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This paper considers the reconstruction of a discrete-valued random vector from possibly underdetermined linear measurements using sum-of-absolute-value (SOAV) optimization. The proposed algorithm, referred to as discreteness-aware approximate message passing (DAMP), is based on the idea of approximate message passing (AMP), which has been originally proposed for compressed sensing. The DAMP algorithm has low computational complexity and its performance in the large system limit can be predicted analytically via state evolution framework, where we provide a condition for the exact reconstruction with DAMP in the noise-free case. From the analysis, we also propose a method to determine the parameters of the SOAV optimization. Moreover, based on the state evolution, we provide Bayes optimal DAMP, which has the minimum mean-square-error at each iteration of the algorithm. Simulation results show that the DAMP algorithms can reconstruct the discrete-valued vector from underdetermined linear measurements and the empirical performance agrees with our theoretical results in large-scale systems. When the problem size is not large enough, the SOAV optimization with the proposed parameters can achieve better performance than the DAMP algorithms for high signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

Endmember extraction is very important in hyperspectral image analysis. The accurate identification of endmembers enables target detection and classification and efficient spectral unmixing. Although a number of endmember extraction algorithms have been proposed, such as two state-of-the-art algorithms-vertex component analysis (VCA) and simplex growing algorithm (SGA)-it is still a rather challenging task. In this paper, a new maximum simplex volume method based on Householder transformation (HT), referred to as maximum volume by HT (MVHT), is presented for endmember extraction. The proposed algorithm provides consistent results with low computational complexity, which overcomes the disadvantage of the inconsistent result of VCA and the shortcoming of the high computational cost of SGA resulted from calculating the simplex volume. A comparative study and analysis are conducted among the three endmember extraction algorithms, VCA, SGA, and MVHT, on both simulated and real hyperspectral data. The obtained experimental results demonstrate that the proposed MVHT algorithm generally provides a competitive or even better performance over VCA and SGA. [ABSTRACT FROM AUTHOR]