A Multiobjective Group Sparse Hyperspectral Unmixing Method With High Correlation Library

Hyperspectral sparse unmixing aims at modeling pixels of hyperspectral image as a linear combination of a subset of a prior spectral library. Over the past years, spectral library has been constantly expanded, including spectra of the same material with intrinsic variability, which may result in the problem of high correlation. Recently, multiobjective sparse unmixing methods presented promising performance in dealing with sparsity via a nonconvex <inline-formula><tex-math notation="LaTeX">$ \mathcal {L}_{0}$</tex-math></inline-formula> norm but are insensitive to identifying endmembers with high correlation. In this article, we propose a multiobjective sparse unmixing method, multiobjective group sparse hyperspectral unmixing (MO-GSU), which integrates a group sparsity structure to address high correlation of the spectral library induced by spectral variability. In order to describe the sparsity within and among groups, MO-GSU develops a mixed norm <inline-formula><tex-math notation="LaTeX">$ \mathcal {L}_{0,q}$</tex-math></inline-formula> instead of the <inline-formula><tex-math notation="LaTeX">$ \mathcal {L}_{0}$</tex-math></inline-formula> norm. During the optimization, we propose two new search strategies: intragroup local search and group-oriented adaptive genetic operator. The intragroup local search strategy is presented in addition to the multiobjective evolutionary algorithm for better exploitation within groups. The group-oriented adaptive genetic operator is designed to maintain the intergroup distribution between generations and further ensure the intragroup exploitation. Moreover, we provide theoretical proof for the advantage of the group operators in exploiting the endmembers within group. To verify the efficiency of the proposed method on high correlation situations, MO-GSU is compared with recently proposed endmember bundle based and multiobjective-based sparse unmixing methods on synthetic and real data with high correlation libraries.

