Many imaging science tasks can be modeled as a discrete linear inverse problem. Solving linear inverse problems is often challenging, with ill-conditioned operators and potentially non-unique solutions. Embedding prior knowledge, such as smoothness, into the solution can overcome these challenges. In this work, we encode prior knowledge using a non-negative patch dictionary, which effectively learns a basis from a training set of natural images. In this dictionary basis, we desire solutions that are non-negative and sparse (i.e., contain many zero entries). With these constraints, standard methods for solving discrete linear inverse problems are not directly applicable. One such approach is the modified residual norm steepest descent (MRNSD), which produces non-negative solutions but does not induce sparsity. In this paper, we provide two methods based on MRNSD that promote sparsity. In our first method, we add an $\ell_1$-regularization term with a new, optimal step size. In our second method, we propose a new non-negative, sparsity-promoting mapping of the solution. We compare the performance of our proposed methods on a number of numerical experiments, including deblurring, image completion, computer tomography, and superresolution. Our results show that these methods effectively solve discrete linear inverse problems with non-negativity and sparsity constraints.

翻译：许多成像科学任务可以建模为离散线性逆问题。由于算子病态且解可能不唯一，求解此类问题通常颇具挑战性。将平滑性等先验知识嵌入解中可克服这些困难。本文利用非负斑块字典编码先验知识，该字典可从自然图像训练集中有效学习基函数。在此字典基下，我们寻求非负且稀疏（即包含大量零元素）的解。受此约束，求解离散线性逆问题的标准方法不再直接适用。修正残差范数最速下降法（MRNSD）虽能产生非负解，但无法诱导稀疏性。本文提供两种基于MRNSD的稀疏性提升方法：第一种方法引入带有新型最优步长的$\ell_1$正则化项；第二种方法提出一种新的非负稀疏性促进映射。我们通过去模糊、图像补全、计算机断层扫描和超分辨率等数值实验对比了所提方法的性能。结果表明，这些方法能有效求解带非负性与稀疏性约束的离散线性逆问题。