Tech Report: A Fast Multiscale Spatial Regularization for ... - arXiv

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Tech Report: A Fast Multiscale Spatial Regularization for Sparse Hyperspectral Unmixing

arXiv:1712.01770v1 [cs.CV] 5 Dec 2017

Ricardo Augusto Borsoi, Tales Imbiriba, Member, IEEE, José Carlos Moreira Bermudez, Senior Member, IEEE, Cédric Richard, Senior Member, IEEE

Abstract—Sparse hyperspectral unmixing from large spectral libraries has been considered to circumvent limitations of endmember extraction algorithms in many applications. This strategy often leads to ill-posed inverse problems, which can benefit from spatial regularization strategies. While existing spatial regularization methods improve the problem conditioning and promote piecewise smooth solutions, they lead to large nonsmooth optimization problems. Thus, efficiently introducing spatial context in the unmixing problem remains a challenge, and a necessity for many real world applications. In this paper, a novel multiscale spatial regularization approach for sparse unmixing is proposed. The method uses a signal-adaptive spatial multiscale decomposition based on superpixels to decompose the unmixing problem into two simpler problems, one in the approximation domain and another in the original domain. Simulation results using both synthetic and real data indicate that the proposed method can outperform state-of-the-art Total Variation-based algorithms with a computation time comparable to that of their unregularized counterparts. Index Terms—Hyperspectral data, sparse unmixing, spatial regularization, multiscale, superpixels.

I. I NTRODUCTION Hyperspectral unmixing is at the core of many remote sensing applications such as land use analysis, mineral detection, environment monitoring, and field surveillance [1], [2], where limited spatial resolution of hyperspectral devices often mixes the spectral contribution of different pure materials, named endmembers, in the scene [3]. The mixing process conceals crucial information relating the endmembers and their spatial disposition. Hyperspectral unmixing (HU) aims to solve this problem by separating the hyperspectral image into a collection of endmembers and their fractional abundances [4]. Notwithstanding the relevance of more complex aspects in modeling the mixing process [5], the simple linear mixing model (LMM) is recognized as an acceptable model for many real-world scenarios [6]. Furthermore, the explosive growth of remote sensing data [7] emphasizes the need for simple, fast and accurate unmixing strategies. The LMM [3] considers that an observed reflectance vector (a pixel) is a linear combination of endmembers, assuming the linear coefficients (abundances) to be proportions of the pure This work has been supported by the National Council for Scientific and Technological Development (CNPq). R.A. Borsoi, T. Imbiriba and J.C.M. Bermudez are with the Department of Electrical Engineering, Federal University of Santa Catarina, Florianópolis, SC, Brazil. e-mail: [email protected]; [email protected]; [email protected]. C. Richard is with the Université Côte d’Azur, Nice, France (e-mail: [email protected]), Lagrange Laboratory (CNRS, OCA). Manuscript received Month day, year; revised Month day, year.

spectra contributions. The proportion connotation imposes positivity and sum-to-one constraints over the abundance vectors, confining the pixels of a hyperspectral image to a simplex whose vertices are the endmembers. The convex geometry of the LMM endows the HU problem with interesting features, especially concerning endmember extraction techniques [6]. The problems of endmember extraction and unmixing are interrelated and addressing them jointly is not always trivial. For instance, most endmember extraction algorithms (EEA) rely on the convex geometry associated with the LMM [6], [8], and often assume either the presence of pure pixels [9], [10] or that the data are not heavily mixed [6], [11]. An interesting strategy to circumvent such issues is to model the observed pixel as a linear combination of a large library of endmembers previously estimated [4], [6]. In this case, the number of endmembers in a given scene is usually much smaller than the size of the spectral library. Hence, the unmixing problem becomes a sparse regression problem that consists of finding a small subset of the library endmembers which best represent all the pixels in the image. This problem is often efficiently solved through the use of sparsity promoting regularizations, resulting in the so-called sparse linear unmixing techniques. Although standard sparse unmixing techniques have shown good performance in many experiments [12], exploiting the spatial correlation between neighboring pixels can improve the unmixing performance [13]. Spatial information has been employed both for endmember extraction [14], [15] and as a regularization term in linear [16]–[20], nonlinear [21], [22], and sparse [23] unmixing solutions. Among the many spatial regularization approaches, Total Variation (TV) regularization deserves special mention since it promotes solutions that are spatially piecewise homogeneous without compromising sharp discontinuities between neighboring pixels. This homogeneity property concurs with the spatial correlation found in many hyperspectral unmixing applications [13]. Moreover, the TV regularization acts as a prior information which improves the conditioning of the underlying inverse problem [24]. The spatial regularization techniques employed so far for HU introduce interdependence among abundance solutions corresponding to different image pixels. This characteristic leads to intricate optimization problems and to a significant increase in computational load as compared to solutions that do not employ spatial regularization. In [23] the authors considered the TV regularization to tackle the sparse unmixing problem. This approach leads to a large non-smooth convex optimization problem, which is solved using the alternating direction method of multipliers (ADMM). The methodology

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used in [23] was later extended, leading to slightly different algorithms. For instance, in [25] TV regularization was considered to perform image deblurring jointly with the unmixing process. In [26] the authors considered second-order spatial derivatives in the regularization penalty while in [27] hyperspectral images corrupted by line strips and shot noise were considered. Spatial regularization strategies exploiting nonlocal redundancy in images were also considered in [28], [29], leading to even larger optimization problems. Despite the success of spatial regularization techniques in sparse unmixing, their adoption has come at the expense of a massive increase in computational cost. This is incompatible with recent demands to process, in a timely fashion, the vast amounts of remotely sensed data required by many real world applications [7], [30]. Thus fast unmixing strategies, that present low computational complexity without significantly compromising performance, must be sought. In this paper a novel multiscale spatial regularization approach for sparse unmixing is proposed. The proposed methodology leads to a fast algorithm, called Sparse unmixing via variable Splitting Augmented Lagrangian and Multiscale decomposition (SUnSAL-M), which promotes piecewise homogeneity in the estimated abundances without compromising sharp discontinuities among neighboring pixels. The proposed method uses a signal-adaptive spatial multiscale decomposition of the linear mixture model. The unmixing problem is decomposed into two different problems in distinct domains: one in an approximation scale representation based on superpixels [31], and another in the original image domain. Spatial contextual information of fractional abundances is initially provided by solving an unregularized sparse unmixing problem in the approximation scale. This information is then mapped back into the original image domain by means of an appropriately defined conjugate transformation of the multiscale decomposition, and is then used to efficiently introduce context-aware spatial regularity into the solution of the original unmixing problem in the form of a computationally simple regularization penalty. Simulation results with both synthetic and real data indicate that the proposed method performs similarly or better than the TV regularization [23], with a computational time comparable to that of the unregularized algorithm [12]. The main contributions of this paper include: 1) a novel methodology to decompose large-scale spatially regularized sparse HU into two simpler problems in different image domains. The two problems can then be solved sequentially at a reasonable computational cost; 2) the use of superpixels in a multiscale transformation, which leads to a simple and efficient strategy to deal with sharp discontinuities between neighboring pixels. The paper is organized as follows. Section II briefly reviews the sparse unmixing problem. In Section III, we introduce the proposed multiscale formulation for the sparse unmixing problem. Simulation results with synthetic and real data are presented in Section IV. Section V presents the conclusions.

