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Integrating Spatial Information in the Normalized P-Linear Algorithm for Nonlinear Hyperspectral Unmixing Maofeng Tang, Student Member, IEEE, Lianru Gao, Member, IEEE, Andrea Marinoni, Senior Member, IEEE, Paolo Gamba, Fellow, IEEE, and Bing Zhang, Senior Member, IEEE

Abstract—To efficiently model high-order nonlinear material mixtures in complex scenery, more and more complex spectral mixing models have been developed, so that over-fitting phenomena more often occur during the unmixing process. Therefore, the accurate and robust inversion of material abundances is a challenging task, especially for low signal-to-noise ratio (SNR) data. In this paper, this task is achieved by inverting the parameters using a hierarchical Bayesian model based on the P-linear mixing model (PLMM). Moreover, spatial information is integrated in the inversion process by considering that similar pixels share the same prior information. Thanks to the fact that PLMM can be translated into a linear model using endmembers and their powers, unmixing is performed by solving a convex optimization problem. Results obtained from synthetic and real data show that the proposed algorithm improves the accuracy of abundance estimation and efficiently reduces over-fitting effects in low SNR data. Index Terms—Hierarchical bayesian, hyperspectral imaging, nonlinear spectral unmixing, spatial information.

I. INTRODUCTION YPERSPECTRAL imaging has recently become an important topic in remote sensing, especially because it can be used to retrieve a thorough characterization of the materials in the considered scene [1], [2]. Hyperspectral unmixing (HSU) aims at separating the target pixel spectrum into a set of constituent spectral signatures, termed “endmembers,” and a set of fractional abundances [2]. However, the high spectral resolution

H

Manuscript received May 30, 2017; revised September 19, 2017; accepted October 24, 2017. This work was supported by the National Natural Science Foundation of China under Grant 41722108, Grant 41571349, and Grant 41325004. (Corresponding author: Lianru Gao.) M. Tang and B. Zhang are with the Key Laboratory of Digital Earth Science, Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing 100094, China, and also with the University of Chinese Academy of Sciences, Beijing 100049, China (e-mail: [email protected]; [email protected]). L. Gao is with the Key Laboratory of Digital Earth Science, Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing 100094, China, and also with the College of Computer Science and Software Engineering, Computer Vision Research Institute, Shenzhen University, Shenzhen 518060, China (e-mail: [email protected]). A. Marinoni and P. Gamba are with the Dipartimento di Ingegneria Industriale e dell’Informazione, Universit`a degli Studi di Pavia, Pavia I-27100, Italy (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTARS.2017.2771482

in hyperspectral images typically comes together with a relatively coarse spatial resolution [3]. Therefore, it is crucial to properly design the spectral unmixing architecture to consider all these issues and achieve reliable outcomes. For HSU, a suitable spectral mixing model is required, and the most common choice is the linear mixing model (LMM) [4]. The LMM is able to efficiently track the combinations of endmembers when light separately interacts with each material, such as in rural and morphologically flat scenarios. Nonetheless, the LMM does not correspond to reality in many complex scenes. Therefore, in recent years, research efforts have been devoted to study and design nonlinear spectral mixing models (NLMMs) [5]. In fact, nonlinear combinations of endmembers are crucial to assess the microscopic and/or macroscopic interactions among materials [5], [6]. So far, many NLMMs have been proposed, some of them considering theoretical analysis and ray tracing to simulate photonic interactions at a microscopic scale or in multilayer mixtures [7], [8]. These models demand adequate knowledge about the environment and sensor geometries, which is difficult to achieve, and results in poor flexibility and scalability. In order to enhance the tradeoff between accuracy in describing the scenes, computational complexity, and prior knowledge of geometric and spectral features, several mixing models have been designed aiming at a more efficient characterization of macroscopic interactions among endmembers, such as the Nascimento model [9], the fan model (FM) [10], the generalized bilinear model (GBM) [11], and the polynomial postnonlinear mixing model (PPNM) [12]. These models aim at improving the description of the macroscopic scale interactions among the constituent elements without prior knowledge of the groundtruth. Specifically, they all focus on bilinear mixing, so that they are actually able to accurately describe interactions between pairs of endmembers in the scene. Additionally, nonlinear models aiming at joint characterizing macroscopic and microscopic scale effects also have been proposed. In [13] and [14], LMM and microscopic scale mixing were combined to characterize linear mixtures of intimate mixtures. Consequently, several optimization schemes have been designed and employed to invert nonlinear mixture models, such as gradient descent, Markov chain Monte Carlo [11], kernel-based SVMs [15]–[21], and manifold learning-based methods [22].

