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Sparse coding techniques for image processing traditionally rely on aprocessing of small overlapping patches separately followed by averaging. Thishas the disadvantage that the reconstructed image no longer obeys the sparsityprior used in the processing. For this purpose convolutional sparse coding hasbeen introduced, where a shift-invariant dictionary is used and the sparsity ofthe recovered image is maintained. Most such strategies target the $\ell_0$"norm" or the $\ell_1$ norm of the whole image, which may create an imbalancedsparsity across various regions in the image. In order to face this challenge,the $\ell_{0,\infty}$ "norm" has been proposed as an alternative that "operateslocally while thinking globally". The approaches taken for tackling thenon-convexity of these optimization problems have been either using a convexrelaxation or local pursuit algorithms. In this paper, we present an efficientgreedy method for sparse coding and dictionary learning, which is specificallytailored to $\ell_{0,\infty}$, and is based on matching pursuit. We demonstratethe usage of our approach in salt-and-pepper noise removal and imageinpainting. A code package which reproduces the experiments presented in thiswork is available at https://web.eng.tau.ac.il/~raja

A Greedy Approach to $\ell_{0,\infty}$ Based Convolutional Sparse Coding