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Variational models with Total Variation (TV) regularization have long been known to preserve image edges and produce high quality reconstruction. On the other hand, recent theory on compressive sensing has shown that it is feasible to accurately reconstruct  images from a few linear measurements via TV regularization. However, in general TV models are difficult to solve due to the nondifferentiability and the universal coupling of variables. In this paper, we propose the use of alternating direction method for image reconstruction from highly incomplete convolution data, where an image is reconstructed as a minimizer of an energy function that sums   a TV term for image regularity and a least squares term for data fitting. Our algorithm, called RecPK, takes advantage of problem structures and has an extremely low per-iteration cost. To demonstrate the efficiency of RecPK, we  compare it with TwIST, a state-of-the-art algorithm for minimizing TV models. Moreover, we also demonstrate the usefulness of RecPK in image zooming.

Image reconstruction from incomplete convolution data via total variation regularization