a sense, i.e., it must contain all the endmembers of the materials in the scene; ii) sparse solutions of the underdetermined linear systems exist. The latter depends on the sparseness degree of the observed spectra over the library (i.e., the number of library members representing the spectral pixels) and the coherence of library members. The coherence of a library can be measured using mutual coherence [169] (or restricted isometric constants (RICs) [170]). A higher mutual coherence requires a lower sparseness degree to achieve a sparse solution. In sparse unmixing, the mutual coherence of the spectral library is very high [152], which is the main drawback, however, the sparseness degree is low (i.e., pixels are mixtures of a few pure spectra). To decrease the mutual coherence, the library must be pruned before applying the sparse regression or throughout the estimation process. Alternatively, the sparsity-inducing prior can be better defined by incorporating spatial information and/or applying segmentation, localization, and group sparsity, which will be discussed in detail in this section.

1) Selection of a Regularizer: In sparse unmixing, many attempts have been made to improve the sparse representation of the abundances over a spectral library by selecting different regularizers. In [171], SunSAL was improved by incorporating spatial information and applying a TV penalty on abundances (SUnSAL-TV) given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \|\mathbf{X}\|_1 + \\ \lambda_2 TV_{ni}(\mathbf{X}) \quad \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (63) $$

where $TV_{ni}(\mathbf{X})$ is a nonisotropic TV function given by

$$ TV_{ni}(\mathbf{X}) = \|\mathbf{X}\mathbf{H}_h^T\|_1 + \|\mathbf{X}\mathbf{H}_v^T\|_1 \qquad (64) $$

One of the main challenges of sparse regression is the high mutual coherence of the spectral library, which might lead to poor abundance estimation [152]. Therefore, in [172], group SUNSAL (GSUNSAL) was suggested by dividing the library into L groups (structured dictionary) and adding a group LASSO (Least Absolute Shrinkage and Selection Operator) [173]

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \\ \sum_{i=1}^{n} \sum_{g=1}^{L} \lambda_1^g \|\mathbf{x}_i^g\|_2 + \lambda_2 \|\mathbf{X}\|_1 \quad \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (65) $$

where $\mathbf{x}^g$ is the gth subvector of abundances. As we mentioned, $\ell_1$ norm conflicts with ASC and therefore $\ell_q$ norm also investigated for sparse unmixing [174]

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda \sum_{i=1}^{n} \|\mathbf{x}_i\|_q^2 \\ \text{s.t.} \quad \mathbf{X} \ge 0, \mathbf{1}_m^T \mathbf{X} = \mathbf{1}_n^T, 0 < q < 1 \qquad (66) $$

where

$$ \|\mathbf{x}\|_q^2 \triangleq \sum_{i=1}^{m} |x_i|^q, \qquad (67) $$

Collaborative sparse unmixing [27] applies $\ell_{2,1}$ (i.e., the sum of $\ell_2$ on the abundances) to promote the sparsity on the abundances.

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda \sum_{i=1}^{m} \|\mathbf{x}_{(i)}\|_2 \\ \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (68) $$

Collaborative sparse unmixing using $\ell_0$ norm was proposed in [175] given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda \|\mathbf{X}\|_{2,0} \\ \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (69) $$

where $\|\mathbf{X}\|_{2,0} = \sum_{i=1}^{m} \mathrm{Id}(\|\mathbf{x}_{(i)}\|_2 > 0)$ and $\mathrm{Id}$ is the Heaviside step indicator function, i.e., it is one when the argument is positive. In [154], the endmember variability was considered using a dictionary of endmember bundles generated from the data. Unlike the methods mentioned above, in [154], the dictionary is extracted from data points. In [154], a framework of sparse regression was proposed given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \\ \lambda \|\mathbf{X}\|_{g,q,d} \quad \text{s.t.} \quad \mathbf{X} \ge 0, \mathbf{1}_m^T \mathbf{X} = \mathbf{1}_n^T, \qquad (70) $$

where the grouped sparse norm is defined as

$$ \|\mathbf{X}\|_{g,q,d} \triangleq \sum_{i=1}^{n} \left( \sum_{g=1}^{L} \|\mathbf{x}_i^g\|_q \right)^{\frac{1}{d}}. \qquad (71) $$

Three cases are compared, i.e., $\{q=1, d=2\}$, $\{q=1, 0 < d < 1\}$, and $\{q=2, d=1\}$. We should note that ASC was used in the endmember bundles method [154], unlike the structured dictionary methods [27], [172].

In [176], a spatial-spectral weighted $\ell_1$ norm used for (63) given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \|\mathbf{W} \odot \mathbf{S}\mathbf{X}\|_1 + \\ \lambda_2 TV_{ni}(\mathbf{X}) \quad \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (72) $$

where $\mathbf{S} = \mathbf{s1}^T$, $s_i = (\|\mathbf{x}_{(i)}\|_1 + \epsilon)^{-1}$, and $\mathbf{W} = (\mathbf{X} + \epsilon)^{-1}$ are the weight matrices updated in every iterations. The weights further promote the sparsity on $\mathbf{X}$.

Local collaborative sparse unmixing (LCSU) [177] and a spectral-spatial weighted sparse unmixing (S$^2$WSU) [29] further incorporate spatial information by adapting the collaborative sparse unmixing and the reweighted $\ell_1$ norm locally to neighboring pixels, respectively. Indeed, S$^2$WSU uses problem (72) locally and when $\lambda_2 = 0$.

In [178], the weighted nuclear norm was added to the weighted $\ell_1$ norm to utilize the low-rankness of the abundances. The proposed sparse regression is given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \|\mathbf{W} \odot \mathbf{X}\|_1 + \\ \lambda_2 \|\mathbf{X}\|_{t,*} \quad \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (73) $$

where the weighted nuclear norm is given by

$$ \|\mathbf{X}\|_{t,*} = \sum_{i=1}^{r} t_i \sigma_i(\mathbf{X}), \qquad (74) $$
a sense, i.e., it must contain all the endmembers of the materials in the scene; ii) sparse solutions of the underdetermined linear systems exist. The latter depends on the sparseness degree of the observed spectra over the library (i.e., the number of library members representing the spectral pixels) and the coherence of library members. The coherence of a library can be measured using mutual coherence [169] (or restricted isometric constants (RICs) [170]). A higher mutual coherence requires a lower sparseness degree to achieve a sparse solution. In sparse unmixing, the mutual coherence of the spectral library is very high [152], which is the main drawback, however, the sparseness degree is low (i.e., pixels are mixtures of a few pure spectra). To decrease the mutual coherence, the library must be pruned before applying the sparse regression or throughout the estimation process. Alternatively, the sparsity-inducing prior can be better defined by incorporating spatial information and/or applying segmentation, localization, and group sparsity, which will be discussed in detail in this section.

1) Selection of a Regularizer: In sparse unmixing, many attempts have been made to improve the sparse representation of the abundances over a spectral library by selecting different regularizers. In [171], SunSAL was improved by incorporating spatial information and applying a TV penalty on abundances (SUnSAL-TV) given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \|\mathbf{X}\|_1 + \\ \lambda_2 TV_{ni}(\mathbf{X}) \quad \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (63) $$

where $TV_{ni}(\mathbf{X})$ is a nonisotropic TV function given by

$$ TV_{ni}(\mathbf{X}) = \|\mathbf{X}\mathbf{H}_h^T\|_1 + \|\mathbf{X}\mathbf{H}_v^T\|_1 \qquad (64) $$

One of the main challenges of sparse regression is the high mutual coherence of the spectral library, which might lead to poor abundance estimation [152]. Therefore, in [172], group SUNSAL (GSUNSAL) was suggested by dividing the library into L groups (structured dictionary) and adding a group LASSO (Least Absolute Shrinkage and Selection Operator) [173]

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \\ \sum_{i=1}^{n} \sum_{g=1}^{L} \lambda_1^g \|\mathbf{x}_i^g\|_2 + \lambda_2 \|\mathbf{X}\|_1 \quad \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (65) $$

where $\mathbf{x}^g$ is the gth subvector of abundances. As we mentioned, $\ell_1$ norm conflicts with ASC and therefore $\ell_q$ norm also investigated for sparse unmixing [174]

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda \sum_{i=1}^{n} \|\mathbf{x}_i\|_q^2 \\ \text{s.t.} \quad \mathbf{X} \ge 0, \mathbf{1}_m^T \mathbf{X} = \mathbf{1}_n^T, 0 < q < 1 \qquad (66) $$

where

$$ \|\mathbf{x}\|_q^2 \triangleq \sum_{i=1}^{m} |x_i|^q, \qquad (67) $$

Collaborative sparse unmixing [27] applies $\ell_{2,1}$ (i.e., the sum of $\ell_2$ on the abundances) to promote the sparsity on the abundances.

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda \sum_{i=1}^{m} \|\mathbf{x}_{(i)}\|_2 \\ \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (68) $$

Collaborative sparse unmixing using $\ell_0$ norm was proposed in [175] given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda \|\mathbf{X}\|_{2,0} \\ \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (69) $$

where $\|\mathbf{X}\|_{2,0} = \sum_{i=1}^{m} \mathrm{Id}(\|\mathbf{x}_{(i)}\|_2 > 0)$ and $\mathrm{Id}$ is the Heaviside step indicator function, i.e., it is one when the argument is positive. In [154], the endmember variability was considered using a dictionary of endmember bundles generated from the data. Unlike the methods mentioned above, in [154], the dictionary is extracted from data points. In [154], a framework of sparse regression was proposed given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \\ \lambda \|\mathbf{X}\|_{g,q,d} \quad \text{s.t.} \quad \mathbf{X} \ge 0, \mathbf{1}_m^T \mathbf{X} = \mathbf{1}_n^T, \qquad (70) $$

where the grouped sparse norm is defined as

$$ \|\mathbf{X}\|_{g,q,d} \triangleq \sum_{i=1}^{n} \left( \sum_{g=1}^{L} \|\mathbf{x}_i^g\|_q \right)^{\frac{1}{d}}. \qquad (71) $$

Three cases are compared, i.e., $\{q=1, d=2\}$, $\{q=1, 0 < d < 1\}$, and $\{q=2, d=1\}$. We should note that ASC was used in the endmember bundles method [154], unlike the structured dictionary methods [27], [172].

In [176], a spatial-spectral weighted $\ell_1$ norm used for (63) given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \|\mathbf{W} \odot \mathbf{S}\mathbf{X}\|_1 + \\ \lambda_2 TV_{ni}(\mathbf{X}) \quad \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (72) $$

where $\mathbf{S} = \mathbf{s1}^T$, $s_i = (\|\mathbf{x}_{(i)}\|_1 + \epsilon)^{-1}$, and $\mathbf{W} = (\mathbf{X} + \epsilon)^{-1}$ are the weight matrices updated in every iterations. The weights further promote the sparsity on $\mathbf{X}$.

Local collaborative sparse unmixing (LCSU) [177] and a spectral-spatial weighted sparse unmixing (S$^2$WSU) [29] further incorporate spatial information by adapting the collaborative sparse unmixing and the reweighted $\ell_1$ norm locally to neighboring pixels, respectively. Indeed, S$^2$WSU uses problem (72) locally and when $\lambda_2 = 0$.

In [178], the weighted nuclear norm was added to the weighted $\ell_1$ norm to utilize the low-rankness of the abundances. The proposed sparse regression is given by

$$ \hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \|\mathbf{W} \odot \mathbf{X}\|_1 + \\ \lambda_2 \|\mathbf{X}\|_{t,*} \quad \text{s.t.} \quad \mathbf{X} \ge 0. \qquad (73) $$

where the weighted nuclear norm is given by

$$ \|\mathbf{X}\|_{t,*} = \sum_{i=1}^{r} t_i \sigma_i(\mathbf{X}), \qquad (74) $$
and $\sigma_i(\mathbf{X})$ are the singular values of $\mathbf{X}$ and the weights are given by $t_i = (\sigma_i(\mathbf{X}) + \epsilon)^{-1}$. In [179], the spatial-spectral weighted $\ell_1$ was used with the weighted nuclear norm. In [180], the weighted Schatten p-norm [181] and a collaborative norm was suggested to induce the low rankness and the sparsity, respectively, given by

$$ \begin{aligned} \hat{\mathbf{X}} = & \arg\min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \sum_{i=1}^m \|\mathbf{x}_{(i)}\|_{2,q}^q + \\ & \lambda_2 \|\mathbf{X}\|_{t,Sp}^q \quad \text{s.t.} \quad \mathbf{X} \ge 0, 0 < q \le 1. \end{aligned} \quad (75) $$

where

$$ \|\mathbf{x}_{(i)}\|_{2,q}^q \triangleq \sum_{i=1}^m \|\mathbf{x}_{(i)}\|_{2}^q, \quad (76) $$

$$ \|\mathbf{X}\|_{t,Sp}^q \triangleq \sum_{i=1}^r t_i \sigma_i^d(\mathbf{X}). \quad (77) $$

Tensor decomposition can be also used for sparse unmixing formulation using the mode-n tensor multiplication [182], [183].

## 2) Segmentation/Super-pixel and Spectral Variability:

Some methods apply segmentation prior to sparse unmixing for grouping. In this way, they remove spectral variation and capture spatial information. A fast Multiscale Sparse Unmixing Algorithm (MUA) was proposed in [28]. MUA first applies segmentation techniques such as a binary partition tree (BPT), simple linear iterative clustering (SLIC), or K-means to group pixels. Then SUNSAL was utilized for unmixing the average pixels of every segment. The coarse fractional abundance matrix estimated using SUNSAL ($\mathbf{X}_{seg.}$) is used as a prior for the final sparse regression given by

$$ \begin{aligned} \hat{\mathbf{X}} = & \arg\min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \|\mathbf{X}\|_1 + \lambda_2 \|\mathbf{X} - \mathbf{X}_{seg.}\|_2^2 \\ & \text{s.t.} \quad \mathbf{X} \ge 0. \end{aligned} \quad (78) $$

This idea is also called "super-pixel" in the literature, and numerous methods have been proposed using this terminology. In [184], SLIC was utilized for the pixel grouping and pixel-based sparse unmixing was performed using superpixel-based graph Laplacian regularization. In [185], the spectral variability was considered using a spectral variability dictionary. The dictionary was obtained by applying PCA on an endmember library extracted from the dataset using a pure pixel-based endmember extraction [186]. The PCs with high variance were used to create the spectral variability library.

## 3) Library pruning:
Library pruning is a necessary pre-processing step in sparse unmixing to reduce the mutual coherence of the spectral library. The pruning step considerably improves the computational time. However, it risks losing endmembers that are scaled versions of each other. Nevertheless, in real-world applications, the library pruning must be done cautiously, particularly when the SNR is low, and the number of endmembers is high [187]. A simple pruning often used before applying sparse unmixing is to compute SAD. Then, spectra with small angles are removed from the library to reduce the mutual coherence. A pruning strategy was proposed in [188]. The library was projected into the

data subspace, and spectra with high projection errors (the normalized Euclidean distance) were removed from the library. Arguably, data-dependent library pruning is more suitable for sparse unmixing. In [189], a library pruning-based sparse unmixing called multiple signal classification collaborative sparse unmixing (MUSIC-CSR) was proposed. The library was pruned (before applying the collaborative sparse unmixing) using an orthogonal projection ($P^\perp = I - \mathbf{V}\mathbf{V}^T$) [190] where $\mathbf{V}$ is the bases of the $\mathbf{Y}$'s subspace obtained by HySime [47] to reduce the noise effect. The normalized projection error of each library endmember $\|P^\perp \mathbf{d}_i\|_2/\|\mathbf{d}_i\|_2$ was sorted by increasing order. Hence, the first $r$ defines the reduced library. A similar idea was used in [191], which formulates the library reduction into a multiobjective optimization w.r.t. a binary vector representing the support of the abundances matrix.

## 4) Library Mismatch:
As mentioned, the endmember library is fixed in the problems and methods discussed, and the ultimate goal is abundance estimation. However, even a pruned and well-selected spectral library cannot represent all the endmembers of materials in a real-world dataset. There are several factors, such as noise, atmospheric effects, illumination variations, and the intrinsic variation of materials, which may change the endmembers and induce scaling factors into the endmembers in the scene compared to the ones from the library [152], [192] known as library mismatches. This problem was partially addressed in [152]. A bandwise scaling $\mathbf{S}\mathbf{Y}$ was suggested to apply to the dataset where the diagonal matrix $\mathbf{S}$ was obtained by solving

$$ \begin{aligned} (\hat{\mathbf{X}}, \hat{\mathbf{S}}) = & \arg\min_{\mathbf{X},\mathbf{S}} \|\mathbf{S}\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F \\ & \text{s.t.} \quad \mathbf{X} \ge 0, 1_m^T \mathbf{X} = 1_n^T, \end{aligned} \quad (79) $$

A cyclic descent solution was suggested with the initialization of $\mathbf{S} = I$ turning the problem into iterative problems of least squares (w.r.t. $\mathbf{S}$) and constrained least squares (w.r.t. $\mathbf{X}$). The nonsparse fractional abundances of $\mathbf{X}$ and the corresponding spectral vectors of $\mathbf{Y}$ were removed after the first iteration to speed up the algorithm. Problem (79) is noncovex, therefore, the solution will be suboptimal. Additionally, the bandwise scaling $\mathbf{S}$ was estimated for a suboptimal sparse $\mathbf{X}$.

In [193], the library mismatch was incorporated into the sparse unmixing problem given by

$$ \begin{aligned} (\hat{\mathbf{X}}, \hat{\mathbf{D}}') = & \arg\min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}'\mathbf{X}\|_F^2 + \lambda \sum_{i=1}^m \|\mathbf{x}_{(i)}\|_{2,q}^q, \\ & \text{s.t.} \quad \mathbf{X} \ge 0, \|\mathbf{d}_i - \mathbf{d}'_i\|_2 < \delta, i = 1, 2, \dots, m. \end{aligned} \quad (80) $$

We should note that a surrogate function was suggested to be minimized instead of (80) to reduce the complexity of the algorithm (see [193]). Additionally, the MUSIC pruning method was adapted (so-called robust MUSIC) to consider the mismatches given by

$$ \arg\min_{\mathbf{z}} \frac{(\mathbf{d}_i - \mathbf{z})^T P^\perp (\mathbf{d}_i - \mathbf{z})}{\|\mathbf{d}_i - \mathbf{z}\|_2^2} \quad \text{s.t.} \quad \|\mathbf{z}\|_2 < \delta. \quad (81) $$

Recently, sparse unmixing using archetypal analysis (SUNAA) was proposed in [30], which addresses the spectral library mismatch from a different viewpoint. Instead of scaling data,
and $\sigma_i(\mathbf{X})$ are the singular values of $\mathbf{X}$ and the weights are given by $t_i = (\sigma_i(\mathbf{X}) + \epsilon)^{-1}$. In [179], the spatial-spectral weighted $\ell_1$ was used with the weighted nuclear norm. In [180], the weighted Schatten p-norm [181] and a collaborative norm was suggested to induce the low rankness and the sparsity, respectively, given by

$$ \begin{aligned} \hat{\mathbf{X}} = & \arg\min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \sum_{i=1}^m \|\mathbf{x}_{(i)}\|_{2,q}^q + \\ & \lambda_2 \|\mathbf{X}\|_{t,Sp}^q \quad \text{s.t.} \quad \mathbf{X} \ge 0, 0 < q \le 1. \end{aligned} \quad (75) $$

where

$$ \|\mathbf{x}_{(i)}\|_{2,q}^q \triangleq \sum_{i=1}^m \|\mathbf{x}_{(i)}\|_{2}^q, \quad (76) $$

$$ \|\mathbf{X}\|_{t,Sp}^q \triangleq \sum_{i=1}^r t_i \sigma_i^d(\mathbf{X}). \quad (77) $$

Tensor decomposition can be also used for sparse unmixing formulation using the mode-n tensor multiplication [182], [183].

## 2) Segmentation/Super-pixel and Spectral Variability:

Some methods apply segmentation prior to sparse unmixing for grouping. In this way, they remove spectral variation and capture spatial information. A fast Multiscale Sparse Unmixing Algorithm (MUA) was proposed in [28]. MUA first applies segmentation techniques such as a binary partition tree (BPT), simple linear iterative clustering (SLIC), or K-means to group pixels. Then SUnSAL was utilized for unmixing the average pixels of every segment. The coarse fractional abundance matrix estimated using SUnSAL ($\mathbf{X}_{seg.}$) is used as a prior for the final sparse regression given by

$$ \begin{aligned} \hat{\mathbf{X}} = & \arg\min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda_1 \|\mathbf{X}\|_1 + \lambda_2 \|\mathbf{X} - \mathbf{X}_{seg.}\|_2^2 \\ & \text{s.t.} \quad \mathbf{X} \ge 0. \end{aligned} \quad (78) $$

This idea is also called "super-pixel" in the literature, and numerous methods have been proposed using this terminology. In [184], SLIC was utilized for the pixel grouping and pixel-based sparse unmixing was performed using superpixel-based graph Laplacian regularization. In [185], the spectral variability was considered using a spectral variability dictionary. The dictionary was obtained by applying PCA on an endmember library extracted from the dataset using a pure pixel-based endmember extraction [186]. The PCs with high variance were used to create the spectral variability library.

### 3) Library pruning:
Library pruning is a necessary pre-processing step in sparse unmixing to reduce the mutual coherence of the spectral library. The pruning step considerably improves the computational time. However, it risks losing endmembers that are scaled versions of each other. Nevertheless, in real-world applications, the library pruning must be done cautiously, particularly when the SNR is low, and the number of endmembers is high [187]. A simple pruning often used before applying sparse unmixing is to compute SAD. Then, spectra with small angles are removed from the library to reduce the mutual coherence. A pruning strategy was proposed in [188]. The library was projected into the

data subspace, and spectra with high projection errors (the normalized Euclidean distance) were removed from the library. Arguably, data-dependent library pruning is more suitable for sparse unmixing. In [189], a library pruning-based sparse unmixing called multiple signal classification collaborative sparse unmixing (MUSIC-CSR) was proposed. The library was pruned (before applying the collaborative sparse unmixing) using an orthogonal projection ($P^\perp = \mathbf{I} - \mathbf{V}\mathbf{V}^T$) [190] where $\mathbf{V}$ is the bases of the $\mathbf{Y}$'s subspace obtained by HySime [47] to reduce the noise effect. The normalized projection error of each library endmember $\|P^\perp \mathbf{d}_i\|_2/\|\mathbf{d}_i\|_2$ was sorted by increasing order. Hence, the first $r$ defines the reduced library. A similar idea was used in [191], which formulates the library reduction into a multiobjective optimization w.r.t. a binary vector representing the support of the abundances matrix.

### 4) Library Mismatch:
As mentioned, the endmember library is fixed in the problems and methods discussed, and the ultimate goal is abundance estimation. However, even a pruned and well-selected spectral library cannot represent all the endmembers of materials in a real-world dataset. There are several factors, such as noise, atmospheric effects, illumination variations, and the intrinsic variation of materials, which may change the endmembers and induce scaling factors into the endmembers in the scene compared to the ones from the library [152], [192] known as library mismatches. This problem was partially addressed in [152]. A bandwise scaling $\mathbf{S}\mathbf{Y}$ was suggested to apply to the dataset where the diagonal matrix $\mathbf{S}$ was obtained by solving

$$ \begin{aligned} (\hat{\mathbf{X}}, \hat{\mathbf{S}}) = & \arg\min_{\mathbf{X},\mathbf{S}} \|\mathbf{S}\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F \\ & \text{s.t.} \quad \mathbf{X} \ge 0, 1_m^T \mathbf{X} = 1_n^T, \end{aligned} \quad (79) $$

A cyclic descent solution was suggested with the initialization of $\mathbf{S} = \mathbf{I}$ turning the problem into iterative problems of least squares (w.r.t. $\mathbf{S}$) and constrained least squares (w.r.t. $\mathbf{X}$). The nonsparse fractional abundances of $\mathbf{X}$ and the corresponding spectral vectors of $\mathbf{Y}$ were removed after the first iteration to speed up the algorithm. Problem (79) is noncovex, therefore, the solution will be suboptimal. Additionally, the bandwise scaling $\mathbf{S}$ was estimated for a suboptimal sparse $\mathbf{X}$.