II. S PARSE LINEAR UNMIXING Let Y ∈ RL×N denote the observed hyperspectral image with L bands and N pixels, and A ∈ RL×M denote a spectral library having M spectral signatures. Instead of extracting the endmembers directly from the hyperspectral image Y , sparse linear unmixing attempts to find an optimal subset of samples from the spectral library A that best represents all the mixed pixels in the image, namely, Y = AX + N ,

(1)

M ×N

where X ∈ R is the fractional abundance matrix, each column of which determines the composition of one image pixel as a linear combination of spectral samples from A, and N ∈ RL×N denotes the joint contribution of modeling errors and noise. The fractional abundance matrix X is frequently subject to physical constraints imposed to the model, such as the non-negativity (i.e. xi,j ≥ 0, ∀i, j, denoted by X ≥ 0) and the sum-to-one constraints (i.e. 1> X = 1> ). Since only a few of the spectral signatures of A are likely to contribute to the observed spectra of each pixel, the matrix X is usually sparse. In this setting, the unmixing problem can be formulated as a convex optimization problem as: b = arg min 1 kY − AXk2 + λkXk1,1 X F X 2 (2) subject to X ≥ 0 > 2 where kXkP F = trace{XX } is the Frobenius norm of X, kXk1,1 = i,j |xi,j |, and λ is a regularization parameter. The sum-to-one constraint is not considered in (2) because it has been shown to be unnecessary when formulating sparse HU as a sparse regression problem. Indeed, in this case, the nonnegativity constraint also enforces a generalized sum-to-one constraint [12]. The optimization problem (2) is usually solved with the alternating direction method of multipliers (ADMM), resulting in the SUnSAL algorithm [32].

A. Total variation for sparse spectral unmixing The SUnSAL algorithm is numerically efficient as it solves the unmixing problem independently for each pixel. However, it fails to account for the correlation between neighboring pixels that may exist in natural images. The addition of a Total Variation regularization penalty to the cost function (2) has been proposed in [23] to promote smooth transitions in the fractional abundance of a same endmember among neighboring pixels. The resulting convex optimization problem is: b = arg min 1 kY − AXk2 + λkXk1,1 + λTV T V (X) X F X 2 subject to X ≥ 0 (3) where λTV is a regularization parameter, and T V (X) is the Total Variation regularizer defined as: X kxi − xj k1 , T V (X) = {i,j}∈N

where N is the set of all pixel neighborhoods in the image. The optimization problem (3) can be solved with the ADMM,

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resulting in the SUnSAL-TV algorithm [23]. This approach yields better reconstruction results, specially in the presence of noise. Note that the huge dimensionality of X, the nonsmooth nature of the constraint and the need to explicitly consider the interdependencies between pixels when implementing the TV regularization make the resolution of (3) time consuming [23]. Even with the use of the ADMM, the computational burden introduced by the spatial regularization term makes the running time of SUnSAL-TV algorithm much higher than that of the SUnSAL algorithm [23]. This problem is emphasized by the huge amount of data that is now available. This recently motivated the development of fast HU algorithms. In the following, we introduce a new regularization procedure based on a multi-scale formulation that introduces spatial regularity into the abundance maps. This algorithm requires a minimum additional computational effort compared to SUnSAL. III. M ULTISCALE REGULARIZATION The use of information over multiple image scales has been explored for a variety of image processing tasks. It often results in considerable improvements, and has become standard practice in some fields, such as optical flow (OF) motion estimation [33], [34], where motion vectors are estimated at a coarse image scale and interpolated to be used as an initialization in the fine-scale motion estimation problem. This allows for the reliable estimation of large displacements, and is considered an important factor contributing to the good performance of modern OF estimation techniques [35]. Besides its ubiquitous presence in motion estimation techniques, multi-scale approaches encounter wide applicability in other areas of image processing as well, most notably in the field of image deblurring or deconvolution. Most deconvolution approaches use image information at coarser scales to construct a reference map. This map increases the regularization strength at the image regions where the coarse estimate is significantly smooth, resulting in smooth images with sharp transitions. For instance, [36]–[40] use local priors for single image deconvolution based on the blurred image to reduce ringing effects. The gradients of both the blurred and reconstructed images are forced to be similar in smooth image regions. The regularization strength is adjusted according to local image features estimated from the blurred observation, or from images previously reconstructed at coarser scales. These methods often include an iterative reconstruction procedure, where the image estimation is performed graduately, one scale at a time, using edge information from the previous scale [41]. Note, however, that these techniques operate by adapting constraints defined on the image gradient. Thus their computational complexity is driven by Tikhnov or Total Variation regularizers, as they introduce coupling between pixels in the optimization problem. To overcome this issue, we propose a novel spatial regularization methodology that is computationally more efficient than TV and yields similar results. It promotes spatial information without directly introducing any coupling between pixels in the optimization problem. This greatly simplifies problem

resolution. This is done by performing HU twice, first in an approximation scale and then in the original image scale. A. Problem formulation The proposed spatially regularized unmixing scheme consists of two steps. First, we transform the original image from the original domain (D) to an approximation (coarse) scale (C) to extract the most relevant inter-pixel contextual information. Then, pixels at the coarse scale are unmixed independently from each other. Next, we apply a conjugate transformation to the abundance estimates obtained at the coarse scale to convert the coarse estimate back to the original image domain. This procedure yields an accurate estimate of the low-level image structures, which is then used to regularize the unmixing process applied to the original image in order to promote the spatial dependency between neighboring pixels. Consider a linear operator W ∈ RN ×K , K < N that implements a spatial transformation of both the HI and the abundance map to the approximation domain. Then, YC = Y W ;