resolution of hyperspectral sensors, allowing precise identification of materials from an image. Informative as an HSI be on the spectral dimension, the spatial resolution of hyperspectral sensors suffers to be relatively low [1]. As a result, there exists mixed pixel, as in one pixel containing multiple materials. In recent years, many hyperspectral unmixing techniques have been developed to identify pure material signatures (called endmembers) in each pixel of the HSI and estimate their proportions (called abundances) at the same time [2], [3], [4]. Linear mixture model is a popular description of the unmixing process. Unlike nonlinear models which take account of multiple reflectance and scattering from various materials [5], [6], linear mixture model assumes the signature of a pixel to be a linear combination of spectra signatures of pure materials. Different approaches have been addressed under the assumption of linear mixture model in the past decades, including categories of geometrical approaches, statistical approaches, nonnegative matrix factorization, and sparse regression approaches [4], [7]. Geometrical approaches consider the linearly mixed signatures with the sum-to-one constraint on abundances in a simplex, of which the vertices represent the endmembers and the volume is the optimization objective [8], [9], [10]. Statistical approaches such as Bayesian methods perform Bayesian inference and parameter estimation on endmembers and abundances [11], [12]. Nonnegative matrix factorization resolves the observed image into endmember and abundance matrices [13], [14], [15]. Sparse regression approaches exploit a redundant prior spectral library as endmember candidates, assuming the observed image a linear combination of pure spectra signatures from a subset of the spectral library [16]. Sparse unmixing is a semisupervised approach including endmember selection and abundance inversion procedures and, thus, has no need for pure pixel assumptions. The method proposed in this article is a sparse unmixing approach.
Nevertheless, sparse unmixing expects the challenge of how to optimize the sparsity of endmembers directly expressed via L 0 norm. The L 0 norm regularization formulates a nonconvex and NP-hard optimization objective. Convex approximation and greedy search are the two common solutions to address L 0 norm. Literature [17] converted the nonconvex problem into a convex form by proposing SUnSAL to relax L 0 norm into L 1 norm. Literature [18] adopted L 0, 1 2 quasi-norm, a better approximation of L 0 norm as penalty and an iterative algorithm for abundance estimation. Literature [19] designed a numerical reweighted least squares algorithm for the constrained sparse l p -l 2 (0 < p < 1) optimization problem. Literature [8] employs This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ sparsity criteria based library pruning and explores standard sparsity measures such as Gini index and pd-norm sparsity. The boosting deep learning methods have provided promising results on many image processing tasks, and literature [20] proposes SUnCNN, a deep convolutional encoder-decoder, to solve sparse unmixing problem.
Multiobjective optimization is an effective solution for overcoming the difficulty induced by the nonconvex optimization [21]. Sparse unmixing task is naturally a bi-objective problem as simultaneously minimizing the reconstruction error of image data and endmember sparsity applied with L 0 norm. The L 0 norm sparsity determines multiobjective sparse unmixing to be an NP-hard problem. Multiobjective evolutionary algorithms (MOEAs) are proved to be available and effective for NP-hard problems. Many multiobjective sparse unmixing algorithms have arisen in recent years [22], [23], [24], [25], [26]. MOSU [27] translated sparse unmixing into a tri-objective problem and employs a cooperative coevolutionary strategy. SMOSU [28] regarded sparse unmixing as a bi-objective problem, applying binary decision vector and a random flipping strategy. PMOSU [29] simultaneously performed library pruning and sparse regression through a gradual compression of search space. Jiang [30] proposed an improved two-phase approach with extra application of spatial-contextual information.
However, the classic multiobjective sparse unmixing methods may suffer the problem of inadequate searching performance due to the high correlation of the spectral library caused by spectral variability. Spectral variability enlarges the scale of prior spectral library and raises correlation. The inadequate search is an inevitable outcome since the classic MOEAs are not capable of simulating the Pareto front (PF) within bearable evaluations when confronting large-scale multiobjective problems (MOPs) [31]. The spectral variability effect [32], namely several spectral signatures representing one single material, occurs attributed to the variation of intrinsic property and extrinsic conditions such as illumination and atmosphere. Methods addressing spectral variability can be divided into two main categories. On the one hand, researchers adjust the conventional linear mixing model (LMM) model to cover the variation of extrinsic conditions [33], [34]. On the other hand, intrinsic variation is expressed via multiple detected instances of one particular material, called endmember bundles [35], [36], [37]. Literature [38] extracted endmember bundles to construct spectral library with high correlation and adopted a mixed norm for class and intraclass sparsity. Dictionary pruning methods eliminate correlation by reducing mutual coherence [39]. Literature [40] presented a recursive PCA approach as well as a mutual coherence reduction method for library pruning.
In this article, we propose a new multiobjective group sparse unmixing (MO-GSU) algorithm to address high correlation within endmember bundles in the spectral library. There are threefold strategies presented to tackle with high correlation caused by spectral variability and the resultant group structure formed of endmember bundles. 1) In order to interpret the sparsity within and among groups, we introduce a mixed norm L 0,q in MO-GSU containing group structure information, where q is a real value between 0 and 1, still formulating an NP-hard problem. The sparsity of material and each endmember underlies in the inherent requirement of hyperspectral unmixing. The proposed MO-GSU tailored for spectral libraries with correlation is then used to optimize the bi-objective problem. 2) An intragroup local search strategy is presented in addition to the MOEA to exploit endmember selection within certain groups [41], [42], [43], significantly improving the effectiveness and efficiency of endmember selection. 3) Furthermore, to balance exploitation and exploration, this article employs a group-oriented adaptive genetic operator, including a crossover operator involving group structure to maintain the distribution within group between generations during the evolutionary process, and an intragroup mutation operator to further ensure the intragroup exploitation.
The contributions of MO-GSU include the following aspects. 1) We propose an advanced multiobjective hyperspectral unmixing method for a mixed L 0,q norm model based on group sparsity, which aims at addressing the high correlation of spectral library. 2) In the proposed MOEA, we design two group-based optimization strategies, intragroup local search and grouporiented adaptive genetic operator, to enhance intragroup exploitation and intergroup exploration. 3) We further provide theoretical proof for the proposed genetic operator of a more sufficient endmember exploitation within group.

