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## Identifiability of the simplex volume minimization criterion for blind hyperspectral unmixing: The no pure-pixel case (2014)

Citations: | 2 - 1 self |

### Citations

7418 | Convex Optimization
- Boyd, Vandenberghe
- 2004
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Citation Context ...enotes the Euclidean norm of a vector; xT denotes the transpose of x and the same applies to matrices; given a set A ⊆ Rn, we denote affA and convA as the affine hull and convex hull of A, resp. (see =-=[23]-=-), intA and bdA as the interior and boundary of A, resp., and volA as the volume of A; the dimension of a set A ⊆ Rn is defined as the affine dimension of affA; x ≥ 0 means that x is elementwise non-n... |

176 | N-findr: An algorithm for fast autonomous spectral end-member determination in hyper-spectral data - Winter - 1999 |

99 | Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches
- Bioucas-Dias, Plaza, et al.
- 2012
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Citation Context ... hyperspectral images, most notably, their high spectral resolutions. In this scope, blind hyperspectral unmixing (HU) is one of the topics that has aroused much interest not only from remote sensing =-=[3]-=-, but also from other communities recently [4–7]. Simply speaking, the problem of blind HU is to solve a problem reminiscent of blind source separation in signal processing, and the desired outcome is... |

83 |
Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization
- Miao, Qi
- 2007
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Citation Context ...rk proposes the use of simplex volume as a metric for blind HU, which is later used in some other blind HU approaches such as simplex volume maximization [11–13] and non-negative matrix factorization =-=[14]-=-. The MVES criterion is to minimize the volume of a simplex, subject to constraints that the simplex encloses all hyperspectral data points. This amounts to a nonconvex optimization problem, and unlik... |

77 |
Minimum-volume transforms for remotely sensed data
- Craig
- 1994
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Citation Context ...nvex geometry structures of the observed hyperspectral data to blindly identify the endmember spectral signatures. In the HU context the MVES concepts were first advocated by Craig back in the 1990’s =-=[9]-=-, although as an aside it is interesting to note an earlier work in mathematical geology [10] which also described the MVES intuitions; see the historical note in [4]. In particular, Craig’s work prop... |

66 | Joint Bayesian endmember extraction and linear unmixing for hyperspectral imagery - Dobigeon, Moussaoui, et al. |

62 | On the complexity of some basic problems in computational convexity: II. Volume and mixed volumes, polyhedra - Gritzmann, Klee - 1994 |

52 | Minimum volume simplex analysis: A fast algorithm to unmix hyperspectral data
- Li, Bioucas-Dias
- 2008
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Citation Context ... question arising from one important blind HU approach, namely, the minimum volume enclosing simplex (MVES) approach. Also called simplex volume minimization or minimum volume simplex analysis (MVSA) =-=[8]-=-, the MVES approach adopts a criterion that exploits the convex geometry structures of the observed hyperspectral data to blindly identify the endmember spectral signatures. In the HU context the MVES... |

40 | A variable splitting augmented Lagrangian approach to linear spectral unmixing
- Bioucas-Dias
- 2009
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Citation Context ... considered very recently [15]. In addition, some recent MVES algorithm designs deal with noise and outlier sensitivity issues by robust formulations, such as the soft constraint formulation in SISAL =-=[16]-=- and the chance-constrained formulation in [17]; the pixel elimination method in [18] should also be noted. We should further mention that MVES also finds application in analytical chemistry [19], and... |

30 | Fast and robust recursive algorithms for separable nonnegative matrix factorization. arXiv preprint arXiv:1208.1237 - Gillis, Vavasis - 2012 |

27 |
A convex analysis based minimum-volume enclosing simplex algorithm for hyperspectral unmixing
- Chan, Chi, et al.
- 2009
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Citation Context ...not seem to have a simple (closed-form) scheme for tackling the MVES problem. However, recent advances in optimization have enabled us to handle MVES implementations efficiently. The works in [8] and =-=[6]-=- independently developed practical MVES optimization algorithms based on iterative linear approximation and alternating linear programming, respectively. The GPU-implementation of the former is also c... |

