# Weighted Group Sparse Regularized Tensor Decomposition for Hyperspectral Image Denoising
## 4 Experimental Results  
In this section, we present the simulation outcomes for assessing the retrieval effectiveness of the novel LRTRDGS technique. The first dataset is the Washington DC Mall data, which is an image of Washington City, an aerial hyperspectral image acquired by the Hydice sensor, with a size of (256, 256, 160). Another dataset is the Pavia City Center dataset. This is an image of the city of Pavia in northern Italy captured by ROSIS sensors, taken by the Reflective Optical System Spectrometer, with dimensions (200, 200, 80) [28]. For comparison, we implement four typical HSI denoising methods, namely, total variation regularized low-rank matrix factorization (LRTV) [25], patchwise low-rank matrix approximation (LRMR) [29], weighted group sparsity-regularized low-rank tensor decomposition (LRTDGS) [16] and an \(L_{1-2}\) spatial-spectral total variation regularized local low-rank tensor recovery model (TLR-\(L_{1-2}\)SSTV) [30]. The experiment applies the parameters proposed in the referenced article or the code authored by the investigator. In this paper, three quantitative image quality indexes are used to evaluate the performance of the comparison methods, including the peak signal-to-noise ratio based on the characteristics of the human visual system (PSNR-HVS), mean structural similarity index measure (MSSIM) and erreur relative globale adimensionnelle desynthese (ERGAS). PSNR-HVS is a quality metric that measures the characteristics of the human visual system in a full-reference manner. MSSSIM assesses the structural consistency to determine the similarity between the original hyperspectral image and the denoised hyperspectral image. Evaluating the fidelity of the denoised hyperspectral image, ERGAS incorporates the MSE of each band by assigning proper weights. To gauge the quality of the resultant denoised image, we employ three metrics:  
\[\text{PSNR-HVS}=10\log\left(\frac{255^{2}}{MSE_{HVS}}\right)\]  
\[MSE_{HVS}=K\sum_{i=1}^{I-7}\sum_{j=1}^{I-7}\sum_{m=1}^{8}\sum_{n=1}^{8}\left( \left(X[m,n]_{ij}-X[m,n]_{ij}^{e}\right)T_{C}[m,n]\right)^{2}.\]  
The PSNR-HVS can be adapted to different block sizes and is not computationally intensive. In this context, the height and width of the image are denoted as \(I\) and \(J\), respectively, and an \(8\times 8\) image block with the upper left corner at \((i,j)\) has \(X_{ij}\) as its DCTcoefficient. The DCT coefficient of the corresponding block in the original image is denoted as \(X_{ij}^{c}\). The JPEG quantization table specified in the JPEG standard is represented as \(T_{c}\), and \(K\) is calculated as \(K=1/\big{[}(I-7)(J-7)64\big{]}\)[31]. \[\text{SSIM}=\frac{\big{(}2\mu_{x}\mu_{y}+C_{1}\big{)}(2\sigma_{xy }+C_{2})}{\big{(}\mu_{x}^{2}+\mu_{y}^{2}+C_{1}\big{)}(\sigma_{x}^{2}+\sigma_{y }^{2}+C_{2})},\] \[\text{MSSIM}(X,Y)=\frac{1}{Q}\sum\limits_{j=1}^{P}\text{SSIM} \big{(}x_{j},y_{j}\big{)},\]  
where the constant \(C_{1}\) represents \(\mu_{x}^{2}+\mu_{y}^{2}\), while the \(C_{2}\) value is equivalent to the \(C_{1}\) value. The standard deviations of \(x\) and \(y\) are symbolized as \(\sigma_{x}\) and \(\sigma_{y}\) correspondingly. The initial image and the altered image are denoted as \(X\) and \(Y\), respectively. The composition of the local window corresponding to the \(j\)th instance is indicated by \(x_{j}\) and \(y_{j}\), whereas the total number of local windows is denoted as Q.  
\[\text{ERGAS}=\sqrt{\frac{1}{P}\sum\limits_{i=1}^{P}\frac{mes(ref_{i},res_{i}) }{Mean_{2}(ref_{i})}},\]  
where \(ref_{i}\) and \(res_{i}\) represent the reference image and recovered image of the \(i\)th band, respectively. \(P\) represents the number of bands. In the experiment, all the reflectance values of the hyperspectral images are normalized to [0,1] for numerical calculation and visualization. To verify the performance of the method under different noise conditions, we add simulated noise to the simulated HSI data. The noise here mainly includes three types of noise: Gaussian noise, impulse noise and deadline noise.