Learning Stable Multilevel Dictionaries for Sparse Representation of Images

Dictionaries adapted to the data provide superior performance when compared to predefined dictionaries in applications involving sparse representations. Algorithmic stability and generalization are desirable characteristics for dictionary learning algorithms that aim to build global dictionaries which can efficiently model any test data similar to the training samples. In this paper, we propose an algorithm to learn dictionaries for sparse representation of image patches, and prove that the proposed learning algorithm is stable and generalizable asymptotically. The algorithm employs a 1-D subspace clustering procedure, the K-lines clustering, in order to learn a hierarchical dictionary with multiple levels. Furthermore, we propose a regularized pursuit scheme for computing sparse representations using a multilevel dictionary. Using simulations, we demonstrate the stability and generalization characteristics of the proposed algorithm with natural image patches. Finally, we employ multilevel dictionaries for compressed recovery and demonstrate improvements in recovery performance using both random and optimized projections when compared to baseline K-SVD dictionaries.