Manifold regularization for sparse unmixing of hyperspectral images
- Junmin Liu^{1}Email author,
- Chunxia Zhang^{1},
- Jiangshe Zhang^{1},
- Huirong Li^{1} and
- Yuelin Gao^{2}
Received: 5 May 2016
Accepted: 10 November 2016
Published: 24 November 2016
Abstract
Background
Recently, sparse unmixing has been successfully applied to spectral mixture analysis of remotely sensed hyperspectral images. Based on the assumption that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance, unmixing of each mixed pixel in the scene is to find an optimal subset of signatures in a very large spectral library, which is cast into the framework of sparse regression. However, traditional sparse regression models, such as collaborative sparse regression, ignore the intrinsic geometric structure in the hyperspectral data.
Results
In this paper, we propose a novel model, called manifold regularized collaborative sparse regression, by introducing a manifold regularization to the collaborative sparse regression model. The manifold regularization utilizes a graph Laplacian to incorporate the locally geometrical structure of the hyperspectral data. An algorithm based on alternating direction method of multipliers has been developed for the manifold regularized collaborative sparse regression model.
Conclusions
Experimental results on both the simulated and real hyperspectral data sets have demonstrated the effectiveness of our proposed model.
Keywords
Hyperspectral images Spectral unmixing Manifold regularizationBackground
In recent years, remotely sensed hyperspectral images have been widely used for various applications ranging from civilian to military purposes (Somers and Delalieux 2009; Settle and Drake 1993; Chang and Heinz 2000; Shimabukuro and Carvalho 1997), since they can provide abundant wavelength information of the land covers with spectral resolution at the micron level (Shaw and Burke 2003). For example, the National Aeronautics and Space Administration (NASA) Hyperion sensor on Earth Observing-1 (EO-1) satellite can provide hyperspectral images with 220 bands and a spectral resolution of the order of 10 nm. Despite high spectral resolution, the relatively low spatial resolution of hyperspectral images leads to mixed pixel problem, i.e., a single pixel usually contains several distinct materials. The existence of mixed pixels seriously restricts the application of hyperspectral images. To cope with mixed pixels, spectral unmixing (Keshava 2003; Bioucas-Dias and Plaza 2012; Meer 2012), which aims at decomposing the observed mixed pixel spectrum into a collection of pure substance spectra, namely endmembers, and their corresponding fractional abundances, has been proposed and widely applied in hyperspectral remote sensing data analysis (Bioucas-Dias and Plaza 2013).
For spectral unmixing, two kinds of models—the linear mixture model (LMM) (Singer et al. 1979) and the non-linear mixture model (NMM) (Hapke 1981)—have been proposed to characterize the mixed pixels. Although the LMM is not as accurate as the NMM to capture the mixing behavior of mixed pixels, it is more popular than the NMM for solving the spectral unmixing problem because of its simplicity and efficiency in most cases (Fan et al. 2009). In addition, the rapidly developed methods in classical signal processing field also provide effective tools to the solution of the LMM (Ma et al. 2014). Nevertheless, modeling the mixed pixels is a very complex and difficult task. In practice, we have to make a compromise between model accuracy and tractability. Therefore, we here focus on the LMM in this study.
The standard LMM used for spectral unmixing assumes that each pixel spectrum is a linear combination of the endmembers present in the scene weighted by the corresponding abundances. From the convex geometry point of view, the LMM forces the mixed pixels to belong to a simplex (or a convex hull), and the vertices of simplex correspond to the endmembers. Based on the geometrical interpretation, many spectral unmixing algorithms have been proposed for endmember extraction such as the N-FINDR (Winter 1999), pixel purity index (PPI) (Boardman et al. 1995), vertex component analysis (VCA) (Nascimento and Bioucas-Dias 2005), simplex growing algorithm (SGA) (Chang 2006) and their variants (Chan et al. 2011; Liu and Zhang 2012; Chang et al. 2010), and for abundance estimation such as the fully constrained least squares (FCLS) (Heinz and Chang 2001), distance geometry-based abundance estimation (DGAE) (Pu et al. 2014), and so on. However, these geometrical-based algorithms are likely to fail when the pixels are highly mixed. As an alternative, the statistical algorithms have been developed by formulating the spectral unmixing as a statistical inference problem. Such algorithms include the dependent component analysis (DECA) (Nascimento and Bioucas-Dias 2012), beta compositional model (BCM) (Zare et al. 2013) and normal compositional model (NCM) (Stein 2003) methods. Although the statistical algorithms have a natural framework for incorporating various priors and endmember variability (Somers et al. 2011; Zare and Ho 2014), it is difficult to derive the close-form expressions of the inference parameters and thus they suffer from high computational complexity.
