  Convolutional Sparse Coding (CSC) is an increasingly popular model in the
signal and image processing communities, tackling some of the limitations of
traditional patch-based sparse representations. Although several works have
addressed the dictionary learning problem under this model, these relied on an
ADMM formulation in the Fourier domain, losing the sense of locality and the
relation to the traditional patch-based sparse pursuit. A recent work suggested
a novel theoretical analysis of this global model, providing guarantees that
rely on a localized sparsity measure. Herein, we extend this local-global
relation by showing how one can efficiently solve the convolutional sparse
pursuit problem and train the filters involved, while operating locally on
image patches. Our approach provides an intuitive algorithm that can leverage
standard techniques from the sparse representations field. The proposed method
is fast to train, simple to implement, and flexible enough that it can be
easily deployed in a variety of applications. We demonstrate the proposed
training scheme for image inpainting and image separation, while achieving
state-of-the-art results.