26 | Hyperspectral remote sensing data analysis and future challenges
- Bioucas-Dias, Plaza, et al.
- 2013
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Citation Context ...s. The theoretical results are verified by numerical simulations. 1 Introduction Signal, image and data processing for hyperspectral imaging has recently received enormous attention in remote sensing =-=[1, 2]-=-, having numerous applications such as environmental monitoring, land mapping and classification, and object detection. Such exciting developments are made possible by exploiting the unique features o... |

24 | A convex analysis framework for blind separation of non-negative sources
- Chan, Ma, et al.
- 2008
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Citation Context ...ressed as aff{e1, . . . ,eN} = {s = Cθ + d | θ ∈ RN−1}, (14) where d = 1 N N∑ i=1 ei = 1 N 1, and C ∈ RN×(N−1) is the first N − 1 principal left singular vectors of R = [ e1 − d, . . . ,eN − d ]; see =-=[6, 25]-=-. We note that R = I − 1 N 11T , which, as a standard matrix result, its first N − 1 principal left singular vector can be shown to be any C such that U = [ C, 1√ N 1 ] (15) is a unitary matrix. Or, e... |

21 | A simplex volume maximization framework for hyperspectral endmember extraction - Chan, Ma, et al. - 2011 |

18 | Hyperspectral unmixing based on mixtures of Dirichlet components
- Nascimento, Bioucas-Dias
- 2012
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Citation Context ...hould also be noted. We should further mention that MVES also finds application in analytical chemistry [19], and that fundamentally MVES has a strong link to stochastic maximum-likelihood estimation =-=[20]-=-. What makes MVES special is that it seems to perform well even in the absence of pure pixels, i.e., pixels that are solely contributed by a single endmember. To be more accurate, extensive simulation... |

14 | A signal processing perspective on hyperspectral unmixing
- Ma, Bioucas-Dias, et al.
- 2014
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Citation Context ...le information to solve the problem, blind HU is a challenging—but also fundamentally intriguing—problem with many possibilities. Readers are referred to some recent articles for overview of blind HU =-=[3, 4]-=-, and here we shall not review the numerous possible ways to perform blind HU. The focus, as well as the contribution, of this paper lie in addressing a fundamental question arising from one important... |

12 |
NIR hyperspectral unmixing based on a minimum volume criterion for fast and accurate chemical characterisation of counterfeit tablets
- Lopes, Wolff, et al.
- 2010
(Show Context)
Citation Context ...ISAL [16] and the chance-constrained formulation in [17]; the pixel elimination method in [18] should also be noted. We should further mention that MVES also finds application in analytical chemistry =-=[19]-=-, and that fundamentally MVES has a strong link to stochastic maximum-likelihood estimation [20]. What makes MVES special is that it seems to perform well even in the absence of pure pixels, i.e., pix... |

11 | Chanceconstrained robust minimum-volume enclosing simplex algorithm for hyperspectral unmixing
- Ambikapathi, Chan, et al.
- 2011
(Show Context)
Citation Context ...me recent MVES algorithm designs deal with noise and outlier sensitivity issues by robust formulations, such as the soft constraint formulation in SISAL [16] and the chance-constrained formulation in =-=[17]-=-; the pixel elimination method in [18] should also be noted. We should further mention that MVES also finds application in analytical chemistry [19], and that fundamentally MVES has a strong link to s... |

11 | NP-hardness of largest contained and smallest containing simplices for V - Packer - 2002 |

10 | The orthocentric simplex as an extreme simplex - Gerber - 1975 |

9 |
Largest j-simplices in npolytopes
- Gritzmann, Klee, et al.
- 1995
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Citation Context ...=1 θibi ∣∣∣∣ θ ≥ 0,1Tθ = 1 } , where b1, . . . , bN ∈ RM , M ≥ N − 1, is called an (N − 1)-dimensional simplex if b1, . . . , bN are affinely independent. The volume of a simplex can be determined by =-=[24]-=- vol(conv{b1, . . . , bN}) = 1 (N − 1)! √ det(B̄T B̄), (1) where B̄ = [ b1 − bN , b2 − bN , . . . , bN−1 − bN ] ∈ RM×(N−1). A simplex is called regular if the distances between any two vertices are th... |