In [193], the library mismatch was incorporated into the sparse unmixing problem given by

$$ \begin{aligned} (\hat{\mathbf{X}}, \hat{\mathbf{D}}') = & \arg\min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}'\mathbf{X}\|_F^2 + \lambda \sum_{i=1}^m \|\mathbf{x}_{(i)}\|_{2,q}^q, \\ & \text{s.t.} \quad \mathbf{X} \ge 0, \|\mathbf{d}_i - \mathbf{d}'_i\|_2 < \delta, i = 1, 2, \dots, m. \end{aligned} \quad (80) $$

We should note that a surrogate function was suggested to be minimized instead of (80) to reduce the complexity of the algorithm (see [193]). Additionally, the MUSIC pruning method was adapted (so-called robust MUSIC) to consider the mismatches given by

$$ \arg\min_{\mathbf{z}} \frac{(\mathbf{d}_i - \mathbf{z})^T P^\perp (\mathbf{d}_i - \mathbf{z})}{\|\mathbf{d}_i - \mathbf{z}\|_2^2} \quad \text{s.t.} \quad \|\mathbf{z}\|_2 < \delta. \quad (81) $$

Recently, sparse unmixing using archetypal analysis (SUnAA) was proposed in [30], which addresses the spectral library mismatch from a different viewpoint. Instead of scaling data,
SUnAA assumes that endmembers can be a convex combination of the library endmembers. Therefore, an extra matrix defines the contributions of the endmember from the library. The proposed linear model in [30] utilizes the advantages of the low-rank model and the sparse and redundant model defined as

$$ 
\begin{aligned}
\mathbf{Y} = \mathbf{DBA} + \mathbf{N}, \quad \text{s.t.} \quad & \mathbf{B} \ge 0, \mathbf{1}_m^T \mathbf{B} = \mathbf{1}_r^T, \\
& \text{and} \quad \mathbf{A} \ge 0, \mathbf{1}_r^T \mathbf{A} = \mathbf{1}_n^T,
\end{aligned} 
\quad (82) $$

where $\mathbf{B} \in \mathbb{R}^{m \times r}$ determines the contributions of the endmembers from the the library $\mathbf{D}$ and $\mathbf{A} \in \mathbb{R}^{r \times n}$ is the (low-rank) abundance matrix. To simultaneously estimate $\mathbf{B}$ and $\mathbf{A}$ SUnAA solves a minimization problem given by

$$ 
\begin{aligned}
(\hat{\mathbf{B}}, \hat{\mathbf{A}}) &= \arg \min_{\mathbf{B},\mathbf{A}} \frac{1}{2} \|\mathbf{Y} - \mathbf{DBA}\|_F^2 \\
\text{s.t.} \mathbf{B} \ge 0, \mathbf{1}_m^T \mathbf{B} = \mathbf{1}_r^T, \text{ and } \mathbf{A} \ge 0, \mathbf{1}_r^T \mathbf{A} = \mathbf{1}_n^T,
\end{aligned} 
\quad (83) $$

SUnAA assumes that the unknown endmembers are a convex combination of the library's endmembers which leads to enforcing non-negativity and sum to one constraint on $\mathbf{B}$. Unlike conventional sparse unmixing, the problem (83) is nonconvex. A cyclic descent algorithm based on an active set method [194] was proposed to solve (83), iteratively. We should note that SUnAA is a parameter-free technique.

Numerous techniques were tailored utilizing a combination of the aforementioned ideas, i.e., spatial regularization (TV), spatial/spectral weighted norm, collaborative norm, low-rank inducing, superpixel (segmentation), structured library/ group norm, and localization, library pruning [195]–[202]. Here, we have drawn the main ideas behind the majority of sparse techniques.

5) Shallow/Deep Neural Networks: There are a few NN-based sparse unmixing approaches proposed in the literature. The pioneer method is called sparse unmixing using an unsupervised convolutional neural network (SUnCNN) [21]. As mentioned, the $\ell_1$ penalties cannot be applied to the abundance while holding the ASC. This was addressed in [21] by implicitly applying an image prior and holding the ASC and ANC. It was shown that the problem of selecting a suitable prior for a sparse regression given by

$$ 
\begin{aligned}
\hat{\mathbf{X}} &= \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda\phi(\mathbf{X}) \\
\text{s.t.} \mathbf{X} \ge 0, \mathbf{1}_m^T \mathbf{X} = \mathbf{1}_n^T,
\end{aligned} 
\quad (84) $$

could be moved to optimizing the parameters of a deep network ($f_\theta$) given by

$$ \hat{\theta} = \arg \min_{\theta} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}f_{\theta}(\mathbf{Z})\|_F^2 \quad \text{s.t.} \quad \hat{\mathbf{X}} = f_{\theta}(\mathbf{Z}), \quad (85) $$

where $f_\theta$ is the deep CNN with parameters $\theta$, and ANC and ASC can be enforced using a softmax layer. In [203], an asymmetric encoder-decoder architecture is used for sparse unmixing. Instead of softmax, a sparse variation of softmax is used to avoid the full support of softmax while enforcing ASC.

Spectral Variability Augmented Two-Stream Network (SVANT) was proposed in [204]. SVATN uses the convolutional encoder-decoder used in SUnCNN for both streams. The

first stream uses a spectral library to estimate the abundances, and the second stream uses a spectral variability library to estimate spectral variation. A correlation-based variability extraction method was proposed to create a spectral variability library. A spectral-spatial feature learning encoder-decoder network was proposed in [205]. The encoder utilizes two branches with spatial and spectral filters to extract the spatial and spatial features, respectively. The concatenated features go through three spectral-spatial residual blocks, which utilize MLP for spectral attention. On the other hand, the decoder is shallow and uses the endmember library to reconstruct data.

Algorithm unrolling-based approaches were also proposed for sparse unmixing. An unrolling-based shallow network was recently proposed for sparse unmixing in [206], [207]. An ADMM-based solution to the nonnegative $\ell_1$ sparse regression problem (so-called SUnSAL), i.e., problem (60), was unrolled. An intermediate convolution was applied to the abundance to incorporate spatial information. A combination of SAD, MSE, and SID was used as the loss function to train the shallow network. In [135], the iterative soft-thresholding algorithm (ISTA) [136] was unrolled to solve the nonnegative $\ell_1$ sparse regression problem. SUnSAL was also unrolled in [134] for sparse unmixing.

## VII. NONLINEAR UNMIXING APPROACHES

The linear mixture model is a simplified model and often fails in the case of intimate mixtures or/and when the light undergoes multiple reflections before reaching the sensor [15]. Alternatively, nonlinear models are used [1], [2].

Bilinear approaches are often used for double scattering. They assume that the received light at the sensor interacts with two materials. This is modeled using an extra mixing term by the Hadamard product between the endmembers of the materials. The Fan model [208], [18] is a variation of this model. One disadvantage of the Fan model is the generalization; therefore, it performs poorly for linearly mixed datasets. Polynomial post nonlinear mixing model (PPNM) [209], generalized bilinear model (GBM) [210], and linear-quadratic model (LQM) [18] were proposed to generalize the Fan model for the linear mixtures. However, they contain hyperparameters to describe the trade-off between the linear and nonlinear terms. In [211], a nonlinear low-rank tensor unmixing algorithm was proposed to solve the GBM. Bilinear models have physical interpretation in some specific applications, such as canopy scenarios; however, they have several disadvantages. They have many parameters, and the estimated abundances are hard to interpret. Additionally, they often do not include self-interactions or consider the reflections from objects outside the instance field of view [18]. Moreover, they are limited to secondary interactions. Therefore, several nonlinear mixing models have been developed, such as the multilinear mixing model (MLM) [212] and the p-linear ($p > 2$) mixture model (pLMM) [213]–[215] were proposed for multiple interactions of the incident light.

Kernel methods have also been employed for nonlinear unmixing [18]. They represent data in a higher dimensional space to linearize the problem. Hence, linear models become
SUnAA assumes that endmembers can be a convex combination of the library endmembers. Therefore, an extra matrix defines the contributions of the endmember from the library. The proposed linear model in [30] utilizes the advantages of the low-rank model and the sparse and redundant model defined as

$$ 
\begin{aligned}
\mathbf{Y} = \mathbf{DBA} + \mathbf{N}, \quad \text{s.t.} \quad & \mathbf{B} \ge 0, \mathbf{1}_m^T \mathbf{B} = \mathbf{1}_r^T, \\
& \text{and} \quad \mathbf{A} \ge 0, \mathbf{1}_r^T \mathbf{A} = \mathbf{1}_n^T,
\end{aligned} 
\quad (82) $$

where $\mathbf{B} \in \mathbb{R}^{m \times r}$ determines the contributions of the endmembers from the the library $\mathbf{D}$ and $\mathbf{A} \in \mathbb{R}^{r \times n}$ is the (low-rank) abundance matrix. To simultaneously estimate $\mathbf{B}$ and $\mathbf{A}$ SUnAA solves a minimization problem given by

$$ 
\begin{aligned}
(\hat{\mathbf{B}}, \hat{\mathbf{A}}) &= \arg \min_{\mathbf{B},\mathbf{A}} \frac{1}{2} \|\mathbf{Y} - \mathbf{DBA}\|_F^2 \\
\text{s.t.} \mathbf{B} \ge 0, \mathbf{1}_m^T \mathbf{B} = \mathbf{1}_r^T, \text{ and } \mathbf{A} \ge 0, \mathbf{1}_r^T \mathbf{A} = \mathbf{1}_n^T,
\end{aligned} 
\quad (83) $$

SUnAA assumes that the unknown endmembers are a convex combination of the library's endmembers which leads to enforcing non-negativity and sum to one constraint on $\mathbf{B}$. Unlike conventional sparse unmixing, the problem (83) is nonconvex. A cyclic descent algorithm based on an active set method [194] was proposed to solve (83), iteratively. We should note that SUnAA is a parameter-free technique.

Numerous techniques were tailored utilizing a combination of the aforementioned ideas, i.e., spatial regularization (TV), spatial/spectral weighted norm, collaborative norm, low-rank inducing, superpixel (segmentation), structured library/ group norm, and localization, library pruning [195]–[202]. Here, we have drawn the main ideas behind the majority of sparse techniques.

5) *Shallow/Deep Neural Networks:* There are a few NN-based sparse unmixing approaches proposed in the literature. The pioneer method is called sparse unmixing using an unsupervised convolutional neural network (SUnCNN) [21]. As mentioned, the $\ell_1$ penalties cannot be applied to the abundance while holding the ASC. This was addressed in [21] by implicitly applying an image prior and holding the ASC and ANC. It was shown that the problem of selecting a suitable prior for a sparse regression given by

$$ 
\begin{aligned}
\hat{\mathbf{X}} &= \arg \min_{\mathbf{X}} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}\mathbf{X}\|_F^2 + \lambda\phi(\mathbf{X}) \\
\text{s.t.} \mathbf{X} \ge 0, \mathbf{1}_m^T \mathbf{X} = \mathbf{1}_n^T,
\end{aligned} 
\quad (84) $$

could be moved to optimizing the parameters of a deep network ($f_\theta$) given by

$$ \hat{\theta} = \arg \min_{\theta} \frac{1}{2} \|\mathbf{Y} - \mathbf{D}f_{\theta}(\mathbf{Z})\|_F^2 \quad \text{s.t.} \quad \hat{\mathbf{X}} = f_{\theta}(\mathbf{Z}), \quad (85) $$

where $f_\theta$ is the deep CNN with parameters $\theta$, and ANC and ASC can be enforced using a softmax layer. In [203], an asymmetric encoder-decoder architecture is used for sparse unmixing. Instead of softmax, a sparse variation of softmax is used to avoid the full support of softmax while enforcing ASC.

Spectral Variability Augmented Two-Stream Network (SVANT) was proposed in [204]. SVATN uses the convolutional encoder-decoder used in SUnCNN for both streams. The

first stream uses a spectral library to estimate the abundances, and the second stream uses a spectral variability library to estimate spectral variation. A correlation-based variability extraction method was proposed to create a spectral variability library. A spectral-spatial feature learning encoder-decoder network was proposed in [205]. The encoder utilizes two branches with spatial and spectral filters to extract the spatial and spatial features, respectively. The concatenated features go through three spectral-spatial residual blocks, which utilize MLP for spectral attention. On the other hand, the decoder is shallow and uses the endmember library to reconstruct data.

Algorithm unrolling-based approaches were also proposed for sparse unmixing. An unrolling-based shallow network was recently proposed for sparse unmixing in [206], [207]. An ADMM-based solution to the nonnegative $\ell_1$ sparse regression problem (so-called SUnSAL), i.e., problem (60), was unrolled. An intermediate convolution was applied to the abundance to incorporate spatial information. A combination of SAD, MSE, and SID was used as the loss function to train the shallow network. In [135], the iterative soft-thresholding algorithm (ISTA) [136] was unrolled to solve the nonnegative $\ell_1$ sparse regression problem. SUnSAL was also unrolled in [134] for sparse unmixing.

## VII. NONLINEAR UNMIXING APPROACHES

The linear mixture model is a simplified model and often fails in the case of intimate mixtures or/and when the light undergoes multiple reflections before reaching the sensor [15]. Alternatively, nonlinear models are used [1], [2].

Bilinear approaches are often used for double scattering. They assume that the received light at the sensor interacts with two materials. This is modeled using an extra mixing term by the Hadamard product between the endmembers of the materials. The Fan model [208], [18] is a variation of this model. One disadvantage of the Fan model is the generalization; therefore, it performs poorly for linearly mixed datasets. Polynomial post nonlinear mixing model (PPNM) [209], generalized bilinear model (GBM) [210], and linear-quadratic model (LQM) [18] were proposed to generalize the Fan model for the linear mixtures. However, they contain hyperparameters to describe the trade-off between the linear and nonlinear terms. In [211], a nonlinear low-rank tensor unmixing algorithm was proposed to solve the GBM. Bilinear models have physical interpretation in some specific applications, such as canopy scenarios; however, they have several disadvantages. They have many parameters, and the estimated abundances are hard to interpret. Additionally, they often do not include self-interactions or consider the reflections from objects outside the instance field of view [18]. Moreover, they are limited to secondary interactions. Therefore, several nonlinear mixing models have been developed, such as the multilinear mixing model (MLM) [212] and the p-linear ($p > 2$) mixture model (pLMM) [213]–[215] were proposed for multiple interactions of the incident light.

Kernel methods have also been employed for nonlinear unmixing [18]. They represent data in a higher dimensional space to linearize the problem. Hence, linear models become
effective in high-dimensional space. In [216], a kernel nonnegative matrix factorization (kernel-NMF) was proposed in [217] for nonlinear unmixing. The FCLSU problem can be solved using support vector machines (SVMs). Pure pixels in data are the common support vectors that span the data simplex and allow the estimation of the abundances of all pixels enclosed within the simplex. Hence, Kernel SVMs were proposed for nonlinear spectral unmixing in [218].

Radiative transfer models are mathematical physic-based mixing models that can be used to reconstruct intimately mixed materials' reflectance spectrum [18]. The inverse problems based on such models are very hard to solve. The Hapke model estimates the areal fractions of the materials within the mixture by transforming the reflectance spectra to their single-scattering albedos (SSA) and applying linear unmixing. The simplified version of the Hapke model was used for predicting intimate mixtures' composition [8], [219]. The proposed nonlinear unmixing methods based on Radiative transfer models often assume that spectral reflectance of the pure materials is available for estimating fractional abundances.

Deep networks were also proposed for nonlinear unmixing. Deep AE-based architectures have been broadly used for linear unmixing. In [220], an AE network was proposed where the encoder utilizes an extra nonlinear layer to capture the nonlinearity of the data. A deep AE with multitask learning was suggested in [221]. A long short-term memory-based autoencoder was proposed for PPNM in [222]. The proposed method in [223] exploits a 3-D CAE-based network for PPNM. A supervised AE was used in [224] for Fan, bilinear, and PPNM. The Radial basis function (RBF) kernels and K-means clustering were used to estimate the number of endmembers and the endmember spectra, respectively. DL-based nonlinear unmixing techniques are often AE-based networks using PPNM, and as we discussed, they may have disadvantages of the bilinear models mentioned above [225]–[227]. GAN architectures were also explored for nonlinear unmixing. In [228], a cycle-consistent loss was used to ensure the reconstruction in addition to two GAN losses. A CNN was designed based on Hapke model (HapkeCNN) in [229] to incorporate the physical model in the learning process.

## VIII. EXPERIMENTAL RESULTS

We use three simulated and one real datasets. The simulated datasets were designed according to different mixing scenarios briefly explained in Table III. We avoid using the widely used benchmark datasets such as Samson and Jasper since their abundances are generated synthetically. For real-world experiments, we use the Cuprite dataset, a well-studied site with geological reference maps. The simulated experiments were carried out for 30 dB SNR. The tuning parameters were fine-tuned for the methods up to some levels. We should note that some methods have several parameters to be tuned; therefore, searching for the optimum is cumbersome. All the results are averaged over 10 experiments, and the standard deviations are shown by error bars. In experiments, we compare 20 unmixing methods from different categories as follows. For supervised methods, we used three endmember

extraction/estimation techniques, i.e., VCA, SiVM, and SISAL with FCLSU and UnDIP. All six combinations of them were considered. For blind unmixing, we use PGMSU, MSNET, CNNAEU, ADMMNet, BayesianSMA, NMFQMV, MiSiC-Net, and EDAA. For sparse unmixing, we used SUnSAL, CLSUnSAL, MUA_SLIC, S²WSU, SUnCNN, and SUnAA. The codes for all those methods were provided in HySUPP for reproducibility.

For the quantitative evaluation, we use the SRE in dB for estimated abundances given by

$$ \text{SRE}(\mathbf{X}, \hat{\mathbf{X}}) = 20 \log_{10} \frac{\|\mathbf{X}\|_F}{\|\mathbf{X} - \hat{\mathbf{X}}\|_F} \quad (86) $$

and the spectral angle distance (SAD) in degrees between the estimated and ground truth endmembers:

$$ \text{SAD}(\mathbf{E}, \hat{\mathbf{E}}) = \frac{1}{r} \sum_{i=1}^{r} \arccos \left( \frac{\langle \mathbf{e}_{(i)}, \hat{\mathbf{e}}_{(i)} \rangle}{\| \mathbf{e}_{(i)} \|_2 \| \hat{\mathbf{e}}_{(i)} \|_2} \right) \frac{180}{\pi}, \quad (87) $$

where $\langle \cdot, \cdot \rangle$ denotes the inner product.

### A. Data Description

1) **Synthetic Datasets with Spatial Structure:** We simulated three data cubes (DC1, DC2, and DC3). DC1 was simulated using a linear mixing model with 5 endmembers selected from the USGS library and 75×75 pixels. The abundance maps are composed of five rows of square regions uniformly distributed over the spatial dimension. This dataset contains pure pixels for all endmembers. DC2 has 100×100 pixels and was simulated using a linear mixing model with 9 endmembers. The abundance maps were sampled from a Dirichlet distribution centered at a Gaussian random field to have piecewise smooth maps with steep transitions. Therefore, DC2 contains spectral variations. For DC1 and DC2, an endmember library $\mathbf{D} \in \mathbb{R}^{188 \times 240}$, composed of 240 spectral signatures were selected from the USGS library with a minimum pair-spectra angle of 4.44°. DC3 was simulated with no pure pixels, and it has two mixed pixels on the facet of the data simplex. DC3 is composed of 105×105 pixels using the linear combination of six endmembers. For DC3, we use a library $\mathbf{D} \in \mathbb{R}^{188 \times 498}$ composed of 498 spectral pixels from the USGS library. Note that we remove the water absorption and noisy bands, and therefore, the final pixels are of dimension $p = 188$.

2) **Cuprite Dataset:** The Cuprite dataset used in this paper contains 250×191 pixels. Cuprite is a well-studied mineral site, and the dominant minerals are demonstrated using a geological ground reference therefore, the estimated abundance maps by different techniques can be compared visually. We use the same library as explained for DC3.

### B. Experimental Results: Synthetic Datasets

Figure 14 demonstrates the unmixing results in terms of abundance SRE in DB for different techniques applied to DC1, DC2, and DC3. The outcomes of the results can be summarized as follows:

* For the pure pixel scenario, supervised techniques perform very well (Fig. 14, DC1). Overall, they perform
effective in high-dimensional space. In [216], a kernel nonnegative matrix factorization (kernel-NMF) was proposed in [217] for nonlinear unmixing. The FCLSU problem can be solved using support vector machines (SVMs). Pure pixels in data are the common support vectors that span the data simplex and allow the estimation of the abundances of all pixels enclosed within the simplex. Hence, Kernel SVMs were proposed for nonlinear spectral unmixing in [218].

Radiative transfer models are mathematical physic-based mixing models that can be used to reconstruct intimately mixed materials' reflectance spectrum [18]. The inverse problems based on such models are very hard to solve. The Hapke model estimates the areal fractions of the materials within the mixture by transforming the reflectance spectra to their single-scattering albedos (SSA) and applying linear unmixing. The simplified version of the Hapke model was used for predicting intimate mixtures' composition [8], [219]. The proposed nonlinear unmixing methods based on Radiative transfer models often assume that spectral reflectance of the pure materials is available for estimating fractional abundances.

Deep networks were also proposed for nonlinear unmixing. Deep AE-based architectures have been broadly used for linear unmixing. In [220], an AE network was proposed where the encoder utilizes an extra nonlinear layer to capture the nonlinearity of the data. A deep AE with multitask learning was suggested in [221]. A long short-term memory-based autoencoder was proposed for PPNM in [222]. The proposed method in [223] exploits a 3-D CAE-based network for PPNM. A supervised AE was used in [224] for Fan, bilinear, and PPNM. The Radial basis function (RBF) kernels and K-means clustering were used to estimate the number of endmembers and the endmember spectra, respectively. DL-based nonlinear unmixing techniques are often AE-based networks using PPNM, and as we discussed, they may have disadvantages of the bilinear models mentioned above [225]–[227]. GAN architectures were also explored for nonlinear unmixing. In [228], a cycle-consistent loss was used to ensure the reconstruction in addition to two GAN losses. A CNN was designed based on Hapke model (HapkeCNN) in [229] to incorporate the physical model in the learning process.

## VIII. EXPERIMENTAL RESULTS

We use three simulated and one real datasets. The simulated datasets were designed according to different mixing scenarios briefly explained in Table III. We avoid using the widely used benchmark datasets such as Samson and Jasper since their abundances are generated synthetically. For real-world experiments, we use the Cuprite dataset, a well-studied site with geological reference maps. The simulated experiments were carried out for 30 dB SNR. The tuning parameters were fine-tuned for the methods up to some levels. We should note that some methods have several parameters to be tuned; therefore, searching for the optimum is cumbersome. All the results are averaged over 10 experiments, and the standard deviations are shown by error bars. In experiments, we compare 20 unmixing methods from different categories as follows. For supervised methods, we used three endmember

extraction/estimation techniques, i.e., VCA, SiVM, and SISAL with FCLSU and UnDIP. All six combinations of them were considered. For blind unmixing, we use PGMSU, MSNET, CNNAEU, ADMMNet, BayesianSMA, NMFQMV, MiSiC-Net, and EDAA. For sparse unmixing, we used SUnSAL, CLSUnSAL, MUA_SLIC, S²WSU, SUnCNN, and SUnAA. The codes for all those methods were provided in HySUPP for reproducibility.

