# Weighted Group Sparse Regularized Tensor Decomposition for Hyperspectral Image Denoising
## 1 Introduction  
Hyperspectral imaging (HSI) is a powerful tool for studying and analyzing data for various applications. HSI uses a combination of visible, infrared and ultraviolet wavelengths to capture images in a way that traditional imaging methods cannot. HSI manifests two essential attributes: spatial and spectral resolutions. Specifically, it possesses two spatial dimensions and one spectral dimension. As a result, it generates an almost uninterrupted reflectance spectrum for each pixel within the scene. This, in turn, facilitates a comprehensive spectral analysis of scene attributes that would otherwise prove challenging to identify using a broader range of multispectral scanners [1]. The Flying Laboratory of Imaging Systems (FLIS) provides a single platform that combines hyperspectral, thermal and laser scanning to produce high-quality remote sensing data [2]. HSI has many applications, such as mining [3], analyzing crops in agriculture [4; 5], astroplanetary exploration [6], medical images [7], mineral exploration [8], the detection of soil-borne viruses [9] and the measurement of water quality [10]. High-spectral-resolution images often contain noise, which can significantly reduce their performance and accuracy. Hence, attenuating noise holds great significance in scrutinizing high-spectral-resolution pictures. This technique aids in augmenting image quality, thereby facilitating the identification and examination of intriguing attributes. With noise reduction algorithms, HSI is more useful and accurate for various applications. Tensor is a powerful mathematical tool for extracting information from HSI images. It allows us to see patterns and relationships between different parts of an image that are not visible with traditional imaging techniques, and it is possible to analyze and interpretthe data in a much more sophisticated way. Many experts use tensor methods, such as Tucker decomposition [11; 12], canonical polyadic (CP) decomposition [13; 14] and tensor singular-value decomposition (t-SVD) [15] for hyperspectral image analysis. For example, a model for the weighted group of low-rank tensor decomposition with regularized sparsity (LRTDGS) based on Tucker decomposition was proposed by [PERSON] et al. [16]. Using Tucker decomposition, [PERSON] et al. developed a hybrid noise removal method for HSI [17]. In their study, [PERSON] et al. introduced a nonlocal low-rank regularized canonical polyadic tensor decomposition method (NLR-CPTD) that relies on two prior information sources: GCS and nonlocal self-similarity [18]. The weighted Schatten \(l_{p}\)-norm low-rank matrix approximation (WSN-LRMA) method was developed by [PERSON] et al. In this method, the eigenvalues were obtained through t-SVD [19]. Despite the satisfactory denoising results obtained through the aforementioned techniques, there is still significant potential for enhancing the efficiency of denoising hyperspectral images. Many scholars have recently proposed a decomposition method based on tensor rings (TRs) using cyclically contracted third-order tensors to represent higher-order tensors [20; 21]. Circular shifting and equivalence can be performed according to the tensor ring factor, which can effectively balance the correlation between the modes. TR decomposition can better approximate higher-order tensors. For tensor completion, refs. [22; 23] verified that the TR method obtains better results than the Tucker and CP decomposition methods. TR is able to improve accuracy and enhance security. It can store and process data with greater accuracy than traditional methods, as the three-dimensional structure of the ring allows for a more precise representation of the data. This improved accuracy helps to ensure accuracy during data processing. Data stored in a tensor ring are more secure than data processed by traditional methods, as the three-dimensional structure of the ring makes it difficult to hack or access the data without authorization.