## 2 Signal Models and Problem Formulation
### Regularization Reformulation  
In hyperspectral unmixing, the well-known [PERSON]'s belief based on convex geometry is that each hyperspectral pixel should lie in the simplex spanned by the ground-truth endmembers \(\{\mathbf{a}_{1},\ldots,\mathbf{a}_{N}\}\); i.e., containing only one underlying material. The [PERSON]'s minimum-volume simplex covers both spectral signature of endmembers and spatial information carried by enclosed pixels (i.e., fractional abundance ). Thus, the minimum-volume belief is well-suited as a regularizer of the original CNMF to enhance hyperspectral image resolution. Because of the high computations of minimum volume, there are some expressions that have been proposed as the surrogates of minimum volume. This article focuses on the term \(\sum_{j=1}^{N}\lVert\mathbf{a}_{j}-\bar{\boldsymbol{\mu}}\rVert_{2}^{2}\), which makes each column (i.e., each endmember) of \(\mathbf{A}\) as close to \(\bar{\boldsymbol{\mu}}\) as possible, as shown in Figure 2. From this figure, we can observe that \(\bar{\boldsymbol{\mu}}\) is the geometric center of the minimum-volume simplex, i.e., \(\bar{\boldsymbol{\mu}}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{a}_{i}\), in which the red triangle is the optimal simplex and \(k\) is the number of iterations. For the convenience of calculation, \(\sum_{j=1}^{N}\lVert\mathbf{a}_{j}-\frac{1}{N}\sum_{i=1}^{N}\mathbf{a}_{i} \rVert_{2}^{2}\) can be reformulated into the form of a matrix-vector product. Therefore, using the relationship between the \(j\)th (or \(i\)th) column \(\mathbf{a}_{j}\) (or \(\mathbf{a}_{i}\)) and the endmember vector \(\mathbf{a}\triangleq\mathbf{vec}(\mathbf{A})\), a matrix \(\mathbf{P}\) is constructed to make the equation true as  
\[\phi_{a}(\mathbf{A})=\frac{1}{2}\sum_{j=1}^{N}\lVert\mathbf{a}_{j}-\frac{1}{ N}\sum_{i=1}^{N}\mathbf{a}_{i}\rVert_{2}^{2}=\frac{1}{2}\lVert\mathbf{Pa} \rVert_{2}^{2}, \tag{8}\]  
where \(\mathbf{P}=\mathbf{I}_{MN}-\frac{1}{N}(\mathbf{1}_{N}\otimes\mathbf{I}_{M})\). The endmember signature describes the spectral reflectance of a substance, which seems to be an intrinsic identity. Regardless of the bands absorbed by vapor, the spectral signature of each endmember is theoretically a smooth curve in Figure 3. It would be piecewise smooth if some bands were to be removed or absorbed by vapor. Equation (7b) aims at smoothing the signature curve, but it is in scalar form. Therefore, we establish a vertical difference matrix \(\bar{\mathbf{D}}_{e}\) using a Toeplitz matrix to promote the smoothing of endmember signature, and then Equation (7b) can be converted as the vertical total variation \(\lVert\bar{\mathbf{D}}_{e}\mathbf{A}\rVert_{1,1}\). When the matrix \(\mathbf{A}\) is vectorized as \(\mathbf{a}\), (7b) can be reformulated as  
\[\phi_{e}(\mathbf{A})=\sum_{j=1}^{N}\sum_{\{m,n\}\in\gamma}\left|\mathbf{a}_{j }(m)-\mathbf{a}_{j}(n)\right|=\lVert\mathbf{D}_{e}\mathbf{a}\rVert_{1} \tag{9}\]  
where \(\mathbf{D}_{e}=\mathbf{I}_{M}\otimes\bar{\mathbf{D}}_{e}\) is the vector-based difference matrix. Figure 2: Schematic diagram of the minimum-volume simplex with three vertices or endmembers \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\) and \(\mathbf{a}_{3}\). The regularizer that we use, \(\sum_{\{m,n\}\in\varepsilon}\|\mathbf{s}_{m}-\mathbf{s}_{n}\|_{1}\), known as anisotropic total variation, is employed to promote image piecewise smoothing and further reduce the impact of structure information loss. Furthermore, anisotropy total variation can be reshaped into a vector-based form, which has two meanings: one is the vectorization of the abundance matrix for separately yielding the horizontal and vertical difference matrices, and the other is the combination of two difference matrices into a block vector. Let \(\mathbf{\tilde{S}}\) be a frontal slice (i.e., single-band image data) of hyperspectral data cube \(\mathbf{\tilde{Z}}\) with the size of \(L_{y}\times L_{x}\) pixels (i.e., \(L_{y}=L_{x}=\sqrt{L}\)).