For the quantitative evaluation, we use the SRE in dB for estimated abundances given by

$$ \text{SRE}(\mathbf{X}, \hat{\mathbf{X}}) = 20 \log_{10} \frac{\|\mathbf{X}\|_F}{\|\mathbf{X} - \hat{\mathbf{X}}\|_F} \quad (86) $$

and the spectral angle distance (SAD) in degrees between the estimated and ground truth endmembers:

$$ \text{SAD}(\mathbf{E}, \hat{\mathbf{E}}) = \frac{1}{r} \sum_{i=1}^{r} \arccos \left( \frac{\langle \mathbf{e}_{(i)}, \hat{\mathbf{e}}_{(i)} \rangle}{\| \mathbf{e}_{(i)} \|_2 \| \hat{\mathbf{e}}_{(i)} \|_2} \right) \frac{180}{\pi}, \quad (87) $$

where $\langle \cdot, \cdot \rangle$ denotes the inner product.

### A. Data Description

1) **Synthetic Datasets with Spatial Structure:** We simulated three data cubes (DC1, DC2, and DC3). DC1 was simulated using a linear mixing model with 5 endmembers selected from the USGS library and 75×75 pixels. The abundance maps are composed of five rows of square regions uniformly distributed over the spatial dimension. This dataset contains pure pixels for all endmembers. DC2 has 100×100 pixels and was simulated using a linear mixing model with 9 endmembers. The abundance maps were sampled from a Dirichlet distribution centered at a Gaussian random field to have piecewise smooth maps with steep transitions. Therefore, DC2 contains spectral variations. For DC1 and DC2, an endmember library $\mathbf{D} \in \mathbb{R}^{188 \times 240}$, composed of 240 spectral signatures were selected from the USGS library with a minimum pair-spectra angle of 4.44°. DC3 was simulated with no pure pixels, and it has two mixed pixels on the facet of the data simplex. DC3 is composed of 105×105 pixels using the linear combination of six endmembers. For DC3, we use a library $\mathbf{D} \in \mathbb{R}^{188 \times 498}$ composed of 498 spectral pixels from the USGS library. Note that we remove the water absorption and noisy bands, and therefore, the final pixels are of dimension $p = 188$.

2) **Cuprite Dataset:** The Cuprite dataset used in this paper contains 250×191 pixels. Cuprite is a well-studied mineral site, and the dominant minerals are demonstrated using a geological ground reference therefore, the estimated abundance maps by different techniques can be compared visually. We use the same library as explained for DC3.

### B. Experimental Results: Synthetic Datasets

Figure 14 demonstrates the unmixing results in terms of abundance SRE in DB for different techniques applied to DC1, DC2, and DC3. The outcomes of the results can be summarized as follows:

* For the pure pixel scenario, supervised techniques perform very well (Fig. 14, DC1). Overall, they perform
<table><caption>TABLE III
SPECIFICATIONS OF THE SYNTHETIC DATASETS USED IN THE EXPERIMENTS.</caption><thead><tr><th></th><th># endmembers (r)</th><th># atoms in D (m)</th><th># pixels (n)</th><th>Main features</th></tr></thead><tbody><tr><td>DC1</td><td>5</td><td>240</td><td>5625 (75 × 75)</td><td>Pure pixels</td></tr><tr><td>DC2</td><td>9</td><td>240</td><td>10000 (100 × 100)</td><td>Pure pixels, Spectral variability</td></tr><tr><td>DC3</td><td>6</td><td>498</td><td>11025 (105 × 105)</td><td>No pure pixels, 2 points on the data simplex facets</td></tr></tbody></table>

better than the semisupervised and blind techniques. This confirms the importance of geometrical information for endmember extraction/ estimation techniques when there are pure pixels in the dataset. This is further confirmed in blind methods where MiSiCNet and NMFQMV, which exploit geometrical information, outperform the other blind techniques and provide competitive results compared to supervised ones. Sparse techniques show moderate results except SUnAA, which outperforms all the other techniques. The results confirm that the sparse unmixing techniques are not suitable when there are pure pixels for the endmembers in the dataset. We should mention that SUnAA does not match the characteristics of conventional sparse techniques. Even though SUnAA relies on a library, it uses a nonconvex optimization to estimate endmembers.

* On the other hand, for DC2, which contains spectral variations, as can be seen from Fig. 14, sparse unmixing techniques outperform the supervised and blind techniques. The results confirm that sparse unmixing techniques are more suitable for capturing the spectral variability. SUnAA outperforms the other technique. Note that SUnAA is a parameter-free technique. Supervised techniques outperform blind techniques due to the presence of pure pixels.

* In the case of no pure pixel scenario (Fig. 14, DC3), blind unmixing techniques that exploit geometrical information outperform the other techniques. Sparse unmixing provides very poor results. Although SUnAA considerably significantly outperforms the performance of sparse unmixing techniques, it is very far from the best performance which is obtained by MiSiCNet. Among supervised techniques, SISAL (+ FCLSU/UnDIP) shows competitive results because it uses geometrical information to estimate the endmembers.

* Table IV compares the SAD values for the endmember estimation/extraction techniques with the blind unmixing techniques for the simulated datasets at 30dB. In the case of pure pixel scenarios, SiVM and VCA outperform the other techniques confirming the advantage of geometrical approaches for endmember extraction when there are pure pixels for all the endmembers. However, in the no-pure pixel scenario, they cannot estimate the endmembers. On the other hand, the blind unmixing methods that incorporate geometrical information i.e., NMFQMV and MiSiCNet outperform the other techniques. Overall, the results reveal the advantage of the geometrical information in accurate endmember extraction/estimation. Finally, we should mention that SAD is scale-invariant

and cannot solely be considered for the evaluation of unmixing models.

### C. Experimental Results: Real Data

We selected three methods per category to conduct unmixing on the Cuprite dataset. Blind unmixing: MiSiCNet, MSNet and NMFQMV. Semi-supervised: SUnAA, MUA_SLIC, S²WSU. Supervised: UnDIP combined with SISAL, SiVM and VCA. We describe the hyperparameters that were fine-tuned for the following techniques. The hyper-parameters set as default are omitted.

* MiSiCNet: $\lambda = 100$, projection=True.

* MSNet: $\alpha = 0.1, \beta = 0.1$.

* MUA_SLIC: $\beta = 30, \lambda_1 = 0.001, \lambda_2 = 0.001, \text{slic_size} = 200$.

* S²WSU: $\lambda = 0.001$.

* SISAL: $\tau = 1e-6$.

Fig. 15 (b) visually compares the estimated abundances for three dominant minerals, i.e., Chalcedony, Alunite, and Kaolinite. The comparison with the geological reference map Fig. 15 (a) reveals that the estimated abundances obtained by semisupervised methods show more resemblance to the reference map for all three minerals. SUnAA visually outperforms the other techniques, particularly in the case of Chalcedony. The blind unmixing methods can better estimate Chalcedony compared to MUA_SLIC and S²WSU. This could be attributed to the mismatch of the endmember with the library's endmembers for this mineral. It is worth mentioning that SUnAA does not entirely rely on the library, and it estimates the endmember. Therefore, it can compensate for such a mismatch. More importantly, SUnAA is a parameter-free technique. We should note that selecting optimum parameters for the unmixing techniques is not a trivial task in real-world applications since the abundance RMSE cannot be computed.

### D. Experimental Results: Processing Time

The processing time of the unmixing methods from the HySUPP are given in Table V in seconds. The processing times were obtained using a computer with an Intel(R) Xeon(R) Silver 4110 processor (2.10 GHz), 32 cores, 64GB of memory, and a NVIDIA GeForce RTX (2080 Ti) graphical processing unit. In supervised unmixing, SISAL+FCLSU is the most efficient method. In semisupervised unmixing, SUnAA is the most efficient one. We should mention that the efficiency of MUA_SLIC considerably drops by increasing the number of pixels which highly affects the segmentation used in that method. For blind unmixing, CNNAEU and MSNet are more efficient than the other methods.
<table><caption>TABLE III
SPECIFICATIONS OF THE SYNTHETIC DATASETS USED IN THE EXPERIMENTS.</caption><thead><tr><th></th><th># endmembers (r)</th><th># atoms in D (m)</th><th># pixels (n)</th><th>Main features</th></tr></thead><tbody><tr><td>DC1</td><td>5</td><td>240</td><td>5625 (75 × 75)</td><td>Pure pixels</td></tr><tr><td>DC2</td><td>9</td><td>240</td><td>10000 (100 × 100)</td><td>Pure pixels, Spectral variability</td></tr><tr><td>DC3</td><td>6</td><td>498</td><td>11025 (105 × 105)</td><td>No pure pixels, 2 points on the data simplex facets</td></tr></tbody></table>

better than the semisupervised and blind techniques. This confirms the importance of geometrical information for endmember extraction/ estimation techniques when there are pure pixels in the dataset. This is further confirmed in blind methods where MiSiCNet and NMFQMV, which exploit geometrical information, outperform the other blind techniques and provide competitive results compared to supervised ones. Sparse techniques show moderate results except SUnAA, which outperforms all the other techniques. The results confirm that the sparse unmixing techniques are not suitable when there are pure pixels for the endmembers in the dataset. We should mention that SUnAA does not match the characteristics of conventional sparse techniques. Even though SUnAA relies on a library, it uses a nonconvex optimization to estimate endmembers.

* On the other hand, for DC2, which contains spectral variations, as can be seen from Fig. 14, sparse unmixing techniques outperform the supervised and blind techniques. The results confirm that sparse unmixing techniques are more suitable for capturing the spectral variability. SUnAA outperforms the other technique. Note that SUnAA is a parameter-free technique. Supervised techniques outperform blind techniques due to the presence of pure pixels.

* In the case of no pure pixel scenario (Fig. 14, DC3), blind unmixing techniques that exploit geometrical information outperform the other techniques. Sparse unmixing provides very poor results. Although SUnAA considerably significantly outperforms the performance of sparse unmixing techniques, it is very far from the best performance which is obtained by MiSiCNet. Among supervised techniques, SISAL (+ FCLSU/UnDIP) shows competitive results because it uses geometrical information to estimate the endmembers.

* Table IV compares the SAD values for the endmember estimation/extraction techniques with the blind unmixing techniques for the simulated datasets at 30dB. In the case of pure pixel scenarios, SiVM and VCA outperform the other techniques confirming the advantage of geometrical approaches for endmember extraction when there are pure pixels for all the endmembers. However, in the no-pure pixel scenario, they cannot estimate the endmembers. On the other hand, the blind unmixing methods that incorporate geometrical information i.e., NMFQMV and MiSiCNet outperform the other techniques. Overall, the results reveal the advantage of the geometrical information in accurate endmember extraction/estimation. Finally, we should mention that SAD is scale-invariant

and cannot solely be considered for the evaluation of unmixing models.

### C. Experimental Results: Real Data

We selected three methods per category to conduct unmixing on the Cuprite dataset. Blind unmixing: MiSiCNet, MSNet and NMFQMV. Semi-supervised: SUnAA, MUA_SLAVE, S²WSU. Supervised: UnDIP combined with SISAL, SiVM and VCA. We describe the hyperparameters that were fine-tuned for the following techniques. The hyper-parameters set as default are omitted.

* MiSiCNet: $\lambda = 100$, projection=True.

* MSNet: $\alpha = 0.1, \beta = 0.1$.

* MUA_SLAVE: $\beta = 30, \lambda_1 = 0.001, \lambda_2 = 0.001, \text{slic_size} = 200$.

* S²WSU: $\lambda = 0.001$.

* SISAL: $\tau = 1e-6$.

Fig. 15 (b) visually compares the estimated abundances for three dominant minerals, i.e., Chalcedony, Alunite, and Kaolinite. The comparison with the geological reference map Fig. 15 (a) reveals that the estimated abundances obtained by semisupervised methods show more resemblance to the reference map for all three minerals. SUnAA visually outperforms the other techniques, particularly in the case of Chalcedony. The blind unmixing methods can better estimate Chalcedony compared to MUA_SLAVE and S²WSU. This could be attributed to the mismatch of the endmember with the library's endmembers for this mineral. It is worth mentioning that SUnAA does not entirely rely on the library, and it estimates the endmember. Therefore, it can compensate for such a mismatch. More importantly, SUnAA is a parameter-free technique. We should note that selecting optimum parameters for the unmixing techniques is not a trivial task in real-world applications since the abundance RMSE cannot be computed.

### D. Experimental Results: Processing Time

The processing time of the unmixing methods from the HySUPP are given in Table V in seconds. The processing times were obtained using a computer with an Intel(R) Xeon(R) Silver 4110 processor (2.10 GHz), 32 cores, 64GB of memory, and a NVIDIA GeForce RTX (2080 Ti) graphical processing unit. In supervised unmixing, SISAL+FCLSU is the most efficient method. In semisupervised unmixing, SUnAA is the most efficient one. We should mention that the efficiency of MUA_SLAVE considerably drops by increasing the number of pixels which highly affects the segmentation used in that method. For blind unmixing, CNNAEU and MSNet are more efficient than the other methods.
<figure><img src="image_15.png" /><figcaption>Fig. 14. Comparing abundance SRE (↑) in dB using different unmixing techniques applied to (from top to bottom) synthetic DC1, DC2, and DC3.</figcaption></figure>
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 14. Comparing abundance SRE (↑) in dB using different unmixing techniques applied to (from top to bottom) synthetic DC1, DC2, and DC3.</figcaption></figure>
<table><caption>TABLE IV
SPECTRAL ANGLE DISTANCE (SAD, IN DEGREES) RESULTS ON THE SIMULATED DATASETS.</caption><thead><tr><th>Dataset</th><th>SNR</th><th>ADMMNet</th><th>BayesianSMA</th><th>CNNAEU</th><th>EDAA</th><th>MSNet</th><th>MiSiCNet</th><th>NMFQMV</th><th>PGMSU</th><th>SISAL</th><th>SiVM</th><th>VCA</th></tr></thead><tbody><tr><td>DC1</td><td>30</td><td>0.84 ± 0.33</td><td>0.49 ± 0.02</td><td>8.12 ± 0.12</td><td>4.93 ± 1.15</td><td>1.77 ± 1.27</td><td><b>0.43</b> ± 0.01</td><td>0.99 ± 0.22</td><td>1.63 ± 0.32</td><td>1.85 ± 0.21</td><td><b>0.43</b> ± 0.02</td><td>0.45 ± 0.01</td></tr><tr><td>DC2</td><td>30</td><td>2.20 ± 0.13</td><td>1.65 ± 0.13</td><td>4.63 ± 0.95</td><td>0.65 ± 0.51</td><td>0.76 ± 0.03</td><td>2.98 ± 0.40</td><td>1.07 ± 0.01</td><td>1.97 ± 0.09</td><td>9.26 ± 0.80</td><td><b>0.45</b> ± 0.05</td><td>0.53 ± 0.04</td></tr><tr><td>DC3</td><td>30</td><td>8.07 ± 3.61</td><td>9.26 ± 2.87</td><td>9.68 ± 0.50</td><td>4.82 ± 0.04</td><td>4.30 ± 0.22</td><td>0.87 ± 0.03</td><td><b>0.45</b> ± 0.08</td><td>6.73 ± 1.08</td><td>2.32 ± 0.17</td><td>7.09 ± 0.36</td><td>4.70 ± 0.75</td></tr></tbody></table>

<figure><img src="image_16.png" /><figcaption>(a) Geological Ref. Map</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(b) Estimate abundances</figcaption></figure>

<caption>Fig. 15. Estimate abundances obtained by applying different unmixing techniques to Cuprite compared with the geological reference map.</caption>

<table><caption>TABLE V
PROCESSING TIMES (IN SECONDS).</caption><thead><tr><th>Method</th><th>DC1 (30dB)</th><th>DC2 (30dB)</th><th>DC3 (30dB)</th><th>Cuprite</th></tr></thead><tbody><tr><td>VCA + FCLSU</td><td><b>3</b></td><td>8</td><td><b>10</b></td><td>92</td></tr><tr><td>SiVM + FCLSU</td><td><b>3</b></td><td>9</td><td><b>10</b></td><td>58</td></tr><tr><td>SISAL + FCLSU</td><td>6</td><td><b>7</b></td><td><b>10</b></td><td><b>54</b></td></tr><tr><td>VCA + UnDIP</td><td>31</td><td>23</td><td>29</td><td>492</td></tr><tr><td>SiVM + UnDIP</td><td>31</td><td>28</td><td>30</td><td>492</td></tr><tr><td>SISAL + UnDIP</td><td>29</td><td>26</td><td>32</td><td>495</td></tr><tr><td>CLSUnSAL</td><td><b>12</b></td><td>23</td><td>56</td><td>457</td></tr><tr><td>MUA_SLAVE</td><td><b>12</b></td><td><b>15</b></td><td><b>36</b></td><td>310</td></tr><tr><td>S<sup>2</sup>WSU</td><td>35</td><td>60</td><td>116</td><td>767</td></tr><tr><td>SUnAA</td><td>80</td><td>178</td><td>146</td><td>1201</td></tr><tr><td>SUnCNN</td><td>75</td><td>65</td><td>97</td><td>492</td></tr><tr><td>SUnSAL</td><td>13</td><td>17</td><td>44</td><td><b>176</b></td></tr><tr><td>ADMMNet</td><td>33</td><td>57</td><td>61</td><td>969</td></tr><tr><td>BayesianSMA</td><td>122</td><td>367</td><td>255</td><td>3593</td></tr><tr><td>CNNAEU</td><td><b>9</b></td><td>22</td><td><b>24</b></td><td><b>20</b></td></tr><tr><td>EDAA</td><td>274</td><td>51</td><td>59</td><td>114</td></tr><tr><td>MiSiCNet</td><td>50</td><td>66</td><td>58</td><td>483</td></tr><tr><td>MSNet</td><td>20</td><td><b>18</b></td><td>25</td><td>33</td></tr><tr><td>NMFQMV</td><td>29</td><td>28</td><td>28</td><td>293</td></tr><tr><td>PGMSU</td><td>43</td><td>41</td><td>49</td><td>135</td></tr></tbody></table>

## IX. CONCLUSION, DISCUSSION, SUMMARY

Assuming that we have captured a spectral dataset and now have an unmixing problem in hand, we need to estimate the abundances of materials. The main question is which method to choose and which group of methods to select to tackle the problem. Indeed, the first step is to evaluate our problem and see if the linear mixing model or its variations are suitable for our problem. This decision needs prior knowledge of the physics of the problem. For instance, if you are dealing with intimate mixtures or close-range and microscopic scenarios,

you should use nonlinear models. If you are dealing with macroscopic Earth observation problems, then linear models or their variations will be suitable. In some research, nonlinear models perform better than linear ones, however, one may pay attention to the selection of the model in a real-world application. Usually, linear models are more general.

Here, we clarify the keys to the success of each category. The success of supervised (or sequential) unmixing lies in the confidence of the endmember measured, extracted, or estimated (pure/no pure scenario). Therefore, we should not use the supervised method if we are not confident about the endmembers. In other words, supervised methods perhaps are the best choice for endmembers with high confidence. When we have prior information on the material in the scene and a well-designed endmember library, semisupervised unmixing could be successful. Semisupervised are also suitable to capture the spectral variability. The success of semisupervised unmixing lies in the quality of the endmember library. Blind unmixing methods should be selected when there is no library, no pure pixels in the data set (including highly mixed scenarios), or the confidence of the measure, selected, or estimated endmembers is low. They should be used with caution, and the estimated endmembers should always go through physical interpretation.

Despite the considerable advances, spectral unmixing can be considered one of the most challenging tasks in hyperspectral analysis. Here, we describe some of the main unmixing challenges.

* The linear models are much more general than the nonlinear models. However, from one dataset to another dataset, their performance may significantly decrease.

* Parameter selection considerably affects the performance of the unmixing methods. The selection of optimum
<table><caption>TABLE IV
SPECTRAL ANGLE DISTANCE (SAD, IN DEGREES) RESULTS ON THE SIMULATED DATASETS.</caption><thead><tr><th>Dataset</th><th>SNR</th><th>ADMMNet</th><th>BayesianSMA</th><th>CNNAEU</th><th>EDAA</th><th>MSNet</th><th>MiSiCNet</th><th>NMFQMV</th><th>PGMSU</th><th>SISAL</th><th>SiVM</th><th>VCA</th></tr></thead><tbody><tr><td>DC1</td><td>30</td><td>0.84 ± 0.33</td><td>0.49 ± 0.02</td><td>8.12 ± 0.12</td><td>4.93 ± 1.15</td><td>1.77 ± 1.27</td><td><strong>0.43</strong> ± 0.01</td><td>0.99 ± 0.22</td><td>1.63 ± 0.32</td><td>1.85 ± 0.21</td><td><strong>0.43</strong> ± 0.02</td><td>0.45 ± 0.01</td></tr><tr><td>DC2</td><td>30</td><td>2.20 ± 0.13</td><td>1.65 ± 0.13</td><td>4.63 ± 0.95</td><td>0.65 ± 0.51</td><td>0.76 ± 0.03</td><td>2.98 ± 0.40</td><td>1.07 ± 0.01</td><td>1.97 ± 0.09</td><td>9.26 ± 0.80</td><td><strong>0.45</strong> ± 0.05</td><td>0.53 ± 0.04</td></tr><tr><td>DC3</td><td>30</td><td>8.07 ± 3.61</td><td>9.26 ± 2.87</td><td>9.68 ± 0.50</td><td>4.82 ± 0.04</td><td>4.30 ± 0.22</td><td>0.87 ± 0.03</td><td><strong>0.45</strong> ± 0.08</td><td>6.73 ± 1.08</td><td>2.32 ± 0.17</td><td>7.09 ± 0.36</td><td>4.70 ± 0.75</td></tr></tbody></table>

<figure><img src="image_18.png" /><figcaption>(a) Geological Ref. Map</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(b) Estimate abundances</figcaption></figure>

<caption>Fig. 15. Estimate abundances obtained by applying different unmixing techniques to Cuprite compared with the geological reference map.</caption>

<table><caption>TABLE V
PROCESSING TIMES (IN SECONDS).</caption><thead><tr><th>Method</th><th>DC1 (30dB)</th><th>DC2 (30dB)</th><th>DC3 (30dB)</th><th>Cuprite</th></tr></thead><tbody><tr><td>VCA + FCLSU</td><td><strong>3</strong></td><td>8</td><td><strong>10</strong></td><td>92</td></tr><tr><td>SiVM + FCLSU</td><td><strong>3</strong></td><td>9</td><td><strong>10</strong></td><td>58</td></tr><tr><td>SISAL + FCLSU</td><td>6</td><td><strong>7</strong></td><td><strong>10</strong></td><td><strong>54</strong></td></tr><tr><td>VCA + UnDIP</td><td>31</td><td>23</td><td>29</td><td>492</td></tr><tr><td>SiVM + UnDIP</td><td>31</td><td>28</td><td>30</td><td>492</td></tr><tr><td>SISAL + UnDIP</td><td>29</td><td>26</td><td>32</td><td>495</td></tr><tr><td>CLSUnSAL</td><td><strong>12</strong></td><td>23</td><td>56</td><td>457</td></tr><tr><td>MUA_SLAVE</td><td><strong>12</strong></td><td><strong>15</strong></td><td><strong>36</strong></td><td>310</td></tr><tr><td>S<sup>2</sup>WSU</td><td>35</td><td>60</td><td>116</td><td>767</td></tr><tr><td>SUnAA</td><td>80</td><td>178</td><td>146</td><td>1201</td></tr><tr><td>SUnCNN</td><td>75</td><td>65</td><td>97</td><td>492</td></tr><tr><td>SUnSAL</td><td>13</td><td>17</td><td>44</td><td><strong>176</strong></td></tr><tr><td>ADMMNet</td><td>33</td><td>57</td><td>61</td><td>969</td></tr><tr><td>BayesianSMA</td><td>122</td><td>367</td><td>255</td><td>3593</td></tr><tr><td>CNNAEU</td><td><strong>9</strong></td><td>22</td><td><strong>24</strong></td><td><strong>20</strong></td></tr><tr><td>EDAA</td><td>274</td><td>51</td><td>59</td><td>114</td></tr><tr><td>MiSiCNet</td><td>50</td><td>66</td><td>58</td><td>483</td></tr><tr><td>MSNet</td><td>20</td><td><strong>18</strong></td><td>25</td><td>33</td></tr><tr><td>NMFQMV</td><td>29</td><td>28</td><td>28</td><td>293</td></tr><tr><td>PGMSU</td><td>43</td><td>41</td><td>49</td><td>135</td></tr></tbody></table>

## IX. CONCLUSION, DISCUSSION, SUMMARY

Assuming that we have captured a spectral dataset and now have an unmixing problem in hand, we need to estimate the abundances of materials. The main question is which method to choose and which group of methods to select to tackle the problem. Indeed, the first step is to evaluate our problem and see if the linear mixing model or its variations are suitable for our problem. This decision needs prior knowledge of the physics of the problem. For instance, if you are dealing with intimate mixtures or close-range and microscopic scenarios,

you should use nonlinear models. If you are dealing with macroscopic Earth observation problems, then linear models or their variations will be suitable. In some research, nonlinear models perform better than linear ones, however, one may pay attention to the selection of the model in a real-world application. Usually, linear models are more general.