Most of the spectral unmixing algorithms based on the standard LMM can not automatically determine the number of endmembers present in the scene. In addition, some endmembers produced by these algorithms are not necessarily present in the image, producing the so-called virtual endmembers (Chen 2011). The virtual endmembers can compensate the approximation of the LMM but will result in unidentifiability of the materials. To tackle these problem, the standard LMM has been extended into a semisupervised version (Liu and Zhang 2014; Iordache et al. 2012; Zhong and Zhang 2014; Feng et al. 2014; Iordache et al. 2011), i.e., by assuming that the endmembers are known in advance. Typically, Iordache et al. (2011) have proposed the sparse regression (SR) model by assuming that the endmembers present in the scene belong to a subset of samples available a priori in a library. The unmixing based on SR is called sparse unmixing. Experimental results have illustrated the potential of sparse unmixing in abundance estimation. The success of sparse unmixing relies crucially on the availability of suitable hyperspectral libraries because the libraries are hardly acquired under the same conditions of the remotely sensed images. Fortunately, this problem can be overcome by a delicate calibration procedure to adapt the library to the image or learning of the libraries directly from the data set without other priori information (Charles et al. 2011). The SR problem can be efficiently solved via the sparse unmixing algorithm via variable splitting and augmented Lagrangian (SUnSAL) (Iordache et al. 2011; Bioucas-Dias and Figueiredo 2010) by exploiting the sparse prior induced by the \(\ell _1\) norm. However, a high correlation of the spectral signatures limits the unmixing accuracy. To mitigate this limitation, Iordache et al. (2014a) have developed a collaborative SR (CSR) model by considering the structured sparsity, which exploits the fact that only a few spectral signatures in the library are active, in other words, only a few lines of abundances collected in a matrix are nonzero. Some modifications of the CSR can be found in Iordache et al. (2014b), Tang et al. (2015). However, the improvements in Iordache et al. (2014b), Tang et al. (2015) are limited since only the spectral information is considered to estimated the abundances.
Generally, the size of spectral library is often large, while the number of endmembers present in the scene is very small. Therefore, the fractional abundances are more likely to reside on a low-dimensional submanifold of the high-dimensional ambient Euclidean space. However, existing sparse unmixing methods only consider the Euclidean structure of the data space while ignoring the intrinsic manifold structure of the hyperspectral data. Many previous studies (Lu et al. 2013; Zheng et al. 2011; Guan et al. 2011; Seung and Lee 2000; He and Niyogi 2004; Belkin et al. 2006) have shown that exploiting the local geometrical structure (i.e., the intrinsic manifold structure) is very important to the model learning and data representation. In this paper, we incorporate the manifold regularization (Belkin et al. 2006) to the CSR and develop a novel model, called manifold regularized collaborative sparse regression (MCSR) model. The manifold regularization is characterized by a Laplacian graph which captures the local geometrical structure of the data manifold such that nearby mixed pixels in the intrinsic geometry of the data space are likely to have similar fractional abundances. By adding an additional manifold structure learning term to CSR, our proposed MCSR model is expected to have higher unmixing accuracy than CSR. To solve the MCSR, an optimization algorithm based on alternating direction method of multipliers (ADMM) is developed. It should be noted that similar works of using the manifold regularization have also been introduced in Lu et al. (2013), Tong et al. (2014), but they are different with ours in the following two main aspects. First, the works in Lu et al. (2013), Tong et al. (2014) are based on the standard LMM while our model is an extension of the SR model. Second, multiplicative iterative algorithm is used to optimize the nonnegative matrix factorization model in Lu et al. (2013), Tong et al. (2014), whereas the ADMM is used to cope with the proposed MCSR model.
Related work
In this section, we first describe the sparse unmixing problem and then briefly review the CSR model.