II. PROPOSED METHOD
In this section, we first introduce some background context on unmixing model, group sparsity with mixed norm, and MOEAs. Then, we elaborate on the proposed MO-GSU method. To start with, we demonstrate the mathematical expression of the GSU problem and the framework of the proposed MO-GSU method. In Section II-A, we first introduce some background context. Section II-B presents the model of MO-GSU. Section II-C describes the optimization process, and we detail the techniques aimed at intragroup correlation orderly in Sections II-D and II-E.
A. Background 1) LMM: LMM assumes each pixel in an HSI as a linear combination of pure spectral signatures: where y i ∈ R L×1 is the reflectance vector of the ith pixel from an observed image over L spectral bands, A ∈ R L×m is the spectral signature matrix associated with m endmembers, x i ∈ R m×1 is the corresponding abundance vector of the m endmembers, and n i ∈ R L×1 is a vector collecting noise and modeling error. Equation (1) can be rewritten considering n pixels in the observed image in the following equation: where Y = [y 1 , y 2 , ..., y n ] ∈ R L×n denotes the reflectance matrix of the whole image, X ∈ R m×n denotes the abundances of m endmembers on n pixels, and N ∈ R L×n represents the noise matrix. The abundance of each pixel satisfies the abundance nonnegative constraint (ANC): x ij ≥ 0, j = 1, 2, ..., n and the abundance sum-to-one constraint (ASC): m i=1 x ij = 1, j = 1, 2, . . ., n. In this article, nonnegative constrained least square method is adopted to inverse the abundance matrix after endmember selection.
2) Endmember Bundles and Mixed Norm: Sparse unmixing methods utilize a prior spectral library as endmember candidates and implement constrained least square regression on the optimal subset for abundance inversion. The prior library that can be represented as A in (2) is designed to adapt to various image scenes. To this end, the library is supposed to be redundant, as in sparsity of nonzero rows of A in (2). The conventional sparse unmixing task is expressed as follows: where δ is the tolerance of reconstruction value, and · row−0 and · F denote the numbers of nonzero rows and the Frobenius norm of a matrix. However, (3) does not exploit the grouping information of the prior library. Due to the spectral variation of extrinsic and intrinsic properties, it is common in a spectral library that several spectra represent a single material. Therefore, different realizations of the same material compose endmember bundles, and the intrinsic similarity within bundles formulates certain group structure. Given such additional group structure G, the L 0 -norm sparsity is translated into group sparsity in L G,p,q norm, 0 < p, q < 1. The sparse unmixing task containing grouping information is expressed via the following equation: where G denotes the grouping strategy, | G | is the number of groups, A g , X g are the corresponding grouping endmembers and abundance of indices from group g, respectively, and · G,p,q denotes the mixed norm of a matrix, 0 < p, q < 1. The L G,p,q norm of a vector x and group structure G is defined in (5) for any real p, q, where | G i | denotes the size of the ith group When (p, q) = (0, 1), the group sparsity is reduced to L 0 -norm in (3). The group sparsity with (p, q) values of (1, 1), (2, 1), (1,2), (1, q), and 0 < q < 1, correspond to regular lasso, group lasso, elitist lasso, and fractional lasso, respectively. In this article, we set (p, q) = (0, q), 0 < q < 1, thus adopting group sparsity of L G,0,q norm, since it is a better reflection of both intergroup and intragroup sparsity than the regular L 0 -norm.

3) Multiobjective Optimization:
A minimization MOP of k objectives can be presented as follows: where s is a feasible solution in the decision space Ω, which is established by inequality and equality constraints h i (s) and h j (s).
MOPs aim at simultaneously optimizing more than one objective function which conflict with each other for most cases. Therefore, the tradeoff relation among objectives is in consideration and a criterion assessing multiple objectives is in need. To this end, the concept of Pareto dominance is introduced as follows. For two feasible solutions s 1 and s 2 of Problem (6), s 1 is said to dominate s 2 if and only if The Pareto optimal set is constituted of feasible solutions not dominated by any other solutions, and the corresponding objective vectors of Pareto set form PF. MOEAs are designed to find a collection of optimal tradeoff solutions so as to approximate PF. In MOEA, commonly, a set of solutions evolve altogether in each generation via crossover, mutation, and nondominated-based selection.

B. Multiobjective Group Sparse Unmixing Model
The endmember bundle based sparse unmixing problem is translated into a bi-objective optimization problem and addressed by the designed algorithm. Considering the prior grouping information, we design the bi-objective minimization of reconstruction error and group sparsity as follows: In this article, we take spatial information into consideration via a uniform endmember selection among all pixels in the image. s ∈ {0, 1} m is a binary vector denoting endmember selection from spectral library with m spectra, in which s i = 1 means the ith material in the library exists in the image Y, and 0 otherwise. f 1 (s) is the reconstruction error of the observed image and is set finite only when the endmember selection satisfies the physical meaning of sparsity. In accordance with [28], we adopt the same algorithm HySime [44] to estimate the number of endmembers k aforehand, which accounts for an indicator in reconstruction error evaluation and an expectation of sparsity. The usage of estimated endmember numbers is a common tactic [28] to constrain the sparsity around a computed positive integer instead of 0 for accuracy. A s is the corresponding subset of library A whose indices are denoted by the nonzero elements in s, and X s is the inverse abundance matrix of the new library A s calculated from nonnegative least squares algorithm is a mixed norm integrating grouping structure of the spectral library, where 0 < q < 1 L G,0,q quasi-norm better expresses both intergroup and intragroup sparsity than the regular L 0 -norm, enforcing sparsity on the endmember selection within bundle as well as bundles per se. The group sparsity structure manifests that it is the materials instead of the spectra that are sparse in a pixel, and describes the sparsity within each bundle, meaning that only one or a few endmembers are supposed to exist. f 1 (s) stands for the reconstruction error, and the quasi-norm denoted by f 2 (s) forms the group sparsity constraint. The two optimization objectives together compose the demands of sparse unmixing tasks. By optimizing (7), tradeoff solutions could be obtained to balance between accuracy and sparsity.