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Nowadays, hyperspectral imaging starts to be employed to analyze the urban environment [23], [24] which is so complex that the NLMMs focusing on the bilinear interactions cannot be considered as adequate. In complex urban scenes, macroscopic scale mixtures are usually due to high-order nonlinear effects [25]. To simulate these high-order interactions, new complex spectral mixing models have been proposed, able to reach better performances than bilinear mixing models. Specifically, in [26] the P-linear mixing model (PLMM) has been proposed to describe p-order nonlinear interactions between endmembers. To describe a complex spectral mixture, the above-mentioned order p should be large, and the number of PLMM coefficients grows geometrically with the model order. Indeed, the higher the order of the model, the more flexible it is, but the more complex nonlinear interactions are. Hence, the flexibility comes at a cost in terms of computational complexity. Furthermore, it may cause over-fitting, especially for low signal-to-noise ratio (SNR), leading to performance degradations in terms of accuracy. This phenomenon is induced when the model is so flexible that its coefficients adapt to the noise [27]. As an additional point, correlations on space, basic features of geospatial data, are usually ignored in current state of the art nonlinear unmixing procedures. Instead, in a complex scene spatial correlations can be regarded as a piecewise smooth constraint for abundance estimation, helping us to retrieve values that are closer to the real spatial distribution of each endmember. Accordingly, in order to interpret and describe the scene in a hyperspectral image, it is crucial to properly consider spatial information. In order to cope with these issues, in this paper we propose a new supervised approach for hyperspectral image nonlinear unmixing which takes the spatial correlation into account, and estimates endmember abundances robustly even in complex scenes. Our approach starts from the PLMM, and takes advantage of a hierarchical Bayesian model to build the relation between coefficients and observed data considering spatial information. The coefficients are then extracted in the framework of maximum a posteriori estimation. By an equivalent transformation, this task can be translated into the minimization of a normalized objective function leading to a convex optimization problem. This paper is organized as follows. Section II introduces the related work about the PLMM which is the start of our study. Section III presents the proposed Super-pixel Segmentation and Density Peaks based Normalized P-Linear Algorithm (SSDPNPLA). Section IV evaluates the experimental performance of the proposed method on synthetic and real hyperspectral datasets. Section V concludes this paper with some remarks. II. PROPOSED METHOD A. P-Linear Mixture Model Let us consider a hyperspectral image described as Y ≡ {y1 , y2 , . . . ,yN }, where N is the total number of pixels. yi = [yi1 , yi2 , . . . , yiL ]T denotes the spectral vector associated with the ith pixel of the image, and L is the number of spectral bands. For supervised unmixing, the spectra of endmembers are assumed to be known, and described by

M = {m1 , m2 , . . . , mR }, where R is the number of endmembers. As mentioned in Section I, unmixing models such as GBM, PPNM, and similar, fail to describe high-order interactions among endmembers, since they can only simulate the process of a photon interacting with not more than two endmembers (not a common phenomenon in reality). Moreover, these nonlinear models are complex and hard to be linearized so that inverting them for the abundance vector requires a substantial computational cost. To tackle this problem, recently attention has been paid to high-order NLMMs, such as the PLMM, written as follows: yl =

R  r=1

αr l mr +

R P   q = 2 r  =1

βr  ,q ,l mqr  + η l

(1)

where βr  ,q ,l is the parameter quantifies the q-order nonlinear contribution of the rth endmember to the lth pixel of image, and η l is the noise of the observed data. The model in (1) is, therefore, able to explain higher order nonlinear interactions and combinations among endmembers. At the same time, it can be properly managed in close form, so that an accurate description of the scene composition can be efficiently retrieved [26], [28]–[30]. B. Normalized P-Linear Algorithm (NPLA) As previously mentioned, the PLMM can simulate high-order nonlinear spectral mixing effects among endmembers. To invert the abundances in this model, the polytope decomposition method (POD) [31] has been proposed. While this method is very efficient, its complexity increases with P because of the increasing number of model parameters. Hence, over-fitting may easily occur when the model order is large, a common case in geometrically complex scenes. Moreover, the POD method assumes band independence, and ignores the correlation in space and in frequency that may be significant in real datasets. Therefore, we introduced a new algorithm aiming at reducing over-fitting when unmixing data with low SNR, and with large P values. At the same time, the proposed algorithm considers the correlation of space and spectral bands to enhance understanding of the scene composition. First, let us note that (1) can be rewritten as ˜ · θl + ηl yl = M

(2)

where θ l = [α1,l , α2,l , . . . , αR ,l , β1,2,l , β2,2,l , . . . , βR ,p,l ] is the ˜ = [m1 , m2 , vector of the parameters to be estimated, and M p p 2 2 2 . . . , mR , m1 , m2 , . . . mR , . . . m1 , . . . mR ] is the spectra of the endmembers and their powers. Equation (2) shows that the mixed spectrum may be considered as the linear combination of columns of the augmented matrix of endmember spectra with respect to the parameters. Let us now introduce a hierarchical Bayesian model, aimed at estimating the parameter vector θ l associated with (2). For the lth pixel of the image, the mixed spectrum and the assumption of Gaussian properties for the noise yields the following likelihood

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function:



yl ∼ N μlpl , σ −1

 ·I

→ p (yl |θ l , σ) =

 σ  L2  σ 2 (3) exp − yl − μlpl  2π 2

˜ · θ l ,  ·  denotes the standard l2 norm such that where μlpl = M √ ˜ are the same X = X T · X, and the definitions of θ l and M l as those defined in (2). In (4), μpl is a linear function with respect to unknown parameters and σ −1 · I is the covariance matrix, which means that the spectrum of each band is independent, and (3) can be expanded according to bands. Introducing the Gaussian noise, the log-likelihood function can be modified as ln p (yl |θ l , σ) = −

σ 2

L 



˜ n θl yn − M

2

n=1

L L ln σ − ln(2π). (4) 2 2 If one assumes that σ is known, by maximizing (4) respect to the θ l , the parameters can be estimated via the maximum likelihood estimation (MLE). Ignoring constants and known terms, this procedure is equivalent to minimize cost function +

θˆl = arg max ln p(yl |θ l , σ) θl



L 2 σ  ˜ n θl yn − M = arg max − θl 2n=1

 L 2 σ  ˜ yn − Mn θ l = arg min θl 2n=1 2 1  ˜ l ˆl − Mθ = arg min y  . θl 2



(5)

Nevertheless, in spectral unmixing, an ML estimator cannot avoid over-fitting because of the absence of testing data, which makes the model selection harder [27]. To solve this issue, let us recall that, in the field of machine learning, Bayesian inference can efficiently reduce a model’s over-fitting [32]. Therefore, to robustly estimate the parameters, an a priori distribution will be considered for the unknown parameters. Marking θ l = [αTl , β Tl ]T where αl = [α1,l , α2,l , . . . , αR ,l ]T quantifies the endmember linear contributions to the mixing spectrum and β l = [β1,2,l , β2,2,l , . . . , βR ,p,l ]T the endmember nonlinear contributions, we consider two Gaussian distributions of the form   p(αl |λl ) = N αl |μlα , λ−1 l I  R /2  T   λl λl  αl − μlα αl − μlα exp − = 2π 2 (6)   p (β l |τl ) = N β l |0 , τl−1 I  τ R ·(P −1)/2  τ l l (7) = exp − β Tl β l 2π 2 −1 where λ−1 l I and τl I are the priori covariance of the abundance vectors and nonlinear parameters, and μlα is the a priori

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abundance average value, especially when the a priori average of the nonlinear parameters are zeros, indicating that the nonlinear effects is small or negligible. Assuming that each unknown parameter is independent, the prior distribution of θ l can be written as p(θ l |λl , τl ) = p(αl |λl )p(β l |τl ).