XC = XW ,

(4)

where Y C ∈ RL×K and XC ∈ RM ×K are the coarse approximations of the original image Y and of the abundance matrix X, respectively. A possible choice for W might be a wavelet transform employing the first K approximation scales of the wavelet decomposition of Y . However, the wavelet transform is feature-agnostic. It does not distinguish between pixels in perceptually different image regions. Its application may result in blurred image edges. Instead, we shall consider a signal-dependent transformation, that is, W ≡ W (Y ), which groups pixels into perceptually meaningful regions (not necessarily uniform), preserving image contours and leading to sharp transitions. Multiplying (1) by W from the right, the unmixing problem can be re-cast into the approximation domain. The resulting unmixing problem is as follows: bC = arg min 1 kY C − AXC k2 + λC kXC k1,1 X F XC 2 (5) subject to XC ≥ 0 . bC to regularize the We shall now use the estimated X original unmixing problem. To this end, we define a conjugate transform W ∗ , which converts images from the approximation domain C back to the original image domain D: bD = X bC W ∗ , X

(6)

bD ∈ RM ×N is the smooth approximation of the where X abundances in the original image domain, which captures correlations between neighboring pixels, and W ∗ ∈ RK×N . Note that transformation W is generally not invertible, that is, W W ∗ 6= I. bD to regularFinally, we use the coarse abundance matrix X ize the original problem (2) as follows: bD − Xk2 b = arg min 1 kY − AXk2 + λkXk1,1 + β kX X F F X 2 2 subject to X ≥ 0 (7)

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where β is a regularization parameter. Note that the proposed regularization process does not require to consider explicitly the dependences between pairs of pixels, unlike the Total Variation or Tikhnov regularization schema. This results in a computationally efficient procedure.

grid that lie inside the corresponding superpixel region. The successive application of both transforms, W W ∗ effectively consists in averaging all pixels inside each superpixel of the input image. The superpixel decomposition of the Cuprite hyperspectral image using the SLIC algorithm is illustrated in Figure 1.

B. Superpixel for multiscale transformation Choosing transformation W approprietly is of paramount importance for the proposed method to achieve a good reconstruction accuracy. Reiterating the objectives of this transform, they can be summarized as 1) grouping image pixels that are semantically similar, that is, that belong to homogeneous regions, and 2) preserves image contours by not grouping pixels that belong to different image structures or features. Additionally, it must be computationally efficient. It follows that the decomposition of an image into regions called superpixels satisfies the desired criteria. Superpixel algorithms group image pixels into regions with contextually similar spatial information [31]. Images are decomposed into a set of contiguous homogeneous regions, whose size and regularity can be controlled by appropriate parameters setting. Superpixels have recently been widely applied to hyperspectral imaging tasks, including classification [42], [43], segmentation [44] and endmember detection [45]. Superpixels capture image redundancies, which helps in recognizing meaningful image features in the presence of noise and reduces the complexity of subsequent image processing tasks. Although several families of algorithms have been proposed to perform superpixel decomposition of an image, fast algorithms reproducing state-of-the-art results are available. This is the case of the Simple Linear Iterative Clustering (SLIC) algorithm [31]. The SLIC algorithm is an adaptation of the k-means algorithm applied to the computation of superpixels. It considers a reduced search space to limit the computational complexity, and a properly defined metric that balances spectral and spatial contributions. The clusters computed with k-means algorithm, which constitute the desired superpixels, are initialized almost uniformly at low-gradient neighborhoods of the image to reduce the influence of noise. The number of clusters Nc is determined as a function of the average superpixel size, which is supplied by the user. A normalized distance function considering both spatial and spectral similarity among pixels is used in the clustering process. The contributions of spatial and spectral components in the metric are balanced via a regularity parameter γ. Parameter γ can be adjusted by the user to obtain either more compact superpixels (with lower area to perimeter ratio) when the spatial distance is heavily weighted, or a tighter adherence to image contours (with more irregular shapes) when spectral distance predominates. We choose the transformation W to be the superpixel decomposition of the image. More precisely, Y W computes the superpixel decomposition of the image Y , and returns the average of all pixels inside each superpixel region. Note that the resulting pixels do not lie on a uniform sampling grid. The conjugate transform, Y C W ∗ , takes each superpixel in Y C and attributes its value to all pixels of the uniform image sampling

Image (bands 50, 80 and 100)

Superpixels

Figure 1: Superpixel decomposition of a section of the Cuprite image for bands 50, 80 and 100 using the algorithm in [31] with Nc = 5 and γ = 0.005. An example of the computation of W ∗ for an image with 5 pixels and 3 superpixels (N = 5 and K = 3), where pixels 1 and 2 belong to superpixel 1, pixel 3 belongs to superpixel 3 and pixels 4 and 5 belong to superpixel 2 is shown in Equation (8). C. The SUnSAL-M algorithm This section presents the proposed sparse Unmixing via variable Splitting Augmented Lagrangian and Multiscale decomposition (SUnSAL-M) algorithm shown in Algorithm 1. The algorithm needs as inputs the hyperspectral image (HI) Y , the spectral library A, the SLIC and regularization parameters, b and returns the estimated fractional abundance matrix X. Algorithm 1: Sparse unmixing via variable Splitting Augmented Lagrangian and Multiscale decomposition (SUnSAL-M) Input : The HI Y , the spectral library A, the SLIC parameters Nc and γ, and the regularization parameters λC , λ, and β. b Output: The estimated abundance matrix X. 1 Compute the superpixel decomposition, Y C of the HI Y using the SLIC algorithm [31]; bC by solving (5); 2 Find X bD using (6); 3 Compute X b 4 Find X by solving (7); b 5 return X; The algorithm starts (line 1) by performing the superpixel decomposition of the original HI Y using the SLIC algorithm.

5



|  xC1 | |

| xC1 |

| xC3 | {z XD

| xC2 |

  | | xC2  =  xC1 | | } |

This leads to the coarse image version Y C . Then (line 2), b C of the coarse approximation X C is estimated the estimate X solving the problem in (5). This problem is a standard sparse unmixing problem and can be solved using the SUnSAL algorithm proposed in [12]. In line 3, the conjugate superpixel b C (see (6)) to obtain the transformation W ∗ is applied to X b smooth abundance approximation XD in the original image domain. In line 4, the estimate of the fractional abundance matrix b is determined by solving the optimization problem (7). This X problem can be efficiently solved using a modified version of the SUnSAL algorithm (see Appendix A). Finally, the b in line 5. algorithm returns X

| xC2 | {z XC

  | 1 xC3  ·  0 | 0 } |

A. Synthetic data sets Three synthetic images with spatial correlation were used for the experiments described in this section. Data cube DC0 (called DC1 in [23]) has 75×75 pixels and contains five endmembers selected from library A1 . Its fractional abundances map is diplayed in Fig. 2. It is composed of square regions distributed uniformly over a background in five rows, containing regions with pure pixels (first row) and mixtures of between two and five endmembers (second to fifth rows, respectively). The background pixels are also mixtures of the same five endmembers, with fractional abundances given by 0.1149, 0.0741, 0.2003, 0.2055 and 0.4051. Data cube DC1 has 75×75 pixels and contains five endmembers selected from library A1 . Its fractional abundances are displayed in Fig. 3. The abundance maps for DC1 were