C. Optimization
In order to handle the nonconvex discrete optimization caused by the L G,0,q norm, we propose a multiobjective evolutionary sparse unmixing method tailored for spectral library with group structure. We apply HySime on image data for endmember number estimation in advance, and k-means on spectral angle distance among the spectral libraries to form the group structure. Note that a predefined grouping structure is preferred, in which case the clustering step is not in need. As is demonstrated in pseudocode in Algorithm 1, the optimization procedure of the proposed MO-GSU adopts a framework similar to NSGA-II, in which the environmental selection via nondominated sorting and crowding distance remains; yet, an additional intragroup local search and a group-oriented adaptive genetic operator are embedded for intragroup exploitation. The procedure of MO-GSU can be described as follows.
1) Initialization: First, we initialize a set of binary vectors with a sparse distribution as the initial population P = {s 1 , s 2 , . . . , s N } with N individuals.
2) Evolution: In the first stage of evolutionary algorithm, the common NSGA-II with the traditional one-point crossover and one-point flipping mutation operators is implemented. In the second stage, the overall population update procedure is exhibited in Fig. 1 for each generation. Three subpopulations are generated to form the updated population: In the left branch of Fig. 1, for sufficient local exploitation, an intragroup local search module designed for selection within bundle is conducted on a selected reference solution s to generate population P LS as the first subpopulation. In the middle branch of 1, the current population P is duplicated as the second subpopulation. In the right branch of 1, a parent population P a consisting 2N individuals are selected via tournament selection from current population P. Then N pairs of parent solutions are drawn without replacement to generate an offspring population O with the proposed grouporiented adaptive genetic operator. The offspring population O with N individuals is the third subpopulation. The combination of the three subpopulations P, O, and P LS are evaluated and sorted via nondominated rank and crowding distance to form the updated population. The number of evaluations denotes the cumulative times of calculating the two objective functions. We

Algorithm 1: MO-GSU.
Input: A (spectral library), Y (image data), N (population size), n g (group number). Output: s (endmember selection), X s (abundance use the number of evaluations as the stopping criterion to limit the computational cost under proper range. 3) Optimal solution selection: After meeting the stopping criterion, we return the current population as the proximate Pareto optimal set. We then select the most compatible solution for the unmixing task in consideration of the estimated endmember number and the knee point. Considering that neglected endmembers cause more inaccuracy than redundant ones, we first keep the solutions containing no less endmember than the estimated number, among which we choose the knee point as the final solution.
4) Abundance inversion: The final solution denotes the optimal endmember selection from the spectral library, and the abundance matrix is then calculated accordingly using nonnegative least squares algorithm. In the next two sections, we detail the local search module and the group-based genetic operator, respectively.

D. Intragroup Local Search
MO-GSU synthesizes the genetic algorithm for its potential on global exploration and a novel intragroup local search strategy based on group structure for sufficient local exploitation. The local search strategy is performed in the second stage of the optimization, when, in general, the global exploration is accomplished.
Algorithm 2 details the local search procedure which requires a local search population size N LS so as to restrict complexity determined by the group size. A random solution s is drawn from the nondominated set of current population as the reference solution. A random group G i is further drawn from the nonzero groups of s. Then a solution set P LS with up to N LS solutions is generated, among which the elements in G i vary while elsewhere are identical to the reference solution. Specifically, if the size of the chosen group G i , |G i |, is smaller than N LS , the reference solution is duplicated for |G i | times. For the jth duplicate, the element of index G ij is set as 1 and other indices in G i as 0, where G ij denotes the jth index of group G i . If |G i | is greater than N LS , the operation holds except that the duplication is carried out for only N LS times, and N LS out of |G i | indices are randomly selected to construct N LS solutions. Fig. 2 provides a graphic demo of the local search module to select reference solution and random group, generating a local search population P LS .
The local search module is proposed to compensate for the failure of evolutionary unmixing to locate the endmember within a bundle. Genetic algorithms determine the primary searching direction quickly due to its competence in global exploration; nevertheless, the local exploitation suffers relatively. As a result, difficulty arises in the more delicate adjustment for the solutions during the optimization process. In the scenario of sparse unmixing on high correlation spectral library, multiobjective evolutionary approaches are able to narrow the selection of endmember down to bundlewise. However the convergence speed tends to slow down when selecting one particular endmember from a group in the experiments. The local search module benefits endmember selection within bundle by conducting a more sufficient exploitation. In Section 3.3, Fig. 4 demonstrates how intragroup local search improves the endmember selection efficiency in experiment.