Here, we clarify the keys to the success of each category. The success of supervised (or sequential) unmixing lies in the confidence of the endmember measured, extracted, or estimated (pure/no pure scenario). Therefore, we should not use the supervised method if we are not confident about the endmembers. In other words, supervised methods perhaps are the best choice for endmembers with high confidence. When we have prior information on the material in the scene and a well-designed endmember library, semisupervised unmixing could be successful. Semisupervised are also suitable to capture the spectral variability. The success of semisupervised unmixing lies in the quality of the endmember library. Blind unmixing methods should be selected when there is no library, no pure pixels in the data set (including highly mixed scenarios), or the confidence of the measure, selected, or estimated endmembers is low. They should be used with caution, and the estimated endmembers should always go through physical interpretation.

Despite the considerable advances, spectral unmixing can be considered one of the most challenging tasks in hyperspectral analysis. Here, we describe some of the main unmixing challenges.

* The linear models are much more general than the nonlinear models. However, from one dataset to another dataset, their performance may significantly decrease.

* Parameter selection considerably affects the performance of the unmixing methods. The selection of optimum
parameters is very challenging when it comes to real-world datasets.

* The performance of the linear unmixing methods often degrades by increasing the number of endmembers. The linear unmixing may fail for a dataset with a higher number of endmembers.

* Spectral variability is still a big challenge and may considerably downgrade the performance of linear unmixing.

* One of the main challenges is the absence of a real dataset with ground truth.

* Multitemporal and multisource spectral unmixing are also challenging tasks.

* Considering the large volume of hyperspectral data, scalable unmixing is the main key for global monitoring.

Finally, we would like to further emphasize the importance of unmixing compared to the other signal and image processing approaches such as feature extraction/selection, clustering, and classification. The estimated features (i.e., abundances) in unmixing have physical meaning and can be attributed to materials. Therefore, estimated abundance maps can be used for quantitative analysis such as estimating vegetation cover, mineral composition, or pollutant concentrations providing quantitative information about the spatial distribution and relative quantities of materials present in the scene. The extracted/estimated endmembers can be used to identify and map different materials or surface types within the scene. By comparing the spectral signatures of unknown pixels with the reference endmembers, it is possible to classify and map land cover types, vegetation species, geological formations, or man-made structures. Temporal changes can be detected more accurately. The estimated abundance maps provide quantitative changes at a subpixel level. The extracted/estimated endmembers can be used for the identification of material changes. Intrinsic changes in the materials can be considered or ignored based on the endmembers assumptions. A future point is to investigate the performance of foundation models for spectral unmixing.

## ACKNOWLEDGMENTS

AZ, JM, and JC were supported by ANR 3IA MIAI@Grenoble Alpes (ANR-19-P3IA-0003). AZ and JM were supported by the ERC grant number 101087696 (APHELAIA project). The authors would like to thank Dr. Michael Arbel (the author of the MLXP package) for the guidance on the MLXP package and for making it adaptable to the Windows operating system.

## APPENDIX

### MINIMUM VOLUME SIMPLEX USING TOTAL VARIATION

The total variation penalty enforces the data simplex to have minimum volume by pulling the endmembers towards each

other. Assuming $\bar{e} = \frac{1}{r}e1_r$ we have

$$ 
\begin{aligned}
TV(\mathbf{E}) &= \sum_{i,j=1}^{r} \frac{1}{2} \|e_i - e_j\|_2^2 = \\
& \frac{1}{2} \sum_{i,j=1}^{r} \|e_i - \bar{e} - e_j + \bar{e}\|_2^2 = \\
& \frac{1}{2} \sum_{i=1}^{r} \|e_i - \bar{e}\|_2^2 + \frac{1}{2} \sum_{j=1}^{r} \|e_j - \bar{e}\|_2^2 - \\
& \left( \sum_{i=1}^{r} (e_i - \bar{e}) \right)^T \sum_{j=1}^{r} (e_j - \bar{e})
\end{aligned}
\quad (88) $$

where the third term is zero. Therefore,

$$ 
\begin{aligned}
TV(\mathbf{E}) &= \sum_{i=1}^{r} \|e_i - \bar{e}\|_2^2 = \\
& \sum_{i=1}^{r} \|e_i - \frac{1}{r}e1_r\|_2^2 = \|\mathbf{E}(I_r - \frac{1}{r}1_r1_r^T)\|_F^2
\end{aligned}
\quad (89) $$

## REFERENCES

[1] J. M. Bioucas-Dias, A. Plaza, N. Dobigeon, M. Parente, Q. Du, P. Gader, and J. Chanussot, "Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 5, no. 2, pp. 354–379, April 2012.

[2] N. Dobigeon, J. Tourneret, C. Richard, J. C. M. Bermudez, S. McLaughlin, and A. O. Hero, "Nonlinear unmixing of hyperspectral images: Models and algorithms," *IEEE Signal Processing Magazine*, vol. 31, no. 1, pp. 82–94, 2014.

[3] R. A. Borsoi, T. Imbiriba, J. C. Moreira Bermudez, C. Richard, J. Chanussot, L. Drumetz, J.-Y. Tourneret, A. Zare, and C. Jutten, "Spectral variability in hyperspectral data unmixing: A comprehensive review," *IEEE Geoscience and Remote Sensing Magazine*, vol. 9, no. 4, pp. 223–270, 2021.

[4] D. Coquelin, B. Rasti, M. Götz, P. Ghamisi, R. Gloaguen, and A. Streit, "Hyde: The first open-source, python-based, gpu-accelerated hyperspectral denoising package," in *2022 12th Workshop on Hyperspectral Imaging and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2022, pp. 1–5.

[5] B. Rasti, P. Scheunders, P. Ghamisi, G. Licciardi, and J. Chanussot, "Noise reduction in hyperspectral imagery: Overview and application," *Remote Sensing*, vol. 10, no. 3, pp. 482, 2018.

[6] B. Rasti, Y. Chang, E. Dalsasso, Loic Denis, and Pedram Ghamisi, "Image Restoration for Remote Sensing: Overview and Toolbox," *IEEE Geoscience and Remote Sensing Magazine*, pp. 2–31, 2021.

[7] B. Hapke, *Theory of Reflectance and Emittance Spectroscopy*, Cambridge University Press, 2 edition, 2012.

[8] B. Hapke, "Bidirectional reflectance spectroscopy: 1. theory," *Journal of Geophysical Research*, vol. 86, pp. 3039–3054, 1981.

[9] N. Keshava, "A Survey of Spectral Unmixing Algorithms," *Lincoln Laboratory Journal*, vol. 14, no. 1, pp. 55–78, 2003.

[10] B. Palsson, J. R. Sveinsson, and M. O. Ulfarsson, "Blind hyperspectral unmixing using autoencoders: A critical comparison," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 1340–1372, 2022.

[11] J. Chen, M. Zhao, X. Wang, C. Richard, and S. Rahardja, "Integration of physics-based and data-driven models for hyperspectral image unmixing: A summary of current methods," *IEEE Signal Processing Magazine*, vol. 40, no. 2, pp. 61–74, 2023.

[12] Ben S., Gregory P. A., Laurent T., and Pol C., "Endmember variability in spectral mixture analysis: A review," *Remote Sensing of Environment*, vol. 115, no. 7, pp. 1603–1616, 2011.

[13] A. Zare and K.C. Ho, "Endmember variability in hyperspectral analysis: Addressing spectral variability during spectral unmixing," *IEEE Signal Processing Magazine*, vol. 31, no. 1, pp. 95–104, 2014.

[14] A. Plaza, G. Martín, J. Plaza, M. Zortea, and S. Sánchez, *Recent Developments in Endmember Extraction and Spectral Unmixing*, pp. 235–267, Springer Berlin Heidelberg, Berlin, Heidelberg, 2011.
parameters is very challenging when it comes to real-world datasets.

* The performance of the linear unmixing methods often degrades by increasing the number of endmembers. The linear unmixing may fail for a dataset with a higher number of endmembers.

* Spectral variability is still a big challenge and may considerably downgrade the performance of linear unmixing.

* One of the main challenges is the absence of a real dataset with ground truth.

* Multitemporal and multisource spectral unmixing are also challenging tasks.

* Considering the large volume of hyperspectral data, scalable unmixing is the main key for global monitoring.

Finally, we would like to further emphasize the importance of unmixing compared to the other signal and image processing approaches such as feature extraction/selection, clustering, and classification. The estimated features (i.e., abundances) in unmixing have physical meaning and can be attributed to materials. Therefore, estimated abundance maps can be used for quantitative analysis such as estimating vegetation cover, mineral composition, or pollutant concentrations providing quantitative information about the spatial distribution and relative quantities of materials present in the scene. The extracted/estimated endmembers can be used to identify and map different materials or surface types within the scene. By comparing the spectral signatures of unknown pixels with the reference endmembers, it is possible to classify and map land cover types, vegetation species, geological formations, or man-made structures. Temporal changes can be detected more accurately. The estimated abundance maps provide quantitative changes at a subpixel level. The extracted/estimated endmembers can be used for the identification of material changes. Intrinsic changes in the materials can be considered or ignored based on the endmembers assumptions. A future point is to investigate the performance of foundation models for spectral unmixing.

## ACKNOWLEDGMENTS

AZ, JM, and JC were supported by ANR 3IA MIAI@Grenoble Alpes (ANR-19-P3IA-0003). AZ and JM were supported by the ERC grant number 101087696 (APHELAIA project). The authors would like to thank Dr. Michael Arbel (the author of the MLXP package) for the guidance on the MLXP package and for making it adaptable to the Windows operating system.

## APPENDIX

### MINIMUM VOLUME SIMPLEX USING TOTAL VARIATION

The total variation penalty enforces the data simplex to have minimum volume by pulling the endmembers towards each

other. Assuming $\bar{e} = \frac{1}{r}e1_r$ we have

$$ 
\begin{aligned}
TV(\mathbf{E}) &= \sum_{i,j=1}^{r} \frac{1}{2} \|e_i - e_j\|_2^2 = \\
& \frac{1}{2} \sum_{i,j=1}^{r} \|e_i - \bar{e} - e_j + \bar{e}\|_2^2 = \\
& \frac{1}{2} \sum_{i=1}^{r} \|e_i - \bar{e}\|_2^2 + \frac{1}{2} \sum_{j=1}^{r} \|e_j - \bar{e}\|_2^2 - \\
& \left( \sum_{i=1}^{r} (e_i - \bar{e}) \right)^T \sum_{j=1}^{r} (e_j - \bar{e})
\end{aligned}
\quad (88) $$

where the third term is zero. Therefore,

$$ 
\begin{aligned}
TV(\mathbf{E}) &= \sum_{i=1}^{r} \|e_i - \bar{e}\|_2^2 = \\
& \sum_{i=1}^{r} \|e_i - \frac{1}{r}e1_r\|_2^2 = \|\mathbf{E}(I_r - \frac{1}{r}1_r1_r^T)\|_F^2
\end{aligned}
\quad (89) $$

## REFERENCES

[1] J. M. Bioucas-Dias, A. Plaza, N. Dobigeon, M. Parente, Q. Du, P. Gader, and J. Chanussot, “Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches,” *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 5, no. 2, pp. 354–379, April 2012.

[2] N. Dobigeon, J. Tourneret, C. Richard, J. C. M. Bermudez, S. McLaughlin, and A. O. Hero, “Nonlinear unmixing of hyperspectral images: Models and algorithms,” *IEEE Signal Processing Magazine*, vol. 31, no. 1, pp. 82–94, 2014.

[3] R. A. Borsoi, T. Imbiriba, J. C. Moreira Bermudez, C. Richard, J. Chanussot, L. Drumetz, J.-Y. Tourneret, A. Zare, and C. Jutten, “Spectral variability in hyperspectral data unmixing: A comprehensive review,” *IEEE Geoscience and Remote Sensing Magazine*, vol. 9, no. 4, pp. 223–270, 2021.

[4] D. Coquelin, B. Rasti, M. Götz, P. Ghamisi, R. Gloaguen, and A. Streit, “Hyde: The first open-source, python-based, gpu-accelerated hyperspectral denoising package,” in *2022 12th Workshop on Hyperspectral Imaging and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2022, pp. 1–5.

[5] B. Rasti, P. Scheunders, P. Ghamisi, G. Licciardi, and J. Chanussot, “Noise reduction in hyperspectral imagery: Overview and application,” *Remote Sensing*, vol. 10, no. 3, pp. 482, 2018.

[6] B. Rasti, Y. Chang, E. Dalsasso, Loic Denis, and Pedram Ghamisi, “Image Restoration for Remote Sensing: Overview and Toolbox,” *IEEE Geoscience and Remote Sensing Magazine*, pp. 2–31, 2021.

[7] B. Hapke, *Theory of Reflectance and Emittance Spectroscopy*, Cambridge University Press, 2 edition, 2012.

[8] B. Hapke, “Bidirectional reflectance spectroscopy: 1. theory,” *Journal of Geophysical Research*, vol. 86, pp. 3039–3054, 1981.

[9] N. Keshava, “A Survey of Spectral Unmixing Algorithms,” *Lincoln Laboratory Journal*, vol. 14, no. 1, pp. 55–78, 2003.

[10] B. Palsson, J. R. Sveinsson, and M. O. Ulfarsson, “Blind hyperspectral unmixing using autoencoders: A critical comparison,” *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 1340–1372, 2022.

[11] J. Chen, M. Zhao, X. Wang, C. Richard, and S. Rahardja, “Integration of physics-based and data-driven models for hyperspectral image unmixing: A summary of current methods,” *IEEE Signal Processing Magazine*, vol. 40, no. 2, pp. 61–74, 2023.

[12] Ben S., Gregory P. A., Laurent T., and Pol C., “Endmember variability in spectral mixture analysis: A review,” *Remote Sensing of Environment*, vol. 115, no. 7, pp. 1603–1616, 2011.

[13] A. Zare and K.C. Ho, “Endmember variability in hyperspectral analysis: Addressing spectral variability during spectral unmixing,” *IEEE Signal Processing Magazine*, vol. 31, no. 1, pp. 95–104, 2014.

[14] A. Plaza, G. Martín, J. Plaza, M. Zortea, and S. Sánchez, *Recent Developments in Endmember Extraction and Spectral Unmixing*, pp. 235–267, Springer Berlin Heidelberg, Berlin, Heidelberg, 2011.
[15] J. M. Bioucas-Dias, A. Plaza, G. Camps-Valls, P. Scheunders, N. Nasrabadi, and J. Chanussot, "Hyperspectral remote sensing data analysis and future challenges," *IEEE Geoscience and Remote Sensing Magazine*, vol. 1, no. 2, pp. 6–36, 2013.

[16] P. Ghamisi, N. Yokoya, J. Li, W. Liao, S. Liu, J. Plaza, B. Rasti, and A. Plaza, "Advances in hyperspectral image and signal processing: A comprehensive overview of the state of the art," *IEEE Geoscience and Remote Sensing Magazine*, vol. 5, no. 4, pp. 37–78, Dec 2017.

[17] M. Wang, D. Hong, Z. Han, J. Li, J. Yao, L. Gao, B. Zhang, and J. Chanussot, "Tensor decompositions for hyperspectral data processing in remote sensing: A comprehensive review," *IEEE Geoscience and Remote Sensing Magazine*, vol. 11, no. 1, pp. 26–72, 2023.

[18] R. Heylen, M. Parente, and P. Gader, "A review of nonlinear hyperspectral unmixing methods," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 7, no. 6, pp. 1844–1868, 2014.

[19] L. I. Jimenez and A. Plaza, "Hypermix: An open source tool for hyperspectral imaging," in *2015 IEEE International Geoscience and Remote Sensing Symposium (IGARSS)*, 2015, pp. 1749–1752.

[20] M. Arbel, "Mlxp: A framework for conducting machine learning experiments in python," Github, 2023.

[21] B. Rasti and B. Koirala, "SUnCNN: Sparse unmixing using unsupervised convolutional neural network," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[22] D. C. Heinz and Chein-I-Chang, "Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 39, no. 3, pp. 529–545, 2001.

[23] R. Heylen, D. Burazerovic, and P. Scheunders, "Fully constrained least squares spectral unmixing by simplex projection," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 11, pp. 4112–4122, Nov 2011.

[24] J. M. Bioucas-Dias, "A variable splitting augmented lagrangian approach to linear spectral unmixing," in *2009 First Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing*, 2009, pp. 1–4.

[25] B. Rasti, B. Koirala, P. Scheunders, and P. Ghamisi, "UnDIP: Hyperspectral unmixing using deep image prior," *IEEE Transactions on Geoscience and Remote Sensing*, pp. 1–15, 2021.

[26] J. Nascimento and J. Bioucas-Dias, "Vertex component analysis: A fast algorithm to extract endmembers spectra from hyperspectral data," in *Pattern Recognition and Image Analysis*, Francisco José Perales, Aurélio J. C. Campilho, Nicolás Pérez de la Blanca, and Alberto Sanfeliu, Eds., Berlin, Heidelberg, 2003, pp. 626–635, Springer Berlin Heidelberg.

[27] M. Iordache, J. M. Bioucas-Dias, and A. Plaza, "Collaborative sparse regression for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 52, no. 1, pp. 341–354, 2014.

[28] R. A. Borsoi, T. Imbiriba, J. C. M. Bermudez, and C. Richard, "A fast multiscale spatial regularization for sparse hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 16, no. 4, pp. 598–602, 2019.

[29] S. Zhang, J. Li, H. Li, C. Deng, and A. Plaza, "Spectral-spatial weighted sparse regression for hyperspectral image unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 56, no. 6, pp. 3265–3276, 2018.

[30] B. Rasti, A. Zouaoui, J. Mairal, and J. Chanussot, "Sunaa: Sparse unmixing using archetypal analysis," *IEEE Geoscience and Remote Sensing Letters*, pp. 1–1, 2023.

[31] J. M. Bioucas-Dias and M. A. T. Figueiredo, "Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing," in *2nd Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing*, 2010, pp. 1–4.

[32] C. Zhou and M. R. Rodrigues, "Admm-based hyperspectral unmixing networks for abundance and endmember estimation," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–18, 2021.

[33] N. Dobigeon, S. Moussaoui, M. Coulon, J.-Y. Tourneret, and A. O. Hero, "Joint bayesian endmember extraction and linear unmixing for hyperspectral imagery," *IEEE Transactions on Signal Processing*, vol. 57, no. 11, pp. 4355–4368, 2009.

[34] B. Palsson, M. O. Ulfarsson, and J. R. Sveinsson, "Convolutional autoencoder for spectral-spatial hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 1, pp. 535–549, 2021.

[35] A. Zouaoui, G. Muhawenayo, B. Rasti, J. Chanussot, and J. Mairal, "Entropic descent archetypal analysis for blind hyperspectral unmixing," *IEEE Transactions on Image Processing*, pp. 1–1, 2023.

[36] B. Rasti, B. Koirala, P. Scheunders, and J. Chanussot, "Misicnet: Minimum simplex convolutional network for deep hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, pp. 1–1, 2022.

[37] Y. Yu, Y. Ma, X. Mei, F. Fan, J. Huang, and H. Li, "Multi-stage convolutional autoencoder network for hyperspectral unmixing," *International Journal of Applied Earth Observation and Information*, vol. 113, pp. 102981, 2022.

[38] L. Zhuang, C. Lin, M. A. T. Figueiredo, and J. M. Bioucas-Dias, "Regularization parameter selection in minimum volume hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 12, pp. 9858–9877, 2019.

[39] S. Shi, M. Zhao, L. Zhang, Y. Altmann, and J. Chen, "Probabilistic generative model for hyperspectral unmixing accounting for endmember variability," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–15, 2022.

[40] B. Rasti, Y. Chang, E. Dalsasso, L. Denis, and P. Ghamisi, "Image restoration for remote sensing: Overview and toolbox," *IEEE Geoscience and Remote Sensing Magazine*, vol. 10, no. 2, pp. 201–230, 2022.

[41] B. Rasti, B. Koirala, P. Scheunders, and P. Ghamisi, "How hyperspectral image unmixing and denoising can boost each other," *Remote Sensing*, vol. 12, no. 11, pp. 1728, May 2020.

[42] T. Ince and T. Dundar, "Simultaneous nonconvex denoising and unmixing for hyperspectral imaging," *IEEE Access*, vol. 7, pp. 124426–124440, 2019.

[43] J. Yang, Y. Zhao, J. C. Chan, and S. G. Kong, "Coupled sparse denoising and unmixing with low-rank constraint for hyperspectral image," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 3, pp. 1818–1833, March 2016.

[44] B. Rasti, J.R. Sveinsson, and M.O. Ulfarsson, "Wavelet-Based Sparse Reduced-Rank Regression for Hyperspectral Image Restoration," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 52, no. 10, pp. 6688–6698, 2014.

[45] B. Rasti, M.O. Ulfarsson, and J. Sveinsson, "Hyperspectral Feature Extraction Using Total Variation Component Analysis," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, 08 2016.

[46] B. Rasti, D. Hong, R. Hang, P. Ghamisi, X. Kang, J. Chanussot, and J. A. Benediktsson, "Feature extraction for hyperspectral imagery: The evolution from shallow to deep (overview and toolbox)," *IEEE Geoscience and Remote Sensing Magazine*, pp. 0–0, 2020.

[47] J.M. Bioucas-Dias and J.M.P. Nascimento, "Hyperspectral subspace identification," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 46, no. 8, pp. 2435–2445, 2008.

[48] B. Rasti, M.O. Ulfarsson, and J.R. Sveinsson, "Hyperspectral subspace identification using SURE," *IEEE Geoscience and Remote Sensing Letter*, vol. 12, no. 12, pp. 2481–2485, Dec. 2015.

[49] C.-I Chang and Q. Du, "Estimation of number of spectrally distinct signal sources in hyperspectral imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 42, no. 3, pp. 608–619, 2004.

[50] J. Harsanyi, W. Farrand, and C.I Chang, "Determining the number and identity of spectral endmembers: An integrated approach using neyman?pearson eigenthresholding and iterative constrained rms error minimization," in *9th Thematic Conf. Geologic Remote Sens.*, 1993.

[51] B. Luo, J. Chanussot, S. Doute, and L. Zhang, "Empirical automatic estimation of the number of endmembers in hyperspectral images," *IEEE Geoscience and Remote Sensing Letters*, vol. 10, no. 1, pp. 24–28, 2013.

[52] M. Berman, H. Kiiveri, R. Lagerstrom, A. Ernst, R. Dunne, and J.F. Huntington, "Ice: a statistical approach to identifying endmembers in hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 42, no. 10, pp. 2085–2095, 2004.

[53] A. Zare and P. Gader, "Solvency promoting iterated constrained endmember detection in hyperspectral imagery," *IEEE Geoscience and Remote Sensing Letters*, vol. 4, no. 3, pp. 446–450, 2007.

[54] A. Ambikapathi, T.-H. Chan, C.-Y. Chi, and K. Keizer, "Hyperspectral data geometry-based estimation of number of endmembers using p-norm-based pure pixel identification algorithm," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 51, no. 5, pp. 2753–2769, May 2013.