(8)

Using Bayes’ theorem, the posterior distribution for θ l is proportional to the product of the prior distribution and the likelihood function p(θ l |yl , M, σl , λl , τl ) ∝ p(yl |θ l , σl )p(θ l |λl , τl ) = p(yl |θ l , σl )p(αl |λl )p(β l |τl ).

(9)

Then by maximizing the posterior distribution, the θ l can be estimated robustly. Taking the negative logarithm of (9) and considering (7) and (8), assuming that all the covariances are known, the maximum of the posterior probability is given by    2 λ  2 τ σ 2 ˜ · θl  θˆl = arg min yl − M  + αl −μlα  + β l  . θl 2 2 2 (10) Because the hyperparameters {σ, λ, τ } are known, (10) can be rewritten as    2 λ  2 τ  1 2 ˜ · θl  θˆl = arg min yl − M  + αl −μlα  + β l  θl 2 2 2 (11) where λ = λ/σ, τ  = τ /σ. In the right-hand side of (11), the first term describes the reconstruction error (RE) of the P-Linear model (the cost function), while the remaining terms can be regarded as the normalization terms. Therefore, this method is called free constraint NPLA. Note that λ and τ  are equivalent to the ratio of the a priori distribution covariance to the conditional distribution covariance. In other words, the normalization parameters contain the abundance vector’s prior information and the pixel spectrum noise information. Assuming that noises are independent and identically distributed, the prior information of the parameters may be controlled by adjusting the normalization parameter while minimizing the objective function. The prior information of parameters can also be used to characterize the spatial information. Thus, since correlated pixels tend to share statistical properties, the spatial and spectral correlation can be both considered in the unmixing procedure, as shown in the following section. C. Super-Pixel Segmentation and Density Peaks (SSDP) In this paper, besides aiming at reducing PLMM over-fitting, we propose to exploit spatial and spectral correlations among the pixels of the hyperspectral image. It is reasonable to assume that endmembers give similar contributions to pixels that are correlated either spatially or spectrally (or both). Formally, after partitioning a hyperspectral image into K classes, the spatial and spectral correlation will be utilized so that pixels in the same class share the same normalized parameters. To this aim, in this paper the SSDP algorithm is used to partition the image.

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The SSDP algorithm segments hyperspectral images into superpixels according to the SLIC method [33]. Then, it computes the density peaks, and search for new super-pixels according to their spectral characteristics in the extracted super-pixel set. Finally, clustering is performed by exploiting relationships among the original pixels and the super-pixel clusters. In this paper, the SLIC-DP algorithm [34] is used to extract the cluster, considering both the spatial and the spectral information, and good performances were obtained. It should be noted that selection of the clustering algorithm was performed according to method already available. Any other cluster or segmentation method which can obtain similar results is equally suitable to this unmixing framework. After clustering, K classes are obtained, denoted as C1 , . . . , CK . Moreover, let IK ⊂ {1, . . . , N } denote the subset of pixel indexes belonging to the kth class (k = 1, . . . , K), i.e., yl ∈ Ck ⇔ l ∈ Ik . The model parameters for pixels in the same class would presumably share the same statistical properties. As a result, the unmixing model based on (11) to analyze the whole hyperspectral image can be written as   2 1 k k ˆ ˜ · θ kl  θ l = arg min yl − M  θl 2      λ k  αkl − μkα ,l 2 + τ k β kl 2 + (12) 2 2

for this model is described, aiming at extracting material abundances. In (12), the observed pixel spectrum yl,k and the aug˜ are known, θ l,k mented matrix of endmembers’ spectrum M is the parameter to be estimated, and the other parameters {λ k , τ  k , μkα ,l } contain the parameter priori information and the data noise information. It is reasonable to determine μkα ,l for each pixel by means of linear unmixing. As for λk and τk , they are the same for all pixels in the same class; meanwhile, for each class they are determined by grid search that aims to minimize cost function defined by (12). Due to physical constraints, the abundance vector satisfies the following nonnegative (ANC) and sum-to-one constraints (ASC):

where l ∈ Ik , k = {1, . . . , K} and λk , τk denote the normalized parameters of the kth class t.

k Especially, the model in (12) is well defined when βr,q ,l < 0, which would increase the total observed reflectance instead of causing a decrease with respect to the LMM result with the same endmembers and abundance. By relaxing the constraints about the nonlinear coefficients, the effect of another factor in the circumstance can be considered [35]. However, the nonlinear effect is usually small. Hence β kl is restricted to the range [–1, 1]. In summary, the SSDP-NPLA algorithm can be described as solving the following optimization problem:   2   1 k k k ˆk ˆ ˜ · θ kl  θl = α ˆ l , β l = arg min yl − M  θl 2      λ k  αkl − μkα ,l 2 + τ k β kl 2 + 2 2 (18)     ⎧ k ⎪ αkl = αr,l , β kl = βrk ,q ,l  ⎪ ⎪ r =1,...,R r =1,...,R ,q =2,...,p ⎨ k s.t. 0 ≤ αr,l ≤ 1 k ⎪ ⎪ ⎪ −1 ≤ βr,q ,l ≤ 1 ⎩ ∀k ∈ {1, . . . , K} , l ∈ {1, . . . , N } .