E. Group-Oriented Adaptive Genetic Operators
For the first stage of the evolutionary process, MO-GSU adopts the ordinary binary genetic operators, namely one-point crossover and flipping mutation, and for the second, we propose a new group-based variation with adaptive mutation probability. Algorithm 3 details the proposed variation. The crossover between two parent individuals is performed in groups uniformly. Elements in the same group are treated as a collective unit, and the units from two parents combine with each other at a given probability to reconstruct a new offspring. Instead of arbitrarily selecting a location and concatenate two segments from two parents, the group-based crossover preserves the endmember selection within a bundle and reassemble bundles as a whole.
The mutation of an offspring is performed with an adaptive flipping probability to sustain the sparsity within group. For the nonzero groups G i in the offspring solution o, the adaptive probabilities in group G i are given by the following equation: where p 1 denotes the probability of an element flip from 1 to 0, and p 0 versa, d is the number of elements in group G i , d 1 is the number of the current nonzero elements of solution o in group G i , and p is the overall flipping probability.
Compared to the ordinary flipping mutation with the same overall flipping probability, the proposed mutation strategy sustains intragroup sparsity. The above formality satisfies the two conditions in (9). The first condition is that the overall flipping probability is equal to p, and the second is that the expected number of nonzero elements after mutation is equal to d 1 Such conditions illustrate that the overall flipping probability of the proposed mutation strategy is the same as general mutation operators. Nevertheless, the number of the nonzero elements within group remains to be d 1 after the mutation. The adaptive flipping probability enhances the intragroup sparsity, which is consistent with the assumption of sparsity within bundle. Considering the intragroup correlation and sparsity, genetic operators are supposed to transfer the position of the current nonzero element to another in the same group. Here, we give two theorems of the superiority of the proposed group operator to implement a more thorough exploitation over the classic onepoint or uniform crossover and flipping mutation.
Theorem 1: For a nonzero group segment g = G i and the Humming distance H(g, h) between two segments g, h, we have P (H(g, g am ) = 2) > P (H(g, g f ) = 2) where g am denotes the group segment after adaptive mutation, and g f denotes that after flipping mutation.
Proof: Due to the intragroup sparsity formed by the objective function, we assume that there is no more than one endmember in each bundle, i.e., g 0 = 1. Under the above circumstance where d is the length of g. According to the assumption, the d 1 in (8) is substituted into 1. Therefore P (H(g, g am ) = 2) P (H(g, g um ) = 2) = Ω(d).
Theorem 2: For the nonzero group segment g = G i in individual p 1 , the segment h of the same group in individual p 2 , and the Humming distance H(g, h) between two segments, we have where g , h denote the segment after mutation and crossover, q denotes the probability of valid intragroup transfer after variation, and the superscripts represent different genetic operators. Specifically, g denotes the proposed group operator, o denotes flipping mutation and one-point crossover, and o denotes flipping mutation and uniform crossover.
Proof: We follow the same assumption in Theorem 1, i.e., g 0 = 1. For convenience, let Hence q g > q o = q u .

Remark:
The ability of an operator to transfer one selected endmember to another of the same bundle is expressed via Hamming distance [45]. The Hamming distance of the group segments before and after variation is equal to two accounts for a valid intragroup transfer, as in choosing a different endmember in the same group while remaining the intragroup sparsity. We evaluate operators by means of the valid transfer probability q, where a higher q accounts for an operator more effective. According to Theorem 1, the proposed group mutation operator with adaptive mutation probability is more likely to implement intragroup transfer than flipping mutation, which indicates that considering mutation alone, group-based operator employs a more thorough exploitation. Theorem 2 evaluates the combination of mutation and crossover operators. We come to the conclusion that group-based operator performs more effectively than flipping mutation combined with one-point or uniform crossover.