0 0 0 1 1 0 {z

W∗

 0 1  0 }

(8)

Table I: SRE results for data cubes DC0, DC1 and DC2 unmixing using SUnSAL, SUnSAL-TV and SUnSAL-M. DC0 data cube SNR

SUnSAL

SUnSAL-TV

SUnSAL-M

20dB

4.54dB λ = 0.7

30dB

8.91dB λ = 0.1

40dB

13.78dB λ = 0.01

9.42dB λ = 0.05 λTV = 0.05 14.44dB λ = 0.007 λTV = 0.01 22.10dB λ = 0.001 λTV = 0.003

11.35dB λC = 0.03, λ = 0.1 β = 30 15.73dB λC = 0.007, λ = 0.05 β = 10 22.34dB λC = 0.001, λ = 0.01 β = 10

IV. R ESULTS We compare now the performances of the proposed SUnSAL-M, the Total Variation (SUnSAL-TV) and the unregularized (SUnSAL) algorithms, both in terms of reconstruction error and computational complexity. The selection of these algorithms comes naturally since the three share the same sparse regression formulation, and SUnSAL-TV is considered as a state-of-the-art library-based sparse unmixing algorithm. We considered a synthetic library A1 ∈ R224×240 generated by selecting a subset of 240 materials from the USGS library, which contains spectral signatures with reflectance values in 224 bands distributed uniformly between 0.4µm and 2.5µm. The subset of materials was selected by removing spectral signatures such that the angle between any pair of signatures was at least 4.44 degrees. The parameters Nc and γ (average size and spatial regularity) of the SLIC superpixel algorithm used for computing W were selected based on the visual conformation of the resulting superpixels with the features of the observed hyperspectral image Y . As shown in the sequel, it appears to us that the performance of SUnSAL-M algorithm is not particularly sensitive to parameter values.

1 0 0

DC1 data cube SNR

SUnSAL

SUnSAL-TV

SUnSAL-M

20dB

4.57dB λ = 0.5

30dB

8.81dB λ = 0.1

40dB

13.30dB λ = 0.01

8.61dB λ = 0.05 λTV = 0.03 12.16dB λ = 0.03 λTV = 0.007 16.89dB λ = 0.003 λTV = 0.003

9.83dB λC = 0.03, λ = 0.1 β = 10 13.31dB λC = 0.007, λ = 0.05 β=3 17.52dB λC = 0.001, λ = 0.01 β = 0.3

DC2 data cube SNR

SUnSAL

SUnSAL-TV

SUnSAL-M

20dB

4.27dB λ = 0.1

30dB

10.48dB λ = 0.01

40dB

17.93dB λ = 0.005

11.61dB λ = 0.01 λTV = 0.03 17.97dB λ = 0.005 λTV = 0.007 21.15dB λ = 0.001 λTV = 0.001

14.88dB λC = 0.007, λ = 0.1 β = 10 18.46dB λC = 0.003, λ = 0.03 β=3 21.00dB λC = 0.001, λ = 0.005 β = 0.3

generated by considering the fractional abundances of each pixel in data cube DC0 as mean vectors of a Dirichlet distribution, from which the new abundance maps were sampled. This extends data cube DC0 to a more realistic scenario in which a slight spatial variability is introduced in the abundance maps while retaining spatial correlation and steep transitions between different regions. Data cube DC2 has 100×100 pixels and contains nine signatures selected from library A1 . The abundance maps are piecewise smooth images showing spatial correlation and steep transitions. They were sampled according to a Dirichlet

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1 0.8 0.6 0.4 0.2 0

Endmember 1

Endmember 2

Endmember 3

Endmember 4

Endmember 5

Figure 2: True fractional abundances of the endmembers 1-5 used in the DC0 data cube.

1 0.8 0.6 0.4 0.2 0

Endmember 1

Endmember 2

Endmember 3

Endmember 4

Endmember 5

Figure 3: True fractional abundances of the endmembers 1-5 used in the DC1 data cube.

1 0.8 0.6 0.4 0.2 0

Endmember 1

Endmember 2

Endmember 4

Endmember 7

Endmember 9

Figure 4: True fractional abundances of the 1st, 2nd, 4th, 7th, and 9th endmembers used in the DC2 data cube. distribution centered at a Gaussian random field1 [46]. This allowed us to introduce a slight spatial variability between the pixels. The true abundance maps for the 1st, 2nd, 4th, 7th, and 9th endmembers are displayed in Fig. 4. For all datacubes, the generated hyperspectral images were contaminated by white Gaussian noise, with signal-to-noise ratios (SNR) of 20, 30, and 40dB. The quality of the reconstruction of the spectral mixtures was evaluated using the signal to reconstruction error, defined as: [12]   E{kXk2F } SRE = 10 log10 . (9) b 2} E{kX − Xk F The parameters of the algorithms used in the simulations with synthetic data are shown in Table I. They were selected by testing all possible combinations of values in a finite subset of a determined range. For instance, the parameters λ (SUnSAL and SUnSAL-TV) and λC (SUnSAL-M) were searched in the range [0.0001, 0.05], λTV (SUnSAL-TV) and λ (SUnSAL-M) were searched from the interval [0.001, 0.1], and β (SUnSAL-M) was searched in the range [0.007, 30]. 1 Generated using the code in http://www.ehu.es/ccwintco/index.php/Hyperspectral_Imagery_Synthesis_tools_for_MATLAB.

