Sparse coding techniques for image processing traditionally rely on processing small overlapping patches separately followed by averaging. This has the disadvantage that the reconstructed image no longer obeys the sparsity prior used in the processing. For this purpose convolutional sparse coding has been introduced, where a shift-invariant dictionary is used and the sparsity of the recovered image is maintained. Most such strategies target the $\ell_0$ ``norm'' of the whole image, which may create an imbalanced sparsity across various regions in the image. In order to face this challenge, the $\ell_{0,\infty}$ ``norm'' has been proposed as an alternative that ``operates locally while thinking globally". The approaches taken for tackling the non-convexity of these optimization problems have been either using a convex relaxation or local pursuit algorithms. In this paper, we present a greedy method for sparse coding and dictionary learning which is specifically tailored to $\ell_{0,\infty}$, and is based on matching pursuit. This technique is based on the convolutional relationship between the local dictionary and the global image, operating locally while taking into account the global nature of the images. We demonstrate the usage of our approach in text-image processing and in texture and cartoon separation.