III. EXPERIMENTS
In this section, the performance of MO-GSU is evaluated through experiments conducted on three synthetic data and one real data. It is compared with state-of-the-art sparse unmixing methods, bundle-based, and multiobjective methods.
The performance of unmixing accuracy is evaluated and quantified via the metric of signal-to-reconstruction error (SRE) where X real is the real abundance matrix and X is the estimated one. The larger value of SRE denotes the unmixing performance more effective.
Considering that MO-GSU is an endmember selection based method, we adopt the same evaluation parameter in [46] for selection accuracy assessment, namely true positive rate (TPR) and false positive rate (FPR). In the scenario of endmember selection, an endmember is positive if selected in the ground truth or the algorithm, and negative otherwise. In general, TPR and FPR are constrained between lower and upper bounds 0 and 1. The TPR equal to 1 and FPR equal to 0 imply a perfectly accurate selection in the algorithm, as in choosing the exact ground truth endmembers. MO-GSU is compared with classic sparse unmixing methods, multiobjective-based, and grouping-based methods: SUn-SAL [17] relaxes the L 0 sparsity into L 1 norm and optimize abundances alternatively. Additionally, a library pruning algorithm MUSIC [47] is executed ahead to alleviate the difficulty of high dimensionality down to the size of 40. RS-FoBa [48] is a greedy method for forward and backward selecting endmember simultaneously. SUnCNN [20] gains abundances via deep convolutional network. MOSU [27], SMOSU [28], and PMOSU [29] are three multiobjective unmixing approaches that transform the unmixing model into an MOP. MOSU adopts a classic nondominance-based algorithm NSGA-II and SMOSU adopts a modified decomposition based algorithm MOEA/D for endmember selection, respectively. PMOSU prunes the library via a gradual compression of search space and performs sparse regression in the same time. The methods presented in [38] incorporate the structure of endmember bundles and present sparsity in mixed norm or double sparsity including three kinds of penalty: group, elitist, and fractional penalty, respectively, corresponding to three quasi-norms, L G,1,2 , L G,2,1 , and L G,1,q .
To verify the efficiency of MO-GSU for dealing with correlation in spectral library, different libraries of high correlation are used in the experiment. In Synthetic data 1 and Synthetic data 2, libraries of different degrees of correlation and grouping structures are first designed based on United States Geological Survey (USGS) library to simulate endmember bundles. In Synthetic data 3, a modified library containing 1543 spectra with complete reflectance of 224 bands is extracted from the USGS library, which also includes massive correlation. All the methods except SUnCNN are conducted using MATLAB R2018b on a desktop with Inter Core i5-8265 U CPU @ 1.60 GHz 1.80 GHz of 8-GB memory running 64-b Windows operation system. SUnCNN is implemented in PyTorch framework running on NVIDIA GeForce RTX3060 Laptop GPU.

A. Synthetic Data 1
First, we generate three synthetic datasets from three designed high correlation spectral library with different degrees of correlation, which are constructed in the same way as [38] and [49]. Twenty spectra are randomly drawn from the USGS spectral library and used to form a library of size 400 including groups of high correlation spectral signals by simulating physical variations. USGS library contains thousands of spectral signals covering 224 bands. Specifically, 20 endmembers are randomly selected from the USGS library, respectively, generating 20 spectra via different degrees of scaling variations and nonlinear quadratic perturbations as a simulation of variability induced by illumination and intrinsic factors. Hence, three prior libraries with increasing degrees of correlation and a naturally predetermined grouping structure are available, denoted as library 1, 2, and 3. Fig. 3 illustrates the spectral library 3 built from the procedure mentioned above.
We generate a spatial-correlated abundance map via spatial low-pass filter and restrict the abundance of each material no larger than 0.7 to increase unmixing difficulty. The image data to be unmixed is obtained by multiplying abundance map with randomly selected endmembers complying with the linear unmixing model, and Gaussian white noises of 20 and 30 dB are added to imitate sensor and measurement deviation.
The proposed MO-GSU is compared with the abovementioned methods, of which the ones marked with an asterisk symbol denote an additional pruning process MUSIC ahead. The three methods from [10], namely "Group", "Elitist" and "Fractional" in Table I, respectively denote approaches integrating group, elitist and fractional penalties. The three penalties respectively correspond to sparsity regularization of the grouping based-sparsity of L G,1,2 , L G,2,1 , and L G,1,q norms. The objective function in MO-GSU adopts a similar sparsity to the fractional penalty. The parameters are set fixed instead of conducting a grid search. By contrast, the multiobjective evolutionary methods require no determination of parameter settings, which is convenient for practical use. MO-GSU is implemented on the basis of PlatEMO [50] with a novel algorithm and a homologous operator tailored for sparse unmixing with the group structure proposed.  I  SRE RESULTS OF SYNTHETIC DATA 1, WHERE THE ASTERISK SUNSAL AND RSFOBA DENOTE CORRESPONDING METHODS COMBINED WITH THE PRUNING  ALGORITHM MUSIC   TABLE II  EXECUTION TIME(S) OF SYNTHETIC DATA 1   Table I shows the SRE results of conjoint endmembers on Synthetic Data 1 with 30-dB noise and different degrees of correlation. Table I demonstrates that the proposed MO-GSU outperforms the classic and grouping-based methods on the highcorrelated spectral library in most cases, especially with higher degrees of correlation. Among the grouping-based methods, the one incorporating fractional norm achieves better performances than elitist and group penalties. Accordingly, the effectiveness of the L G,1,q norm is testified since the optimization approaches are alike. Compared to the L G,1,q norm fractional penalty, the superiority gained by the inspired L G,0,q norm-based MO-GSU derives from the distinction of evolution optimization approach with group operators. In spite of the better performance MUSIC-SUnSAL obtains on 9 and 10 endmembers, MO-GSU operates on the original high correlation library with ten times more spectra than the pruned and correlation eliminated library. When the prior library is fixed to the original one, the SRE results of SUnSAL are diminished. Besides, the advantages brought by the pruning process tend to reduce when the correlation increases due to the possibility to prune the true endmembers. Table II displays the average execution time of MO-GSU as well as multiobjective and grouping-based unmixing methods.
The multiobjective methods are implemented with the fixed evaluation number of 20 000 and the grouping methods with default settings. Compared to other multiobjective methods, the proposed MO-GSU can achieve better SRE results under limited time. Besides, the comparable time consumption indicates that the proposed local search and group-oriented adaptive genetic operator module do not improve unmixing effect by consuming more computational time.