Finally, the parameters leading to the best SRE results for each method were selected. The SRE achieved by the SUnSAL, SUnSAL-TV and SUnSAL-M are displayed in Table I for all tested SNR values. Samples of the reconstructed abundance maps for the three data cubes and all SNRs are shown in Figs. 5, 6 and 7 for a qualitative comparison. The computational complexity of the algorithms was evaluated through their execution time, which are displayed in Table II. The algorithms were implemented in Matlab on a desktop computer equipped with an Intel Core I7 processor with 4.2Ghz and 16Gb RAM. SUnSAL, SUnSAL-TV and SLIC were implemented using the codes made available by the respective authors. 1) Discussion: It can be seen from Table I that the SRE performance of the SUnSAL-M algorithm for all three data cubes was similar to that of SUnSAL-TV for the highest SNR, and significantly better for lower SNR values, when the use of spatial regularization proves to be most important. The SRE improvements that were obtained range from 0.5 to 1.2 dB for a 30 dB SNR, and from 1.2 to 3.2 dB for a 20 dB SNR. Figs. 5, 6 and 7 show the abundance maps estimated by the

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Table II: Execution time (in seconds) of the unmixing algorithms, averaged for all SNR values considered SUnSAL

SUnSAL-TV

SUnSAL-M

3.39 s 2.66 s 5.78 s 184.8 s

81.86 s 79.16 s 130.37 s 1145.8 s

3.24 s 2.53 s 4.95 s 101.5 s

1 0.8

DC0 DC1 DC2 Real Image

0.6 0.4 0.2 0 1

0.8

three unmixing methods for the 5th endmember selected from data cubes DC0 and DC1, and for the 1st endmember selected from data cube DC2, for different noise levels. It can be seen that the use of the spatial regularization terms promoted a notable improvement in the unmixing performance over the unregularized SUnSAL algorithm for low SNR values. The results of the SUnSAL-M algorithm were qualitatively similar to those the SUnSAL-TV algorithm for high SNR. For lower SNR values, SUnSAL-M performed significantly better, especially for the data cube DC2, where the resulting abundance maps are much closer to the ground truth than those estimated by the SUnSAL-TV algorithm. In terms of computational cost, SUnSAL-M performed significantly better than SUnSAL-TV, with execution times comparable to those of SUnSAL, and 25 to 31 times smaller than those of SUnSAL-TV. Moreover, the average execution time was even smaller than that of SUnSAL. This happened because introducing spatial regularization generally improves the conditioning of the inverse problem, resulting in a faster convergence. These results illustrate the effectiveness of the proposed regularization method.

0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0

Figure 6: Abundance maps estimated by the different unmixing methods for the 5th endmember of data cube DC1. From left to right: SUnSAL, SUnSAL-TV and SUnSAL-M. From top to bottom: SNR of 20, 30 and 40dB.

1 0.8 1

0.6 0.8

0.4 0.6

0.2 0.4 0.2 0 1

0 1 0.8 0.6

0.8 0.4 0.6 0.2 0.4 0.2 0 1

0 1 0.8 0.6

0.8 0.4 0.6 0.2 0.4 0 0.2 0

Figure 5: Abundance maps estimated by the different unmixing methods for the 5th endmember of data cube DC0. From left to right: SUnSAL, SUnSAL-TV and SUnSAL-M. From top to bottom: SNR of 20, 30 and 40dB.

Figure 7: Abundance maps estimated by the different unmixing methods for the 5th endmember of data cube DC2. From left to right: SUnSAL, SUnSAL-TV and SUnSAL-M. From top to bottom: SNR of 20, 30 and 40dB.

8

16

14

14 SRE [dB]

16

12 10

8 10 0

10 -2

λC

10 -4

λT V

β

10 -4

10 -4

10 -2

10 0

λ

18

SRE [dB]

16 14 12 10 10 10 0

8 6 SLIC size

4

10 -5

SLIC regularity

Figure 10: SUnSAL-M SRE as a function of SLIC parameters N/Nc and γ, with the regularization parameters fixed at the values in Table I.

were selected empirically for SUnSAL-M, and set identically to those reported in [23] for SUnSAL and SUnSAL-TV. Table III: Parameters used in the Cuprite image simulations. SUnSAL

14

14

λ = 10−3

11 -100 -50 0 50 100 Percent change about the optimum parameter value [%]

10 -2

Figure 9: SRE as a function of the regularization parameters for SUnSAL-M (left) and SUnSAL-TV (right). The parameter λ for SUnSAL-M was fixed at its optimal value.

15

12

10 2

10 0

10 -2

15

λC λ β

10

10 0

16

13

12

8

16

SRE [dB]

SRE [dB]

We analyze now the sensitivity of the SUnSAL-M and SUnSAL-TV SRE performances to variations of the their respective parameters. For SUnSAL-M, we consider the regularization and SLIC parameters separately in order to provide a better comparison with SUnSAL-TV.2 We initially set all regularization parameters equal to their optimal values in Table I. Then, we varied one parameter at a time within a range from −80% to +80% of its optimal value. Fig. 8 presents the resulting SRE obtained by varying each parameter for both SUnSAL-M (left) and SUnSAL-TV (right). It can be seen that small variations of the parameter around the optimal values do not significantly affect the SRE of both methods. Furthermore, the SRE obtained by the proposed SUnSAL-M is consistently higher than that of the SUnSALTV algorithm for the whole parameter range. Given the very low sensitivity of the SUnSAL-M SRE to changes in λ, we fixed this parameter at its optimal value to plot the SRE variation for a range of values for λC and β. This plot is compared in Fig. 9 with that of SUnSAL-TV SRE for a range of values for parameters λ and λT V . Like in Fig. 8, it can be seen that SUnSAL-M yields consistently higher SRE values when compared to SUnSAL-TV. To evaluate the sensitivity of the SUnSAL-M performance to the SLIC parameter values, Fig. 10 shows the SRE as a function of the average superpixel size (N/Nc ) and regularity parameter (γ), with the regularization parameters λC , λ and β fixed at their optimal values. It can be seen that the SRE performance is not significantly sensitive to the SLIC parameters, as the surface is relatively flat and retains a large SRE for a wide range of parameter values.

SRE [dB]

B. Sensitivity analysis

SUnSAL-TV λ = 10−3 λTV = 10−3

SUnSAL-M λC = 0.001, λ = 0.001 β=3

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Figure 8: SRE variation due to relative changes in each parameter value about its optimal value for SUnSAL-M (left) and SUnSAL-TV (right).

C. Simulations with real image In this experiment, we consider a well-known region of the Cuprite data set with 250×191 pixels. Spectral bands presenting water absorption and low SNR were removed from the image, resulting in 188 spectral bands. The spectral signatures from the USGS library with the corresponding bands removed were used as the spectral library A ∈ R188×498 . The parameters of the algorithms are shown in Table III. They 2 For conciseness, we present only the results for the DC0 data cube with a 30dB SNR. The results for the remaining data cubes and SNRs are included in Appendix B, and corroborate the conclusions presented in this section.

Since the true abundance maps are unavailable for this hyperspectral image, we make a qualitative assessment of the recovered abundance maps based on reference maps of minerals known to be present in prominent fashion in the Cuprite mining district3 . A qualitative comparison of the fractional abundance maps of three dominant materials (Alunite, Buddingtonite, and Chalcedony) estimated using the three algorithms is shown in Fig. 11. Although all algorithms estimated considerable abundances for the materials known to be present in the scene, the use of spatial regularization resulted in more spatially consistent estimates, with less outliers resulting from the influence of measurement noise. This can be observed in the Buddingtonite abundance maps, where the SUnSAL results contain regions only sparsely filled with non-zero abundances, whereas SUnSAL-TV and SUnSAL-M provided more homogeneous estimates of the abundances in those regions where the material is present. 3 http://speclab.cr.usgs.gov/cuprite95.tgif.2.2um_map.gif

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Figure 11: Fractional abundance maps estimated for the Cuprite image. From left to right: Alunite, Buddingtonite, and Chalcedony. From top to bottom: SUnSAL, SUnSAL-TV and SUnSAL-M.

