Efficient convolutional sparse coding

When applying sparse representation techniques to images, the standard approach is to independently compute the representations for a set of overlapping image patches. This method performs very well in a variety of applications, but the independent sparse coding of each patch results in a representation that is not optimal for the image as a whole. A recent development is convolutional sparse coding, in which a sparse representation for an entire image is computed by replacing the linear combination of a set of dictionary vectors by the sum of a set of convolutions with dictionary filters. A disadvantage of this formulation is its computational expense, but the development of efficient algorithms has received some attention in the literature, with the current leading method exploiting a Fourier domain approach. The present paper introduces a new way of solving the problem in the Fourier domain, leading to substantially reduced computational cost.