D. SSDP-NPLA Unmixing Strategy 1) SSDP Algorithm: To integrate the spatial and spectral information into HSU, the proposed method implements SSDP. In SSDP, K super-pixels {wi }K i = 1 are first obtained by using the SLIC algorithm. Then, the local density ρi and distance of cluster δi for the ith cluster are computed dij = wi − wj 2 , ρi =



−

e

d

ij dc

2

(13)

j

δi = min (dij ) j :ρ i > ρ j

(14)

where dc is the weight parameter known in advance. Then, we use ρ˜i and δ˜i , the normalized versions of ρi and δi , to compute the selection index γi γi = ρ˜i × δ˜i .

(15)

Moreover, the spectra of the clusters {cj }p can be selected by sorting the index of each cluster, and p is the number of the land-cover classes which is less than the number of super-pixels. Finally, by computing the spectral distance between each pixel and cluster and labeling the pixel according to the nearest class, we resort to the final classification map, where each pixel has been labeled according to its class. 2) Parameter Inversion in SSDP-NPLA: So far, a nonlinear unmixing model that considers spatial and spectral correlations has been introduced. In this section, an inversion procedure

k ≥ 0, r = 1, . . . , R, αr,l

R 

k αr,l = 1.

(16)

r

Nevertheless, in this paper, the endmembers’ contribution is characterized by α and β, which jointly correspond to linear and nonlinear parts. So we free the ASC and redefine the constraints about parameters with k ≤ 1, 0 ≤ αr,l

r = 1, . . . , R

k − 1 ≤ βr,q ,l ≤ 1,

r = 1, . . . , R, q = 2, . . . , p.

(17)

The problem is a convex optimization problem, hence a global optimum value can be found. To solve this problem, the CVX toolkit [36], [37] is employed. 3) Abundance Estimation via SSDP-NPLA: In general, the abundances can be obtained by estimating the linear parameters of spectral mixing model such as αkl in (16). Nevertheless, in this paper we propose a new idea to estimate the abundances, exploiting the both linear and nonlinear contributions in the model. To this aim, let us first rewrite (1) as a linear combination

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of endmember spectra yl =

R 

αr,l mr +

r=1

=

R 

 mr 

R P   q = 2 r  =1

βr  ,q ,l mqr  + η l

αr,l · IL ×1 +

r=1

=

R 

P 

 βr,q ,l mqr − 1

+ ηl

q=2

mr  Ar,l + η l

(19)

r=1

where Ar,l is a L × 1 vector that implies the rth endmember’s contribution to the lth pixel. Different from the common practice, Ar,l is not a scalar and each of its elements implies the contribution to corresponding band, i.e., Ar,l,n is the nth element of Ar,l and represents rth endmember’s contribution to the lth pixel on the nth band. Denoting A˜r,l as the mean contribution on all bands L 

A˜r,l =

n=1

Fig. 1.

Endmember spectra used in synthetic data.

TABLE I ABUNDANCE RMSE OF ABUNDANCE ON SYNTHETICINDEPENDENT DATA SPECTRA SET Unmixing SNR Mixing (db)

FCLS

FM

GBM

PPNM NPLA-2 NPLA-6

LMM

0.1343 0.0621 0.0105 0.3585 0.3511 0.3509 0.4028 0.3544 0.3317 0.4188 0.4163 0.4103

0.2873 0.2872 0.2800 0.2878 0.0605 0.0147 0.2257 0.1344 0.1241 0.3153 0.3116 0.3109

0.1342 0.0644 0.0369 2.0000 0.3511 0.3502 0.4015 0.3523 0.3376 0.4186 0.4131 0.4122

0.1629 0.1047 0.0105 0.2520 0.0645 0.0754 0.2124 0.0913 0.0109 0.2446 0.2599 0.1593

Ar,l,n

. (20) L The abundance of the rth endmember may be estimated as A˜r,l . α ˜r = R  ˜ Ar,l

(21)

r=1

Blinear PNMM PLMM-2

15 30 60 15 30 60 15 30 60 15 30 60

0.1312 0.0620 0.0095 0.2230 0.0597 0.0618 0.2235 0.0892 0.0811 0.1663 0.1124 0.0143

0.1613 0.0732 0.0166 0.2249 0.0833 0.0993 0.2405 0.1035 0.0794 0.2014 0.1207 0.0145

III. EXPERIMENTS The performance of the SSDP-NPLA algorithm for nonlinear unmixing is evaluated on synthetic and real hyperspectral data. For comparison, other well established supervised unmixing algorithms, such as LMM, FM, GBM, and PPNM, are considered. The average RE and the abundance root mean square error (RMSE) are used for the evaluation    N  N 1  1  2  yl − y ˆl 2 RMSE =  αl − α ˆ l 22 RE = LN N l=1

l=1

(22) where y ˆl identifies the reconstructed spectral signature of the lth pixel, and α ˆ r is the lth pixel estimated abundance vector. A. Experiments on Synthetic Data 1) Test for NPLA Using Independent Pixels: We first tested the performance of NPLA applied to independent synthetic data. The algorithm was tested on 100 pixels generated by different mixing models of five endmembers on 420 bands selected from the USGS spectral library [38]. The materials we have considered are Alunite, Calcite, Epidote, Kaolinite, and Buddingtonite, whose spectra are shown in Fig. 1. To test the algorithm’s robustness, we add Gaussian white noise to the synthetic data with different SNR values (15 db/30 db/60 db), where SNR is defined as   E ylT yl (23) SNR = 10 log10  T  E ηl ηl

