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Nonlinear spectral mixture analysis (NSMA) is a widely used unmixing algorithm. It can fit the mixed spectra adequately, but collinearity effect among true and virtual endmembers will decrease the retrieval accuracies of endmember fractions. Use of linear spectral mixture analysis (LSMA) can effectively reduce the degree of collinearity in the NSMA. However, the inadequate modeling of mixed spectra in the LSMA will also yield retrieval errors, especially for the cases where the multiple scattering is not ignorable. In this study, a generalized spectral unmixing scheme based on a spectral shape measure, i.e. spectral information divergence (SID), was applied to overcome the limitations of the conventional NSMA and LSMA. Two simulation experiments were undertaken to test the performances of the SID, LSMA and NSMA in the mixture cases of treesoil, tree-concrete and tree-grass. Results demonstrated that the SID yielded higher accuracies than the LSMA for almost all the mixture cases in this study. On the other hand, performances of the SID method were comparable with the NSMA for the tree-soil and tree-grass mixture cases, but significantly better than the NSMA for the tree-concrete mixture case. All the results indicate that the SID method is fairly effective to circumvent collinearity effect within the NSMA, and compensate the inadequate modeling of mixed spectra within the LSMA.

Land surfaces are inherently heterogeneous at large scales, and mixed pixels exist widely in remotely sensed images due to coarse spatial resolutions. The spectral mixture not only results in errors for the materials’ discrimination and classification, but also greatly hinders the development of quantitative remote sensing. On the other hand, understanding abundances (fractions) of components (endmembers) will greatly benefit the modeling of biogeochemical cycles and climate at both global and regional scales [

The simplest SMA method is to model a mixed spectrum as a linear combination of the pure spectra of the endmembers weighted by their fractional coverage (i.e. Linear SMA, LSMA) [

Accordingly, a number of Nonlinear SMA (NSMA) methods have been developed [

To overcome the dilemmas for applying LSMA and NSMA to spectral mixture cases with significant multiple scattering, we applied a generalized spectral unmixing scheme proposed by [

1) LSMA: The LSMA is reasonable when multiple photon interactions are believed to be negligible, in which the reflectance of a pixel is modeled as a weighted sum of the reflectance of each endmember within a pixel:

where

member i; f_{i} is the fraction of the endmember, M is the number of endmembers, and

Model fit is usually assessed by the root-mean-square-error (RMSE):

2) NSMA: The NSMA accounts for the multiple photon interactions by introducing virtual endmembers into the LSMA. The virtual endmembers are presented by the cross-products of the true endmembers [

where

c_{i}, and c_{i,j} are the contribution factors of the true and virtual endmembers. Reference [

where

The model fit is also assessed by RMSE calculated as Equation (3).

In the mixture cases that the three dimensional (3-D) objects are dominant, the NSMA can fit the mixed spectrum better than the LSMA with a smaller value of RMSE [

To circumvent the collinearity effect resulted from the interaction terms, a generalized unmixing framework was utilized, in which spectrum of a target pixel was matched with a modeled spectrum, and the matching criterion can be either spectral-shape-based or spectral-magnitude-based measures [

s.t.

where R is the target spectrum; R_{mod} is the spectrum modeled as the weighted linear sum of the endmembers’ spectra (same as the LSMA); G is the objective function, which is the spectral matching criterion selected to measure the difference between R and R_{mod}; the fractions of the endmembers,

To compensate the modeling errors caused by multiple scattering, the spectral information divergence (SID), which is a spectral-shape-based measure, is used in this study because of its high effectiveness [

where p_{i} and q_{i},

In order to carry out the simulation analysis for evaluating the performances of the unmimxing methods, reflectance spectra of endmembers were collected from the spectral library contained in the software ENVI. Specifically, reflectance spectra of tree, soil, concrete and grass were derived from the John Hopkins University spectral library. All the spectra data sets were resampled to the spectral range of 400 - 2400 nm with an interval of 1 nm.

The spectra were grouped as tree-soil mixture, tree-concrete mixture and tree-grass mixture as shown in Figures 1(a)-(c). Mixed pixels were generated according to the method in [

with

For each two-endmember combination, two groups of mixed spectra were synthesized with different levels of Gaussian noise (σ) and nonlinear mixing intensities (i.e. c_{12}).

The Group I data were generated to test the sensitivity of the unmixing methods to the Gaussian noise contained in the spectra, in which c_{12} was set as a constant value of 0.15 for each spectrum. Values of f_{1} increased from 0% to 100% with steps of 1%, and correspondingly

The Group II data were generated to test the sensitivity of the unmixing methods to the nonlinear mixing intensities. Values of f_{1} and f_{2} were determined as in the first dataset, while the c_{12} was changed from 0 to 0.2 with steps of 0.02. Gaussian white noises were added with the constant standard deviation of 0.05. Corresponding to each c_{12}, the Gaussian noises were also run 500 times with different random numbers.

