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A Fast Hyperplane-Based Minimum-Volume Enclosing 1946 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 64, NO. 8, APRIL 15, 2016 A Fast Hyperplane-Based Minimum-Volume Enclosing Simplex Algorithm for Blind Hyperspectral Unmixing Chia-Hsiang Lin, Chong-Yung Chi, Senior Member, IEEE, Yu-Hsiang Wang, and Tsung-Han Chan Abstract—Hyperspectral unmixing (HU) is a crucial signal well as their corresponding fractions (or abundances) present processing procedure to identify the underlying materials (or end- in a scene of interest from observed hyperspectral data, having members) and their corresponding proportions (or abundances) various applications such as planetary exploration, land map- from an observed hyperspectral scene. A well-known blind HU criterion, advocated by Craig during the early 1990s, considers ping and classification, environmental monitoring, and mineral the vertices of the minimum-volume enclosing simplex of the data identification and quantification [5]–[7]. The observed pixels in cloud as good endmember estimates, and it has been empirically the hyperspectral imaging data cube are often spectral mixtures and theoretically found effective even in the scenario of no pure of multiple substances, the so-called mixed pixel phenomenon pixels. However, such kinds of algorithms may suffer from heavy simplex volume computations in numerical optimization, etc. In [8], owing to the limited spatial resolution of the hyperspectral this paper, without involving any simplex volume computations, sensor (usually equipped on board the satellite or aircraft) uti- by exploiting a convex geometry fact that a simplest simplex of lized for recording the electromagnetic scattering patterns of the vertices can be defined by associated hyperplanes, we propose underlying materials in the observed hyperspectal scene over a fast blind HU algorithm, for which each of the hyperplanes about several hundreds of narrowly spaced (typically, 5–10 nm) associated with the Craig’s simplex of vertices is constructed from affinely independent data pixels, together with an wavelengths that contiguously range from visible to near-in- endmember identifiability analysis for its performance support. frared bands. Occasionally, the mixed pixel phenomenon can Without resorting to numerical optimization, the devised algo- also be present in hyperspectral scenes where the underlying rithm searches for the active data pixels via simple materials are intimately mixed with each other [9] even though linear algebraic computations, accounting for its computational efficiency. Monte Carlo simulations and real data experiments are the spatial resolution is high enough. Hyperspectral unmixing provided to demonstrate its superior efficacy over some bench- (HU) [8], [10], an essential procedure of extracting individual mark Craig-criterion-based algorithms in both computational spectral signatures of the underlying materials in the captured efficiency and estimation accuracy. scene from these measured spectral mixtures, is therefore of Index Terms—Hyperspectral unmixing, Craig’s criterion, paramount importance in the HRS context. convex geometry, minimum-volume enclosing simplex, hyper- Blind HU, or unsupervised HU, involves two core stages, plane. namely endmember extraction and abundance estimation, without (or with very limited) prior knowledge about the end- members’ nature or the mixing mechanism. Some endmember I. INTRODUCTION extraction algorithms (EEAs), such as alternating projected YPERSPECTRAL remote sensing (HRS) [2]–[4], also subgradients (APS) [11], joint Bayesian approach (JBA) [12], H known as imaging spectroscopy, is a crucial technology and iterated constrained endmembers (ICE) [13] (also the to the identification of material substances (or endmembers) as sparsity promoting ICE (SPICE) [14]), can simultaneously determine the associated abundance fractions while extracting Manuscript received April 01, 2015; revised October 02, 2015; accepted De- the endmember signatures. Nevertheless, some EEAs perform cember 01, 2015. Date of publication December 17, 2015; date of current ver- endmember estimation, followed by abundance estimation sion February 26, 2016. The associate editor coordinating the review of this using such as the fully constrained least squares (FCLS) [15] to manuscript and approving it for publication was Prof. Andre de Almeida. This work is supported partly by the National Science Council, R.O.C., under Grant complete the entire HU processing. NSC 102-2221-E-007-035-MY2, and partly by Ministry of Science and Tech- The pure-pixel assumption has been exploited in devising nology, R.O.C., under Grant MOST 104-2221-E-007-069-MY3. Part of this fast blind HU algorithms to search for the purest pixels over work was presented at IEEE ICASSP 2015 [1]. (Corresponding author: Chia-Hsiang Lin.) the data set as the endmember estimates, and such searching C.