Blind spectral unmixing based on sparse component analysis for hyperspectral remote sensing imagery

Abstract

Recently, many blind source separation (BSS)-based techniques have been applied to hyperspectral unmixing. In this paper, a new blind spectral unmixing method based on sparse component analysis (BSUSCA) is proposed to solve the problem of highly mixed data. The BSUSCA algorithm consists of an alternative scheme based on two-block alternating optimization, by which we can simultaneously obtain the endmember signatures and their corresponding fractional abundances. According to the spatial distribution of the endmembers, the sparse properties of the fractional abundances are considered in the proposed algorithm. A sparse component analysis (SCA)-based mixing matrix estimation method is applied to update the endmember signatures, and the abundance estimation problem is solved by the alternating direction method of multipliers (ADMM). SCA is utilized for the unmixing due to its various advantages, including the unique solution and robust modeling assumption. The robustness of the proposed algorithm is verified through simulated experimental study. The experimental results using both simulated data and real hyperspectral remote sensing images confirm the high efficiency and precision of the proposed algorithm.

Introduction

Hyperspectral remote sensing technology can obtain abundant spectral information to identify and distinguish spectrally unique materials, and it is able to provide a large amount of images for various thematic applications (Bioucas-Dias et al., 2013). Hyperspectral unmixing is one of the most prominent research topics for hyperspectral remote sensing. Due to the limited spatial resolution of the sensor, a single pixel may consist of various materials in the scene, which leads to the existence of mixed pixels in hyperspectral remote sensing imagery. The hyperspectral unmixing technique is an effective method of solving the mixed pixel problem, by which the measured mixed spectrum of a pixel is decomposed into a set of pure spectra called endmember signatures and a set of corresponding fractional abundances that indicate the proportion of each endmember present in the pixel (Keshava and Mustard, 2002). The linear mixture model (LMM) has been widely used in the past few decades to solve hyperspectral unmixing problems. Most of the existing unmixing algorithms are based on the LMM, which assumes that the pixels are mixed as a linear combination of the endmember signatures with the corresponding abundances. Due to the physical meaning of the abundances, there are two constraints, which are termed the abundance sum-to-one constraint (ASC) and the abundance nonnegative constraint (ANC). Despite the fact that the LMM is not always true, it is generally recognized as an acceptable model for real-world applications.

Geometrical-based linear spectral unmixing approaches have been proposed and used in many different fields because of their low computational complexity and specific physical meaning (Bioucas-Dias et al., 2012, Geng et al., 2013). The geometrical-based approaches, such as N-FINDR (Winter, 1999), vertex component analysis (VCA) (Nascimento and Dias, 2005b), the simplex growing algorithm (SGA) (Chang et al., 2006), and the minimum-volume enclosing simplex (MVES) (Chan et al., 2009), exploit the fact that the observed spectral vectors will be contained in a simplex after affine transformation, where the endmembers are located in the vertexes. However, in the case of real scene, the results of some of the geometrical-based methods may not be satisfactory since there are no pure pixels near the vertexes of the simplex. In order to improve the accuracy, particle swarm optimization (PSO) (Eberhart and Kennedy, 1995), which is a global optimal searching algorithm based on swarm intelligence theory, has been introduced to unmixing, where the endmember extraction is described as a combinational optimization problem (Zhang et al., 2011, Zhuang et al., 2015). The most commonly used abundance estimation method is the fully constrained least squares (FCLS) (Heinz and Chang, 2001) algorithm. However, the precision of the estimated abundances always hinges on the performance of the endmember extraction.

