## 2 Signal Models and Problem Formulation
### Signal Models  
Based on linear mixed model (LMM) shown in Figure 1, the hyperspectral data \(\mathbf{Z}\) can be factorized into the product of two matrices [2], i.e.,  
\[\mathbf{Z}=\mathbf{A}\mathbf{S}, \tag{1}\]  
where \(\mathbf{A}\succeq\mathbf{0}_{M\times N}\) is the endmember signature matrix containing \(N\) endmembers; \(\mathbf{S}\succeq\mathbf{0}_{N\times L}\) is the abundance matrix containing the fractions of endmembers for each pixel [37]. Besides the non-negativity condition, \(\mathbf{S}\) also satisfies the sum-to-one constraint \(1_{N}^{T}\mathbf{S}=1_{L}^{T}\); that is, the sum of the abundance of substances in each pixel is equal to 1. However, the \(L_{1}\)-norm of \(\mathbf{S}\) is equal to a constant when \(\mathbf{S}\) is sparse, which makes the sum-to-one constraint unnecessary. Furthermore, according to observation model, the desired data \(\mathbf{Z}\) can also be degraded respectively spatially and spectrally to yield the observed hyperspectral data \(\mathbf{Y}_{h}\in\mathbb{R}^{M\times L_{h}}\) and multispectral data \(\mathbf{Y}_{m}\in\mathbb{R}^{M_{m}\times L}\), where \(L_{h}\) denotes the number of pixels in \(\mathbf{Y}_{h}\) and \(M_{m}\) represents the number of spectral bands in \(\mathbf{Y}_{m}\) \begin{table}
\begin{tabular}{c l} \hline \hline
**Notations** & **Explanations** \\ \hline \(\mathbb{R},\mathbb{R}^{n},\mathbb{R}^{m\times n}\) & Set of real number, \(n\)-vector, \(m\times n\) matrices \\ \(\mathbb{R}_{+},\mathbb{R}_{+}^{n},\mathbb{R}_{+}^{m\times n}\) & Set of non-negative real number, \(n\)-vector, \(m\times n\) matrices \\ \(\|\cdot\|\) & Frobenius-norm \\
**0, 1 and I** & All-zero vector, all-one vector, identity matrix \\ \(\mathbf{e}_{i}^{(m)}\) & \(i\)th \(m\)-dimensional unit vector \\ \(\text{conv}\left(\mathcal{S}\right)\) & Convex hull of the set \(\mathcal{S}\) \\
**vec\((\mathbf{X})\)** & Vector formed by stacking the columns of the matrix \(\mathbf{X}\) \\ \(\otimes\) & Kronecker product \\ \([]_{+}\) & Orthogonal projection onto the non-negative orthant \\ \(\succeq\) & Component-wise inequality operation \\ \(\mathbf{Z}^{(3)}\) & Mode-n product of \(\mathbf{Z}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. Figure 1: Diagram of LMM and spatial degradation matrix \(\mathbf{G}\) with the parameter \(\mathbf{g}\in\mathbb{R}^{25}\) (\(r=5\)). Combining LMM with the observation model, we have  
\[\mathbf{Y}_{h} =\mathbf{Z}\mathbf{G}+\mathbf{E}_{h}=\mathbf{AS}\mathbf{G}+\mathbf{E} _{h}, \tag{2}\] \[\mathbf{Y}_{m} =\mathbf{F}\mathbf{Z}+\mathbf{E}_{m}=\mathbf{F}\mathbf{AS}+\mathbf{ E}_{m}, \tag{3}\]  
where \(\mathbf{E}_{h}\) and \(\mathbf{E}_{m}\) are the residuals of observed HSI and MSI data, respectively. The matrix \(\mathbf{F}\in\mathbb{R}^{M_{m}\times M}\), constructed by the uniform spectral response of two observed sensors, downsamples the hyperspectral bands of \(\mathbf{Z}\) to yield the multispectral data \(\mathbf{Y}_{m}\). The matrix \(\mathbf{G}=\mathbf{I}_{L_{h}}\otimes\mathbf{g}\in\mathbb{R}^{L\times L_{h}}\), derived from a point spread function, degrades the desired data \(\mathbf{Z}\) to obtain the observed hyperspectral data \(\mathbf{Y}_{h}\)[25; 36; 38]. Using a Guassian kernel with a blurring factor of \(r\triangleq\sqrt{L/L_{h}}\) shown in Figure 1, \(\mathbf{G}\) is defined as  
\[\mathbf{G}=\mathbf{I}_{L_{h}}\otimes\mathbf{g}\in\mathbb{R}^{L\times L_{h}}, \tag{4}\]  
where \(\mathbf{g}\in\mathbb{R}^{r^{2}}\) is a Gaussian kernel vector. In this article, \(\mathbf{F}\) and \(\mathbf{G}\) can be estimated from the datasets for the fusion method [25; 32]. From Equations (2) and (3), the objective function of original CNMF proposed in [25] is formulated as two coupled data fidelity terms; i.e.,  
\[ \begin{split}\min_{\mathbf{A},\mathbf{S}}& \mathbf{C}(\mathbf{A},\mathbf{S})\triangleq\|\mathbf{Y}_{h}-\mathbf{AS} \mathbf{G}\|_{F}^{2}+\|\mathbf{Y}_{m}-\mathbf{F}\mathbf{S}\|_{F}^{2}\\ \text{s.t.}&\mathbf{A}\succeq\mathbf{0}_{M\times N },\;\mathbf{S}\succeq\mathbf{0}_{N\times L}.\end{split}