Fast convolutional sparse coding with separable filters

Convolutional sparse representations (CSR) of images are receiving increasing attention as an alternative to the usual independent patch-wise application of standard sparse representations. For CSR the dictionary is a filter bank of non-separable 2D filters, and the representation itself can be viewed as the synthesis dual of the analysis representation provided by a single level of a convolutional neural network (CNN). The current state-of-the-art convolutional sparse coding (CSC) algorithms achieve their computational efficiency by applying the convolutions in the frequency domain. It has been shown that any given 2D non-separable filter bank can be approximated as a linear combination of a relatively small number of separable filters. This approximation has been exploited for computationally efficient CNN implementations, but has thus far not been considered for convolutional sparse coding. In this paper we propose a computationally efficient algorithm, that apply the convolution in the spatial domain, to solve the CSC problem when the corresponding dictionary filters are separable. Our algorithm, based on the ISTA framework, use a two-term penalty function to attain competitive results when compared to the state-of-the-art methods in terms of computational performance, sparsity and reconstruction quality.