  In recent work (Soltani, Kilmer, Hansen, BIT 2016), an algorithm for
non-negative tensor patch dictionary learning in the context of X-ray CT
imaging and based on a tensor-tensor product called the $t$-product (Kilmer and
Martin, 2011) was presented. Building on that work, in this paper, we use of
non-negative tensor patch-based dictionaries trained on other data, such as
facial 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 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 non-negative tensor
coefficients given a non-negative 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 non-negative tensor dictionaries in image
deblurring. We show that tensor treatment of the deblurring problem coupled
with non-negative tensor patch dictionaries can give superior restorations as
compared to standard treatment of the non-negativity constrained deblurring
problem.