and yl is the N-dimensional spectral signature of the lth pixel generated by the spectral mixing model. Specifically, synthetic spectra were generated according to LMM, bilinear mixture model, postnonlinear mixture model (PNMM) and PLMM-2 order with the abundance generated uniformly and randomly from the unit simplex (abundance = [0.15, 0.25, 0.3, 0.1, 0.2]). Then, the mixture spectrum was unmixed with different approaches including the fully constrained least square method (FCLS), the FM-based unmixing method, the gradient-based generalized bilinear unmixing method (GBM), PPNM, NPLA-2 order (NPLA-2) and NPLA-6 order (NPLA-6). Table I reports the performances in terms of RMSE and abundances. In case of large SNRs, there is no unmixing algorithm that can always obtain the best performance for all different mixture models. For example, with 60 db SNR and data mixed by PNMM, the PPNM algorithm can get the best performance. However, it is worth noting that the method proposed in this work always obtains suboptimal performances, very close to the optimal ones. When the SNR is small, NPLA is almost always the best technique. Indeed, algorithms whose model assumptions are too strong may be not the best choice for data mixed by the corresponding mixture model. For example, with SNR = 15 db/30 db, the FM and PPNM algorithm are not the best methods for data mixed by bilinear and PNMM model, and the method proposed in this work achieves the best performance. This effect is caused by the strength of the model assumption,

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TABLE II RE OF RECONSTUCTION ON INDEPENDENT SYNTHETIC DATA SPECTRA SET Unmixing SNR Mixing (db)

FCLS

FM

GBM

PPNM NPLA-2 NPLA-6

LMM

0.1139 0.0226 0.0006 0.1724 0.0977 0.0838 0.1996 0.1005 0.0651 0.6475 1.0211 1.3542

0.1231 0.0409 0.0353 0.1511 0.0274 0.0008 0.1554 0.0280 0.0048 0.4639 0.8414 1.1800

0.1139 0.0226 0.0008 0.1700 0.0912 0.0826 0.1995 0.0993 0.0638 0.6487 1.0211 1.35

0.1139 0.0225 0.0006 0.1529 0.0274 0.0011 0.1537 0.0278 0.0008 0.06 0.0165 0.0214

Blinear PNMM PLMM-2

15 30 60 15 30 60 15 30 60 15 30 60

0.1139 0.0226 0.0006 0.1927 0.0278 0.0034 0.1521 0.0274 0.0026 0.0407 0.0112 0.0002

0.1138 0.0226 0.0006 0.1933 0.0269 0.0039 0.1507 0.0273 0.0031 0.0621 0.0112 0.0004

Fig. 4.

Abundance maps of the synthetic image generated by GBM.

Fig. 5.

Abundance maps of the synthetic image generated by PNMM.

Fig. 6.

Comparison of the abundance RMSE values for the synthetic datasets.

Fig. 2. Clustering of synthetic image using LMM in on the left, using GBM in the middle, and finally using PNMM on the right.

Fig. 3.

Abundance maps of the synthetic image generated by LMM.

which enhances the model pertinence while reducing the generalization and causing over-fitting. Table II provides a similar analysis with respect to the RE. Even with respect to this measure, when the SNR is large, the algorithm fitting the mixture model obtains the best performance. For example, when SNR = 60 db, the PPNM algorithm is the best performing one on data mixed by PNMM. Compared to Table I, the RE and RMSE are minimum simultaneously, which means that the accuracy of the model is the main factor affecting the unmixing performance when the SNR is large. Instead, when the SNR is low, RE and RMSE do not match. For ex-

ample, when SNR = 15 db, the PPNM RE is smaller than the NPLA-2 one on data mixed by bilinear mixing. 2) Test SSDP-NPLA Using Synthetic Image: In this part, we test the performances of the proposed SSDP-NPLA by using correlated synthetic data, i.e., the same spectra as before, but

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Fig. 7.

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Comparison of the RE values for the synthetic datasets.

TABLE III MCR OF COMPARING ALGORITHM IN THE SYNTHETIC DATA (SNR = 30 db) Unmixing MCR

FM

GBM

PPNM

NPLA-2

8.1428

4.1800

5.3680

3.3028

Fig. 9.

Fig. 10.

Fig. 8. Computing time (in seconds) required for unmixing the test datasets in Section III-A-2 (SNR = 30 db).

arranged into an image. Accordingly, the endmembers are the same as those used in Section IV-A-1, but a spatially correlated abundance map has been generated for the synthetic image. The generation of the synthetic image follows the procedure given in [39]. The synthetic hyperspectral cube contains pixels mixed by the same five endmembers as in Fig. 1. The mixed regions are distributed spatially in the form of squared patches which involve two to five endmembers, while the background pixels are mixtures of all endmembers with a known abundance vector equal to [0.1149, 0.0741, 0.2003, 0.2055, 0.4051]T as in [40]. To validate the generalization properties of the proposed method, we compared its performances on different samples generated by the three linear/nonlinear mixture models de-

Segmentation map for the Cuprite dataset obtained by SSDP.

RE for Cuprite obtained from SSDP-NPLA.

scribed earlier. Moreover, these synthetic images have been corrupted by a zero-mean white Gaussian noise with an SNR of 30 db. The cluster results for different dataset are illustrated in Fig. 2 Compared with the real abundance maps, Fig. 2 illustrates that the more nonlinear the datasets are, the more complex the relationships among pixels are. After clustering, the hyperparameter initial values for each class are computed using the average spectrum for each class. Then, the hyper-parameters values for pixels in the same class are selected with the same initial value, and the prior average of the parameters is updated during the search. Fig. 3 illustrates that the above unmixing approaches can efficiently invert the abundances of linear mixture data. However, except for SSDP-NPLA, the other algorithms cannot represent spatial correlations, as apparent in the background of the abundance maps, since only the backgrounds of the abundance maps of the bottom row are smoother, and approximate the real abundance distribution. In Fig. 4, we report the abundance maps of an image generated by bilinear model, and the FCLS estimates are wrong, while the maps obtained by GBM and PPNM are affected by impulse noise in the background.

