reconstructions in a matter of seconds on a standard desktop computer is presented.