Performance of the unmixing method was assessed by comparing the estimated and true tree fractions. For the first dataset, RMSE corresponding to each level of σ was calculated as follows:

in which f_{tree,k} and f_{esti,k} are the true and estimated tree fraction. There would be 500 values of RMSE corresponding to 500 runs of Gaussian noises for each σ. Consequently, the mean and standard deviation (SD) of these RMSEs were calculated. The same calculation was performed in the second dataset for each level of nonlinear mixing intensities (i.e. c_{12}).

The degree of collinearity among the endmembers in spectral mixture models can be quantified by using the so-called Variance Inflation Factor (VIF, [

The SID method was first applied to the Group I simulation dataset. For comparison, the conventional LSMA and NSMA as demonstrated in [

Figures 3(a)-(c) show the mean RMSE of tree-fraction estimation against the standard deviation of Gaussian noises (σ) for mixture cases of tree-soil, tree-concrete, and tree-grass, respectively. The error bars denote standard

deviation for the 500 RMSEs corresponding to each level of σ. It was found that the SID method outperformed the conventional LSMA significantly for all mixture cases. This is because the conventional LSMA is implemented using brightness-based measure, and then the spectral mixture modeling errors will largely affect the retrieval accuracies. In contrast, the SID method applying the spectral-shape-based measure can compensate the modeling errors to some extent. Compared with the conventional NSMA, the SID method yielded similarly small RMSE (<0.05) for the tree-soil mixture, except that when the σ is quite small, the RMSE for NSMA is relatively lower (

The SID method, as well as conventional LSMA and NSMA were then applied to the Group II simulation dataset, in which the parameter c_{12} denotes the interactive intensity in the spectral mixture model. _{12}) when the level of Gaussian noise is fixed (σ = 0.05).

Results showed that RMSEs of the conventional LSMA increased rapidly when the c_{12} increased in all mixture cases, while the SID method showed relatively stable patterns against the increasing c_{12} (Figures 4(a)-(c)). The RMSEs of the SID were slightly higher than the conventional LSMA when c_{12} was low; when the c_{12} was larger than a certain value (e.g., about 0.04 for the tree-soil mixture), the SID method yielded noticeably higher accuracy than the conventional LSMA. Compared with the conventional NSMA, the SID method yielded similar accuracies for the mixture cases of tree-soil and tree-grass. While for the tree-concrete mixture, the SID method significantly outperformed the conventional NSMA, totally because of the dramatically increased degree of collinearity among the endmembers in the NSMA.

Advantages of the spectral-shape-based unmixing method, SID, are mainly in two aspects. First, the modeling

errors results from the linear mixture model can be compensated by using the spectral-shape-based measure as the matching criterion. This explains the better performance of the SID than the conventional LSMA when the interactive intensities are relatively high. Second, the increased degree of collinearity due to the interaction terms in the NSMA can be simply circumvented by using the linear mixture model. The merit of using the linear model was apparently demonstrated in the case of tree-concrete mixture, in which the SID significantly outperformed the conventional NSMA. It should be noted that the SID cannot overcome the collinearity between true endmembers (e.g., tree and grass), while its accuracy is comparable with the NSMA even for that case.

Another potential merit of the SID method is that the spectral-shape-based measure is less sensitive to magnitude variations of the mixed spectra. Therefore, it has lower requirement for exact atmospheric correction than the conventional brightness-based algorithms, which can avoid the difficulties in obtaining all the atmospheric parameters [

The simulation experiments in this study are designed to imitate the simplest two-endmember mixtures with 3-D structures (i.e. the tree), showing three typical variations of collinearity effect between the LSMA and NSMA. These are: 1) the collinearity of LSMA and NSMA are both low, i.e. tree-soil mixture; 2) the LSMA is low, but in contrast the NSMA is very high, i.e. tree-concrete mixture; and 3) the LSMA and NSMA are both high, i.e. tree-grass mixture. Natural circumstances would be much more complicated than the simulated cases. However, it can be deduced that more complicated experiments will not change the characteristics of collinearity for the LSMA and NSMA, and consequently will not yield different results from this study. Additionally, application of the SID in more field measurements and/or satellite images will be investigated as future works.

In this study, the SID was applied to circumvent the collinearity effect in nonlinear mixture model. Simulation data sets with different levels of Gaussian noises and nonlinear mixing intensities were generated to test the performances of the SID in mixture cases of tree-soil, tree-grass, and tree-concrete. Results demonstrated that the SID method is generally more flexible than the conventional LSMA and NSMA. The SID outperforms the conventional LSMA for almost all the mixture cases, and performs significantly better than the NSMA for the tree-concrete mixture. For the mixture cases of tree-soil and tree-grass, the performance of the SID method is comparable with the NSMA. The results indicate the potential of SID method in circumventing the collinearity effect in NSMA caused by the virtual endmembers. In future works, the SID method will be applied to field measurements and satellite images to make further validations.

This research was conducted under the JAXA GCOM Research Announcement (the 4th RA 102 and the 6th RA 111).

Wei Yang,Akihiko Kondoh, (2016) Toward Circumventing Collinearity Effect in Nonlinear Spectral Mixture Analysis by Using a Spectral Shape Measure. Advances in Remote Sensing,05,183-191. doi: 10.4236/ars.2016.53015