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Fig. 11.

Fig. 12.

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Abundance maps for a few endmembers by different unmixing algorithms.

Map of error distribution on Cuprite obtained from SSDP-NPLA.

In Fig. 5, we analyze a synthetic image obtained by PNMM. Obviously, FCLS and GBM work poorly on this kind of data. PPNM obtains a better performance than FCLS and GBM, because of its good generalization capacity. Nonetheless, PPNM cannot efficiently restore the background where the spatial correlation is strong. NPLA provides an even better performance; nonetheless, there is always a residual bias on the background

abundance for endmember 1, because NPLA does not take spatial correlation into account. Instead, SSDP-NPLA performs well in all scenarios, and restores the spatial distribution information. Figs. 6 and 7 report the RMSE and RE values for all the tests, which support the previous analyses. Indeed, the results of the proposed approach are the best ones (or very close to the best) in all cases. 3) Computational Costs: The computational load of the proposed algorithm depends on two processing steps: 1) the search for the hyper-parameters; 2) the optimization of the cost function. Taking the spatial information into account, similar pixels share the same hyper-parameters and the computational cost of this step can be reduced significantly. Therefore, the optimization of cost function dominates the computational load. Indeed, as many other nonlinear unmixing algorithms, the proposed method deals with the image pixel by pixel. In order to assess the complexity of the proposed architecture, we measured latency and memory allocation required to process the dataset in Section III-A-II. To this aim, we considered the performance of SSDP-NPLA and the other algorithms

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Fig. 13.

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Abundance maps for the Washington DC subset using SSDP-NPLA.

unmixing architecture. Moreover, we estimated the latency of the frameworks by calculating the execution time. The results are illustrated in Table III and Fig. 8, respectively. From Table III, NPLA outperforms the other unmixing algorithms in terms of MCR. Therefore, we can assume that the computational cost of proposed unmixing framework mainly depends on the cluster method. Furthermore, we can appreciate that the proposed normalized p-linear mixture approach is able to enhance the resource allocation in operating systems, so that the computational load is reduced with respect to other state-of-the-art methods in technical literature. Moreover, considering Fig. 8, we can state that SSDP-NPLA is more efficient than PPNM and GBM when dealing with nonlinear mixing data. Additionally, Fig. 8 shows that the computing efficiency changes according to the data mixing model. As expected, the more complex a mixture is, the longer the corresponding computational time is. B. Experiments on Real Datasets Fig. 14. False-color image of testing site of Washington DC (R-band 64, G-band 52, B-band 36).

as achieved over a laptop with 64 bits Intel i7-6700 U 3.4 GHz CPU and 8 GB RAM. Specifically, we evaluated the memory allocation by computing the memory consumption rate (MCR), i.e., the ratio between the total memory that must be allocated to perform unmixing and memory that must be allocated in order to store the image to be analyzed. As such, MCR reports an estimate of the memory overhead that is required by each

1) Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) Cuprite Dataset: To complete the analysis, tests on real datasets were performed. The first real image used to test the proposed method refers to the Cuprite mining site in Nevada, and was acquired by the AVIRIS in 1997. This dataset contains 350 × 350 pixels with a spatial resolution of 20 m. The number of spectral bands is 189, in the wavelength range is 0.4–2.5 μm. In this paper, we use endmember spectral signatures extracted using the vertex component analysis to be compliant with [41], and the number of endmembers is estimated to be nine by

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Fig. 15. scene.

Extracted endmember spectra for the Washington DC HYDICE

Fig. 16.

RE for the Washington DC HYDICE scene.

Fig. 17. Classification map of the Washington DC test site based on extended morphological profiles [43].

HySime [42]. The nine endmembers were identified by locating the spectra of minimal spectral angle in the USGS spectral database. Specifically, the extracted endmembers are Kaolinite, Alunite, Muscovite, Calcite, Halloysite, Buddingtonite, Nontronite, Montmorillonite, and Chlorite. The unmixing results of the proposed method were compared with those by FCLS, FM, GBM, and PPNM. Fig. 9 illustrates the segmentation results based on the spatial and spectral information, while Fig. 10 reports the RE for the compared algorithms. Fig. 10 shows that the proposed method can restore the image with the smallest residual error due to its smoothing capacity, which exploits the spatial correlation inside the extracted re-

gions. Because of the absence of real abundance values for the endmembers no other quantitative comparison can be performed, but the distribution of the RE supports this point. Fig. 11 reports the abundance maps and Fig. 12 shows the error images from different algorithms for the whole scene. In Fig. 12, the error maps for FCLS, FM, GBM, and PPNM present always bright points, which means they contain a few points with large error. Instead, the error maps for NPLA and SSDP-NPLA (especially the latter) are smoother. And Fig. 11 reports the abundance maps of endmembers which show the different endmembers’ spatial distribution. Comparing them, we can find that the abundance estimation used the SSDP-NPLA has remained more details for the spatial distributions, at the same time the results from the method keep accuracy. 2) Hyperspectral Digital Imagery Collection Experiment (HYDICE) Washington DC Dataset: The second real image dataset used in experiments was collected by the (HYDICE) sensor over Washington DC, and a subset of pixels was extracted from the original image for this experiment. In this dataset, there are 210 bands covering the range of 0.4 − 2.5 μm. Low-SNR and water-vapor absorption bands were removed in advance, leaving 197 bands. Fig. 13 shows the endmembers abundance maps from SSDP-NPLA. Fig. 14 shows the false-color image of the testing site. Although there are six distinct materials in the image [43], in our experiment we combine “trail” and “road” and consider five endmembers only. Before unmixing, we select typical regions for each endmember and compute the average spectra. These computed averages are regarded as the endmember spectra, and are shown in Fig. 15. Similar to part A of Section III-B-1, we compared the proposed approach with different unmixing method. Fig. 16 reports the RE and shows that the proposed method obtains the best performance. Fig. 17 shows the classification map for the test site. Compared with the Fig. 13, it is obvious that the distribution of each endmember by NPLA corresponds well with the classification result, and this comparison supports the efficiency of the method proposed in this paper. IV. CONCLUSION In this paper, we developed a new nonlinear HSU algorithm called SSDP-NPLA. By taking advantage of Bayesian estimation, this approach reduces the effect of over-fitting for highorder nonlinear spectral mixtures. Additionally, by controlling the hyper-parameters using the SSDP algorithm, the approach takes spatial correlations into account. Finally, the abundance vectors can be estimated by minimizing the normalized cost function in a convex framework. Experiments with synthetic and real hyperspectral dataset demonstrate that the proposed method can achieve better accuracy values and lower computational costs than other state-of-the-art unmixing methods. Future works will be devoted to develop a flexible and efficient method to estimate the hyper-parameters. Moreover, we will explore the meaning of abundance in nonlinear unmixing to search for more reliable abundance estimation methods.