[55] J. Boardman, F. A. Kruse, and R. Green, "Mapping target signatures via partial unmixing of aviris data: in summaries," in *JPL Airborne Earth Sci. Workshop*, 1995, pp. 23–26.

[56] J.C. Harsanyi and C.-I. Chang, "Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 32, no. 4, pp. 779–785, 1994.
[15] J. M. Bioucas-Dias, A. Plaza, G. Camps-Valls, P. Scheunders, N. Nasrabadi, and J. Chanussot, "Hyperspectral remote sensing data analysis and future challenges," *IEEE Geoscience and Remote Sensing Magazine*, vol. 1, no. 2, pp. 6–36, 2013.

[16] P. Ghamisi, N. Yokoya, J. Li, W. Liao, S. Liu, J. Plaza, B. Rasti, and A. Plaza, "Advances in hyperspectral image and signal processing: A comprehensive overview of the state of the art," *IEEE Geoscience and Remote Sensing Magazine*, vol. 5, no. 4, pp. 37–78, Dec 2017.

[17] M. Wang, D. Hong, Z. Han, J. Li, J. Yao, L. Gao, B. Zhang, and J. Chanussot, "Tensor decompositions for hyperspectral data processing in remote sensing: A comprehensive review," *IEEE Geoscience and Remote Sensing Magazine*, vol. 11, no. 1, pp. 26–72, 2023.

[18] R. Heylen, M. Parente, and P. Gader, "A review of nonlinear hyperspectral unmixing methods," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 7, no. 6, pp. 1844–1868, 2014.

[19] L. I. Jimenez and A. Plaza, "Hypermix: An open source tool for hyperspectral imaging," in *2015 IEEE International Geoscience and Remote Sensing Symposium (IGARSS)*, 2015, pp. 1749–1752.

[20] M. Arbel, "Mlxp: A framework for conducting machine learning experiments in python," Github, 2023.

[21] B. Rasti and B. Koirala, "SUnCNN: Sparse unmixing using unsupervised convolutional neural network," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[22] D. C. Heinz and Chein-I-Chang, "Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 39, no. 3, pp. 529–545, 2001.

[23] R. Heylen, D. Burazerovic, and P. Scheunders, "Fully constrained least squares spectral unmixing by simplex projection," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 11, pp. 4112–4122, Nov 2011.

[24] J. M. Bioucas-Dias, "A variable splitting augmented lagrangian approach to linear spectral unmixing," in *2009 First Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing*, 2009, pp. 1–4.

[25] B. Rasti, B. Koirala, P. Scheunders, and P. Ghamisi, "UnDIP: Hyperspectral unmixing using deep image prior," *IEEE Transactions on Geoscience and Remote Sensing*, pp. 1–15, 2021.

[26] J. Nascimento and J. Bioucas-Dias, "Vertex component analysis: A fast algorithm to extract endmembers spectra from hyperspectral data," in *Pattern Recognition and Image Analysis*, Francisco José Perales, Aurélio J. C. Campilho, Nicolás Pérez de la Blanca, and Alberto Sanfeliu, Eds., Berlin, Heidelberg, 2003, pp. 626–635, Springer Berlin Heidelberg.

[27] M. Iordache, J. M. Bioucas-Dias, and A. Plaza, "Collaborative sparse regression for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 52, no. 1, pp. 341–354, 2014.

[28] R. A. Borsoi, T. Imbiriba, J. C. M. Bermudez, and C. Richard, "A fast multiscale spatial regularization for sparse hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 16, no. 4, pp. 598–602, 2019.

[29] S. Zhang, J. Li, H. Li, C. Deng, and A. Plaza, "Spectral-spatial weighted sparse regression for hyperspectral image unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 56, no. 6, pp. 3265–3276, 2018.

[30] B. Rasti, A. Zouaoui, J. Mairal, and J. Chanussot, "Sunaa: Sparse unmixing using archetypal analysis," *IEEE Geoscience and Remote Sensing Letters*, pp. 1–1, 2023.

[31] J. M. Bioucas-Dias and M. A. T. Figueiredo, "Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing," in *2nd Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing*, 2010, pp. 1–4.

[32] C. Zhou and M. R. Rodrigues, "Admm-based hyperspectral unmixing networks for abundance and endmember estimation," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–18, 2021.

[33] N. Dobigeon, S. Moussaoui, M. Coulon, J.-Y. Tourneret, and A. O. Hero, "Joint bayesian endmember extraction and linear unmixing for hyperspectral imagery," *IEEE Transactions on Signal Processing*, vol. 57, no. 11, pp. 4355–4368, 2009.

[34] B. Palsson, M. O. Ulfarsson, and J. R. Sveinsson, "Convolutional autoencoder for spectral-spatial hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 1, pp. 535–549, 2021.

[35] A. Zouaoui, G. Muhawenayo, B. Rasti, J. Chanussot, and J. Mairal, "Entropic descent archetypal analysis for blind hyperspectral unmixing," *IEEE Transactions on Image Processing*, pp. 1–1, 2023.

[36] B. Rasti, B. Koirala, P. Scheunders, and J. Chanussot, "Misicnet: Minimum simplex convolutional network for deep hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, pp. 1–1, 2022.

[37] Y. Yu, Y. Ma, X. Mei, F. Fan, J. Huang, and H. Li, "Multi-stage convolutional autoencoder network for hyperspectral unmixing," *International Journal of Applied Earth Observation and Information*, vol. 113, pp. 102981, 2022.

[38] L. Zhuang, C. Lin, M. A. T. Figueiredo, and J. M. Bioucas-Dias, "Regularization parameter selection in minimum volume hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 12, pp. 9858–9877, 2019.

[39] S. Shi, M. Zhao, L. Zhang, Y. Altmann, and J. Chen, "Probabilistic generative model for hyperspectral unmixing accounting for endmember variability," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–15, 2022.

[40] B. Rasti, Y. Chang, E. Dalsasso, L. Denis, and P. Ghamisi, "Image restoration for remote sensing: Overview and toolbox," *IEEE Geoscience and Remote Sensing Magazine*, vol. 10, no. 2, pp. 201–230, 2022.

[41] B. Rasti, B. Koirala, P. Scheunders, and P. Ghamisi, "How hyperspectral image unmixing and denoising can boost each other," *Remote Sensing*, vol. 12, no. 11, pp. 1728, May 2020.

[42] T. Ince and T. Dundar, "Simultaneous nonconvex denoising and unmixing for hyperspectral imaging," *IEEE Access*, vol. 7, pp. 124426–124440, 2019.

[43] J. Yang, Y. Zhao, J. C. Chan, and S. G. Kong, "Coupled sparse denoising and unmixing with low-rank constraint for hyperspectral image," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 3, pp. 1818–1833, March 2016.

[44] B. Rasti, J.R. Sveinsson, and M.O. Ulfarsson, "Wavelet-Based Sparse Reduced-Rank Regression for Hyperspectral Image Restoration," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 52, no. 10, pp. 6688–6698, 2014.

[45] B. Rasti, M.O. Ulfarsson, and J. Sveinsson, "Hyperspectral Feature Extraction Using Total Variation Component Analysis," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, 08 2016.

[46] B. Rasti, D. Hong, R. Hang, P. Ghamisi, X. Kang, J. Chanussot, and J. A. Benediktsson, "Feature extraction for hyperspectral imagery: The evolution from shallow to deep (overview and toolbox)," *IEEE Geoscience and Remote Sensing Magazine*, pp. 0–0, 2020.

[47] J.M. Bioucas-Dias and J.M.P. Nascimento, "Hyperspectral subspace identification," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 46, no. 8, pp. 2435–2445, 2008.

[48] B. Rasti, M.O. Ulfarsson, and J.R. Sveinsson, "Hyperspectral subspace identification using SURE," *IEEE Geoscience and Remote Sensing Letter*, vol. 12, no. 12, pp. 2481–2485, Dec. 2015.

[49] C.-I Chang and Q. Du, "Estimation of number of spectrally distinct signal sources in hyperspectral imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 42, no. 3, pp. 608–619, 2004.

[50] J. Harsanyi, W. Farrand, and C.I Chang, "Determining the number and identity of spectral endmembers: An integrated approach using neyman?pearson eigenthresholding and iterative constrained rms error minimization," in *9th Thematic Conf. Geologic Remote Sens.*, 1993.

[51] B. Luo, J. Chanussot, S. Doute, and L. Zhang, "Empirical automatic estimation of the number of endmembers in hyperspectral images," *IEEE Geoscience and Remote Sensing Letters*, vol. 10, no. 1, pp. 24–28, 2013.

[52] M. Berman, H. Kiiveri, R. Lagerstrom, A. Ernst, R. Dunne, and J.F. Huntington, "Ice: a statistical approach to identifying endmembers in hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 42, no. 10, pp. 2085–2095, 2004.

[53] A. Zare and P. Gader, "Solvency promoting iterated constrained endmember detection in hyperspectral imagery," *IEEE Geoscience and Remote Sensing Letters*, vol. 4, no. 3, pp. 446–450, 2007.

[54] A. Ambikapathi, T.-H. Chan, C.-Y. Chi, and K. Keizer, "Hyperspectral data geometry-based estimation of number of endmembers using p-norm-based pure pixel identification algorithm," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 51, no. 5, pp. 2753–2769, May 2013.

[55] J. Boardman, F. A. Kruse, and R. Green, "Mapping target signatures via partial unmixing of aviris data: in summaries," in *JPL Airborne Earth Sci. Workshop*, 1995, pp. 23–26.

[56] J.C. Harsanyi and C.-I. Chang, "Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 32, no. 4, pp. 779–785, 1994.
[57] M. E. Winter, "N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data," in *Imaging Spectrometry V*, Michael R. Descour and Sylvia S. Shen, Eds. International Society for Optics and Photonics, 1999, vol. 3753, pp. 266–275, SPIE.

[58] A. Bateson and B. Curtiss, "A method for manual endmember selection and spectral unmixing," *Remote Sensing of Environment*, vol. 55, no. 3, pp. 229–243, 1996.

[59] T. Bajjouk, J. Populus, and B. Guillaumont, "Quantification of subpixel cover fractions using principal component analysis and a linear programming method: Application to the coastal zone of rocoff (france)," *Remote Sensing of Environment*, vol. 64, no. 2, pp. 153–165, 1998.

[60] C.A. Bateson, G.P. Asner, and C.A. Wessman, "Endmember bundles: a new approach to incorporating endmember variability into spectral mixture analysis," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 38, no. 2, pp. 1083–1094, 2000.

[61] B. Somers, M. Zortea, A. Plaza, and G. P. Asner, "Automated extraction of image-based endmember bundles for improved spectral unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 5, no. 2, pp. 396–408, 2012.

[62] A. Plaza, P. Martinez, R. Perez, and J. Plaza, "Spatial/spectral endmember extraction by multidimensional morphological operations," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 40, no. 9, pp. 2025–2041, 2002.

[63] D.M. Rogge, B. Rivard, J. Zhang, A. Sanchez, J. Harris, and J. Feng, "Integration of spatial-spectral information for the improved extraction of endmembers," *Remote Sensing of Environment*, vol. 110, no. 3, pp. 287–303, 2007.

[64] M. Xu, L. Zhang, and B. Du, "An image-based endmember bundle extraction algorithm using both spatial and spectral information," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 8, no. 6, pp. 2607–2617, 2015.

[65] X. Shen, W. Bao, and K. Qu, "Spatial-spectral hyperspectral endmember extraction using a spatial energy prior constrained maximum simplex volume approach," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 13, pp. 1347–1361, 2020.

[66] M. Zortea and A. Plaza, "Spatial preprocessing for endmember extraction," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 47, no. 8, pp. 2679–2693, 2009.

[67] T.-H. Chan, C.-Y. Chi, Y.-M. Huang, and W.-K. Ma, "A convex analysis-based minimum-volume enclosing simplex algorithm for hyperspectral unmixing," *IEEE Transactions on Signal Processing*, vol. 57, no. 11, pp. 4418–4432, 2009.

[68] M.D. Craig, "Minimum-volume transforms for remotely sensed data," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 32, no. 3, pp. 542–552, 1994.

[69] C.-H. Lin, R. Wu, W.-K. Ma, C.-Y. Chi, and Y. Wang, "Maximum volume inscribed ellipsoid: A new simplex-structured matrix factorization framework via facet enumeration and convex optimization," *SIAM Journal on Imaging Sciences*, vol. 11, no. 2, pp. 1651–1679, 2018.

[70] J. Li and J. M. Bioucas-Dias, "Minimum volume simplex analysis: A fast algorithm to unmix hyperspectral data," in *IGARSS 2008 - 2008 IEEE International Geoscience and Remote Sensing Symposium*, 2008, vol. 3, pp. III – 250–III – 253.

[71] J. Eckstein and D. P. Bertsekas, "On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators," *Mathematical Programming*, vol. 55, pp. 293–318, 1992.

[72] C.-I. Chang, X.-L. Zhao, M.L.G. Althouse, and J. J. Pan, "Least squares subspace projection approach to mixed pixel classification for hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 36, no. 3, pp. 898–912, 1998.

[73] J.J. Settle and N. Drake, "Linear mixing and the estimation of ground cover proportions," *International Journal of Remote Sensing - INT J REMOTE SENS*, vol. 14, pp. 1159–1177, 04 1993.

[74] Chein-I Chang and D. C. Heinz, "Constrained subpixel target detection for remotely sensed imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 38, no. 3, pp. 1144–1159, 2000.

[75] Y.E. Shimabukuro and J.A. Smith, "The least-squares mixing models to generate fraction images derived from remote sensing multispectral data," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 29, no. 1, pp. 16–20, 1991.

[76] Y.E. Shimabukuro, *Shade Images Derived from Linear Mixing Models of Multispectral Measurements of Forested Areas*, Colorado State University, 1987.

[77] R. Heylen, D. Burzerovic, and P. Scheunders, "Fully constrained least squares spectral unmixing by simplex projection," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 11, pp. 4112–4122, 2011.

[78] F. Khajehrayeni and H. Ghassemian, "Hyperspectral unmixing using deep convolutional autoencoders in a supervised scenario," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 13, pp. 567–576, 2020.

[79] D. Ulyanov, A. Vedaldi, and V. Lempitsky, "Deep image prior," *International Journal of Computer Vision*, vol. 128, no. 7, pp. 1867–1888, Mar 2020.

[80] D. Ulyanov, A. Vedaldi, and V. Lempitsky, "Deep image prior," in *Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, June 2018.

[81] Z. Han, D. Hong, L. Gao, B. Zhang, and J. Chanussot, "Deep half-siamese networks for hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 18, no. 11, pp. 1996–2000, 2021.

[82] D. Hong, L. Gao, J. Yao, N. Yokoya, J. Chanussot, U. Heiden, and B. Zhang, "Endmember-guided unmixing network (egu-net): A general deep learning framework for self-supervised hyperspectral unmixing," *IEEE Transactions on Neural Networks and Learning Systems*, vol. 33, no. 11, pp. 6518–6531, 2022.

[83] M. Berman, H. Kiiveri, R. Lagerstrom, A. Ernst, R. Dunne, and J.F. Huntington, "Ice: a statistical approach to identifying endmembers in hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 42, no. 10, pp. 2085–2095, 2004.

[84] L. Miao and H. Qi, "Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 45, no. 3, pp. 765–777, 2007.

[85] J. Li, J. M. Bioucas-Dias, and A. Plaza, "Collaborative nonnegative matrix factorization for remotely sensed hyperspectral unmixing," in *IEEE International Geoscience and Remote Sensing Symposium (IGARSS)*, July 2012, pp. 3078–3081.

[86] J. Li, J. M. Bioucas-Dias, A. Plaza, and L. Liu, "Robust collaborative nonnegative matrix factorization for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 10, pp. 6076–6090, 2016.

[87] Y. Qian, S. Jia, J. Zhou, and A. Robles-Kelly, "Hyperspectral unmixing via $l_{1/2}$ sparsity-constrained nonnegative matrix factorization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 11, pp. 4282–4297, 2011.

[88] D. Lee and H. S. Seung, "Algorithms for non-negative matrix factorization," in *Advances in Neural Information Processing Systems*, T. Leen, T. Dietterich, and V. Tresp, Eds. 2000, vol. 13, p. 556–562, MIT Press.

[89] P.O. Hoyer, "Non-negative sparse coding," in *Proceedings of the 12th IEEE Workshop on Neural Networks for Signal Processing*, 2002, pp. 557–565.

[90] J. Sigurdsson, M. O. Ulfarsson, and J. R. Sveinsson, "Hyperspectral unmixing with $l_q$ regularization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 52, no. 11, pp. 6793–6806, 2014.

[91] J. Sigurdsson, M.O. Ulfarsson, J.R. Sveinsson, and J.A. Benediktsson, "Smooth spectral unmixing using total variation regularization and a first order roughness penalty," in *IEEE International Geoscience and Remote Sensing Symposium (IGARSS)*, July 2013, pp. 2160–2163.

[92] W. He, H. Zhang, and L. Zhang, "Total variation regularized reweighted sparse nonnegative matrix factorization for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 55, no. 7, pp. 3909–3921, 2017.

[93] J. Yao, D. Meng, Q. Zhao, W. Cao, and Z. Xu, "Nonconvex-sparsity and nonlocal-smoothness-based blind hyperspectral unmixing," *IEEE Transactions on Image Processing*, vol. 28, no. 6, pp. 2991–3006, 2019.

[94] C. G. Tsinos, A. A. Rontogiannis, and K. Berberidis, "Distributed blind hyperspectral unmixing via joint sparsity and low-rank constrained non-negative matrix factorization," *IEEE Transactions on Computational Imaging*, vol. 3, no. 2, pp. 160–174, 2017.

[95] Z. Yang, G. Zhou, S. Xie, S. Ding, J.-M. Yang, and J. Zhang, "Blind spectral unmixing based on sparse nonnegative matrix factorization," *IEEE Transactions on Image Processing*, vol. 20, no. 4, pp. 1112–1125, 2011.

[96] S. Zhang, G. Zhang, F. Li, C. Deng, S. Wang, A. Plaza, and J. Li, "Spectral-spatial hyperspectral unmixing using nonnegative matrix factorization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–13, 2022.

[97] M. Zhao, X. Wang, J. Chen, and W. Chen, "A plug-and-play priors framework for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–13, 2022.
[57] M. E. Winter, "N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data," in *Imaging Spectrometry V*, Michael R. Descour and Sylvia S. Shen, Eds. International Society for Optics and Photonics, 1999, vol. 3753, pp. 266–275, SPIE.

[58] A. Bateson and B. Curtiss, "A method for manual endmember selection and spectral unmixing," *Remote Sensing of Environment*, vol. 55, no. 3, pp. 229–243, 1996.

[59] T. Bajjouk, J. Populus, and B. Guillaumont, "Quantification of subpixel cover fractions using principal component analysis and a linear programming method: Application to the coastal zone of rocoff (france)," *Remote Sensing of Environment*, vol. 64, no. 2, pp. 153–165, 1998.

[60] C.A. Bateson, G.P. Asner, and C.A. Wessman, "Endmember bundles: a new approach to incorporating endmember variability into spectral mixture analysis," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 38, no. 2, pp. 1083–1094, 2000.

[61] B. Somers, M. Zortea, A. Plaza, and G. P. Asner, "Automated extraction of image-based endmember bundles for improved spectral unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 5, no. 2, pp. 396–408, 2012.

[62] A. Plaza, P. Martinez, R. Perez, and J. Plaza, "Spatial/spectral endmember extraction by multidimensional morphological operations," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 40, no. 9, pp. 2025–2041, 2002.

[63] D.M. Rogge, B. Rivard, J. Zhang, A. Sanchez, J. Harris, and J. Feng, "Integration of spatial-spectral information for the improved extraction of endmembers," *Remote Sensing of Environment*, vol. 110, no. 3, pp. 287–303, 2007.

[64] M. Xu, L. Zhang, and B. Du, "An image-based endmember bundle extraction algorithm using both spatial and spectral information," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 8, no. 6, pp. 2607–2617, 2015.

[65] X. Shen, W. Bao, and K. Qu, "Spatial-spectral hyperspectral endmember extraction using a spatial energy prior constrained maximum simplex volume approach," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 13, pp. 1347–1361, 2020.

[66] M. Zortea and A. Plaza, "Spatial preprocessing for endmember extraction," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 47, no. 8, pp. 2679–2693, 2009.

[67] T.-H. Chan, C.-Y. Chi, Y.-M. Huang, and W.-K. Ma, "A convex analysis-based minimum-volume enclosing simplex algorithm for hyperspectral unmixing," *IEEE Transactions on Signal Processing*, vol. 57, no. 11, pp. 4418–4432, 2009.

[68] M.D. Craig, "Minimum-volume transforms for remotely sensed data," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 32, no. 3, pp. 542–552, 1994.

[69] C.-H. Lin, R. Wu, W.-K. Ma, C.-Y. Chi, and Y. Wang, "Maximum volume inscribed ellipsoid: A new simplex-structured matrix factorization framework via facet enumeration and convex optimization," *SIAM Journal on Imaging Sciences*, vol. 11, no. 2, pp. 1651–1679, 2018.

[70] J. Li and J. M. Bioucas-Dias, "Minimum volume simplex analysis: A fast algorithm to unmix hyperspectral data," in *IGARSS 2008 - 2008 IEEE International Geoscience and Remote Sensing Symposium*, 2008, vol. 3, pp. III – 250–III – 253.

[71] J. Eckstein and D. P. Bertsekas, "On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators," *Mathematical Programming*, vol. 55, pp. 293–318, 1992.

[72] C.-I. Chang, X.-L. Zhao, M.L.G. Althouse, and J. J. Pan, "Least squares subspace projection approach to mixed pixel classification for hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 36, no. 3, pp. 898–912, 1998.

[73] J.J. Settle and N. Drake, "Linear mixing and the estimation of ground cover proportions," *International Journal of Remote Sensing - INT J REMOTE SENS*, vol. 14, pp. 1159–1177, 04 1993.

[74] Chein-I Chang and D. C. Heinz, "Constrained subpixel target detection for remotely sensed imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 38, no. 3, pp. 1144–1159, 2000.

[75] Y.E. Shimabukuro and J.A. Smith, "The least-squares mixing models to generate fraction images derived from remote sensing multispectral data," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 29, no. 1, pp. 16–20, 1991.

[76] Y.E. Shimabukuro, *Shade Images Derived from Linear Mixing Models of Multispectral Measurements of Forested Areas*, Colorado State University, 1987.

[77] R. Heylen, D. Burzerovic, and P. Scheunders, "Fully constrained least squares spectral unmixing by simplex projection," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 11, pp. 4112–4122, 2011.

[78] F. Khajehrayeni and H. Ghassemian, "Hyperspectral unmixing using deep convolutional autoencoders in a supervised scenario," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 13, pp. 567–576, 2020.

[79] D. Ulyanov, A. Vedaldi, and V. Lempitsky, "Deep image prior," *International Journal of Computer Vision*, vol. 128, no. 7, pp. 1867–1888, Mar 2020.

[80] D. Ulyanov, A. Vedaldi, and V. Lempitsky, "Deep image prior," in *Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, June 2018.

[81] Z. Han, D. Hong, L. Gao, B. Zhang, and J. Chanussot, "Deep half-siamese networks for hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 18, no. 11, pp. 1996–2000, 2021.

[82] D. Hong, L. Gao, J. Yao, N. Yokoya, J. Chanussot, U. Heiden, and B. Zhang, "Endmember-guided unmixing network (egu-net): A general deep learning framework for self-supervised hyperspectral unmixing," *IEEE Transactions on Neural Networks and Learning Systems*, vol. 33, no. 11, pp. 6518–6531, 2022.

[83] M. Berman, H. Kiiveri, R. Lagerstrom, A. Ernst, R. Dunne, and J.F. Huntington, "Ice: a statistical approach to identifying endmembers in hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 42, no. 10, pp. 2085–2095, 2004.

