Separable dictionary learning for convolutional sparse coding via split updates

Abstract

The increasing ubiquity of Convolutional Sparse Representation techniques for several image processing tasks (such as object recognition and classification, as well as image denoising) has recently sparked interest in the use of separable 2D dictionary filter banks (as alternatives to standard nonseparable dictionaries) for efficient Convolutional Sparse Coding (CSC) implementations. However, existing methods approximate a set of K non-separable filters via a linear combination of R (R << K) separable filters, which puts an upper bound on the latter’s quality. Furthermore, this implies the need to learn first the whole set of non-separable filters, and only then compute the separable set, which is not optimal from a computational perspective. In this context, the purpose of the present work is to propose a method to directly learn a set of K separable dictionary filters from a given image training set by drawing ideas from standard Convolutional Dictionary Learning (CDL) methods. We show that the separable filters obtained by the proposed method match the performance of an equivalent number of non-separable filters. Furthermore, the computational performance of this learning method is shown to be substantially faster than a state-of-the-art non-separable CDL method when either the image training set or the filter set are large. The method and results presented here have been published [1] at the 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2018). Furthermore, a preliminary approach (mentioned at the end of Chapter 2) was also published at ICASSP 2017 [2]. The structure of the document is organized as follows. Chapter 1 introduces the problem of interest and outlines the scope of this work. Chapter 2 provides the reader with a brief summary of the relevant literature in optimization, CDL and previous use of separable filters. Chapter 3 presents the details of the proposed method and some implementation highlights. Chapter 4 reports the attained computational results through several simulations. Chapter 5 summarizes the attained results and draws some final conclusions.