Separable Dictionary Learning

Abstract

Many techniques in computer vision, machine learning, and statistics rely on the fact that a signal of interest admits a sparse representation over some dictionary. Dictionaries are either available analytically, or can be learned from a suitable training set. While analytic dictionaries permit to capture the global structure of a signal and allow a fast implementation, learned dictionaries often perform better in applications as they are more adapted to the considered class of signals. In imagery, unfortunately, the numerical burden for (i) learning a dictionary and for (ii) employing the dictionary for reconstruction tasks only allows to deal with relatively small image patches that only capture local image information.The approach presented in this paper aims at overcoming these drawbacks by allowing a separable structure on the dictionary throughout the learning process. On the one hand, this permits larger patch-sizes for the learning phase, on the other hand, the dictionary is applied efficiently in reconstruction tasks. The learning procedure is based on optimizing over a product of spheres which updates the dictionary as a whole, thus enforces basic dictionary properties such as mutual coherence explicitly during the learning procedure. In the special case where no separable structure is enforced, our method competes with state-of-the-art dictionary learning methods like K-SVD.

Patch-based models and algorithms for image processing: a review of the basic principles and methods, and their application in computed tomography

“…This dictionary reduced the denoising time for 3D CT images by factors of around 30, while also improving the denoising performance. A similar idea is the separable dictionary where D is the Kronecker product of two smaller dictionaries [36]. This dictionary reduces the complexity of sparse coding from O(n) to O( √ n).…”

Learning Fast Dictionaries for Sparse Representations Using Low-Rank Tensor Decompositions

“…The Kronecker structure was introduced in the Dictionary Learning domain by [8,13] both addressing only 2-dimensional data (i.e. 2-KS dictionaries).…”

Single image super-resolution via 2D sparse representation

“…The 2D sparse model as well as 2D dictionary learning method [13] helps to capture the intrinsic 2D structure and local correlation within the images and has been successfully applied to image denoising with promising results. However, the 2D sparse model cannot be directly applied to image SR.…”