[84] L. Miao and H. Qi, "Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 45, no. 3, pp. 765–777, 2007.

[85] J. Li, J. M. Bioucas-Dias, and A. Plaza, "Collaborative nonnegative matrix factorization for remotely sensed hyperspectral unmixing," in *IEEE International Geoscience and Remote Sensing Symposium (IGARSS)*, July 2012, pp. 3078–3081.

[86] J. Li, J. M. Bioucas-Dias, A. Plaza, and L. Liu, "Robust collaborative nonnegative matrix factorization for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 10, pp. 6076–6090, 2016.

[87] Y. Qian, S. Jia, J. Zhou, and A. Robles-Kelly, "Hyperspectral unmixing via $l_{1/2}$ sparsity-constrained nonnegative matrix factorization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 11, pp. 4282–4297, 2011.

[88] D. Lee and H. S. Seung, "Algorithms for non-negative matrix factorization," in *Advances in Neural Information Processing Systems*, T. Leen, T. Dietterich, and V. Tresp, Eds. 2000, vol. 13, p. 556–562, MIT Press.

[89] P.O. Hoyer, "Non-negative sparse coding," in *Proceedings of the 12th IEEE Workshop on Neural Networks for Signal Processing*, 2002, pp. 557–565.

[90] J. Sigurdsson, M. O. Ulfarsson, and J. R. Sveinsson, "Hyperspectral unmixing with $l_q$ regularization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 52, no. 11, pp. 6793–6806, 2014.

[91] J. Sigurdsson, M.O. Ulfarsson, J.R. Sveinsson, and J.A. Benediktsson, "Smooth spectral unmixing using total variation regularization and a first order roughness penalty," in *IEEE International Geoscience and Remote Sensing Symposium (IGARSS)*, July 2013, pp. 2160–2163.

[92] W. He, H. Zhang, and L. Zhang, "Total variation regularized reweighted sparse nonnegative matrix factorization for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 55, no. 7, pp. 3909–3921, 2017.

[93] J. Yao, D. Meng, Q. Zhao, W. Cao, and Z. Xu, "Nonconvex-sparsity and nonlocal-smoothness-based blind hyperspectral unmixing," *IEEE Transactions on Image Processing*, vol. 28, no. 6, pp. 2991–3006, 2019.

[94] C. G. Tsinos, A. A. Rontogiannis, and K. Berberidis, "Distributed blind hyperspectral unmixing via joint sparsity and low-rank constrained non-negative matrix factorization," *IEEE Transactions on Computational Imaging*, vol. 3, no. 2, pp. 160–174, 2017.

[95] Z. Yang, G. Zhou, S. Xie, S. Ding, J.-M. Yang, and J. Zhang, "Blind spectral unmixing based on sparse nonnegative matrix factorization," *IEEE Transactions on Image Processing*, vol. 20, no. 4, pp. 1112–1125, 2011.

[96] S. Zhang, G. Zhang, F. Li, C. Deng, S. Wang, A. Plaza, and J. Li, "Spectral-spatial hyperspectral unmixing using nonnegative matrix factorization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–13, 2022.

[97] M. Zhao, X. Wang, J. Chen, and W. Chen, "A plug-and-play priors framework for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–13, 2022.
[98] A. Cutler and L. Breiman, "Archetypal analysis," Technometrics, vol. 36, no. 4, pp. 338-347, 1994.

[99] G. Zhao, X. Jia, and C. Zhao, "Multiple endmembers based unmixing using archetypal analysis," in *2015 IEEE International Geoscience and Remote Sensing Symposium (IGARSS)*, 2015, pp. 5039-5042.

[100] C. Zhao, G. Zhao, and X. Jia, "Hyperspectral image unmixing based on fast kernel archetypal analysis," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 10, no. 1, pp. 331-346, 2017.

[101] M. Xu, Z. Yang, G. Ren, H. Sheng, S. Liu, W. Liu, and C. Ye, "L1 sparsity-constrained archetypal analysis algorithm for hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1-5, 2022.

[102] Q. Zhang, H. Wang, R. Plemmons, and V. P. Pauca, "Spectral unmixing using nonnegative tensor factorization," in *Proceedings of the 45th Annual Southeast Regional Conference, New York, NY, USA, 2007*, ACM-SE 45, p. 531-532, Association for Computing Machinery.

[103] A. Huck and M. Guillaume, "Estimation of the hyperspectral tucker ranks," in *2009 IEEE International Conference on Acoustics, Speech and Signal Processing*, 2009, pp. 1281-1284.

[104] L. De Lathauwer, "Decompositions of a higher-order tensor in block terms—part ii: Definitions and uniqueness," *SIAM Journal on Matrix Analysis and Applications*, vol. 30, no. 3, pp. 1033-1066, 2008.

[105] Y. Qian, F. Xiong, S. Zeng, J. Zhou, and Y. Y. Tang, "Matrix-vector nonnegative tensor factorization for blind unmixing of hyperspectral imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 55, no. 3, pp. 1776-1792, 2017.

[106] Pan Zheng, Hongjun Su, and Qian Du, "Sparse and low-rank constrained tensor factorization for hyperspectral image unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 14, pp. 1754-1767, 2021.

[107] P. Yang, Ti-Z. Huang, J. Huang, and J.-J. Wang, "Efficient weighted-adaptive sparse constrained nonnegative tensor factorization for hyperspectral unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 10113-10130, 2022.

[108] F. Xiong, Y. Qian, J. Zhou, and Y. Y. Tang, "Hyperspectral unmixing via total variation regularized nonnegative tensor factorization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 4, pp. 2341-2357, 2019.

[109] M. Jouni, M. D. Mura, L. Drumetz, and P. Comon, "Multihu-td: Multifeature hyperspectral unmixing based on tensor decomposition," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 61, pp. 1-21, 2023.

[110] M.A. Veganzones, L. Drumetz, G. Tochon, M. Dalla Mura, A. Plaza, J. Bioucas-Dias, and J. Chanussot, "A new extended linear mixing model to address spectral variability," in *2014 6th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2014, pp. 1-4.

[111] L. Drumetz, M.-A. Veganzones, S. Henrot, R. Phlypo, J. Chanussot, and C. Jutten, "Blind hyperspectral unmixing using an extended linear mixing model to address spectral variability," *IEEE Transactions on Image Processing*, vol. 25, no. 8, pp. 3890-3905, 2016.

[112] T. Imbiriba, R. A. Borsoi, and J. C. Moreira Bermudez, "Generalized linear mixing model accounting for endmember variability," in *2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*, 2018, pp. 1862-1866.

[113] R. A. Borsoi, T. Imbiriba, and J. C. Moreira Bermudez, "Improved hyperspectral unmixing with endmember variability parametrized using an interpolated scaling tensor," in *ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*, 2019, pp. 2177-2181.

[114] P.-A. Thouvenin, N. Dobigeon, and J.-Y. Tourneret, "Hyperspectral unmixing with spectral variability using a perturbed linear mixing model," *IEEE Transactions on Signal Processing*, vol. 64, no. 2, pp. 525-538, 2016.

[115] P.-A. Thouvenin, N. Dobigeon, and J.-Y. Tourneret, "Online unmixing of multitemporal hyperspectral images accounting for spectral variability," *IEEE Transactions on Image Processing*, vol. 25, no. 9, pp. 3979-3990, 2016.

[116] R. A. Borsoi, T. Imbiriba, P. Closas, J. C. M. Bermudez, and C. Richard, "Kalman filtering and expectation maximization for multitemporal spectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1-5, 2022.

[117] J. Yao, D. Hong, L. Xu, D. Meng, J. Chanussot, and Z. Xu, "Sparsity-enhanced convolutional decomposition: A novel tensor-based paradigm for blind hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-14, 2021.

[118] D. Hong, N. Yokoya, J. Chanussot, and X. Zhu, "An augmented linear mixing model to address spectral variability for hyperspectral unmixing," *IEEE Transactions on Image Processing*, vol. 28, no. 4, pp. 1923-1938, 2018.

[119] D. Hong and X. X. Zhu, "Sulora: Subspace unmixing with low-rank attribute embedding for hyperspectral data analysis," *IEEE Journal of Selected Topics in Signal Processing*, vol. 12, no. 6, pp. 1351-1363, 2018.

[120] S. Ozkan, B. Kaya, and G. B. Akar, "Endnet: Sparse autoencoder network for endmember extraction and hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 1, pp. 482-496, 2019.

[121] Y. Su, A. Marinoni, J. Li, J. Plaza, and P. Gamba, "Stacked nonnegative sparse autoencoders for robust hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 15, no. 9, pp. 1427-1431, 2018.

[122] Y. Su, J. Li, A. Plaza, A. Marinoni, P. Gamba, and S. Chakravortty, "Daen: Deep autoencoder networks for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 7, pp. 4309-4321, 2019.

[123] Y. Qu and H. Qi, "udas: An untied denoising autoencoder with sparsity for spectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 3, pp. 1698-1712, 2019.

[124] Z. Hua, X. Li, Q. Qiu, and L. Zhao, "Autoencoder network for hyperspectral unmixing with adaptive abundance smoothing," *IEEE Geoscience and Remote Sensing Letters*, vol. 18, no. 9, pp. 1640-1644, 2021.

[125] X. Zhang, Y. Sun, J. Zhang, P. Wu, and L. Jiao, "Hyperspectral unmixing via deep convolutional neural networks," *IEEE Geoscience and Remote Sensing Letters*, vol. 15, no. 11, pp. 1755-1759, 2018.

[126] B. Palsson, J. R. Sveinsson, and M. O. Ulfarsson, "Spectral-spatial hyperspectral unmixing using multitask learning," *IEEE Access*, vol. 7, pp. 148861-148872, 2019.

[127] B. Palsson, M. O. Ulfarsson, and J. R. Sveinsson, "Convolutional autoencoder for spectral-spatial hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, pp. 1-15, 2020.

[128] L. Gao, Z. Han, D. Hong, B. Zhang, and J. Chanussot, "Cycunet: Cycle-consistency unmixing network by learning cascaded autoencoders," *IEEE Transactions on Geoscience and Remote Sensing*, pp. 1-14, 2021.

[129] P. Ghosh, S. K. Roy, B. Koirala, B. Rasti, and P. Scheunders, "Hyperspectral unmixing using transformer network," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-16, 2022.

[130] Qiwen J., Yong M., F. Fan, J. Huang, X. Mei, and J. Ma, "Adversarial autoencoder network for hyperspectral unmixing," *IEEE Transactions on Neural Networks and Learning Systems*, pp. 1-15, 2021.

[131] R. A. Borsoi, T. Imbiriba, and J. C. M. Bermudez, "Deep generative endmember modeling: An application to unsupervised spectral unmixing," *IEEE Transactions on Computational Imaging*, vol. 6, pp. 374-384, 2020.

[132] S. Shi, L. Zhang, Y. Altmann, and J. Chen, "Deep generative model for spatial-spectral unmixing with multiple endmember priors," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-14, 2022.

[133] W.-K. Lin, Ch.-H. and Ma, W.-Ch. Li, Ch.-Y. Chi, and A. Ambikapathi, "Identifiability of the simplex volume minimization criterion for blind hyperspectral unmixing: The no-pure-pixel case," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 53, no. 10, pp. 5530-5546, 2015.

[134] C. Zhou and M. R. D. Rodrigues, "Admm-based hyperspectral unmixing networks for abundance and endmember estimation," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-18, 2022.

[135] Y. Qian, F. Xiong, Q. Qian, and J. Zhou, "Spectral mixture model inspired network architectures for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 58, no. 10, pp. 7418-7434, 2020.

[136] I. Daubechies, M. Defrise, and C. De Mol, "An iterative thresholding algorithm for linear inverse problems with a sparsity constraint," *Communications on Pure and Applied Mathematics*, vol. 57, no. 11, pp. 1413-1457, 2004.

[137] F. Xiong, J. Zhou, S. Tao, J. Lu, and Y. Qian, "Snmf-net: Learning a deep alternating neural network for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-16, 2022.

[138] W. Zuo, D. Meng, L. Zhang, X. Feng, and D. Zhang, "A generalized iterated shrinkage algorithm for non-convex sparse coding," in 2013
[98] A. Cutler and L. Breiman, "Archetypal analysis," Technometrics, vol. 36, no. 4, pp. 338-347, 1994.

[99] G. Zhao, X. Jia, and C. Zhao, "Multiple endmembers based unmixing using archetypal analysis," in *2015 IEEE International Geoscience and Remote Sensing Symposium (IGARSS)*, 2015, pp. 5039-5042.

[100] C. Zhao, G. Zhao, and X. Jia, "Hyperspectral image unmixing based on fast kernel archetypal analysis," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 10, no. 1, pp. 331-346, 2017.

[101] M. Xu, Z. Yang, G. Ren, H. Sheng, S. Liu, W. Liu, and C. Ye, "L1 sparsity-constrained archetypal analysis algorithm for hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1-5, 2022.

[102] Q. Zhang, H. Wang, R. Plemmons, and V. P. Pauca, "Spectral unmixing using nonnegative tensor factorization," in *Proceedings of the 45th Annual Southeast Regional Conference, New York, NY, USA, 2007*, ACM-SE 45, p. 531-532, Association for Computing Machinery.

[103] A. Huck and M. Guillaume, "Estimation of the hyperspectral tucker ranks," in *2009 IEEE International Conference on Acoustics, Speech and Signal Processing*, 2009, pp. 1281-1284.

[104] L. De Lathauwer, "Decompositions of a higher-order tensor in block terms—part ii: Definitions and uniqueness," *SIAM Journal on Matrix Analysis and Applications*, vol. 30, no. 3, pp. 1033-1066, 2008.

[105] Y. Qian, F. Xiong, S. Zeng, J. Zhou, and Y. Y. Tang, "Matrix-vector nonnegative tensor factorization for blind unmixing of hyperspectral imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 55, no. 3, pp. 1776-1792, 2017.

[106] Pan Zheng, Hongjun Su, and Qian Du, "Sparse and low-rank constrained tensor factorization for hyperspectral image unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 14, pp. 1754-1767, 2021.

[107] P. Yang, Ti-Z. Huang, J. Huang, and J.-J. Wang, "Efficient weighted-adaptive sparse constrained nonnegative tensor factorization for hyperspectral unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 10113-10130, 2022.

[108] F. Xiong, Y. Qian, J. Zhou, and Y. Y. Tang, "Hyperspectral unmixing via total variation regularized nonnegative tensor factorization," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 4, pp. 2341-2357, 2019.

[109] M. Jouni, M. D. Mura, L. Drumetz, and P. Comon, "Multihu-td: Multifeature hyperspectral unmixing based on tensor decomposition," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 61, pp. 1-21, 2023.

[110] M.A. Veganzones, L. Drumetz, G. Tochon, M. Dalla Mura, A. Plaza, J. Bioucas-Dias, and J. Chanussot, "A new extended linear mixing model to address spectral variability," in *2014 6th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2014, pp. 1-4.

[111] L. Drumetz, M.-A. Veganzones, S. Henrot, R. Phlypo, J. Chanussot, and C. Jutten, "Blind hyperspectral unmixing using an extended linear mixing model to address spectral variability," *IEEE Transactions on Image Processing*, vol. 25, no. 8, pp. 3890-3905, 2016.

[112] T. Imbiriba, R. A. Borsoi, and J. C. Moreira Bermudez, "Generalized linear mixing model accounting for endmember variability," in *2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*, 2018, pp. 1862-1866.

[113] R. A. Borsoi, T. Imbiriba, and J. C. Moreira Bermudez, "Improved hyperspectral unmixing with endmember variability parametrized using an interpolated scaling tensor," in *ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*, 2019, pp. 2177-2181.

[114] P.-A. Thouvenin, N. Dobigeon, and J.-Y. Tourneret, "Hyperspectral unmixing with spectral variability using a perturbed linear mixing model," *IEEE Transactions on Signal Processing*, vol. 64, no. 2, pp. 525-538, 2016.

[115] P.-A. Thouvenin, N. Dobigeon, and J.-Y. Tourneret, "Online unmixing of multitemporal hyperspectral images accounting for spectral variability," *IEEE Transactions on Image Processing*, vol. 25, no. 9, pp. 3979-3990, 2016.

[116] R. A. Borsoi, T. Imbiriba, P. Closas, J. C. M. Bermudez, and C. Richard, "Kalman filtering and expectation maximization for multitemporal spectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1-5, 2022.

[117] J. Yao, D. Hong, L. Xu, D. Meng, J. Chanussot, and Z. Xu, "Sparsity-enhanced convolutional decomposition: A novel tensor-based paradigm for blind hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-14, 2021.

[118] D. Hong, N. Yokoya, J. Chanussot, and X. Zhu, "An augmented linear mixing model to address spectral variability for hyperspectral unmixing," *IEEE Transactions on Image Processing*, vol. 28, no. 4, pp. 1923-1938, 2018.

[119] D. Hong and X. X. Zhu, "Sulora: Subspace unmixing with low-rank attribute embedding for hyperspectral data analysis," *IEEE Journal of Selected Topics in Signal Processing*, vol. 12, no. 6, pp. 1351-1363, 2018.

[120] S. Ozkan, B. Kaya, and G. B. Akar, "Endnet: Sparse autoencoder network for endmember extraction and hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 1, pp. 482-496, 2019.

[121] Y. Su, A. Marinoni, J. Li, J. Plaza, and P. Gamba, "Stacked nonnegative sparse autoencoders for robust hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 15, no. 9, pp. 1427-1431, 2018.

[122] Y. Su, J. Li, A. Plaza, A. Marinoni, P. Gamba, and S. Chakravortty, "Daen: Deep autoencoder networks for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 7, pp. 4309-4321, 2019.

[123] Y. Qu and H. Qi, "udas: An untied denoising autoencoder with sparsity for spectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 57, no. 3, pp. 1698-1712, 2019.

[124] Z. Hua, X. Li, Q. Qiu, and L. Zhao, "Autoencoder network for hyperspectral unmixing with adaptive abundance smoothing," *IEEE Geoscience and Remote Sensing Letters*, vol. 18, no. 9, pp. 1640-1644, 2021.

[125] X. Zhang, Y. Sun, J. Zhang, P. Wu, and L. Jiao, "Hyperspectral unmixing via deep convolutional neural networks," *IEEE Geoscience and Remote Sensing Letters*, vol. 15, no. 11, pp. 1755-1759, 2018.

[126] B. Palsson, J. R. Sveinsson, and M. O. Ulfarsson, "Spectral-spatial hyperspectral unmixing using multitask learning," *IEEE Access*, vol. 7, pp. 148861-148872, 2019.

[127] B. Palsson, M. O. Ulfarsson, and J. R. Sveinsson, "Convolutional autoencoder for spectral-spatial hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, pp. 1-15, 2020.

[128] L. Gao, Z. Han, D. Hong, B. Zhang, and J. Chanussot, "Cycunet: Cycle-consistency unmixing network by learning cascaded autoencoders," *IEEE Transactions on Geoscience and Remote Sensing*, pp. 1-14, 2021.

[129] P. Ghosh, S. K. Roy, B. Koirala, B. Rasti, and P. Scheunders, "Hyperspectral unmixing using transformer network," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-16, 2022.

[130] Qiwen J., Yong M., F. Fan, J. Huang, X. Mei, and J. Ma, "Adversarial autoencoder network for hyperspectral unmixing," *IEEE Transactions on Neural Networks and Learning Systems*, pp. 1-15, 2021.

[131] R. A. Borsoi, T. Imbiriba, and J. C. M. Bermudez, "Deep generative endmember modeling: An application to unsupervised spectral unmixing," *IEEE Transactions on Computational Imaging*, vol. 6, pp. 374-384, 2020.

[132] S. Shi, L. Zhang, Y. Altmann, and J. Chen, "Deep generative model for spatial-spectral unmixing with multiple endmember priors," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-14, 2022.

[133] W.-K. Lin, Ch.-H. and Ma, W.-Ch. Li, Ch.-Y. Chi, and A. Ambikapathi, "Identifiability of the simplex volume minimization criterion for blind hyperspectral unmixing: The no-pure-pixel case," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 53, no. 10, pp. 5530-5546, 2015.

[134] C. Zhou and M. R. D. Rodrigues, "Admm-based hyperspectral unmixing networks for abundance and endmember estimation," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-18, 2022.

[135] Y. Qian, F. Xiong, Q. Qian, and J. Zhou, "Spectral mixture model inspired network architectures for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 58, no. 10, pp. 7418-7434, 2020.

[136] I. Daubechies, M. Defrise, and C. De Mol, "An iterative thresholding algorithm for linear inverse problems with a sparsity constraint," *Communications on Pure and Applied Mathematics*, vol. 57, no. 11, pp. 1413-1457, 2004.

[137] F. Xiong, J. Zhou, S. Tao, J. Lu, and Y. Qian, "Snmf-net: Learning a deep alternating neural network for hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-16, 2022.

[138] W. Zuo, D. Meng, L. Zhang, X. Feng, and D. Zhang, "A generalized iterated shrinkage algorithm for non-convex sparse coding," in 2013
IEEE International Conference on Computer Vision, 2013, pp. 217–224.

[139] L. C. Parra, C. Spence, P. Sajda, A. Ziehe, and K.-R. Müller, “Unmixing hyperspectral data,” in Advances in Neural Information Processing Systems 12. 2000, pp. 942–948, MIT Press.

[140] N. Dobigeon, S. Moussaoui, M. Coulon, J. Tourneret, and A. O. Hero, “Joint bayesian endmember extraction and linear unmixing for hyperspectral imagery,” *IEEE Transactions on Signal Processing*, vol. 57, no. 11, pp. 4355–4368, 2009.

[141] M. Elad, P. Milanfar, and Ron Rubinstein, “Analysis versus synthesis in signal priors,” *Inverse Problems*, vol. 23, no. 3, pp. 947–968, apr 2007.

[142] C. Jutten and J. Herault, “Blind separation of sources, part 1: An adaptive algorithm based on neuromimetic architecture,” *Sig. Process.*, vol. 24, no. 1, pp. 1–10, Aug. 1991.

[143] J. D. Bayliss, J. A. Gualtieri, and R. F. Cromp, “Analyzing hyperspectral data with independent component analysis,” in *26th AIPR Workshop: Exploiting New Image Sources and Sensors*, J. Michael Selander, Ed. International Society for Optics and Photonics, 1998, vol. 3240, pp. 133–143, SPIE.

[144] C. H. Chen and X. Zhang, “Independent component analysis for remote sensing study,” in *Image and Signal Processing for Remote Sensing V, Sebastiano Bruno Serpico*, Ed. International Society for Optics and Photonics, 1999, vol. 3871, pp. 150–158, SPIE.

[145] J.M.P. Nascimento and J.M.B. Dias, “Does independent component analysis play a role in unmixing hyperspectral data?,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 43, no. 1, pp. 175–187, 2005.

[146] J. M. P. Nascimento and J. M. Bioucas-Dias, “Hyperspectral unmixing algorithm via dependent component analysis,” in *2007 IEEE International Geoscience and Remote Sensing Symposium*, 2007, pp. 4033–4036.

[147] J. M. P. Nascimento and J. M. Bioucas-Dias, “Hyperspectral unmixing based on mixtures of dirichlet components,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 50, no. 3, pp. 863–878, 2012.

[148] N. Dobigeon, S. Moussaoui, J.-Y. Tourneret, and C. Carteret, “Bayesian separation of spectral sources under non-negativity and full additivity constraints,” *Signal Processing*, vol. 89, no. 12, pp. 2657–2669, 2009, Special Section: Visual Information Analysis for Security.

[149] B. Somers, L. Titş, D. Roberts, and E. Wetherley, “Chapter 17 - endmember library approaches to resolve spectral mixing problems in remotely sensed data: Potential, challenges, and applications,” in *Resolving Spectral Mixtures*, Cyril Ruckebusch, Ed., vol. 30 of Data Handling in Science and Technology, pp. 551–577. Elsevier, 2016.

