# Hyperspectral Image Super-Resolution Based on Spatial Group Sparsity Regularization Unmixing
## 3 Proposed Method
### Problem Solution  
As suggested above, if \(E\) (of the HSI) and \(A\) (of the conventional image) are known, a super-resolution HSI can be estimated. This subsection introduces a solution for sparse unmixing based on spatial group sparsity regularization. Suppose \(Y\in\mathbb{R}^{L\times N}\) denotes an HSI with \(L\) spectral bands and \(N\) pixels, \(E=[e_{1},\cdots,e_{P}]\in\mathbb{R}^{L\times P}\) denotes the endmember matrix, and the abundance vector is \(A=[a_{1},\cdots,a_{N}]\in\mathbb{R}^{P\times N}\), where \(P\) is the number of endmembers and \(n\) represents the additive noise. Then, the unmixing can be formulated as follows:  
\[Y=EA+n. \tag{6}\]  
In the sparse unmixing model, hyperspectral vectors can be approximated by a linear combination of a small number of spectral signatures in the spectral library and, so, the non-zero abundance should appear in only a few lines, which indicates sparsity along the pixels of an HSI [33]. Traditional sparse unmixing approaches usually adopt the least square error function; however, it is vulnerable to noise or outliers, as the error term is squared, which means that even small disturbances play a significant role in the objective function. Furthermore, the abundance has a collaborative sparse property [13], which means it can be characterized by the \(\ell_{2,1}\) norm. This can also impose sparsity, which is more rotation invariant than the \(\ell_{1}\) norm. Sparse unmixing is equivalent to solving the following optimization problem:  
\[\min_{A}||EA-Y||_{2,1}+\lambda||A||_{2,1}\quad s.t.\ A\geq 0, \tag{7}\]  
where the \(\ell_{2,1}\) norm is described as:  
\[||A||_{2,1}=\sum_{i=1}^{L}\sqrt{\sum_{j=1}^{P}A_{ij}^{2}}. \tag{8}\]  
According to geography theory, the pixels in a local spatial group have a similar sparse model. This section presents the process of taking advantage of the spatial information and sparsity priors by coupling a spatial group sparsity regularizer with sparse unmixing. To improve the dependence between adjacent pixels and make full use of the spatial information, the original HSI needs to be transformed to a superpixel scale before the unmixing process. Let \(T\in\mathbb{R}^{M\times K}\) denote a spatial transform operator for the HSI and abundance library, and and \(E_{s}\in\mathbb{R}^{P\times K}\) be the superpixel scale approximation of the original HSI \(Y\) and the original abundance \(A\), respectively. The transformation result is inserted into Equation (7):  
\[Y_{s}=YT\ A_{s}=AT \tag{9}\]  
The goal of the superpixel transformation is to cluster spatially adjacent pixels, which are structurally similar and share the same sparse model. We introduce a representative gradient ascent-based algorithm, SLIC [34], to obtain superpixels, which also have the advantage of preserving the structures or features of non-clustered pixels. The SLIC superpixel algorithm was originally designed for color images, but has been applied in HSI analysis in recent years. SLIC has a similar function to k-means clustering, but it is applied in a feature domain consisting of spatial and color features. To adapt it to hyperspectral data, color features are replaced by spectral features. We used an improved SLIC method to produce the superpixels, which is summarized as follows:  
Step 1: Create a feature vector \(\Phi(x,y)\):  
\[\Phi(x,y)= \begin{bmatrix}\lambda_{x}\\ \lambda_{y}\\ I(x,y))\end{bmatrix} , \tag{10}\]  
where \((x,y)\) is the spatial location of the pixel, \(\lambda=m/S\) is a parameter denoting the trade-off between the spatial and spectral priors, and \(I(x,y)\) contains the values of each spatial band. IN \(\lambda\), \(m\) is a parameter related to superpixel regularity and \(S\) is the size of superpixels. Step 2: Construct a set of cluster centers \(C_{k}=\Phi(x_{k},y_{k})\). Traditional SLIC for a color image uses a square grid to overlap the whole image.