Sparse unmixing
Collaborative sparse regression
In fact, sparse unmixing, as a semi-supervised model, is a typical underdetermined linear system. To solve it, sparsity prior for the fractional abundance of each individual pixel is imposed in (6). Although the high level of sparsity of fractional abundances can enhance the recovery ability of the \(\ell _1\)-minimization problem (6), the highly correlated samples in library still restrict the capability of model (6) to obtain a desirable solution. To tackle this problem, the CSR model has been recently proposed in Iordache et al. (2014a). Unlike the \(\ell _1\)-minimization problem (6), the CSR model simultaneously (or collaboratively) imposes a sparsity to all pixels in the data set by exploiting the fact that pixels in a scene should share the same set of active endmembers, and thus only a few rows of the abundance matrix are nonzero.
Methods
In this section, we introduce an enhanced CSR model, called manifold regularized CSR (MCSR) model, by incorporating a manifold regularization to CSR, and then an alternating direction method is developed to solve the resulting optimization problem.
MCSR
As for CSR problem (8), the data fitting term \(\Vert {Y}-{AX}\Vert _F^2\) is useful for learning the Euclidean structures in the hyperspectral data space. As we have previously mentioned, the size of spectral library is usually very large, while the number of endmembers present in the scene is very small. From a geometric viewpoint, the fractional abundances are more likely to reside on a low-dimensional submanifold of the high-dimensional ambient Euclidean space. Recent studies (Lu et al. 2013; Zheng et al. 2011; Guan et al. 2011; Seung and Lee 2000; He and Niyogi 2004; Belkin et al. 2006) have shown that intrinsic geometric structures on manifolds are very important to the data representation and many manifold learning methods (Tenenbaum et al. 2000; Roweis and Saul 2000; Donoho and Grimes 2003; Belkin and Niyogi 2003; Lin and Zha 2008) have been proposed to recover the geometry of a data set. In the literature of spectral unmixing, many existing methods (Lu et al. 2013) only explore the Euclidean structure while fail to discover the intrinsic geometry structure of the data manifold. Therefore, we want to explore the ability of the intrinsic geometry structure of the hyperspectral data in improving the unmixing accuracy.
To preserve the intrinsic geometry structure of the data, a natural assumption, referred to the manifold assumption (He and Niyogi 2004), is that nearby data points are also nearby points in their low-dimensional representations. However, modeling the global geometric structures of the data is a very big challenge due to the insufficient number of samples and the high dimensionality of the ambient space. In practice, a nearest neighbor graph on the data points is often used to characterize the underlying local geometric structures.
Application of ADMM to MCSR
In this subsection, we propose to apply the alternating direction method of multipliers (ADMM) (Afonso et al. 2011; Yang and Zhang 2011) method to the MCSR problem (10). The ADMM method has recently attracted more attention because it can decouple the variables, and it is usually used to solve the problems of a convex, non-smooth objective function with structured linear constraints.
The convergence of ADMM is guaranteed by the following theorem given in Eckstein and Bertsekas (1992) (see Theorem 8).
Theorem 1
Consider problem (13) with \({G}\) having full columns rank and f, g being closed, proper, convex functions. Then, for arbitrary \(\mu >0\) and \({x}^0, {v}^0, {d}^0\), if problem (13) has a solution, the sequences \(\{{x}^t,{v}^t,{d}^t\}\) generated by Algorithm 1 converges to it; otherwise, at least one of the sequences \(\{{d}^t\}\) and \(\{({x}^t,{v}^t)\}\) diverges.
The convergence of Algorithm 2 is guaranteed by Theorem 1, since it can be expressed as an instance of problem (13). \({G}\) is a full column rank matrix, and functions f, g are closed, proper, convex. These meet the conditions in Theorem 1, and hence the convergence of Algorithm 2 is guaranteed.
Experiments
Synthetic image experiments
Our experiments are first performed on two synthetic images, which were generated from the USGS library \({A}\in {\mathscr {R}}^{224\times 445}\) that contains 445 materials with each having 224 spectral bands that uniformly distribute in the interval 0.4–2.5 μm.
