SIAM Journal on Imaging Sciences, Volume 13, Issue 3, Page 1084-1112, January 2020.
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.

中文翻译：

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用于图像压缩和去模糊的非负张量补丁字典方法

SIAM影像科学杂志，第13卷，第3期，第1084-1112页，2020年1月。
在最近的工作中[S. Soltani，M. Kilmer和PC Hansen，BIT，56（2016）]，一种用于非负张量补丁字典学习的算法，基于X射线CT成像，并基于称为t乘积的张量-张量积[ME]介绍了Kilmer和CD Martin，《线性代数应用》，435（2011），第641--658页]。在此工作的基础上，本文使用基于非负张量补丁的字典对其他数据（例如面部图像数据）进行训练，以实现压缩或图像去模糊的目的。我们从分析开始，在该分析中我们解决了一些问题，例如基于张量的方法相对于基于矩阵的方法的适用性，字典大小和补丁大小，以平衡计算效率和定性表示。下一个，我们开发了一种算法，该算法能够在给定非负张量字典的情况下恢复非负张量系数。该算法基于改进的残差范数最速下降方法的一种变体。我们展示了如何增强算法以在张量系数中实施稀疏性，并注意到该方法具有更广泛的适用性，因为它也可以应用于矩阵情况。我们说明了令人惊讶的结果，即在一类图像数据上训练的字典可以成功地用于表示和压缩不同类别和不同分辨率的图像数据。最后，我们解决了在图像去模糊中使用非负张量字典的问题。