Approximate message passing with a colored aliasing model for variable density Fourier sampled images

The Approximate Message Passing (AMP) algorithm eciently reconstructs signals which have been sampled with large i.i.d. sub-Gaussian sensing matrices. However, when Fourier coecients of a signal with non-uniform spectral density are sampled, such as in Magnetic Resonance Imaging (MRI), the aliasing is intrinsically colored. Consequently, AMP’s i.i.d. state evolution is no longer accurate and the algorithm encounters convergence problems. In response, we propose an algorithm based on Orthogonal Approximate Message Passing (OAMP) that uses the wavelet domain to model the colored aliasing. We present empirical evidence that a structured state evolution occurs, where the e↵ective noise covariance matrix is diagonal with one unique entry per subband. A benefit of state evolution is that Stein’s Unbiased Risk Estimate (SURE) can be e↵ectively implemented, yielding an algorithm with no free parameters. We empirically evaluate the e↵ectiveness of the parameterfree algorithm on a synthetic image with three variable density sampling schemes and find that it converges in over 20x fewer iterations than optimally tuned Fast Iterative Shrinkage-Thresholding (FISTA).