B. Synthetic Data 2
In order to testify the effectiveness of the proposed method on high correlation spectral library, we build three 2000 size libraries with different correlation level in the same way as Synthetic Data 1. In Synthetic Data 1, n group randomly selected endmembers generate s group spectra respectively, presenting n group high correlation spectral groups with the size of s group . Synthetic Data 2 constructs three libraries of the same size but employing different settings of n group , s group , which are (100, 20), (40,50), and (20,100), enforcing a progressively enhancive correlation within each group. Synthetic Data 2 generates abundance map  based on Dirichlet distribution which is compatible with ANC and ASC constraints and bears no spatial correlation. Table III shows the TPR, FPR, and the SRE results of endmember selection of different combination of three levels, namely noise, endmember number, and library correlation. The selection accuracy of MO-GSU on large-scale spectral library is validated attributing to most TPR and FPR values equal to 1 and 0, which is an indication of perfectly accurate selection. In general, the selection difficulty increases in response to the growth of endmember numbers and noises. Three spectral libraries with progressively strong correlation examine the capacity of MO-GSU for intragroup selection. Table III, MO-GSU reaches perfect selection in most cases.

As in
From Table III, the effectiveness of MO-GSU suffers with a larger number of groups, seemingly a negative outcome. Note that such outcome is more likely the aftermath of the inevitable drawback of multiobjective unmixing assessing criteria rather than MO-GSU's inability of extremely complicated combination. For instance, in the experiment on 20-dB noise, 10 endmembers and group parameter (n group , s group ) = (40,50), the TPR and FPR of the final solution provided by MO-GSU are 0.90 and 5.03 × 10 −4 . In this case, the failure of the proposed algorithm to correct selection is a systematic drawback, since the objective vector of MO-GSU's solution is (48.0344, 10) and of the correct selection is (48.0346, 10). Therefore, in Pareto-dominant wise, MO-GSU's solution is a better solution than the correct selection. Hence, it can be safely inferred that the correct selection has been visited yet discarded in the evolutionary process.