Although the unmixing results for SUnSAL-TV and SUnSAL-M were similar, it can be observed that the TV regularization tends to yield an over-smooth visual effect. This is not observed in the results using SUnSAL-M, which produces spatially consistent abundance maps without compromising the fine variability and the intricate structures in the image. These results again indicate the effectiveness of the proposed spatial regularization. The computational times are shown in Table II, and illustrate again the considerably lower complexity of SUnSAL-M when compared to SUnSAL-TV. It also runs significantly faster than the SUnSAL algorithm, certainly due to the faster convergence rate achieved with the use of proposed regularization.

data showed that the proposed method outperforms both, SUnSAL and SUnSAL-TV, in almost all scenarios, with Execution Time comparable to the SUnSAL. In fact, the proposed method solved the problems 10 to 31 times faster then the SUnSALTV depending on factors such as the number of pixels, the number of bands, and problem conditioning. Simulations also showed that the improvement in the problem conditioning, when using the proposed methodology, provided Execution Times that can be even smaller than the SUnSAL algorithm.

V. C ONCLUSIONS

min f1 (x) + f2 (Gx) ,

In this paper, we presented a novel multiscale methodology to couple with the spatial regularization of sparse HU problems. The presented approach allowed one to decompose the spatially regularized unmixing problem into two simple, lowcost problems in different domains. Although many domain transformation could be considered with the proposed strategy, the superpixel decomposition captured the spatial contextual information of the fractional abundances at reasonable computational cost. Simulation results with both synthetic and real

where f1 : Rn → R+ ∪ {∞} and f2 : Rp → R+ ∪ {∞} are closed, proper and convex functions, and G has full column rank. The Alternating Direction Method of Multipliers (ADMM) method decomposes a problem in the form (10) into a sequence of simpler problems, which can be solved efficiently [47]. The optimization problem (7) can be written in the equivalent form β b 1 2 min kY − AXk2F + kX D − XkF + λkXk1,1 + ι+ (X) , X 2 2

A PPENDIX A S OLUTION TO THE OPTIMIZATION SUBPROBLEM Consider an unconstrained optimization problem of the form x

(10)

10

m×N where ι+ (·) is the indicator function of the set R+ , that is, ι+ (X) = 0 if X ≥ 0 and ι+ (X) = ∞ otherwise. We can select the functions f1 , f2 and G in (10) as

1 β b 2 kY − AXk2F + kX D − XkF 2 2 f2 ≡ λkXk1,1 + ι+ (X)

f1 ≡

(11)

G≡I.

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Algorithm 2: ADMM method for solving (7) Input : The HI Y , the spectral library A, regularization parameters λ and β, parameter µ > 0 and matrices U 0 , V 0 ∈ Rm×N . b Output: The estimated abundance matrix X. 1 Set i = 0 ; 2 while stopping criterion is not satisfied do b ; 3 Ω = A> Y + µ(U i + V i ) + β X
−1 D > 4 X i+1 = A A + (µ + β)I Ω; 5 U i+1 = max{0, soft(X i+1 − V i , λ/µ)} ; 6 V i+1 = V i − (X i+1 − U i+1 ) ; 7 i=i+1 ; 8 end b = X i+1 ; 9 return X

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The ADMM method can then be used to solve (7), with the resulting procedure detailed in Algorithm 2 [32], [47], where soft denotes the component-wise soft thresholding operator soft(y, τ ) = sign(y) max{|y| − τ, 0} [48].

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The simulations discussed in Section IV.B in the manuscript are replicated here for all datasets DC0, DC1 and DC2, and all SNR values 20, 30 and 40dB.

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A PPENDIX B S UPPLEMENTARY S ENSIBILITY S IMULATIONS

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A. DC0 Figures 12 to 14 presents the sensibility results for the DC0 synthetic dataset considering SNRs of 20, 30 and 40dB respectively. The results corroborate the discussion presented in IV.B of the manuscript. B. DC1 Figures 15 to 17 presents the sensibility results for the DC1 synthetic dataset considering SNRs of 20, 30 and 40dB respectively. The results corroborate the discussion presented in IV.B of the manuscript.

Figure 12: Sensibility of the SRE with respect to variations in the parameter values around their optimal values for DC0 and SNR of 20dB. SRE for the regularization parameters for the SUnSAL-M (a) and SUnSAL-TV (b) algorithms. 2D plot of the SRE as a function of the regularization parameters for the SUnSAL-M (c) and SUnSAL-TV (d) algorithms (the parameter λ for the SUnSAL-M algorithm was fixed at its optimal value). (e) SRE as a function of the SLIC parameters N/Nc and γ (the regularization parameters were fixed at the values in Table I of the manuscript).

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Figure 13: Sensibility of the SRE with respect to variations in the parameter values around their optimal values for DC0 and SNR of 30dB. SRE for the regularization parameters for the SUnSAL-M (a) and SUnSAL-TV (b) algorithms. 2D plot of the SRE as a function of the regularization parameters for the SUnSAL-M (c) and SUnSAL-TV (d) algorithms (the parameter λ for the SUnSAL-M algorithm was fixed at its optimal value). (e) SRE as a function of the SLIC parameters N/Nc and γ (the regularization parameters were fixed at the values in Table I of the manuscript).

Figure 14: Sensibility of the SRE with respect to variations in the parameter values around their optimal values for DC0 and SNR of 40dB. SRE for the regularization parameters for the SUnSAL-M (a) and SUnSAL-TV (b) algorithms. 2D plot of the SRE as a function of the regularization parameters for the SUnSAL-M (c) and SUnSAL-TV (d) algorithms (the parameter λ for the SUnSAL-M algorithm was fixed at its optimal value). (e) SRE as a function of the SLIC parameters N/Nc and γ (the regularization parameters were fixed at the values in Table I of the manuscript).

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Figure 15: Sensibility of the SRE with respect to variations in the parameter values around their optimal values for DC1 and SNR of 20dB. SRE for the regularization parameters for the SUnSAL-M (a) and SUnSAL-TV (b) algorithms. 2D plot of the SRE as a function of the regularization parameters for the SUnSAL-M (c) and SUnSAL-TV (d) algorithms (the parameter λ for the SUnSAL-M algorithm was fixed at its optimal value). (e) SRE as a function of the SLIC parameters N/Nc and γ (the regularization parameters were fixed at the values in Table I of the manuscript).

