Nonnegative Tensor Patch Dictionary Approaches for Image Compression and Deblurring Applications

In recent work [S. Soltani, M. Kilmer, and P. C. Hansen, BIT, 56 (2016)], an algorithm for nonnegative tensor patch dictionary learning in the context of X-ray CT imaging and based on a tensor-tensor product called the t-product [M. E. Kilmer and C. D. Martin, Linear Algebra Appl., 435 (2011), pp. 641--658] was presented. Building on that work, in this paper, we use nonnegative tensor patch--based dictionaries trained on other data, such as facial image data, for the purpose of either compression or image deblurring. We begin with an analysis in which we address issues such as suitability of the tensor-based approach relative to a matrix-based approach, dictionary size, and patch size to balance computational efficiency and qualitative representations. Next, we develop an algorithm that is capable of recovering nonnegative tensor coefficients given a nonnegative tensor dictionary. The algorithm is based on a variant of the modified residual norm steepest descent method. We show how to augment the algorithm to enforce sparsity in the tensor coefficients and note that the approach has broader applicability since it can be applied to the matrix case as well. We illustrate the surprising result that dictionaries trained on image data from one class can be successfully used to represent and compress image data from different classes and across different resolutions. Finally, we address the use of nonnegative tensor dictionaries in image deblurring. We show that tensor treatment of the deblurring problem coupled with nonnegative tensor patch dictionaries can give superior restorations as compared to standard treatment of the nonnegativity constrained deblurring problem.