C. Synthetic Data 3
USGS library stores spectral signal data with group structure naturally formulated by bundles of multiple endmembers per material. Hence, it is naturally a high correlation library with endmembers varying in shape and different degrees of correlation. Synthetic Data 3 conducts abundance and image data generation on a subset of the USGS spectral library version seven (Splib07). Some spectra from the original USGS library have different bands of data missing from the 224 bands, which are excluded from the prior library. The modified spectral library contains 1543 endmembers with complete reflectance of 224 bands. Synthetic Data 3 adopts the same abundance and image generation methods as Synthetic Data 1, while the spectral libraries vary. Fig. 4 is a demonstration of how endmember selection evolves in MOSU and MO-GSU. Fig. 4 depicts that classic genetic algorithm is capable of locating the right group efficiently, yet ineffective to adjust intra-group selection for a long time. On the contrary, after a certain amount of regular evolution by NSGA-II, MO-GSU is more flexible and approximates both the correct group and the endmember within, showing a promising potential. Fig. 5 is a graphic illustration of abundance maps generated from the above methods on the image of six endmembers and 30-dB noise. Some subgraphs are pure dark, meaning the corresponding endmember is not chosen by the algorithm. Compared with the abundance from other methods, MO-GSU generates a figure of sharper edge, more similar to that of the ground truth. Fig. 5 visually examines the effectiveness of MO-GSU by a more accurate outcome.  difficult circumstances with 20 and 30 dB are presented. Table IV demonstrates that the proposed method is still valid when the group structure of the library is not predefined but gained by clustering method. We choose fractional penalty as the representative of bundle-based methods, since the sparsity regularization form is the same as MO-GSU. MO-GSU is compared to multiobjective methods with the same amount of evaluations and population size. MO-GSU is conducted on the original prior library of 1543 spectra, while some of the comparison methods perform on the pruned library of size 40. MO-GSU outperforms the remaining methods in most cases. MOSU shows potential when there are more true endmembers, which implies a better exploration than MO-GSU, given limited evaluations. The incompetence of perfectly accurate selection when MO-GSU faces endmember number increasing speaks the unbalance between local and global search. In particular, the convergence of local search within group outweighs global variation among groups immediately after the local search module starts and consequently the recombination of groups is harder. Such a problem may be relieved by setting the starting point of local search adaptive according to estimated endmember numbers. More specifically, the starting point of the second stage is proportional to the endmember number to guarantee a more thorough exploration.

D. Real Data
We choose Cuprite data acquired by AVIRIS with the size of 250 × 190 pixels for real data experiment. The original image comprises 224 bands ranging from 370 to 2480 nm. From the 224 bands, a few noisy and water absorption ones are discarded and the final image size is 250 × 190 × 188. The library is the first chapter of splib06a with 498 endmembers, including the signatures of the above-mentioned materials and the noisy bands in the spectral library are removed accordingly. Fig. 6 provides a qualitative reference of the ground truth materials. Note that each pixel in Fig. 6 is assigned to a single material. Fig. 7 illustrates the abundance maps of Alunite, Jarosite, Kaolin+Smectite, Buddingtonite, and Chalcedony from the top down obtained by sparse unmixing methods. In practice, the quantitative criteria of the ground truth are hard to achieve. The subimages of Fig. 7 provide a qualitative and visualized assessment of the effectiveness of the proposed method. Compared to Fig. 6, MO-GSU is able to select endmembers more accurately and achieves a clearer outline reflecting structural information, so that the performance on real-world application is testified. Some abundance maps of the grouping method with fractional norm appears to be average distribution among pixels, which indicates that some noise may be introduced into the abundance map.

IV. CONCLUSION
In this article, we propose a multiobjective sparse unmixing method based on group structure. The high correlation within endmember bundle formulates a group structure inside spectral library. Inspired by the fractional sparsity penalty represented by mixed norm, the proposed MO-GSU replaces the L 0 sparsity from classic multiobjective sparse unmixing methods with a L 0,q norm involving group structure G. In order to minimize the reconstruction error of the image and the L 0,q norm sparsity, we adopt the framework of NSGA-II. For the purpose of accurate endmember identification within each group, an intragroup local search strategy is proposed to exploit intragroup selection. We design a group-oriented adaptive genetic operator to further ensure the balance between intergroup exploration and intragroup exploitation. We provide theoretical proof for the proposed group operator of a more sufficient intragroup endmember exploitation. Experimental results on synthetic and real data show that the proposed method is valid for strong correlation spectral libraries. The main shortcoming of evolutionary algorithms lies in the great time consumption and incompetence for guaranteed convergence, especially facing the ever-expanding spectral library. In the future, we would mainly focus on speeding up convergence by introducing gradient-based orientation, as well as adjusting multigradient descend to solve unmixing problem.
Yanyi Wei received the B.S. degree in information and numerical science from Nankai University, Tianjin, China, in 2021, where she is currently working toward the M.S. degree with the School of Statistics and Data Science.
Her research interests include hyperspectral unmixing, multiobjective optimization, and remote sensing image processing. Since 2019, he has been an Associate Professor with the School of Statistics and Data Science, Nankai University, Tianjin, China. His research interests include machine learning, remote sensing image processing, and multiobjective optimization.