Sparsity and low rankness are essential properties of real-world data. The sparsity attribute mentions that the data contain very few non-zero elements. On the other hand, the low-rank property characterizes redundancy in the data. The concept of sparsity and low rankness is central to numerous signal processing tasks such as signal inversion, data compression, and noise removal. With the advent of multi-sensor and multi-dimensional data, we frequently encounter multi-way data or tensors with sparse and low-rank structure. Prevalent vector or matrix-based sparsity estimation criterias cannot conveniently characterize the sparsity structure of multi-way data. Hence, we require efficient criteria to quantify the sparsity structure of multi-way data. Sparsity plays a crucial aspect in hyperspectral image processing tasks such as unmixing and denoising. Hyperspectral images contain an inherent multi-way structure; we perform multi-way sparsity estimation in hyperspectral image processing. We introduce an efficient criterion to estimate multi-way sparsity, facilitating signal inversion, and noise removal tasks. The proposed sparsity criteria combine the sparsity of the core tensor obtained by Tucker tensor decomposition, the low rankness of the tensor, and the approximated rank. Since the core tensor obtained by tucker decomposition is highly sparse, the core tensor sparsity reflects the sparsity of the whole tensor indirectly. On the other hand, the nuclear norm and Stein’s unbiased risk estimate quantifies the low rankness of the tensor. Unlike other approaches, our proposed measure takes both low rankness and sparsity attribute into account. In addition, our proposed sparsity measure also satisfies some of the desirable properties of ideal sparsity measures. We demonstrate the efficacy of our proposed sparsity quantification measure in applications such as hyperspectral image denoising, hyperspectral unmixing tasks as demonstrated in the real image experiments.

中文翻译：

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高光谱图像处理的高效多路稀疏度估计

稀疏性和低等级性是现实世界数据的基本属性。稀疏属性提到数据包含很少的非零元素。另一方面，低等级属性表示数据中的冗余。稀疏性和低等级性的概念对于许多信号处理任务（例如信号倒置，数据压缩和噪声消除）至关重要。随着多传感器和多维数据的出现，我们经常遇到稀疏，低秩结构的多路数据或张量。流行的基于矢量或矩阵的稀疏性估计标准无法方便地表征多路数据的稀疏性结构。因此，我们需要有效的标准来量化多路数据的稀疏性结构。稀疏性在高光谱图像处理任务（例如解混和去噪）中起着至关重要的作用。高光谱图像包含固有的多路结构；我们在高光谱图像处理中执行多方向稀疏度估计。我们引入了一种有效的标准来估计多向稀疏度，促进信号反转和噪声消除任务。拟议的稀疏性标准结合了通过Tucker张量分解获得的核心张量的稀疏性，张量的低秩和近似秩。由于通过塔克分解获得的核心张量非常稀疏，因此核心张量稀疏性间接反映了整个张量的稀疏性。另一方面，核规范和斯坦因的无偏风险估计量化了张量的低秩。与其他方法不同，我们提出的措施同时考虑了低等级和稀疏属性。此外，我们提出的稀疏性度量还满足了理想稀疏性度量的一些理想属性。我们在实际图像实验中证明了我们提出的稀疏量化度量在诸如高光谱图像降噪，高光谱分解任务等应用中的功效。