Fourier domain structured low-rank matrix priors are emerging as powerfulalternatives to traditional image recovery methods such as total variation andwavelet regularization. These priors specify that a convolutional structuredmatrix, i.e., Toeplitz, Hankel, or their multi-level generalizations, builtfrom Fourier data of the image should be low-rank. The main challenge inapplying these schemes to large-scale problems is the computational complexityand memory demand resulting from lifting the image data to a large scalematrix. We introduce a fast and memory efficient approach called the GenericIterative Reweighted Annihilation Filter (GIRAF) algorithm that exploits theconvolutional structure of the lifted matrix to work in the original un-lifteddomain, thus considerably reducing the complexity. Our experiments on therecovery of images from undersampled Fourier measurements show that theresulting algorithm is considerably faster than previously proposed algorithms,and can accommodate much larger problem sizes than previously studied.
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