An Ensemble Learning Based Method for the Improvement of Nonnegative Matrix Factorization (NMF)

Images material identification plays an important role in remotely sensed image processing and its applications. Hyperspectral images, which contain a lot of narrow spectral bands of the electromagnetic spectrum, have a great potential for information extraction from remotely sensed images and material identification. Due to the low spatial resolution of hyperspectral cameras, material mixing, and light multi scattering, these images usually contain a lot of mixed pixels which face material identification with many problems. Recently, hyperspectral unmixing methods have been widely considered by the researchers as a powerful tool for identifying materials in the mixed pixels. Some of the algorithms, proposed for hyperspectral unmixing, are based on the linear mixture model (LMM) and the others are based on nonlinear mixture model (NLMM). LMM-based algorithms are more simple and commonly used methods for hyperspectral unmixing. Among various algorithms, based on LMM, non-negative matrix factorization (NMF) has attracted the most attention due to essentially implying non-negativity of the endmembers and their corresponding abundances, and moreover, simultaneously extracting spectral signature and abundances of the endmembers. In spite of these capabilities, NMF leads to local minima due to its non-convex objective function. In this regards, various studies have attempted to lead NMF results to the global optimum by imposing some additional constraint to the main objective function of NMF. However, NMF-based methods still suffer from the problem of falling into local minima. To tackle this problem, an iterative post-processing procedure, based on an ensemble learning technique, has been presented in this paper.  The main goal of this paper is to demonstrate the ability of ensemble learning to improve the hyperspectral unmixing results in a simple and non-parametric manner. To this end, NMF with  sparse constraint is performed in several iterations, and then, the results of each of these iterations are weightened on the basis of identifying a primary endmember that certainly exists in the image. Weightening is done with calculating spectral angle distance (SAD) metric between the true and extracted spectral signatures of the primary endmember. Usually, there is prior information about the hyperspectral images such as some existed materials or the number of materials in the images. Therefore, it is always possible to find a primary endmember in an image. The accuracy of identifying primary endmember is extended the accuracy of identifying other endmembers of their corresponding abundances. Final mixing and abundances matrices are determined using weighted combinations of the mixing and abundances matrices, extracted from each of the iterations. The proposed procedure is nonparametric and mathematically clear which can be extended to more advanced algorithms of hyperspectral unmixing. The performance of the proposed method for extraction of endmembers and their corresponding abundances has been evaluated using various synthetic and real hyperspectral data sets. Synthetic hyperspectral images constructed using USGS spectral library with several numbers of endmembers in the different signal to noise ratio (SNR) levels. Cuprite hyperspectral image, acquired by the AVIRIS sensor in 1997 from the Nevada Desert in the United States, has been used in this study as a real hyperspectral data set. The results of the experiments on both types of hyperspectral data illustrate the superiority of the proposed method over state-of-the-art competing methods.