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For certain sensing matrices, the Approximate Message Passing (AMP) algorithmefficiently reconstructs undersampled signals. However, in Magnetic ResonanceImaging (MRI), where Fourier coefficients of a natural image are sampled withvariable density, AMP encounters convergence problems. In response we presentan algorithm based on Orthogonal AMP constructed specifically for variabledensity partial Fourier sensing matrices. For the first time in this setting astate evolution has been observed. A practical advantage of state evolution isthat Stein's Unbiased Risk Estimate (SURE) can be effectively implemented,yielding an algorithm with no free parameters. We empirically evaluate theeffectiveness of the parameter-free algorithm on simulated data and find thatit converges over 5x faster and to a lower mean-squared error solution thanFast Iterative Shrinkage-Thresholding (FISTA).

An Approximate Message Passing Algorithm for Rapid Parameter-Free Compressed Sensing MRI