# Hyperspectral Image Super-Resolution Based on Spatial Group Sparsity Regularization Unmixing
## 3 Proposed Method
### Problem Formulation  
Our final goal is to estimate a high spatial resolution hyperspectral image \(S_{3D}\in\mathbb{R}^{M\times N\times L}\) from a low resolution hyperspectral image \(H_{3D}\in\mathbb{R}^{m\times n\times L}\) and a high spatial resolution conventional image \(C_{3D}\in\mathbb{R}^{M\times N\times l}\), where \(M\times N\) and \(m\times n\) represent the spatial dimensions, and \(L\) and \(l\) denote the number of the spectral bands. For the proposed problem, \(m\times n\) is generally several times lower than \(M\times N\), \(L\) for the hyperspectral image is in the range of 100 to 200 bands, and \(l\ll L\). This makes the problem severely ill-posed.  
HSIs and MSIs are 3D image cubes. The 3D image cubes are resized into 2D matrices. Each column of the matrix corresponds to the spectrum of a pixel and each row corresponds to the complete image in a specific spectral band. Thus, \(S\in\mathbb{R}^{L\times N_{c}}\), \(H\in\mathbb{R}^{L\times N_{h}}\), and \(C\in\mathbb{R}^{l\times N_{c}}\), where \(N_{c}=M\times N\), \(N_{h}=m\times n\).  
The number of endmembers in an HSI scene is lower than the total number of spectral bands. The linear mixing model indicates that the mixed spectrum in a given HSI is a linear combination of abundances. Thus, a pixel \(s\in\mathbb{R}^{L}\) of the HSI \(S\) can be approximated as:  
\[s\approx\sum_{j=1}^{P}e_{j}a_{j}, \tag{1}\]  
where \(e_{j}\in\mathbb{R}^{L}\) represents the reflectance of the \(j^{\text{th}}\) endmember and \(a_{j}\) is the corresponding abundance of the endmember. \(P\) is the highest possible number of endmembers in the scene. Equation (1) can be written, in matrix form, as follows:  
\[S\approx EA, \tag{2}\]  
where \(E=[e_{1},e_{2},\cdots,e_{P}]\) and \(A=[a_{1},a_{2},\cdots,a_{N_{c}}]\). The endmember matrix \(E\) is an non-orthogonal basis which can represent \(S\) in a lower dimensional space \(\mathbb{R}^{L\times P}\).  
The input HSI \(H\) and the conventional image \(C\) describe the same scene as \(S\). However, \(H\) has a smaller spatial size, which can be seen as a spatially downsampled version of \(S\). Similarly, \(C\) is a downsampled version of \(S\) in the spectral dimension. They are formulated as follows:  
\[H\approx ST_{h}=EAT_{h}=E\hat{A}, \tag{3}\]\[C\approx T_{c}S=T_{c}EA=\hat{E}A, \tag{4}\]  
where \(\hat{A}=AT_{h}\) is the abundance with lower spatial resolution and \(T_{h}\in\mathbb{R}^{N_{h}\times N_{c}}\) is the downsampling operator for spatial pixels. Similarly, \(\hat{E}=T_{c}E\) is the spectrally downsampled endmembers and \(T_{c}\in\mathbb{R}^{l\times L}\) is the downsampling operator for spectral bands. \(T_{h}\) and \(T_{c}\) are respectively determined by the cameras.  
If \(E\) and \(A\) are known, the super-resolution hyperspectral image can be estimated, which means that the HSI super-resolution problem can be transformed into an unmixing problem. Thus, two physical constraints should be considered: The abundance vector obeys the non-negative constraint and sum-to-one constraint, given by:  
\[a_{i}\geq 0\quad\forall a_{i}=1,\cdots,L, \tag{5}\] \[\sum_{i=1}^{L}a_{i}=1\]  
where \(a_{i}\) are the elements of \(A\). These constraints include the desired sparsity of the abundances.