[150] D. Roberts, M. Gardner, R. Church, S. Ustin, G. Scheer, and R.O. Green, “Mapping chaparral in the santa monica mountains using multiple endmember spectral mixture models,” *Remote Sensing of Environment*, vol. 65, 09 1998.

[151] S. Tompkins, J. F. Mustard, C. M. Pieters, and D. W. Forsyth, “Optimization of endmembers for spectral mixture analysis,” *Remote Sensing of Environment*, vol. 59, no. 3, pp. 472–489, 1997.

[152] M. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Sparse unmixing of hyperspectral data,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 6, pp. 2014–2039, 2011.

[153] C. Quintano, A. Fernandez-Manso, and D. A. Roberts, “Multiple endmember spectral mixture analysis (mesma) to map burn severity levels from landsat images in mediterranean countries,” *Remote Sensing of Environment*, vol. 136, pp. 76–88, 2013.

[154] L. Drumetz, T. R. Meyer, J. Chanussot, A. L. Bertozzi, and C. Jutten, “Hyperspectral image unmixing with endmember bundles and group sparsity inducing mixed norms,” *IEEE Transactions on Image Processing*, vol. 28, no. 7, pp. 3435–3450, 2019.

[155] L. Titş, B. Somers, and P. Coppin, “The potential and limitations of a clustering approach for the improved efficiency of multiple endmember spectral mixture analysis in plant production system monitoring,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 50, no. 6, pp. 2273–2286, 2012.

[156] S. Jacquemoud and F. Baret, “Prospect: A model of leaf optical properties spectra,” *Remote Sensing of Environment*, vol. 34, no. 2, pp. 75–91, 1990.

[157] R. A. Borsoi, T. Imbiriba, J. C. M. Bermudez, and C. Richard, “Deep generative models for library augmentation in multiple endmember spectral mixture analysis,” *IEEE Geoscience and Remote Sensing Letters*, vol. 18, no. 10, pp. 1831–1835, 2021.

[158] R. Heylen, A. Zare, P. Gader, and P. Scheunders, “Hyperspectral unmixing with endmember variability via alternating angle minimization,”

IEEE Transactions on Geoscience and Remote Sensing, vol. 54, no. 8, pp. 4983–4993, 2016.

[159] Philip E. D., Kerry Q. H., and Dar A. R., “A comparison of error metrics and constraints for multiple endmember spectral mixture analysis and spectral angle mapper,” *Remote Sensing of Environment*, vol. 93, no. 3, pp. 359–367, 2004.

[160] J.-Ph. Combe, S. Le MoüÄlic, C. Sotin, A. Gendrin, J.F. Mustard, L. Le Deit, P. Launeau, J.-P. Bibring, B. Gondet, Y. Langevin, and P. Pinet, “Analysis of omega/mars express data hyperspectral data using a multiple-endmember linear spectral unmixing model (melsum): Methodology and first results,” *Planetary and Space Science*, vol. 56, no. 7, pp. 951–975, 2008.

[161] D.A. Roberts, P.E. Dennison, M.E. Gardner, Y. Hetzel, S.L. Ustin, and C.T. Lee, “Evaluation of the potential of hyperion for fire danger assessment by comparison to the airborne visible/infrared imaging spectrometer,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 41, no. 6, pp. 1297–1310, 2003.

[162] Carmen Q., Alfonso F.-M., and Dar A. R., “Multiple endmember spectral mixture analysis (mesma) to map burn severity levels from landsat images in mediterranean countries,” *Remote Sensing of Environment*, vol. 136, pp. 76–88, 2013.

[163] Philip E. D. and Dar A. R., “Endmember selection for multiple endmember spectral mixture analysis using endmember average rmse,” *Remote Sensing of Environment*, vol. 87, no. 2, pp. 123–135, 2003.

[164] R. Heylen, A. Zare, P. Gader, and P. Scheunders, “Hyperspectral unmixing with endmember variability via alternating angle minimization,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 8, pp. 4983–4993, 2016.

[165] L. Titş, R. Heylen, B. Somers, P. Scheunders, and P. Coppin, “A geometric unmixing concept for the selection of optimal binary endmember combinations,” *IEEE Geoscience and Remote Sensing Letters*, vol. 12, no. 1, pp. 82–86, 2015.

[166] J. M. Bioucas-Dias and M. A. T. Figueiredo, “Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing,” *arXiv*, 2012.

[167] J. M. P. Nascimento, J. M. Bioucas-Dias, J. M. Rodríguez Alves, V. Silva, and A. Plaza, “Parallel hyperspectral unmixing on gpus,” *IEEE Geoscience and Remote Sensing Letters*, vol. 11, no. 3, pp. 666–670, 2014.

[168] Y. C. Pati, R. Rezaiifar, and P. Krishnaprasad, “Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition,” in *Proc. 27th Annu. Asilomar Conf. Signals, Syst., Comput., Los Alamitos, CA*, 1993, p. 40–44.

[169] D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via $l_1$ minimization,” *Proceedings of the National Academy of Sciences of the United States of America*, vol. 100, no. 5, pp. 2197–2202, 2003.

[170] E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” *Communications on Pure and Applied Mathematics*, vol. 59, no. 8, pp. 1207–1223, 2006.

[171] M. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Total variation spatial regularization for sparse hyperspectral unmixing,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 50, no. 11, pp. 4484–4502, 2012.

[172] M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Hyperspectral unmixing with sparse group lasso,” in *2011 IEEE International Geoscience and Remote Sensing Symposium*, 2011, pp. 3586–3589.

[173] L. Meier, S. V. D. Geer, and P. Bühlmann, “The group lasso for logistic regression,” *Journal of the Royal Statistical Society. Series B*, vol. 70, no. 1, pp. 53–71, 2008.

[174] F. Chen and Y. Zhang, “Sparse hyperspectral unmixing based on constrained lp - l2 optimization,” *IEEE Geoscience and Remote Sensing Letters*, vol. 10, no. 5, pp. 1142–1146, 2013.

[175] Z. Shi, T. Shi, M. Zhou, and X. Xu, “Collaborative sparse hyperspectral unmixing using $l_0$ norm,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 56, no. 9, pp. 5495–5508, 2018.

[176] R. Wang, H.-C. Li, A. Pizurica, J. Li, A. Plaza, and W. J. Emery, “Hyperspectral unmixing using double reweighted sparse regression and total variation,” *IEEE Geoscience and Remote Sensing Letters*, vol. 14, no. 7, pp. 1146–1150, 2017.

[177] S. Zhang, J. Li, K. Liu, C. Deng, L. Liu, and A. Plaza, “Hyperspectral unmixing based on local collaborative sparse regression,” *IEEE Geoscience and Remote Sensing Letters*, vol. 13, no. 5, pp. 631–635, 2016.

[178] P. V. Giampouras, K. E. Themelis, A. A. Rontogiannis, and K. D. Koutroumbas, “Simultaneously sparse and low-rank abundance matrix
IEEE International Conference on Computer Vision, 2013, pp. 217–224.

[139] L. C. Parra, C. Spence, P. Sajda, A. Ziehe, and K.-R. Müller, “Unmixing hyperspectral data,” in Advances in Neural Information Processing Systems 12. 2000, pp. 942–948, MIT Press.

[140] N. Dobigeon, S. Moussaoui, M. Coulon, J. Tourneret, and A. O. Hero, “Joint bayesian endmember extraction and linear unmixing for hyperspectral imagery,” *IEEE Transactions on Signal Processing*, vol. 57, no. 11, pp. 4355–4368, 2009.

[141] M. Elad, P. Milanfar, and Ron Rubinstein, “Analysis versus synthesis in signal priors,” *Inverse Problems*, vol. 23, no. 3, pp. 947–968, apr 2007.

[142] C. Jutten and J. Herault, “Blind separation of sources, part 1: An adaptive algorithm based on neuromimetic architecture,” *Sig. Process.*, vol. 24, no. 1, pp. 1–10, Aug. 1991.

[143] J. D. Bayliss, J. A. Gualtieri, and R. F. Cromp, “Analyzing hyperspectral data with independent component analysis,” in *26th AIPR Workshop: Exploiting New Image Sources and Sensors*, J. Michael Selander, Ed. *International Society for Optics and Photonics*, 1998, vol. 3240, pp. 133–143, SPIE.

[144] C. H. Chen and X. Zhang, “Independent component analysis for remote sensing study,” in *Image and Signal Processing for Remote Sensing V, Sebastiano Bruno Serpico*, Ed. *International Society for Optics and Photonics*, 1999, vol. 3871, pp. 150–158, SPIE.

[145] J.M.P. Nascimento and J.M.B. Dias, “Does independent component analysis play a role in unmixing hyperspectral data?,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 43, no. 1, pp. 175–187, 2005.

[146] J. M. P. Nascimento and J. M. Bioucas-Dias, “Hyperspectral unmixing algorithm via dependent component analysis,” in *2007 IEEE International Geoscience and Remote Sensing Symposium*, 2007, pp. 4033–4036.

[147] J. M. P. Nascimento and J. M. Bioucas-Dias, “Hyperspectral unmixing based on mixtures of dirichlet components,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 50, no. 3, pp. 863–878, 2012.

[148] N. Dobigeon, S. Moussaoui, J.-Y. Tourneret, and C. Carteret, “Bayesian separation of spectral sources under non-negativity and full additivity constraints,” *Signal Processing*, vol. 89, no. 12, pp. 2657–2669, 2009, Special Section: Visual Information Analysis for Security.

[149] B. Somers, L. Titş, D. Roberts, and E. Wetherley, “Chapter 17 - endmember library approaches to resolve spectral mixing problems in remotely sensed data: Potential, challenges, and applications,” in *Resolving Spectral Mixtures*, Cyril Ruckebusch, Ed., vol. 30 of Data Handling in Science and Technology, pp. 551–577. Elsevier, 2016.

[150] D. Roberts, M. Gardner, R. Church, S. Ustin, G. Scheer, and R.O. Green, “Mapping chaparral in the santa monica mountains using multiple endmember spectral mixture models,” *Remote Sensing of Environment*, vol. 65, 09 1998.

[151] S. Tompkins, J. F. Mustard, C. M. Pieters, and D. W. Forsyth, “Optimization of endmembers for spectral mixture analysis,” *Remote Sensing of Environment*, vol. 59, no. 3, pp. 472–489, 1997.

[152] M. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Sparse unmixing of hyperspectral data,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 6, pp. 2014–2039, 2011.

[153] C. Quintano, A. Fernandez-Manso, and D. A. Roberts, “Multiple endmember spectral mixture analysis (mesma) to map burn severity levels from landsat images in mediterranean countries,” *Remote Sensing of Environment*, vol. 136, pp. 76–88, 2013.

[154] L. Drumetz, T. R. Meyer, J. Chanussot, A. L. Bertozzi, and C. Jutten, “Hyperspectral image unmixing with endmember bundles and group sparsity inducing mixed norms,” *IEEE Transactions on Image Processing*, vol. 28, no. 7, pp. 3435–3450, 2019.

[155] L. Titş, B. Somers, and P. Coppin, “The potential and limitations of a clustering approach for the improved efficiency of multiple endmember spectral mixture analysis in plant production system monitoring,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 50, no. 6, pp. 2273–2286, 2012.

[156] S. Jacquemoud and F. Baret, “Prospect: A model of leaf optical properties spectra,” *Remote Sensing of Environment*, vol. 34, no. 2, pp. 75–91, 1990.

[157] R. A. Borsoi, T. Imbiriba, J. C. M. Bermudez, and C. Richard, “Deep generative models for library augmentation in multiple endmember spectral mixture analysis,” *IEEE Geoscience and Remote Sensing Letters*, vol. 18, no. 10, pp. 1831–1835, 2021.

[158] R. Heylen, A. Zare, P. Gader, and P. Scheunders, “Hyperspectral unmixing with endmember variability via alternating angle minimization,”

*IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 8, pp. 4983–4993, 2016.

[159] Philip E. D., Kerry Q. H., and Dar A. R., “A comparison of error metrics and constraints for multiple endmember spectral mixture analysis and spectral angle mapper,” *Remote Sensing of Environment*, vol. 93, no. 3, pp. 359–367, 2004.

[160] J.-Ph. Combe, S. Le Moüālic, C. Sotin, A. Gendrin, J.F. Mustard, L. Le Deit, P. Launeau, J.-P. Bibring, B. Gondet, Y. Langevin, and P. Pinet, “Analysis of omega/mars express data hyperspectral data using a multiple-endmember linear spectral unmixing model (melsum): Methodology and first results,” *Planetary and Space Science*, vol. 56, no. 7, pp. 951–975, 2008.

[161] D.A. Roberts, P.E. Dennison, M.E. Gardner, Y. Hetzel, S.L. Ustin, and C.T. Lee, “Evaluation of the potential of hyperion for fire danger assessment by comparison to the airborne visible/infrared imaging spectrometer,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 41, no. 6, pp. 1297–1310, 2003.

[162] Carmen Q., Alfonso F.-M., and Dar A. R., “Multiple endmember spectral mixture analysis (mesma) to map burn severity levels from landsat images in mediterranean countries,” *Remote Sensing of Environment*, vol. 136, pp. 76–88, 2013.

[163] Philip E. D. and Dar A. R., “Endmember selection for multiple endmember spectral mixture analysis using endmember average rmse,” *Remote Sensing of Environment*, vol. 87, no. 2, pp. 123–135, 2003.

[164] R. Heylen, A. Zare, P. Gader, and P. Scheunders, “Hyperspectral unmixing with endmember variability via alternating angle minimization,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 8, pp. 4983–4993, 2016.

[165] L. Titş, R. Heylen, B. Somers, P. Scheunders, and P. Coppin, “A geometric unmixing concept for the selection of optimal binary endmember combinations,” *IEEE Geoscience and Remote Sensing Letters*, vol. 12, no. 1, pp. 82–86, 2015.

[166] J. M. Bioucas-Dias and M. A. T. Figueiredo, “Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing,” *arXiv*, 2012.

[167] J. M. P. Nascimento, J. M. Bioucas-Dias, J. M. Rodríguez Alves, V. Silva, and A. Plaza, “Parallel hyperspectral unmixing on gpus,” *IEEE Geoscience and Remote Sensing Letters*, vol. 11, no. 3, pp. 666–670, 2014.

[168] Y. C. Pati, R. Rezaiifar, and P. Krishnaprasad, “Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition,” in *Proc. 27th Annu. Asilomar Conf. Signals, Syst., Comput., Los Alamitos, CA*, 1993, p. 40–44.

[169] D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via $l_1$ minimization,” *Proceedings of the National Academy of Sciences of the United States of America*, vol. 100, no. 5, pp. 2197–2202, 2003.

[170] E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” *Communications on Pure and Applied Mathematics*, vol. 59, no. 8, pp. 1207–1223, 2006.

[171] M. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Total variation spatial regularization for sparse hyperspectral unmixing,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 50, no. 11, pp. 4484–4502, 2012.

[172] M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Hyperspectral unmixing with sparse group lasso,” in *2011 IEEE International Geoscience and Remote Sensing Symposium*, 2011, pp. 3586–3589.

[173] L. Meier, S. V. D. Geer, and P. Bühlmann, “The group lasso for logistic regression,” *Journal of the Royal Statistical Society. Series B*, vol. 70, no. 1, pp. 53–71, 2008.

[174] F. Chen and Y. Zhang, “Sparse hyperspectral unmixing based on constrained lp - l2 optimization,” *IEEE Geoscience and Remote Sensing Letters*, vol. 10, no. 5, pp. 1142–1146, 2013.

[175] Z. Shi, T. Shi, M. Zhou, and X. Xu, “Collaborative sparse hyperspectral unmixing using $l_0$ norm,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 56, no. 9, pp. 5495–5508, 2018.

[176] R. Wang, H.-C. Li, A. Pizurica, J. Li, A. Plaza, and W. J. Emery, “Hyperspectral unmixing using double reweighted sparse regression and total variation,” *IEEE Geoscience and Remote Sensing Letters*, vol. 14, no. 7, pp. 1146–1150, 2017.

[177] S. Zhang, J. Li, K. Liu, C. Deng, L. Liu, and A. Plaza, “Hyperspectral unmixing based on local collaborative sparse regression,” *IEEE Geoscience and Remote Sensing Letters*, vol. 13, no. 5, pp. 631–635, 2016.

[178] P. V. Giampouras, K. E. Themelis, A. A. Rontogiannis, and K. D. Koutroumbas, “Simultaneously sparse and low-rank abundance matrix
estimation for hyperspectral image unmixing,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 8, pp. 4775–4789, 2016.

[179] F. Li, S. Zhang, B. Liang, C. Deng, C. Xu, and S. Wang, "Hyperspectral sparse unmixing with spectral-spatial low-rank constraint," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 14, pp. 6119–6130, 2021.

[180] H. Han, G. Wang, M. Wang, J. Miao, S. Guo, L. Chen, M. Zhang, and K. Guo, "Hyperspectral unmixing via nonconvex sparse and low-rank constraint," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 13, pp. 5704–5718, 2020.

[181] Y. Xie, S. Gu, Y. Liu, W. Zuo, W. Zhang, and L. Zhang, "Weighted schatten p-norm minimization for image denoising and background subtraction," *IEEE Transactions on Image Processing*, vol. 25, no. 10, pp. 4842–4857, 2016.

[182] L. Sun, F. Wu, T. Zhan, W. Liu, J. Wang, and B. Jeon, "Weighted nonlocal low-rank tensor decomposition method for sparse unmixing of hyperspectral images," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 13, pp. 1174–1188, 2020.

[183] J. Huang, T.-Z. Huang, X.-L. Zhao, and L.-J. Deng, "Nonlocal tensor-based sparse hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 8, pp. 6854–6868, 2021.

[184] T. Ince, "Superpixel-based graph laplacian regularization for sparse hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, pp. 1–5, 2020.

[185] G. Zhang, S. Mei, B. Xie, M. Ma, Y. Zhang, Y. Feng, and Q. Du, "Spectral variability augmented sparse unmixing of hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–13, 2022.

[186] S. Mei, M. He, Z. Wang, and D. Feng, "Spatial purity based endmember extraction for spectral mixture analysis," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 48, no. 9, pp. 3434–3445, 2010.

[187] M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, "Potential and limitations of band selection and library pruning in sparse hyperspectral unmixing," in *2015 7th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2015, pp. 1–4.

[188] M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, "Dictionary pruning in sparse unmixing of hyperspectral data," in *2012 4th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2012, pp. 1–4.

[189] M.-D. Iordache, J. M. Bioucas-Dias, A. Plaza, and B. Somers, "Music-csr: Hyperspectral unmixing via multiple signal classification and collaborative sparse regression," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 52, no. 7, pp. 4364–4382, 2014.

[190] R. Schmidt, "Multiple emitter location and signal parameter estimation," *IEEE Transactions on Antennas and Propagation*, vol. 34, no. 3, pp. 276–280, 1986.

[191] X. Xu, B. Pan, Z. Chen, Z. Shi, and T. Li, "Simultaneously multiobjective sparse unmixing and library pruning for hyperspectral imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 4, pp. 3383–3395, 2021.

[192] M. Parente and M.-D. Iordache, "Sparse unmixing of hyperspectral data: The legacy of sunsal," in *2021 IEEE International Geoscience and Remote Sensing Symposium IGARSS*, 2021, pp. 21–24.

[193] X. Fu, W.-K. Ma, J. M. Bioucas-Dias, and T.-H. Chan, "Semiblind hyperspectral unmixing in the presence of spectral library mismatches," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 9, pp. 5171–5184, 2016.

[194] Y. Chen, J. Mairal, and Z. Harchaoui, "Fast and robust archetypal analysis for representation learning," *2014 IEEE Conference on Computer Vision and Pattern Recognition*, pp. 1478–1485, 2014.

[195] H. Su, C. Jia, P. Zheng, and Q. Du, "Superpixel-based weighted collaborative sparse regression and reweighted low-rank representation for hyperspectral image unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 393–408, 2022.

[196] L. Sun, F. Wu, C. He, T. Zhan, W. Liu, and D. Zhang, "Weighted collaborative sparse and 11/2 low-rank regularizations with superpixel segmentation for hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[197] T. Chen, Y. Liu, Y. Zhang, B. Du, and A. Plaza, "Superpixel-based collaborative and low-rank regularization for sparse hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–16, 2022.

[198] L. Qi, J. Li, Y. Wang, Y. Huang, and X. Gao, "Spectral-spatial-weighted multiview collaborative sparse unmixing for hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 58, no. 12, pp. 8766–8779, 2020.

[199] X. Shen, H. Liu, X. Zhang, K. Qin, and X. Zhou, "Superpixel-guided local sparsity prior for hyperspectral sparse regression unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[200] H. Li, R. Feng, L. Wang, Y. Zhong, and L. Zhang, "Superpixel-based reweighted low-rank and total variation sparse unmixing for hyperspectral remote sensing imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 1, pp. 629–647, 2021.

[201] L. Ren, Z. Ma, F. Bovolo, and L. Bruzzone, "A nonconvex framework for sparse unmixing incorporating the group structure of the spectral library," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–19, 2022.

[202] D. Zhang, T. Wang, S. Yang, Y. Jia, and F. Li, "Spectral reweighting and spectral similarity weighting for sparse hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[203] Y. Lin and P. Gader, "Addressing spectral variability in hyperspectral unmixing with unsupervised neural networks," in *2022 12th Workshop on Hyperspectral Imaging and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2022, pp. 1–5.

[204] G. Zhang, S. Mei, B. Xie, Y. Feng, and Q. Du, "Spectral variability augmented two-stream network for hyperspectral sparse unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[205] F. Kong, M. Chen, Y. Li, and D. Li, "A global spectral-spatial feature learning network for semisupervised hyperspectral unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 3190–3203, 2022.

[206] Q. Qian, F. Xiong, and J. Zhou, "Deep unfolded iterative shrinkage-thresholding model for hyperspectral unmixing," in *IGARSS 2019 - 2019 IEEE International Geoscience and Remote Sensing Symposium*, 2019, pp. 2151–2154.

[207] Y. Shao, Q. Liu, and L. Xiao, "Iviu-net: Implicit variable iterative unrolling network for hyperspectral sparse unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 16, pp. 1756–1770, 2023.

[208] W. Fan, B. Hu, J. Miller, and M. Li, "Comparative study between a new nonlinear model and common linear model for analysing laboratory simulated-forest hyperspectral data," *International Journal of Remote Sensing*, vol. 30, no. 11, pp. 2951–2962, 2009.

[209] Y. Altmann, A. Halimi, N. Dobigeon, and J.Y. Tourneret, "Supervised nonlinear spectral unmixing using a postnonlinear mixing model for hyperspectral imagery," *IEEE Transactions on Image Processing*, vol. 21, no. 6, pp. 3017–3025, 2012.

[210] A. Halimi, Y. Altmann, N. Dobigeon, and J. Tourneret, "Nonlinear unmixing of hyperspectral images using a generalized bilinear model," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 11, pp. 4153–4162, 2011.

[211] L. Gao, Z. Wang, L. Zhuang, H. Yu, B. Zhang, and J. Chanussot, "Using low-rank representation of abundance maps and nonnegative tensor factorization for hyperspectral nonlinear unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–17, 2022.

[212] R. Heylen and P. Scheunders, "A multilinear mixing model for nonlinear spectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 1, pp. 240–251, Jan 2016.

[213] A. Marinoni and P. Gamba, "A novel approach for efficient p-linear hyperspectral unmixing," *IEEE Journal of Selected Topics in Signal Processing*, vol. 9, no. 6, pp. 1156–1168, Sep. 2015.

[214] A. Marinoni, J. Plaza, A. Plaza, and P. Gamba, "Nonlinear hyperspectral unmixing using nonlinearity order estimation and polytope decomposition," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 8, no. 6, pp. 2644–2654, June 2015.

[215] A. Marinoni, A. Plaza, and P. Gamba, "Harmonic mixture modeling for efficient nonlinear hyperspectral unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 9, no. 9, pp. 4247–4256, Sep. 2016.