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[24] A. Marinoni and P. Gamba, “Big data for human-environment interaction assessment: Challenges and opportunities,” in Proc. ESA Big Data Space Conf., Frascati, Italy, Nov. 2014, pp. 1–4. [25] A. Marinoni and P. Gamba, “On the effect of nonlinear mixing in hyperspectral images of human settlements,” in Proc. Joint Urban Remote Sens. Event, Lausanne, Switzerland, 2015, pp. 1–4. [26] A. Marinoni and P. Gamba, “A novel approach for efficient p-linear hyperspectral unmixing,” IEEE J. Sel. Topics Signal Process., vol. 9, no. 6, pp. 1156–1168, Sep. 2015. [27] C. M. Bishop, “Pattern recognition,” Mach. Learn., vol. 128, pp. 1–58, Feb. 2006. [28] A. Marinoni, J. Plaza, A. Plaza, and P. Gamba, “Nonlinear hyperspectral unmixing using nonlinearity order estimation and polytope decomposition,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens., vol. 8, no. 6, pp. 2644–2654, Jun. 2015. [29] A. Marinoni, A. Plaza, and P. Gamba, “Harmonic mixture modeling for efficient nonlinear hyperspectral unmixing,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens., vol. 9, no. 9, pp. 4247–4256, Sep. 2016. [30] A. Marinoni and P. Gamba, “Accurate detection of anthropogenic settlements in hyperspectral images by higher order nonlinear unmixing,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens., vol. 9, no. 5, pp. 1792– 1801, May 2016. [31] A. Marinoni and P. Gamba, “Non-linear hyperspectral unmixing by polytope decomposition,” in Proc. IEEE Workshop Hyperspectral Image Signal Process. Evol. Remote Sens., Lausanne, Switzerland, Jun. 2014, pp. 1–4. [32] C. M. Bishop, Pattern Recognition and Machine Learning. New York, NY, USA: Springer, 2006. [33] R. Achanta, A. Shaji, K. Smith, A. Lucchi, P. Fua, and S. S¨usstrunk, “SLIC superpixels compared to state-of-the-art superpixel methods,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 34, no. 11, pp. 2274–2282, Nov. 2012. [34] W. Yu, Z. Wang, S. Li, and X. Sun, “Hyperspectral image clustering based on density peaks and superpixel segmentation,” J. Image Graph., vol. 21, no. 10, pp. 1402–1410, Dec. 2016. [35] R. Heylen and P. Scheunders, “A multilinear mixing model for nonlinear spectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 54, no. 1, pp. 240–251, Jan. 2016. [36] S. Boyd and L. Vandenberghe, Convex Optimization. New York, NY, USA: Cambridge Univ. Press, 2004. [37] M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab software for disciplined convex programming,” 2008. [Online]. Available: http://cvxr.com/cvx/ [38] R. N. Clark, G. A. Swayze, A. J. Gallagher, T. V. King, and W. M. Calvin, “The US geological survey, digital spectral library: Version 1 (0.2 to 3.0 μm),” USGS, Reston, VA, USA, Open File Rep. 93–552, 1993. [39] J. Chen, C. Richard, and P. Honeine, “Nonlinear estimation of material abundances in hyperspectral images with l1 -Norm spatial regularization,” IEEE Trans. Geosci. Remote Sens., vol. 52, no. 5, pp. 2654–2665, May 2014. [40] M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Total variation spatial regularization for sparse hyperspectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 11, pp. 4484–4502, Nov. 2012. [41] J. M. Nascimento and J. M. Dias, “Vertex component analysis: A fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 4, pp. 898–910, Nov. 2005. [42] J. M. Bioucas-Dias and J. M. Nascimento, “Hyperspectral subspace identification,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 8, pp. 2435–2445, Aug. 2008. [43] J. A. Benediktsson, J. A. Palmason, and J. R. Sveinsson, “Classification of hyperspectral data from urban areas based on extended morphological profiles,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 3, pp. 480–491, Mar. 2005.

Maofeng Tang (S’17) received the B.S. degree in geodesy and geomatics in engineering from Wuhan University, Wuhan, China, in 2015. He is currently working toward the M.S. degree on the thesis of hyperspectral image nonlinear unmixing at the Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing, China. His research interests include remote sensing information extraction, Bayesian inference, nonlinear inverse problems, machine learning, and deep learning.