Synthetic Image 1 (SI-1)
Synthetic Image 2 (SI-2)
Results
SREs obtained by the MCSUnADMM, CLSUnSAl, TVSUnSAL, and SUnSAL algorithms on different SNRs
SNR (dB) | 15 | 25 | 35 | 45 |
---|---|---|---|---|
SI-1 | ||||
SUnSAL | 2.302 | 4.262 | 8.851 | 16.035 |
CLSUnSAL | 4.595 | 10.419 | 14.411 | 16.598 |
TVSUnSAL | 2.104 | 4.212 | 8.564 | 15.091 |
MCSUnADMM | 6.519 | 13.924 | 19.763 | 21.144 |
SI-2 | ||||
SUnSAL | 1.516 | 2.415 | 5.184 | 13.032 |
CLSUnSAL | 2.234 | 4.748 | 10.253 | 35.193 |
TVSUnSAL | 2.670 | 6.115 | 15.926 | 38.323 |
MCSUnADMM | 2.673 | 6.095 | 16.023 | 38.517 |
Times in second for the MCSUnADMM, CLSUnSAl, TVSUnSAL, and SUnSAL algorithms on different SNRs
Time (s) | 15 | 25 | 35 | 45 |
---|---|---|---|---|
SI-1 | ||||
SUnSAL | 677 | 405 | 194 | 133 |
CLSUnSAL | 575 | 225 | 213 | 203 |
TVSUnSAL | 763 | 556 | 334 | 256 |
MCSUnADMM | 208 | 192 | 195 | 194 |
SI-2 | ||||
SUnSAL | 1136 | 1168 | 1091 | 1322 |
CLSUnSAL | 1092 | 1136 | 1066 | 1916 |
TVSUnSAL | 645 | 618 | 523 | 642 |
MCSUnADMM | 531 | 504 | 487 | 467 |
Times in second for the SI-1 and SI-2 experiments by the MCSUnADMM, CLSUnSAl, TVSUnSAL, and SUnSAL algorithms when setting the maximum iteration number as the stopping criterion
SUnSAL | CLSUnSAL | TVSUnSAL | MCSUnADMM | |
---|---|---|---|---|
SI-1 | 436 | 439 | 4242 | 4749 |
SI-2 | 754 | 763 | 20,496 | 20,512 |
Real hyperspectral image experiments
In this section, we apply the proposed MCSUnADMM algorithm to real hyperspectral data collected by the Airborne Visible/InfRared Imaging Spectrometer (AVIRIS). The AVIRIS instrument can cover a spectral region from 0.41 to 2.45 μm in 224 bands with a 10 nm bandwidth.
Data sets
Results analysis
We conduct our experiment on a subscene with a size of \(250\times 190\) pixels and 224 bands. However, the bands 1–2, 104–113, 148–167, and 221–224 were removed due to water absorption and noise, and thus only a total of 188 bands were used in the experiment. In addition, the corresponding water absorption and noise bands are also removed from the spectral library \({A}\).
Conclusions
This paper presents a novel sparse unmixing model, which incorporate the manifold regularization into the collaborative sparse regression. The manifold regularization is induced by a Laplacian graph, which can characterize the locally geometrical structure of the hyperspectral data. In this way, the proposed manifold regularized collaborative sparse (MCS) model can consider both the Euclidean structures and underlying manifold structures. To solve the proposed model, an efficient algorithm based on ADMM, called MCSUnADMM, has been developed. And the convergence of the proposed MCSUnADMM algorithm can be guaranteed based on the framework of ADMM. Experimental results with both simulated and real hyperspectral datasets demonstrate the effectiveness of the proposed model and algorithm. However, the efficiency of the proposed MCSUnADMM algorithm is affected by two regularization parameters. Therefore, our future work will focus on designing a strategy to adaptively set these parameters. In addition, further experiments with different scenes of real hyperspectral images are need to investigate the performances of our proposed model and method.
Declarations
Authors' contributions
JL and JZ gave the basic idea. CZ and HL participated in the collection and analysis of the data. JL and YG participated in the design of the study and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was supported in part by the National Basic Research Program of China (973) (Grant No. 2013CB329404), in part by the National Natural Science Foundation of China (Grant Nos. 11401465, 61572393, 11601415, 11671317, and 61561001), the Projects funded by China Postdoctoral Science Foundation (Grant Nos. 2014M560781, 2016M590934), and by the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2014JM2-6098).
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
the USGS library can be available at http://speclab.cr.usgs.gov/spectral.lib06, the synthetic image is produced by a hyperspectral imagery synthesis tools at http://www.ehu.es/ccwintco/index.php, the real hyperspectral data sets can be available at http://aviris.jpl.nasa.gov/html/aviris.freedata.html, and the mineral map for the real hyperspectral image is in http://speclab.cr.usgs.gov/cuprite95.tgif.2.2um_map.gif.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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