[216] J. Broadwater, R. Chellappa, A. Banerjee, and P. Burlina, "Kernel fully constrained least squares abundance estimates," in *2007 IEEE International Geoscience and Remote Sensing Symposium*, 2007, pp. 4041–4044.

[217] X. Wu, X. Li, and L. Zhao, "A kernel spatial complexity-based nonlinear unmixing method of hyperspectral imagery," in *International Conference on Intelligent Computing for Sustainable Energy and Environment*, Berlin, Heidelberg, 2010, LSMS/ICSEE'10, p. 451–458, Springer-Verlag.

[218] M. Brown, H.G. Lewis, and S.R. Gunn, "Linear spectral mixture models and support vector machines for remote sensing," *IEEE*
estimation for hyperspectral image unmixing,” *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 8, pp. 4775–4789, 2016.

[179] F. Li, S. Zhang, B. Liang, C. Deng, C. Xu, and S. Wang, "Hyperspectral sparse unmixing with spectral-spatial low-rank constraint," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 14, pp. 6119–6130, 2021.

[180] H. Han, G. Wang, M. Wang, J. Miao, S. Guo, L. Chen, M. Zhang, and K. Guo, "Hyperspectral unmixing via nonconvex sparse and low-rank constraint," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 13, pp. 5704–5718, 2020.

[181] Y. Xie, S. Gu, Y. Liu, W. Zuo, W. Zhang, and L. Zhang, "Weighted schatten p-norm minimization for image denoising and background subtraction," *IEEE Transactions on Image Processing*, vol. 25, no. 10, pp. 4842–4857, 2016.

[182] L. Sun, F. Wu, T. Zhan, W. Liu, J. Wang, and B. Jeon, "Weighted nonlocal low-rank tensor decomposition method for sparse unmixing of hyperspectral images," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 13, pp. 1174–1188, 2020.

[183] J. Huang, T.-Z. Huang, X.-L. Zhao, and L.-J. Deng, "Nonlocal tensor-based sparse hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 8, pp. 6854–6868, 2021.

[184] T. Ince, "Superpixel-based graph laplacian regularization for sparse hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, pp. 1–5, 2020.

[185] G. Zhang, S. Mei, B. Xie, M. Ma, Y. Zhang, Y. Feng, and Q. Du, "Spectral variability augmented sparse unmixing of hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–13, 2022.

[186] S. Mei, M. He, Z. Wang, and D. Feng, "Spatial purity based endmember extraction for spectral mixture analysis," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 48, no. 9, pp. 3434–3445, 2010.

[187] M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, "Potential and limitations of band selection and library pruning in sparse hyperspectral unmixing," in *2015 7th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2015, pp. 1–4.

[188] M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, "Dictionary pruning in sparse unmixing of hyperspectral data," in *2012 4th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2012, pp. 1–4.

[189] M.-D. Iordache, J. M. Bioucas-Dias, A. Plaza, and B. Somers, "Music-csr: Hyperspectral unmixing via multiple signal classification and collaborative sparse regression," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 52, no. 7, pp. 4364–4382, 2014.

[190] R. Schmidt, "Multiple emitter location and signal parameter estimation," *IEEE Transactions on Antennas and Propagation*, vol. 34, no. 3, pp. 276–280, 1986.

[191] X. Xu, B. Pan, Z. Chen, Z. Shi, and T. Li, "Simultaneously multiobjective sparse unmixing and library pruning for hyperspectral imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 4, pp. 3383–3395, 2021.

[192] M. Parente and M.-D. Iordache, "Sparse unmixing of hyperspectral data: The legacy of sunsal," in *2021 IEEE International Geoscience and Remote Sensing Symposium IGARSS*, 2021, pp. 21–24.

[193] X. Fu, W.-K. Ma, J. M. Bioucas-Dias, and T.-H. Chan, "Semiblind hyperspectral unmixing in the presence of spectral library mismatches," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 9, pp. 5171–5184, 2016.

[194] Y. Chen, J. Mairal, and Z. Harchaoui, "Fast and robust archetypal analysis for representation learning," *2014 IEEE Conference on Computer Vision and Pattern Recognition*, pp. 1478–1485, 2014.

[195] H. Su, C. Jia, P. Zheng, and Q. Du, "Superpixel-based weighted collaborative sparse regression and reweighted low-rank representation for hyperspectral image unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 393–408, 2022.

[196] L. Sun, F. Wu, C. He, T. Zhan, W. Liu, and D. Zhang, "Weighted collaborative sparse and 11/2 low-rank regularizations with superpixel segmentation for hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[197] T. Chen, Y. Liu, Y. Zhang, B. Du, and A. Plaza, "Superpixel-based collaborative and low-rank regularization for sparse hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–16, 2022.

[198] L. Qi, J. Li, Y. Wang, Y. Huang, and X. Gao, "Spectral-spatial-weighted multiview collaborative sparse unmixing for hyperspectral images," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 58, no. 12, pp. 8766–8779, 2020.

[199] X. Shen, H. Liu, X. Zhang, K. Qin, and X. Zhou, "Superpixel-guided local sparsity prior for hyperspectral sparse regression unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[200] H. Li, R. Feng, L. Wang, Y. Zhong, and L. Zhang, "Superpixel-based reweighted low-rank and total variation sparse unmixing for hyperspectral remote sensing imagery," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 1, pp. 629–647, 2021.

[201] L. Ren, Z. Ma, F. Bovolo, and L. Bruzzone, "A nonconvex framework for sparse unmixing incorporating the group structure of the spectral library," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–19, 2022.

[202] D. Zhang, T. Wang, S. Yang, Y. Jia, and F. Li, "Spectral reweighting and spectral similarity weighting for sparse hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[203] Y. Lin and P. Gader, "Addressing spectral variability in hyperspectral unmixing with unsupervised neural networks," in *2022 12th Workshop on Hyperspectral Imaging and Signal Processing: Evolution in Remote Sensing (WHISPERS)*, 2022, pp. 1–5.

[204] G. Zhang, S. Mei, B. Xie, Y. Feng, and Q. Du, "Spectral variability augmented two-stream network for hyperspectral sparse unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1–5, 2022.

[205] F. Kong, M. Chen, Y. Li, and D. Li, "A global spectral-spatial feature learning network for semisupervised hyperspectral unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 3190–3203, 2022.

[206] Q. Qian, F. Xiong, and J. Zhou, "Deep unfolded iterative shrinkage-thresholding model for hyperspectral unmixing," in *IGARSS 2019 - 2019 IEEE International Geoscience and Remote Sensing Symposium*, 2019, pp. 2151–2154.

[207] Y. Shao, Q. Liu, and L. Xiao, "Iviu-net: Implicit variable iterative unrolling network for hyperspectral sparse unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 16, pp. 1756–1770, 2023.

[208] W. Fan, B. Hu, J. Miller, and M. Li, "Comparative study between a new nonlinear model and common linear model for analysing laboratory simulated-forest hyperspectral data," *International Journal of Remote Sensing*, vol. 30, no. 11, pp. 2951–2962, 2009.

[209] Y. Altmann, A. Halimi, N. Dobigeon, and J.Y. Tourneret, "Supervised nonlinear spectral unmixing using a postnonlinear mixing model for hyperspectral imagery," *IEEE Transactions on Image Processing*, vol. 21, no. 6, pp. 3017–3025, 2012.

[210] A. Halimi, Y. Altmann, N. Dobigeon, and J. Tourneret, "Nonlinear unmixing of hyperspectral images using a generalized bilinear model," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 11, pp. 4153–4162, 2011.

[211] L. Gao, Z. Wang, L. Zhuang, H. Yu, B. Zhang, and J. Chanussot, "Using low-rank representation of abundance maps and nonnegative tensor factorization for hyperspectral nonlinear unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–17, 2022.

[212] R. Heylen and P. Scheunders, "A multilinear mixing model for nonlinear spectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 54, no. 1, pp. 240–251, Jan 2016.

[213] A. Marinoni and P. Gamba, "A novel approach for efficient p-linear hyperspectral unmixing," *IEEE Journal of Selected Topics in Signal Processing*, vol. 9, no. 6, pp. 1156–1168, Sep. 2015.

[214] A. Marinoni, J. Plaza, A. Plaza, and P. Gamba, "Nonlinear hyperspectral unmixing using nonlinearity order estimation and polytope decomposition," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 8, no. 6, pp. 2644–2654, June 2015.

[215] A. Marinoni, A. Plaza, and P. Gamba, "Harmonic mixture modeling for efficient nonlinear hyperspectral unmixing," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 9, no. 9, pp. 4247–4256, Sep. 2016.

[216] J. Broadwater, R. Chellappa, A. Banerjee, and P. Burlina, "Kernel fully constrained least squares abundance estimates," in *2007 IEEE International Geoscience and Remote Sensing Symposium*, 2007, pp. 4041–4044.

[217] X. Wu, X. Li, and L. Zhao, "A kernel spatial complexity-based nonlinear unmixing method of hyperspectral imagery," in *International Conference on Intelligent Computing for Sustainable Energy and Environment*, Berlin, Heidelberg, 2010, LSMS/ICSEE'10, p. 451–458, Springer-Verlag.

[218] M. Brown, H.G. Lewis, and S.R. Gunn, "Linear spectral mixture models and support vector machines for remote sensing," *IEEE*
*Transactions on Geoscience and Remote Sensing*, vol. 38, no. 5, pp. 2346–2360, 2000.

[219] B. Hapke, R. Nelson, and W. Smythe, "The opposition effect of the moon: Coherent backscatter and shadow hiding," *Icarus*, vol. 133, no. 1, pp. 89-97, 1998.

[220] M. Wang, M. Zhao, J. Chen, and S. Rahardja, "Nonlinear unmixing of hyperspectral data via deep autoencoder networks," *IEEE Geoscience and Remote Sensing Letters*, vol. 16, no. 9, pp. 1467-1471, 2019.

[221] Y. Su, X. Xu, J. Li, H. Qi, P. Gamba, and A. Plaza, "Deep autoencoders with multitask learning for bilinear hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 10, pp. 8615-8629, 2021.

[222] M. Zhao, L. Yan, and J. Chen, "Lstm-dnn based autoencoder network for nonlinear hyperspectral image unmixing," *IEEE Journal of Selected Topics in Signal Processing*, vol. 15, no. 2, pp. 295-309, 2021.

[223] M. Zhao, M. Wang, J. Chen, and S. Rahardja, "Hyperspectral unmixing for additive nonlinear models with a 3-d-cnn autoencoder network," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-15, 2022.

[224] K. T. Shahid and I. D. Schizas, "Spatial-aware hyperspectral nonlinear unmixing autoencoder with endmember number estimation," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 20-41, 2022.

[225] H. Li, R. A. Borsoi, T. Imbiriba, P. Closas, J. C. M. Bermudez, and D. Erdoğmuş, "Model-based deep autoencoder networks for nonlinear hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1-5, 2022.

[226] K. T. Shahid and I. D. Schizas, "Unsupervised hyperspectral unmixing via nonlinear autoencoders," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-13, 2022.

[227] Q. Lyu and X. Fu, "Identifiability-guaranteed simplex-structured post-nonlinear mixture learning via autoencoder," *IEEE Transactions on Signal Processing*, vol. 69, pp. 4921-4936, 2021.

[228] M. Tang, Y. Qu, and H. Qi, "Hyperspectral nonlinear unmixing via generative adversarial network," in *IGARSS 2020 - 2020 IEEE International Geoscience and Remote Sensing Symposium*, 2020, pp. 2404-2407.

[229] B. Rasti, B. Koirala, and P. Scheunders, "HapkeCNN: Blind nonlinear unmixing for intimate mixtures using hapke model and convolutional neural network," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-15, 2022.

<figure><img src="image_18.png" /><figcaption>tral imagery.</figcaption></figure>

**Alexandre Zouaoui** received his M.Sc. degree in computer science from Télécom Paris, Paris, France, in 2019. He received his Ph.D. in applied mathematics at Université Grenoble Alpes, Grenoble, France, in 2024 on Sparse and Archetypal Decomposition Algorithms for Hyperspectral Images Restoration and Spectral Unmixing. His research interests include computer vision, interpretable machine learning and remote sensing image processing. In 2024, he joined Data Science Experts as a research engineer focusing on applications related to hyperspectral imagery.

<figure><img src="image_19.png" /></figure>

**Julien Mairal** received the graduate degree from Ecole Polytechnique, Palaiseau, France, in 2005, and the Ph.D. degree from Ecole Normale Superieure, Cachan, France, in 2010. After that, he joined the statistics department at UC Berkeley as a post-doctoral researcher. In 2012, he joined Inria, Grenoble, France, where he is currently a research director and head of the Thoth team. His research interests include machine learning, computer vision, mathematical optimization, and statistical image and signal processing. He received a starting grant and a consolidator grant from the European Research Council, respectively in 2016 and 2022. He was awarded the Cor Baayen prize in 2013, the IEEE PAMI young research award in 2017 and the test-of-time award at ICML 2019.

<figure><img src="image_20.png" /></figure>

**Jocelyn Chanussot (M'04-SM'04-F'12)** received the M.Sc. degree in electrical engineering from the Grenoble Institute of Technology (Grenoble INP), Grenoble, France, in 1995, and the Ph.D. degree from the Université de Savoie, Annecy, France, in 1998. From 1999 to 2023, he has been with Grenoble INP, where he was a Professor of signal and image processing. He is currently a Research Director with INRIA, Grenoble. His research interests include image analysis, hyperspectral remote sensing, data fusion, machine learning and artificial intelligence.

He has been a visiting scholar at Stanford University (USA), KTH (Sweden) and NUS (Singapore). Since 2013, he is an Adjunct Professor of the University of Iceland. In 2015-2017, he was a visiting professor at the University of California, Los Angeles (UCLA). He holds the AXA chair in remote sensing and is an Adjunct professor at the Chinese Academy of Sciences, Aerospace Information research Institute, Beijing. Dr. Chanussot is the founding President of IEEE Geoscience and Remote Sensing French chapter (2007-2010) which received the 2010 IEEE GRS-S Chapter Excellence Award. He has received multiple outstanding paper awards. He was the Vice-President of the IEEE Geoscience and Remote Sensing Society, in charge of meetings and symposia (2017-2019). He was the General Chair of the first IEEE GRSS Workshop on Hyperspectral Image and Signal Processing, Evolution in Remote sensing (WHISPERS). He was the Chair (2009-2011) and Cochair of the GRS Data Fusion Technical Committee (2005-2008). He was a member of the Machine Learning for Signal Processing Technical Committee of the IEEE Signal Processing Society (2006-2008) and the Program Chair of the IEEE International Workshop on Machine Learning for Signal Processing (2009). He is an Associate Editor for the IEEE Transactions on Geoscience and Remote Sensing, the IEEE Transactions on Image Processing and the Proceedings of the IEEE. He was the Editor-in-Chief of the IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing (2011-2015). In 2014 he served as a Guest Editor for the IEEE Signal Processing Magazine. He is a Fellow of the IEEE, an ELLIS Fellow, a Fellow of the Asia-Pacific Artificial Intelligence Association, a member of the Institut Universitaire de France (2012-2017) and a Highly Cited Researcher (Clarivate Analytics/Thomson Reuters, since 2018).

<figure><img src="image_21.png" /></figure>

**Behnood Rasti (M'12-SM'19)** received the B.Sc. and M.Sc. degrees in electronics and electrical engineering from the Electrical Engineering Department, University of Guilan, Rasht, Iran, in 2006 and 2009, respectively, and the Ph.D. degree in electrical and computer engineering from the University of Iceland, Reykjavik, Iceland, in 2014. He was a Valedictorian as an M.Sc. Student in 2009. In 2015 and 2016, he worked as a Post-Doctoral Researcher with the Electrical and Computer Engineering Department, University of Iceland. From 2016 to 2019,

he was a Lecturer with the Center of Engineering Technology and Applied Sciences, Department of Electrical and Computer Engineering, University of Iceland. Dr. Rasti was a Humboldt Research Fellow in 2020 and 2021. From 2022 to 2023, he was a Principal Research Associate with Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Dresden, Germany. He is currently a Senior Research Scientist with the Faculty of Electrical Engineering and Computer Science, Technische Universität Berlin. His research interests include signal and image processing, machine/deep learning, remote sensing, and artificial intelligence.

Dr. Rasti won the Doctoral Grant of the University of Iceland Research Fund "The Eimskip University Fund" and the "Alexander von Humboldt Research Fellowship Grant" in 2013 and 2019, respectively. He serves as an Associate Editor for the IEEE GEOSCIENCE AND REMOTE SENSING LETTERS (GRSL).
*Transactions on Geoscience and Remote Sensing*, vol. 38, no. 5, pp. 2346–2360, 2000.

[219] B. Hapke, R. Nelson, and W. Smythe, "The opposition effect of the moon: Coherent backscatter and shadow hiding," *Icarus*, vol. 133, no. 1, pp. 89-97, 1998.

[220] M. Wang, M. Zhao, J. Chen, and S. Rahardja, "Nonlinear unmixing of hyperspectral data via deep autoencoder networks," *IEEE Geoscience and Remote Sensing Letters*, vol. 16, no. 9, pp. 1467-1471, 2019.

[221] Y. Su, X. Xu, J. Li, H. Qi, P. Gamba, and A. Plaza, "Deep autoencoders with multitask learning for bilinear hyperspectral unmixing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 59, no. 10, pp. 8615-8629, 2021.

[222] M. Zhao, L. Yan, and J. Chen, "Lstm-dnn based autoencoder network for nonlinear hyperspectral image unmixing," *IEEE Journal of Selected Topics in Signal Processing*, vol. 15, no. 2, pp. 295-309, 2021.

[223] M. Zhao, M. Wang, J. Chen, and S. Rahardja, "Hyperspectral unmixing for additive nonlinear models with a 3-d-cnn autoencoder network," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-15, 2022.

[224] K. T. Shahid and I. D. Schizas, "Spatial-aware hyperspectral nonlinear unmixing autoencoder with endmember number estimation," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 15, pp. 20-41, 2022.

[225] H. Li, R. A. Borsoi, T. Imbiriba, P. Closas, J. C. M. Bermudez, and D. Erdoğmuş, "Model-based deep autoencoder networks for nonlinear hyperspectral unmixing," *IEEE Geoscience and Remote Sensing Letters*, vol. 19, pp. 1-5, 2022.

[226] K. T. Shahid and I. D. Schizas, "Unsupervised hyperspectral unmixing via nonlinear autoencoders," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-13, 2022.

[227] Q. Lyu and X. Fu, "Identifiability-guaranteed simplex-structured post-nonlinear mixture learning via autoencoder," *IEEE Transactions on Signal Processing*, vol. 69, pp. 4921-4936, 2021.

[228] M. Tang, Y. Qu, and H. Qi, "Hyperspectral nonlinear unmixing via generative adversarial network," in *IGARSS 2020 - 2020 IEEE International Geoscience and Remote Sensing Symposium*, 2020, pp. 2404-2407.

[229] B. Rasti, B. Koirala, and P. Scheunders, "HapkeCNN: Blind nonlinear unmixing for intimate mixtures using hapke model and convolutional neural network," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1-15, 2022.

<figure><img src="image_20.png" /><figcaption>tral imagery.</figcaption></figure>

**Alexandre Zouaoui** received his M.Sc. degree in computer science from Télécom Paris, Paris, France, in 2019. He received his Ph.D. in applied mathematics at Université Grenoble Alpes, Grenoble, France, in 2024 on Sparse and Archetypal Decomposition Algorithms for Hyperspectral Images Restoration and Spectral Unmixing. His research interests include computer vision, interpretable machine learning and remote sensing image processing. In 2024, he joined Data Science Experts as a research engineer focusing on applications related to hyperspectral imagery.

<figure><img src="image_21.png" /></figure>

**Julien Mairal** received the graduate degree from Ecole Polytechnique, Palaiseau, France, in 2005, and the Ph.D. degree from Ecole Normale Superieure, Cachan, France, in 2010. After that, he joined the statistics department at UC Berkeley as a post-doctoral researcher. In 2012, he joined Inria, Grenoble, France, where he is currently a research director and head of the Thoth team. His research interests include machine learning, computer vision, mathematical optimization, and statistical image and signal processing. He received a starting grant and a consolidator grant from the European Research Council, respectively in 2016 and 2022. He was awarded the Cor Baayen prize in 2013, the IEEE PAMI young research award in 2017 and the test-of-time award at ICML 2019.

<figure><img src="image_22.png" /></figure>

**Jocelyn Chanussot (M'04-SM'04-F'12)** received the M.Sc. degree in electrical engineering from the Grenoble Institute of Technology (Grenoble INP), Grenoble, France, in 1995, and the Ph.D. degree from the Université de Savoie, Annecy, France, in 1998. From 1999 to 2023, he has been with Grenoble INP, where he was a Professor of signal and image processing. He is currently a Research Director with INRIA, Grenoble. His research interests include image analysis, hyperspectral remote sensing, data fusion, machine learning and artificial intelligence.

He has been a visiting scholar at Stanford University (USA), KTH (Sweden) and NUS (Singapore). Since 2013, he is an Adjunct Professor of the University of Iceland. In 2015-2017, he was a visiting professor at the University of California, Los Angeles (UCLA). He holds the AXA chair in remote sensing and is an Adjunct professor at the Chinese Academy of Sciences, Aerospace Information research Institute, Beijing. Dr. Chanussot is the founding President of IEEE Geoscience and Remote Sensing French chapter (2007-2010) which received the 2010 IEEE GRS-S Chapter Excellence Award. He has received multiple outstanding paper awards. He was the Vice-President of the IEEE Geoscience and Remote Sensing Society, in charge of meetings and symposia (2017-2019). He was the General Chair of the first IEEE GRSS Workshop on Hyperspectral Image and Signal Processing, Evolution in Remote sensing (WHISPERS). He was the Chair (2009-2011) and Cochair of the GRS Data Fusion Technical Committee (2005-2008). He was a member of the Machine Learning for Signal Processing Technical Committee of the IEEE Signal Processing Society (2006-2008) and the Program Chair of the IEEE International Workshop on Machine Learning for Signal Processing (2009). He is an Associate Editor for the IEEE Transactions on Geoscience and Remote Sensing, the IEEE Transactions on Image Processing and the Proceedings of the IEEE. He was the Editor-in-Chief of the IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing (2011-2015). In 2014 he served as a Guest Editor for the IEEE Signal Processing Magazine. He is a Fellow of the IEEE, an ELLIS Fellow, a Fellow of the Asia-Pacific Artificial Intelligence Association, a member of the Institut Universitaire de France (2012-2017) and a Highly Cited Researcher (Clarivate Analytics/Thomson Reuters, since 2018).

<figure><img src="image_23.png" /></figure>

**Behnood Rasti (M'12-SM'19)** received the B.Sc. and M.Sc. degrees in electronics and electrical engineering from the Electrical Engineering Department, University of Guilan, Rasht, Iran, in 2006 and 2009, respectively, and the Ph.D. degree in electrical and computer engineering from the University of Iceland, Reykjavik, Iceland, in 2014. He was a Valedictorian as an M.Sc. Student in 2009. In 2015 and 2016, he worked as a Post-Doctoral Researcher with the Electrical and Computer Engineering Department, University of Iceland. From 2016 to 2019,

he was a Lecturer with the Center of Engineering Technology and Applied Sciences, Department of Electrical and Computer Engineering, University of Iceland. Dr. Rasti was a Humboldt Research Fellow in 2020 and 2021. From 2022 to 2023, he was a Principal Research Associate with Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Dresden, Germany. He is currently a Senior Research Scientist with the Faculty of Electrical Engineering and Computer Science, Technische Universität Berlin. His research interests include signal and image processing, machine/deep learning, remote sensing, and artificial intelligence.

Dr. Rasti won the Doctoral Grant of the University of Iceland Research Fund "The Eimskip University Fund" and the "Alexander von Humboldt Research Fellowship Grant" in 2013 and 2019, respectively. He serves as an Associate Editor for the IEEE GEOSCIENCE AND REMOTE SENSING LETTERS (GRSL).