Hyperspectral Anomaly Detection via Sparsity of Core Tensor Under Gradient Domain

A hyperspectral anomaly detection (AD) task is a typical binary classification problem, and utilizing background prior knowledge is a key technique to solving such problems. The two most commonly used priors for hyperspectral images (HSIs) are low-rank and local-smooth properties. Most traditional matrix-based methods use two regularizations to model these two types of priors and integrate them into one model, which makes these two regularizations unable to maximize their effectiveness. In addition, the matrix method also destroys the structure of the HSIs. To address these issues, this study identified a unique sparsity property in the gradient tensor of HSI. Specifically, the core tensor resulting from the Tucker decomposition of the gradient tensor was observed to exhibit sparsity. This sparsity property, referred to as the sparsity on the core tensor of the gradient map (GCS), effectively captures the structural information of HSI and improves detection performance. The GCS regularization offers the following advantages: 1) GCS regularization uses one term to simultaneously capture both low rankness and local smoothness, the size of the core tensor represents the low rank prior to the background, and the $\ell _{1}$ norm describes the sparsity of gradient map, i.e., the local smoothness of the original data and 2) GCS is a constrained regularization, allowing for the full utilization of information from different dimensions of the HSI when updating the core tensor, i.e., utilizing the spatial and spectral information carried by three-factor matrices of the Tucker decomposition. Finally, extensive experiments validate the superiority of our proposed methods.