  The Convolutional Sparse Coding (CSC) model has recently gained considerable
traction in the signal and image processing communities. By providing a global,
yet tractable, model that operates on the whole image, the CSC was shown to
overcome several limitations of the patch-based sparse model while achieving
superior performance in various applications. Contemporary methods for pursuit
and learning the CSC dictionary often rely on the Alternating Direction Method
of Multipliers (ADMM) in the Fourier domain for the computational convenience
of convolutions, while ignoring the local characterizations of the image. A
recent work by Papyan et al. suggested the SBDL algorithm for the CSC, while
operating locally on image patches. SBDL demonstrates better performance
compared to the Fourier-based methods, albeit still relying on the ADMM. In
this work we maintain the localized strategy of the SBDL, while proposing a new
and much simpler approach based on the Block Coordinate Descent algorithm -
this method is termed Local Block Coordinate Descent (LoBCoD). Furthermore, we
introduce a novel stochastic gradient descent version of LoBCoD for training
the convolutional filters. The Stochastic-LoBCoD leverages the benefits of
online learning, while being applicable to a single training image. We
demonstrate the advantages of the proposed algorithms for image inpainting and
multi-focus image fusion, achieving state-of-the-art results.