9 | received the B.S. degree in electrical engineering from the National Tsing Hua University (NTHU - Lin - 2010 |

8 | End-member extraction for hyperspectral image analysis - Du, Raksuntorn, et al. - 2008 |

7 |
EXTENDED QMODEL— objective definition of external endmembers in the analysis of mixtures
- Full, Ehrlich, et al.
- 1981
(Show Context)
Citation Context ...r spectral signatures. In the HU context the MVES concepts were first advocated by Craig back in the 1990’s [9], although as an aside it is interesting to note an earlier work in mathematical geology =-=[10]-=- which also described the MVES intuitions; see the historical note in [4]. In particular, Craig’s work proposes the use of simplex volume as a metric for blind HU, which is later used in some other bl... |

4 |
On endmember identification in hyperspectral images without pure pixels: A comparison of algorithms
- Plaza, Hendrix, et al.
- 2012
(Show Context)
Citation Context ... by a single endmember. To be more accurate, extensive simulations found that MVES may obtain the ground truth quite accurately in the noiseless case and without the pure-pixel assumption; see, e.g., =-=[6, 20, 21]-=-. At this point we should mention that while the pure-pixel assumption is elegant and has been exploited by some other approaches, such as simplex volume maximization (also [7] for a more recent work ... |

4 | received the B.S. degree in electrical engineering from the National Tsing Hua University - Li - 2009 |

3 | is currently serving or has served as Associate Editor and Guest Editor of several journals, which include - Ma |

3 | is a senior member of IEEE. He has been a Technical Program Committee member for many IEEE sponsored and co-sponsored workshops, symposiums and conferences on signal processing and wireless communications, including Co-organizer and - Chi - 2009 |

3 | was an Associate Editor (AE) of - He |

2 | Multi-GPU implementation of the minimum volume simplex analysis algorithm for hyperspectral unmixing
- Agathos, Li, et al.
- 2014
(Show Context)
Citation Context ... practical MVES optimization algorithms based on iterative linear approximation and alternating linear programming, respectively. The GPU-implementation of the former is also considered very recently =-=[15]-=-. In addition, some recent MVES algorithm designs deal with noise and outlier sensitivity issues by robust formulations, such as the soft constraint formulation in SISAL [16] and the chance-constraine... |

2 |
A new minimum-volume enclosing algorithm for endmember identification and abundance sstimation in hyperspectral data
- Hendrix, Garcia, et al.
- 2012
(Show Context)
Citation Context ...with noise and outlier sensitivity issues by robust formulations, such as the soft constraint formulation in SISAL [16] and the chance-constrained formulation in [17]; the pixel elimination method in =-=[18]-=- should also be noted. We should further mention that MVES also finds application in analytical chemistry [19], and that fundamentally MVES has a strong link to stochastic maximum-likelihood estimatio... |

1 |
On the endmember identifiability of Craig’s criterion for hyperspectral unmixing: A statistical analysis for three-source case
- Lin, Ambikapathi, et al.
- 2013
(Show Context)
Citation Context ...ioned questions through analysis. Previously, identifiability analysis for MVES was done only for the pure-pixel case in [6], and for the three endmember case in the preliminary version of this paper =-=[22]-=-. This paper considers the no pure-pixel case for any number of endmembers. We prove that MVES can indeed guarantee exact recovery of the endmembers. The key condition for attaining such exact 2 ident... |

1 | A signal processing perspective on hyperspectral unmixing - Plaza, Chi - 2014 |

1 | Bayesian endmember extraction and linear unmixing for hyperspectral imagery - Hero - 2009 |

1 | M’01-SM’11) received the B.Eng. degree in electrical and electronic engineering from the University of Portsmouth - Ma - 1995 |

1 | to 1988, he was with the Jet Propulsion - From - 1983 |