-H. Lin, C.-Y. Chi, and Y.-H. Wang are with the Institute of procedure can always be carried out through simple linear al- Communications Engineering and the Department of Electrical En- gineering, National Tsing Hua University, Hsinchu, Taiwan 30013, gebraic formulations; see, e.g., pixel purity index (PPI) [16] R.O.C. (e-mail: [email protected]; [email protected]; and vertex component analysis (VCA) [17]. An important blind [email protected]). HU criterion, called Winter’s criterion [18], also based on the T.-H. Chan is with MediaTek Inc., National Science Park, Hsinchu, Taiwan 30013, R.O.C. (e-mail: [email protected]). pure-pixel assumption, is to identify the vertices of the max- Color versions of one or more of the figures in this paper are available online imum-volume simplex inscribed in the observed data cloud as at http://ieeexplore.ieee.org. endmember estimates. HU algorithms in this category include Digital Object Identifier 10.1109/TSP.2015.2508778 1053-587X © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. LIN et al.: A FAST HYPERPLANE-BASED MINIMUM-VOLUME ENCLOSING SIMPLEX ALGORITHM FOR BLIND HU 1947 N-finder (N-FINDR) [18], simplex growing algorithm (SGA) by this observation, we naturally raise a question: Can Craig’s [19] (also the real-time implemented SGA [20]), and worst case minimum-volume simplex be identified by simply searching for alternating volume maximization (WAVMAX) [21], to name a a specific set of pixels in the data set regardless of the existence few. However, the pure-pixel assumption could be seriously in- of pure pixels? The answer is affirmative and will be given in fringed in practical scenarios especially when the pixels are in- this paper. timately mixed, for instance, the hyperspectral imaging data for Based on the convex geometry fact that a simplest simplex retinal analysis in the ophthalmology [9]. In these scenarios, HU of vertices can be characterized by the associated hy- algorithms in this category could completely fail; actually, it is perplanes, this paper proposes an efficient and effective unsu- proven that perfect endmember identifiability is impossible for pervised Craig-criterion-based HU algorithm, together with an Winter-criterion-based algorithms if the pure-pixel assumption endmember identifiability analysis. Each hyperplane, parame- is violated [21]. terized by a normal vector and an inner product constant [26], Without relying on the existence of pure pixels, another can then be estimated from affinely independent pixels promising blind HU approach, advocated by Craig in early in the data set via simple linear algebraic formulations. The re- 1990’s [22], exploits the simplex structure of hyperspectral sulting hyperplane-based CSI (HyperCSI) algorithm, based on data, and believes that the vertices of the minimum-volume the above pixel search scheme, can withstand the no pure-pixel data-enclosing simplex can yield good endmember estimates, scenario, and can yield deterministic, non-negative, and, most and algorithms developed accordingly include such as min- importantly, accurate endmember estimates. After endmember imum-volume transform (MVT) [22], minimum-volume con- estimation, a closed-form expression in terms of the identified strained nonnegative matrix factorization (MVC-NMF) [23], hyperplanes’ parameters is derived for abundance estimation. and minimum-volume-based elimination strategy (MINVEST) Then some Monte Carlo numerical simulations and real hyper- [24]. Moreover, some linearization-based methods have also spectral data experiments are presented to demonstrate the su- been reported to practically identify Craig’s minimum-volume perior efficacy of the proposed HyperCSI algorithm over some simplex, e.g., the iterative linear approximation in min- benchmark Craig-criterion-based HU algorithms in both esti- imum-volume simplex analysis (MVSA) [25] (also its fast mation accuracy and computational efficiency. implementation using the interior-point method [26], termed The remaining part of this paper is organized as follows. as ipMVSA [27]), and the alternating linear programming in In Section II, we briefly review some essential convex geom- minimum-volume enclosing simplex (MVES) [28]. Empirical etry concepts, followed by the signal model and dimension re- evidences do well support that this minimum-volume approach duction. Section III focuses on the HyperCSI algorithm devel- is resistant to lack of pure pixels, and can recover ground truth opment, and in Section IV, some simulation results are pre- endmembers quite accurately
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