When the observed data are mixed and without pure pixels in the scene, the statistical methods are a powerful alternative (Ma et al., 2014). Statistically, the spectral unmixing problem can be considered as a typical blind source separation (BSS) problem, where the mixed pixels, endmembers, and abundances in the hyperspectral unmixing can be considered as the observed signals, mixing matrix, and source signals, respectively (Yang et al., 2011). Independent component analysis (ICA) is a powerful tool to solve the BSS problem. It assumes that the source signals (abundances) are mutually independent, and has been utilized for hyperspectral unmixing (Moussaoui et al., 2008, Nascimento and Dias, 2005a, Wang and Chang, 2006, Xia et al., 2011). However, the mutually independent assumption of ICA can have a negative effect on the abundance estimation, since the ASC implies that the source signals are statistically dependent. Taking the ANC into consideration, another statistical-based approach—nonnegative matrix factorization (NMF)—has been introduced to blind spectral unmixing. NMF decomposes the observation data matrix into two low-dimensional nonnegative matrices, serving as the endmembers and abundances, respectively. Although the NMF-based methods naturally ensure the nonnegative property of the abundances, the cost function of NMF is nonconvex. In order to guarantee a unique solution, the NMF-based methods always need additional constraints (Zhu et al., 2014). In recent years, a large number of constrained NMF methods have been proposed. Minimum-volume-constrained NMF (MVCNMF) (Miao and Qi, 2007), as one of the typical constrained NMF algorithms, is a combination of a convex geometry assumption together with NMF, which enforces the minimization of the simplex volume in the iterative process. Abundance separation and smoothness NMF (ASSNMF) adds the abundance separation and abundance smoothness constraints into NMF to take effect in the spectral and spatial domains, respectively (Liu et al., 2011). Since the NMF-based methods can simultaneously obtain the endmember matrix and the abundance matrix, the overall accuracy is better than that of the two-step based methods, which are usually affected by cumulative errors.

Recently, to take advantage of compressive sensing, many sparse unmixing methods have been proposed in a semi-supervised fashion based on a known endmember spectral library. These methods include sparse unmixing by variable splitting and augmented Lagrangian (SUnSAL) (Bioucas-Dias and Figueiredo, 2010) and the newly developed algorithm called collaborative SUnSAL (CLSUnSAL) (Iordache et al., 2014). However, due to the differences in the spectra acquisition conditions, the endmember spectral dictionaries need to be calibrated before use in real applications, and these methods always have a high computational complexity. Since the sparsity is an important feature of the ground object distribution and a powerful constraint, it has also been introduced into some blind spectral unmixing algorithms, such as piecewise smoothness NMF with sparseness constraint (PSNMFSC) (Jia and Qian, 2009), $l_{1/2}$ sparsity-constrained NMF ($l_{1/2}$ NMF) (Qian et al., 2011), and manifold regularized sparse NMF (Lu et al., 2013). These various algorithms have all demonstrated the promising performance of sparsity in hyperspectral unmixing.

In this paper, to consider the properties of sparsity for hyperspectral remote sensing imagery, a novel blind spectral unmixing method based on sparse component analysis (BSUSCA) is proposed. The sparse component analysis (SCA) algorithm is robust to noise and can obtain a unique solution by a proper permutation and scaling of the mixing matrix, and has been successfully applied in many fields, such as pattern recognition (Wright et al., 2010), feature extraction (Li et al., 2006), image processing (Elad et al., 2010), speech separation (Yu-jing and Yu, 2011), and compressed sensing (Stojnic et al., 2008).

Considering the complexity and homogeneity of the ground object distribution, the number of distinct materials in a scene will be much larger than in a single pixel (Keshava and Mustard, 2002). Consequently, the endmembers will be sparsely distributed in the original data, and the corresponding abundance matrix will also be sparse. In BSUSCA, the sparsity is incorporated into the endmember update and sparse coding processes as the assumption and the constraint, respectively. The performance is further improved by introducing two significant iterative operations, one of which is used to obtain the optimal endmember set, and the other is used to maximize the sparsity of the corresponding abundances. The contributions of this paper can be summarized as follows:

• The BSUSCA method is proposed to solve the problem of mixed data without pure pixels. A robust SCA-based mixing matrix estimation algorithm is utilized to sequentially update the endmember signatures. It provides a convenient way to divide a matrix, which perhaps contains some information of interest, into simpler and meaningful pieces.

• In order to make full use of the relevant information of the mixing matrix and source signals, and to improve the unmixing performance, an efficient alternating iteration operation is utilized to simultaneously update the endmembers and the corresponding abundances.

• The least absolute shrinkage and selection operator (LASSO) problem is reformulated in the sparse coding stage in this algorithm. The ADMM is also utilized to accelerate and simplify the problem because of its benefits of dual decomposition and the augmented Lagrangian method for the constrained optimization. In addition, the sparse properties of the abundance matrix are preserved and maximized in the process of the sparse coding. The sparse structure of each spatial source (abundance) is also impacted by a sequence of corresponding endmember spectra, instead of a single spectrum.

The rest of the paper is organized as follows. In Section 2, the LMM and the original SCA algorithm are presented. Section 3 describes the proposed BSUSCA algorithm in detail. In Section 3, the experimental results using both simulated and real hyperspectral remote sensing images are presented and analyzed. Finally, the conclusions are summarized in Section 5.