On the complexity of some basic problems in computational convexity: I.: containment problems
Discrete Mathematics - Special issue: trends in discrete mathematics
Algorithms for a Minimum Volume Enclosing Simplex in Three Dimensions
SIAM Journal on Computing
Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications
Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications
Convex Optimization
Remote Sensing and Image Interpretation
Remote Sensing and Image Interpretation
Joint Bayesian endmember extraction and linear unmixing for hyperspectral imagery
IEEE Transactions on Signal Processing
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Convex Analysis Framework for Blind Separation of Non-Negative Sources
IEEE Transactions on Signal Processing - Part II
A lattice matrix method for hyperspectral image unmixing
Information Sciences: an International Journal
Unmixing of Hyperspectral Images using Bayesian Non-negative Matrix Factorization with Volume Prior
Journal of Signal Processing Systems
LVA/ICA'12 Proceedings of the 10th international conference on Latent Variable Analysis and Signal Separation
Hybrid computational methods for hyperspectral image analysis
HAIS'12 Proceedings of the 7th international conference on Hybrid Artificial Intelligent Systems - Volume Part II
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Hyperspectral unmixing aims at identifying the hidden spectral signatures (or endmembers) and their corresponding proportions (or abundances) from an observed hyperspectral scene. Many existing hyperspectral unmixing algorithms were developed under a commonly used assumption that pure pixels exist. However, the pure-pixel assumption may be seriously violated for highly mixed data. Based on intuitive grounds, Craig reported an unmixing criterion without requiring the pure-pixel assumption, which estimates the endmembers by vertices of a minimum-volume simplex enclosing all the observed pixels. In this paper, we incorporate convex analysis and Craig's criterion to develop a minimum-volume enclosing simplex (MVES) formulation for hyperspectral unmixing. A cyclic minimization algorithm for approximating the MVES problem is developed using linear programs (LPs), which can be practically implemented by readily available LP solvers. We also provide a non-heuristic guarantee of our MVES problem formulation, where the existence of pure pixels is proved to be a sufficient condition for MVES to perfectly identify the true endmembers. Some Monte Carlo simulations and real data experiments are presented to demonstrate the efficacy of the proposed MVES algorithm over several existing hyperspectral unmixing methods.