Objective

2

During hyperspectral remote sensing imaging

each captured pixel in the image always has mixed spectrum of several pure constituent spectra due to the low spatial resolution of hyperspectral cameras and the diversity of spectral signatures in nature scenes. The mixed pixels limit the application of hyperspectral images (HSI)

such as target detection

classification

and change detection

to a great extent. In many real-world applications

subpixel level accuracy is often required to improve the performance. Thus

the unmixing is very essential in the HSI analysis. In most scenarios

the spectral signatures in HSI

also termed endmembers

are unknown in advance. Therefore

the unmixing has to be performed in an unsupervised way

usually involving two steps

namely

endmember extraction and abundance coefficient estimation. In this study

we assume that the generation of mixed pixels in HSI is based on a linear mixing model. Moreover

the observed HSI

endmembers in the image

and the corresponding abundances are all non-negative

according to their physical meaning. In recent years

non-negative matrix factorization (NMF) based approaches has received extensive attention and become a research hotspot because of making full use of the HSI sparsity. However

existing non-negative matrix factorization-based approaches are not robust enough for noisy HSI data. The main reason is that a least square loss function

which is sensitive to noise and prone to large deviations

is always used for endmember extraction in these approaches. To overcome the drawbacks of NMF-based approaches

we can improve the robustness of the unmixing approaches by choosing a new loss function

which is robust. In this study

we utilize more robust L

1

loss function for NMF when performing the endmember extraction and proposed a novel unsupervised hyperspectral unmixing method based on robust NMF.

Method

2

To perform the unsupervised hyperspectral unmixing

with the endmembers unknown

both endmember extraction and the corresponding abundance fraction estimation are needed to be solved. The hyperspectral unmixing problem can be modeled as a NMF due to the similarity of underlying mathematical models. In real scenes

hyperspectral image data often contains noise and missing values

and objective functions used for endmember extraction in existing NMF-based approaches are sensitive to noise and prone to large deviations. Therefore

we use the L

1

norm on the reconstruction error term instead of the common L

2

norm to construct a new objective function. However

the new objective function is nonconvex. We can obtain the global optimal solution of one variable when the other is fixed. In addition

considering the nonsmooth L

1

norm term

to solve the optimization problem in the proposed unmixing approach is challenging. Thus

we propose an efficient multiplicative updating algorithm with the theory of the iterative reweighted least squares. In this way

each iteration can guarantee positive result. Finally

we can obtain the extracted endmembers and estimated abundances when iterative convergences.

Result

2

To verify the effectiveness and competitiveness of the proposed unsupervised unmixing

synthetic and real-world dataset are used in the experiments. The performance of the proposed approach is comprehensively evaluated with visual observations and quantitative measures. For quantitative comparison

three categories of evaluation indexes are used in this study. The indexes to measure the quality of the endmember extraction are the spectral angle distance (SAD) and the spectral information divergence (SID). Similarly

the performance discriminators to evaluate the accuracy of the abundance estimation are the abundance angle distance (AAD) and the abundance information divergence (AID). Moreover

the metrics to evaluate the performance of the reconstruction of spectral mixtures are the root mean square error (RMSE) and the signal to reconstruction error (SRE).With the synthetic dataset

we compare the proposed method with five representative methods. Among them

vertex component analysis (VCA) is a classic geometrical-based method and also the initialization of the proposed method. Minimum volume constrained nonnegative matrix factorization (MVCNMF)

robust collaborative NMF (RCoNMF)

and total variation regularized reweighted sparse NMF (TVWSNMF) are NMF-based approaches

and an untied denoising autoencoder with sparsity for spectral unmixing (uDAS) is deep learning-based approach. The experimental results showed that

when signal-to-noise ratio (SNR) is 20 dB

the SAD (less is better) of the proposed algorithm decreased by 41.3%

43%

65.6%

10.5%

and 58.0% compared with the VCA

MVCNMF

RCoNMF

TVWSNMF

and uDAS; the SID (less is better) decreased by 68.9%

83.6%

95.5%

62.2%

and 97.0%; the AAD (less is better) decreased by 28.5%

12.5%

45.8%

8.4%

and 42.5%; the AID (less is better) decreased by 38.1%

28.7%

51.1%

17.2%

and 49.7%; the RMSE (less is better) decreased by 48.3%

31.2%

68.9%

33.5%

and 67.8%; and SRE (dB) (higher is better) increased by 23.7%

14.5%

60.0%

9.3%

and 47.3%

respectively. With the real world data set

the ability of the proposed approach to extract endmembers can be evaluated using the corresponding USGS(United States Geological Survey) library signatures. Moreover

given that the ground truth abundance maps are unknown exactly

the abundance estimation results can only be evaluated qualitatively. The experimental results demonstrated that the extracted endmembers achieved a good match with the ground truth signatures. In addition

the proposed approach achieved piecewise smooth abundance maps with high spatial consistency.

Conclusion

2

In this study

we proposed a novel NMF-based unmixing approach to perform the unsupervised hyperspectral unmixing in noisy environments. Different from the current NMF-based unmixing approaches

robust L

1

norm is used as the reconstruction error term in the objective function to improve the accuracy of the endmember extraction and abundance estimation. Compared with the state-of-the-art approaches

experimental results of synthetic data set and real world data set illustrate that the proposed approach performs well under different noise conditions

especially in low SNR.