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Lianru Gao (M’12) received the B.S. degree in civil engineering from Tsinghua University, Beijing, China, in 2002, and the Ph.D. degree in cartography and geographic information system from the Institute of Remote Sensing Applications, Chinese Academy of Sciences (CAS), Beijing, in 2007. He is currently a Professor in the Key Laboratory of Digital Earth Science, Institute of Remote Sensing and Digital Earth, CAS. He also has been a Visiting Scholar at the University of Extremadura, C´aceres, Spain, in 2014, and at the Mississippi State University, Starkville, USA, in 2017. In last ten years, he was the PI of ten scientific research projects at national and ministerial levels, including projects by the National Natural Science Foundation of China (2010–2012, 2016–2019, 2018– 2020), by the Key Research Program of the CAS (2013–2015), etc. He has published more than 110 peer-reviewed papers, and there are 47 journal papers included by Science Citation Index. He was a coauthor of an academic book entitled Hyperspectral Image Classification and Target Detection. He obtained 12 National Invention Patents and 4 Software Copyright Registrations in China. His research focuses on models and algorithms for hyperspectral image processing, analysis, and applications. Dr. Gao received the Outstanding Science and Technology Achievement Prize of the CAS in 2016, and the China National Science Fund for Excellent Young Scholars in 2017. He received the recognition of the Best Reviewers of the IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS and Remote Sensing in 2015.

Andrea Marinoni (S’07–M’11–SM‘16) received the B.Sc., M.Sc. “cum laude,” and Ph.D. degrees in electronic engineering from the University of Pavia, Pavia, Italy, in 2005, 2007, and 2011, respectively. Since 2011, he has been a Postdoctoral Fellow in the Telecommunications and Remote Sensing Laboratory, Department of Electrical, Computer and Biomedical Engineering, University of Pavia. In 2009, he was a Visiting Researcher at the University of California, Los Angeles, CA. In 2015, he was a short-term Visiting Researcher at Earth and Planetary Image Facility, Ben-Gurion University of the Negev, Be’er Sheva, Israel. In 2016, he was a Visiting Researcher in the School of Geography and Planning, Sun Yat-Sen University, Guangzhou, China, and a short-term Visiting Researcher in the School of Computer Science, Fudan University, Shanghai, China. In 2017, is a Visiting Researcher at the Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing, China. His research activities are focused on efficient nonlinear signal processing applied to hyperspectral unmixing, astroinformatics, and big data mining, and analysis and management for human–environment interaction assessment. Dr. Marinoni has been an invited speaker at 2010 Workshop on Application of Communication Theory to Emerging Memory Technologies; IEEE Global Communications conference (GLOBECOM), Miami, FL; 2014 New Challenges in Astro- and Environmental Informatics in the Big Data Era Workshop, Szombathely, Hungary; Big Data Methodologies in Remote Sensing and Astronomy special session at 2015 IEEE Geoscience and Remote Sensing Symposium, Milan, Italy; and 2015 Astroinformatics, Dubrovnik, Croatia. In 2011, he received the two-year “Dote Ricerca Applicata” award, sponsored by the Region of Lumbardy, Italy, and STMicroelectronics N.V.

Paolo Gamba (SM’00–F’13) received the Laurea degree in electronic engineering “cum laude” and the Ph.D. degree in electronic engineering from the University of Pavia, Pavia, Italy, in 1989 and 1993, respectively. He is a Professor of telecommunications at the University of Pavia, where he leads the Telecommunications and Remote Sensing Laboratory and serves as the Deputy Coordinator of the Ph.D. School in Electronics and Computer Science. He has been invited to give keynote lectures and tutorials in several occasions about urban remote sensing, data fusion, EO data for physical exposure, and risk management. He published more than 140 papers in international peer-reviewed journals and presented nearly 300 research works in workshops and conferences. Dr. Gamba was as the Editor-in-Chief of the IEEE GEOSCIENCE AND REMOTE SENSING LETTERS from 2009 to 2013, and the Chair of the Data Fusion Committee of the IEEE Geoscience and Remote Sensing Society from October 2005 to May 2009. He currently serves as the GRSS Executive Vice President. He has been the organizer and Technical Chair of the biennial GRSS/ISPRS Joint Workshops on “Remote Sensing and Data Fusion over Urban Areas” from 2001 to 2015. He also served as Technical Co-Chair of the 2010 and 2015 IGARSS conferences, in Honolulu, Hawaii, and in Milan Italy, respectively. He has been the Guest Editor of special issues of IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, IEEE JOURNAL OF SELECTED TOPICS IN REMOTE SENSING APPLICATIONS, ISPRS Journal of Photogrammetry and Remote Sensing, International Journal of Information Fusion, and Pattern Recognition Letters on the topics of Urban Remote Sensing, Remote Sensing for Disaster Management, and Pattern Recognition in Remote Sensing Applications.

Bing Zhang (M’11–SM’12) received the B.S. degree in geography from Peking University, Beijing, China, and the M.S. and Ph.D. degrees in remote sensing from the Institute of Remote Sensing Applications, Chinese Academy of Sciences (CAS), Beijing, in 1991,1994 and 2003, respectively. He is currently a Full Professor and the Deputy Director of the Institute of Remote Sensing and Digital Earth, where he has been leading key scientific projects in the area of hyperspectral remote sensing for more than 20 years. He has authored more than 300 publications, including more than 170 journal papers. He has edited six books/contributed book chapters on hyperspectral image processing and subsequent applications. He has developed five software systems in the image processing and applications. His research interests include the development of mathematical and physical models and image processing software for the analysis of hyperspectral remote sensing data in many different areas. Dr. Zhang received ten important prizes from Chinese government, and special government allowances of the Chinese State Council. He received the National Science Foundation for Distinguished Young Scholars of China in 2013, the 2016 Outstanding Science and Technology Achievement Prize of the Chinese Academy of Sciences, and the highest level of Awards for the CAS scholars. He is currently serving as the Associate Editor for IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING and IEEE GEOSCIENCE AND REMOTE SENSING LETTERS. He has been serving as the Technical Committee Member of IEEE Workshop on Hyperspectral Image and Signal Processing since 2011, and as the President of hyperspectral remote sensing committee of China National Committee of International Society for Digital Earth since 2012. He is the Student Paper Competition Committee member in IGARSS 2015, 2016, and 2017.