Figure 16: Sensibility of the SRE with respect to variations in the parameter values around their optimal values for DC1 and SNR of 30dB. SRE for the regularization parameters for the SUnSAL-M (a) and SUnSAL-TV (b) algorithms. 2D plot of the SRE as a function of the regularization parameters for the SUnSAL-M (c) and SUnSAL-TV (d) algorithms (the parameter λ for the SUnSAL-M algorithm was fixed at its optimal value). (e) SRE as a function of the SLIC parameters N/Nc and γ (the regularization parameters were fixed at the values in Table I of the manuscript).

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Figure 17: Sensibility of the SRE with respect to variations in the parameter values around their optimal values for DC1 and SNR of 40dB. SRE for the regularization parameters for the SUnSAL-M (a) and SUnSAL-TV (b) algorithms. 2D plot of the SRE as a function of the regularization parameters for the SUnSAL-M (c) and SUnSAL-TV (d) algorithms (the parameter λ for the SUnSAL-M algorithm was fixed at its optimal value). (e) SRE as a function of the SLIC parameters N/Nc and γ (the regularization parameters were fixed at the values in Table I of the manuscript).

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C. DC2 Figures 18 to 20 presents the sensibility results for the DC2 synthetic dataset considering SNRs of 20, 30 and 40dB respectively. The results corroborate the discussion presented in IV.B of the manuscript.

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Figure 18: Sensibility of the SRE with respect to variations in the parameter values around their optimal values for DC2 and SNR of 20dB. SRE for the regularization parameters for the SUnSAL-M (a) and SUnSAL-TV (b) algorithms. 2D plot of the SRE as a function of the regularization parameters for the SUnSAL-M (c) and SUnSAL-TV (d) algorithms (the parameter λ for the SUnSAL-M algorithm was fixed at its optimal value). (e) SRE as a function of the SLIC parameters N/Nc and γ (the regularization parameters were fixed at the values in Table I of the manuscript). R EFERENCES [1] 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, 1(2):6–36, 2013. [2] D. Manolakis. Detection algorithms for hyperspectral imaging applications. IEEE Signal Processing Magazine, 19(1):29–43, 2002. [3] N. Keshava and J. F. Mustard. Spectral unmixing. IEEE Signal Processing Magazine, 19(1):44–57, 2002. [4] 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, 5(2):354– 379, 2012. [5] N. Dobigeon, J.-Y. 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, 31(1):82–94, Jan 2014.

Figure 19: Sensibility of the SRE with respect to variations in the parameter values around their optimal values for DC2 and SNR of 30dB. SRE for the regularization parameters for the SUnSAL-M (a) and SUnSAL-TV (b) algorithms. 2D plot of the SRE as a function of the regularization parameters for the SUnSAL-M (c) and SUnSAL-TV (d) algorithms (the parameter λ for the SUnSAL-M algorithm was fixed at its optimal value). (e) SRE as a function of the SLIC parameters N/Nc and γ (the regularization parameters were fixed at the values in Table I of the manuscript).

[6] W.-K. Ma, J. M. Bioucas-Dias, Tsung-Han Chan, N. Gillis, P. Gader, A. J. Plaza, A. Ambikapathi, and C.-Y. Chi. A signal processing perspective on hyperspectral unmixing: Insights from remote sensing. IEEE Signal Processing Magazine, 31(1):67–81, January 2014. [7] Yan Ma, Haiping Wu, Lizhe Wang, Bormin Huang, Rajiv Ranjan, Albert Zomaya, and Wei Jie. Remote sensing big data computing: Challenges and opportunities. Future Generation Computer Systems, 51:47–60, 2015. [8] T. Imbiriba, J. C. M. Bermudez, C. Richard, and J.-Y. Tourneret. Nonparametric detection of nonlinearly mixed pixels and endmember estimation in hyperspectral images. IEEE Transactions on Image Processing, 25(3):1136–1151, March 2016. [9] Hsuan Ren and Chein-I Chang. Automatic spectral target recognition in hyperspectral imagery. IEEE Transactions on Aerospace and Electronic Systems, 39(4):1232–1249, 2003. [10] Tsung-Han Chan, Wing-Kin Ma, ArulMurugan Ambikapathi, and Chong-Yung Chi. A simplex volume maximization framework for hyperspectral endmember extraction. IEEE Transactions on Geoscience and Remote Sensing, 49(11):4177–4193, 2011. [11] Tsung-Han Chan, Chong-Yung Chi, Yu-Min Huang, and Wing-Kin Ma. A convex analysis-based minimum-volume enclosing simplex algorithm for hyperspectral unmixing. IEEE Transactions on Signal Processing, 57(11):4418–4432, 2009. [12] Marian-Daniel Iordache, José M Bioucas-Dias, and Antonio Plaza. Sparse unmixing of hyperspectral data. IEEE Transactions on Geoscience and Remote Sensing, 49(6):2014–2039, 2011.

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Figure 20: Sensibility of the SRE with respect to variations in the parameter values around their optimal values for DC2 and SNR of 40dB. SRE for the regularization parameters for the SUnSAL-M (a) and SUnSAL-TV (b) algorithms. 2D plot of the SRE as a function of the regularization parameters for the SUnSAL-M (c) and SUnSAL-TV (d) algorithms (the parameter λ for the SUnSAL-M algorithm was fixed at its optimal value). (e) SRE as a function of the SLIC parameters N/Nc and γ (the regularization parameters were fixed at the values in Table I of the manuscript).

[13] Chen Shi and Le Wang. Incorporating spatial information in spectral unmixing: A review. Remote Sensing of Environment, 149:70–87, 2014. [14] Maciel Zortea and Antonio Plaza. Spatial preprocessing for endmember extraction. IEEE Transactions on Geoscience and Remote Sensing, 47(8):2679–2693, 2009. [15] Maria C Torres-Madronero and Miguel Velez-Reyes. Integrating spatial information in unsupervised unmixing of hyperspectral imagery using multiscale representation. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 7(6):1985–1993, 2014. [16] A Zymnis, S-J Kim, J Skaf, M Parente, and S Boyd. Hyperspectral image unmixing via alternating projected subgradients. In Signals, Systems and Computers, 2007. ACSSC 2007. Conference Record of the Forty-First Asilomar Conference on, pages 1164–1168. IEEE, 2007. [17] Olivier Eches, Nicolas Dobigeon, and Jean-Yves Tourneret. Enhancing hyperspectral image unmixing with spatial correlations. IEEE Transactions on Geoscience and Remote Sensing, 49(11):4239–4247, 2011. [18] Peng Chen, James DB Nelson, and Jean-Yves Tourneret. Toward a sparse bayesian markov random field approach to hyperspectral unmixing and classification. IEEE Transactions on Image Processing, 26(1):426–438, 2017. [19] Yoann Altmann, Steve McLaughlin, and Alfred Hero. Robust linear spectral unmixing using anomaly detection. IEEE Transactions on Computational Imaging, 1(2):74–85, 2015. [20] Rita Ammanouil, André Ferrari, Cédric Richard, and David Mary. Blind and fully constrained unmixing of hyperspectral images. IEEE Transactions on Image Processing, 23(12):5510–5518, 2014.

