Chance Constrained Robust Unmixing Algorithms and Estimation of Number of Endmembers in Hyperspectral Images

Abstract

Abstract Hyperspectral remote sensing is a powerful technique to identify the materials and their composition in an area by exploiting the spectral diversity of the observed hyperspectral data. The analysis of hyperspectral images obtained for the purpose of mineral identification and quantification is considered in this thesis. The limited spatial resolution of the sensor used for hyperspectral imaging and the presence of noise in the measured hyperspectral data demand an effective hyperspectral unmixing (HU) scheme to extract the underlying endmember signatures and the associated abundance maps distributed over a scene of interest. Existing HU algorithms are basically devised under either of the two famous unmixing criteria, namely Winter’s criterion and Craig’s criterion. However, the presence of additive Gaussian noise in the observations expands the actual data cloud and as a consequence, the endmember estimates obtained by applying either Winter’s or Craig’s criterion based algorithms to the noisy data may no longer be in close proximity to the true endmember signatures. Hence, we propose two robust algorithms, they are Winter’s criterion based robust alternating volume maximization (RAVMAX) algorithm and Craig’s criterion based robust minimum volume enclosing simplex (RMVES) algorithm. The robust algorithms account for the noise effects in the observations by employing chance constraints, and employ the notion of alternating optimization to handle the resulting non-convex optimization problems. In RAVMAX algorithm, the subproblems involved in each alternating optimization turn out to be convex problems and they can be effectively solved using available convex optimization solvers. On the other hand, the subproblems involved in RMVES algorithm are non-convex and are hence dealt using available sequential quadratic programming solvers. The HU results can be completely interpretable, only when the number of substances (or endmembers) present in that area is given a priori, which however is unknown in practice. Considering the linear mixing model, we propose a hyperspectral data geometry based approach for estimating the number of endmembers by utilizing a successive endmember extraction algorithm (EEA). The approach is fulfilled by two novel algorithms, namely geometry based estimation of number of endmembers - convex hull (GENE-CH) algorithm and affine hull (GENE-AH) algorithm. The GENE-CH and GENE-AH algorithms are based on the fact that all the observed pixel vectors lie in the convex hull and affine hull of the endmember signatures, respectively. The proposed GENE algorithms estimate the number of endmembers by using the Neyman-Pearson hypothesis testing over the endmember estimates provided by a successive EEA until the estimate of the number of endmembers is obtained. Since the estimation accuracies of the proposed GENE algorithms depend on the performance of the EEA used, a reliable, reproducible, and successive EEA, called p-norm based pure pixel identification (TRI-P) algorithm is then proposed. Monte-Carlo simulations and real data experiments on AVIRIS hyperspectral data obtained over the Cuprite mining site, Nevada are performed to demonstrate the efficacy of the proposed RAVMAX, RMVES, GENE, and TRI-P algorithms. We believe that the proposed chance constrained robust algorithms for hyperspectral unmixing, and data geometry based algorithms for estimating the number of endmembers, will provide a new dimension in analyzing hyperspectral data where noise is always present.