In this study, the problem of computing a sparse representation of multi-dimensional visual data is considered. In general, such data e.g., hyperspectral images, color images or video data consists of signals that exhibit strong local dependencies. A new computationally efficient sparse coding optimization problem is derived by employing regularization terms that are adapted to the properties of the signals of interest. Exploiting the merits of the learnable regularization techniques, a neural network is employed to act as structure prior and reveal the underlying signal dependencies. To solve the optimization problem Deep unrolling and Deep equilibrium based algorithms are developed, forming highly interpretable and concise deep-learning-based architectures, that process the input dataset in a block-by-block fashion. Extensive simulation results, in the context of hyperspectral image denoising, are provided, which demonstrate that the proposed algorithms outperform significantly other sparse coding approaches and exhibit superior performance against recent state-of-the-art deep-learning-based denoising models. In a wider perspective, our work provides a unique bridge between a classic approach, that is the sparse representation theory, and modern representation tools that are based on deep learning modeling.

翻译：本研究探讨了多维视觉数据稀疏表示的计算问题。通常，此类数据（例如高光谱图像、彩色图像或视频数据）由具有强局部依赖性的信号组成。通过采用适应目标信号特性的正则化项，推导出一种新的计算高效的稀疏编码优化问题。利用可学习正则化技术的优势，引入神经网络作为结构先验，以揭示潜在的信号依赖性。为求解该优化问题，开发了基于深度展开和深度平衡的算法，形成了高度可解释且简洁的深度学习架构，能够以逐块方式处理输入数据集。在高光谱图像去噪背景下的大量仿真结果表明，所提算法显著优于其他稀疏编码方法，并且相较于近期最先进的基于深度学习的去噪模型展现出卓越性能。从更广阔的视角看，我们的工作为经典方法（即稀疏表示理论）与现代表示工具（基于深度学习建模）之间建立了独特的桥梁。