A Convex Analysis-Based Minimum-Volume Enclosing Simplex Algorithm for Hyperspectral Unmixing

• Published in: IEEE transactions on signal processing Vol. 57; no. 11; pp. 4418 - 4432
• Main Authors: CHAN, Tsung-Han, CHI, Chong-Yung, HUANG, Yu-Min, MA, Wing-Kin
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.11.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Applied sciences; Approximation; Computer simulation; Convex analysis; convex optimization; Councils; Criteria; Data mining; Exact sciences and technology; Formulations; Hyperspectral imaging; Hyperspectral sensors; hyperspectral unmixing; Information, signal and communications theory; Layout; linear programming; Minimization methods; minimum-volume enclosing simplex; Miscellaneous; Monitoring; Monte Carlo methods; Pixels; Principal component analysis; Signal processing; Signal processing algorithms; Solvers; Studies; Telecommunications and information theory; Monte Carlo method; minimum-volume enclosing simplex; Linear programming; Algorithm; Convex programming; Implementation; Spectral signature; Sufficient condition; Volume; Simplex method; Heuristic method; Signal processing; Numerical simulation; convex optimization; hyperspectral unmixing; Convex analysis
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2009.2025802

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.