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In recent work (Soltani, Kilmer, Hansen, BIT 2016), an algorithm fornon-negative tensor patch dictionary learning in the context of X-ray CTimaging and based on a tensor-tensor product called the $t$-product (Kilmer andMartin, 2011) was presented. Building on that work, in this paper, we use ofnon-negative tensor patch-based dictionaries trained on other data, such asfacial image data, for the purposes of either compression or image deblurring.We begin with an analysis in which we address issues such as suitability of thetensor-based approach relative to a matrix-based approach, dictionary size andpatch size to balance computational efficiency and qualitative representations.Next, we develop an algorithm that is capable of recovering non-negative tensorcoefficients given a non-negative tensor dictionary. The algorithm is based ona variant of the Modified Residual Norm Steepest Descent method. We show how toaugment the algorithm to enforce sparsity in the tensor coefficients, and notethat the approach has broader applicability since it can be applied to thematrix case as well. We illustrate the surprising result that dictionariestrained on image data from one class can be successfully used to represent andcompress image data from different classes and across different resolutions.Finally, we address the use of non-negative tensor dictionaries in imagedeblurring. We show that tensor treatment of the deblurring problem coupledwith non-negative tensor patch dictionaries can give superior restorations ascompared to standard treatment of the non-negativity constrained deblurringproblem.

Non-negative Tensor Patch Dictionary Approaches for Image Compression and Deblurring Applications