[21] Jie Chen, Cédric Richard, and Paul Honeine. Nonlinear estimation of material abundances in hyperspectral images with `1 -norm spatial regularization. IEEE Transactions on Geoscience and Remote Sensing, 52(5):2654–2665, 2014. [22] Rita Ammanouil, André Ferrari, Cédric Richard, and Sandrine Mathieu. Nonlinear unmixing of hyperspectral data with vector-valued kernel functions. IEEE Transactions on Image Processing, 26(1):340–354, 2017. [23] Marian-Daniel Iordache, José M Bioucas-Dias, and Antonio Plaza. Total variation spatial regularization for sparse hyperspectral unmixing. IEEE Transactions on Geoscience and Remote Sensing, 50(11):4484–4502, 2012. [24] Manya V Afonso, José M Bioucas-Dias, and Mário AT Figueiredo. An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Transactions on Image Processing, 20(3):681–695, 2011. [25] Xi-Le Zhao, Fan Wang, Ting-Zhu Huang, Michael K Ng, and Robert J Plemmons. Deblurring and sparse unmixing for hyperspectral images. IEEE Transactions on Geoscience and Remote Sensing, 51(7):4045– 4058, 2013. [26] Sebastian Bauer, Johannes Stefan, Matthias Michelsburg, Thomas Laengle, and Fernando Puente León. Robustness improvement of hyperspectral image unmixing by spatial second-order regularization. IEEE Transactions on Image Processing, 23(12):5209–5221, 2014. [27] Hemant Kumar Aggarwal and Angshul Majumdar. Hyperspectral unmixing in the presence of mixed noise using joint-sparsity and total variation. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 9(9):4257–4266, 2016. [28] Yanfei Zhong, Ruyi Feng, and Liangpei Zhang. Non-local sparse unmixing for hyperspectral remote sensing imagery. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 7(6):1889–1909, 2014. [29] Rui Wang, Heng-Chao Li, Wenzhi Liao, Xin Huang, and Wilfried Philips. Centralized collaborative sparse unmixing for hyperspectral images. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 10(5):1949–1962, 2017. [30] Mingmin Chi, Antonio Plaza, Jón Atli Benediktsson, Zhongyi Sun, Jinsheng Shen, and Yangyong Zhu. Big data for remote sensing: challenges and opportunities. Proceedings of the IEEE, 104(11):2207– 2219, 2016. [31] Radhakrishna Achanta, Appu Shaji, Kevin Smith, Aurelien Lucchi, Pascal Fua, and Sabine Süsstrunk. SLIC superpixels compared to stateof-the-art superpixel methods. IEEE transactions on pattern analysis and machine intelligence, 34(11):2274–2282, 2012. [32] José M Bioucas-Dias and Mário AT Figueiredo. Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing. In Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), 2010 2nd Workshop on, pages 1–4. IEEE, 2010. [33] Padmanabhan Anandan. A computational framework and an algorithm for the measurement of visual motion. International Journal of Computer Vision, 2(3):283–310, 1989. [34] James Bergen, Patrick Anandan, Keith Hanna, and Rajesh Hingorani. Hierarchical model-based motion estimation. In Computer Vision—ECCV’92, pages 237–252. Springer, 1992. [35] Deqing Sun, Stefan Roth, and Michael J Black. Secrets of optical flow estimation and their principles. In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, pages 2432–2439. IEEE, 2010. [36] Qi Shan, Jiaya Jia, and Aseem Agarwala. High-quality motion deblurring from a single image. In Acm transactions on graphics (tog), volume 27, page 73. ACM, 2008. [37] Wang Yongpan, Feng Huajun, Xu Zhihai, Li Qi, and Dai Chaoyue. An improved richardson–lucy algorithm based on local prior. Optics & Laser Technology, 42(5):845–849, 2010. [38] Jong-Ho Lee and Yo-Sung Ho. High-quality non-blind image deconvolution with adaptive regularization. Journal of Visual Communication and Image Representation, 22(7):653–663, 2011. [39] Hsin-Che Tsai and Jiunn-Lin Wu. An improved adaptive deconvolution algorithm for single image deblurring. Mathematical Problems in Engineering, 2014, 2014. [40] Wen Li, Jun Zhang, and Qionghai Dai. Exploring aligned complementary image pair for blind motion deblurring. In Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, pages 273–280. IEEE, 2011.

16

[41] Lu Yuan, Jian Sun, Long Quan, and Heung-Yeung Shum. Progressive inter-scale and intra-scale non-blind image deconvolution. In Acm Transactions on Graphics (TOG), volume 27, page 74. ACM, 2008. [42] Sen Jia, Bin Deng, Jiasong Zhu, Xiuping Jia, and Qingquan Li. Superpixel-based multitask learning framework for hyperspectral image classification. IEEE Transactions on Geoscience and Remote Sensing, 55(5):2575–2588, 2017. [43] Tanu Priya, Saurabh Prasad, and Hao Wu. Superpixels for spatially reinforced bayesian classification of hyperspectral images. IEEE Geoscience and Remote Sensing Letters, 12(5):1071–1075, 2015. [44] Arun M Saranathan and Mario Parente. Uniformity-based superpixel segmentation of hyperspectral images. IEEE Transactions on Geoscience and Remote Sensing, 54(3):1419–1430, 2016. [45] David R Thompson, Lukas Mandrake, Martha S Gilmore, and Rebecca Castaño. Superpixel endmember detection. IEEE Transactions on Geoscience and Remote Sensing, 48(11):4023–4033, 2010. [46] Boris Kozintsev. Computations with Gaussian random fields. PhD thesis, University of Maryland, 1999. [47] Jonathan Eckstein and Dimitri P Bertsekas. On the douglas—rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming, 55(1):293–318, 1992. [48] Scott Shaobing Chen, David L Donoho, and Michael A Saunders. Atomic decomposition by basis pursuit. SIAM review, 43(1):129–159, 2001.