### Citations

543 | Robust principal component analysis?
- Candes, Li, et al.
- 2011
(Show Context)
Citation Context ...nts of A, ‖x‖1 = ∑ k |xk| and ‖ · ‖2,1 is the so-called `2,1-norm defined by ‖R‖2,1 = ‖e‖1 = P∑ p=1 ‖rp‖2 . (6) Eq. (5) defines a robust NMF problem. Robust NMF is a nonnegative variant of robust PCA =-=[18]-=- which has appeared in different forms in the literature. In [19], the outlier term R is nonnegative and penalized by the `1 norm. In [20] and [21], R is real-valued and penalized by `1 and `1,2 norms... |

219 |
Spectral unmixing
- Keshava, Mustard
- 2002
(Show Context)
Citation Context ... [y1, . . . ,yP ] into a collection of K individual spectra M = [m1, . . . ,mK ], called endmembers, and estimating their relative proportions (or abundances) A = [a1, . . . ,aP ] in each observation =-=[5]-=-, [6]. Most of the hyperspectral unmixing algorithms proposed in the signal & image processing and geoscience literatures rely on the commonly admitted linear mixing model (LMM), Y ≈MA. Indeed, LMM pr... |

192 | Bioucas-Dias, “Vertex Component Analysis: A fast algorithm to unmix hyperspectral data
- Nascimento, M
- 2005
(Show Context)
Citation Context ...ated in a second step, given the endmember estimates (in a so-called “inversion” step). In contrast, rNMF performs a joint estimation of M and A (and R). We considered vertex component analysis (VCA) =-=[42]-=- coupled with fully constrained least squares (FCLS) [43]. VCA and FCLS are standard endmember extraction and inversion methods designed for the LMM. Besides, we considered the nonlinear endmember ext... |

180 | Monaural sound source separation by nonnegative matrix factorization with temporal continuity and sparness criteria,”
- Virtanen
- 2007
(Show Context)
Citation Context ...ew objective function in (21) – in particular because it can no longer be easily decomposed as a convex part and concave part. Instead, we resort to a heuristic commonly used in NMF, see, e.g., [37], =-=[39]-=-, as follows. As it appears, the gradient of C(U) can be expressed as the difference of two nonnegative functions such that ∇ukpC(U) = ∇+ukpC(U)−∇−ukpC(U). (22) The heuristic algorithm simply writes u... |

166 | Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery
- Heinz, Chang
- 2001
(Show Context)
Citation Context ...a so-called “inversion” step). In contrast, rNMF performs a joint estimation of M and A (and R). We considered vertex component analysis (VCA) [42] coupled with fully constrained least squares (FCLS) =-=[43]-=-. VCA and FCLS are standard endmember extraction and inversion methods designed for the LMM. Besides, we considered the nonlinear endmember extraction technique proposed in [44], denoted as Heylen’s a... |

155 |
Nonnegative matrix factorization with the itakura-saito divergence. with application to music analysis
- Févotte, Bertin, et al.
(Show Context)
Citation Context ... then we have dβ(λx|λy) = λβdβ(x|y). This means that the data-fitting term will rely more heavily on large (resp., small) coefficients in Y for β > 0 (resp., β < 0); see a more detailed discussion in =-=[37]-=-. 3III. BLOCK-COORDINATE DESCENT ALGORITHM In order to solve the rNMF minimization problem defined at Eq. (5), we present an iterative block-coordinate descent algorithm that updates each of the param... |

117 |
The Theory of Dispersion Models
- Jørgensen
- 1997
(Show Context)
Citation Context ...ake the measure of fit in Eq. (5) to be such that D(A|B) = ∑ ij d(aij |bij), (7) where d(x|y) is the β-divergence between positive scalars x and y. The β-divergence was introduced in various forms in =-=[28]-=-–[30] and has become a standard measure of fit in NMF, see, e.g., [31]–[34]. In this paper we use the following definition: dβ(x|y) def= xβ β (β−1) + yβ β − x y β−1 β−1 β ∈ R\{0, 1} x log xy − x... |

99 | Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches,”
- Bioucas-Dias, Plaza, et al.
- 2012
(Show Context)
Citation Context ... . . . ,yP ] into a collection of K individual spectra M = [m1, . . . ,mK ], called endmembers, and estimating their relative proportions (or abundances) A = [a1, . . . ,aP ] in each observation [5], =-=[6]-=-. Most of the hyperspectral unmixing algorithms proposed in the signal & image processing and geoscience literatures rely on the commonly admitted linear mixing model (LMM), Y ≈MA. Indeed, LMM provide... |

93 | Robust and efficient estimation by minimising a density power divergence - BASU, HARRIS, et al. - 1998 |

87 | Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization - Miao, Qi - 2007 |

81 |
Nonnegative matrix factorization for spectral data analysis,” Linear Algebra and its Applications 416(1
- Pauca, Piper, et al.
- 2006
(Show Context)
Citation Context ...uclidean distance, for the computation of a regular NMF Y ≈MA. Note finally that other articles that have addressed hyperspectral unmixing with regular NMF (i.e., in the standard linear model), e.g., =-=[24]-=-–[27]. To the best of our knowledge, the formulation of robust NMF described by Eq. (5), where R is nonnegative and penalized by the `2,1 norm (and where the abundances sum to one), is entirely novel.... |

77 | Csiszar’s divergence for nonnegative matrix factorization: family of new algorithms - Cichocki, Zdunek, et al. - 2006 |

52 | An index which distinguishes between some important exponential families. Pages 579–604 of - Tweedie - 1984 |

51 |
Sparse coding and NMF
- Eggert, Körner
- 2004
(Show Context)
Citation Context ...rned into the new optimization problem min U C(U) = D ( Y |M [ u1 ‖u1‖1 , . . . , uP ‖uP ‖1 ] +R ) s.t. U ≥ 0 (21) which is free from the sum-to-one constraint. This approach has been used for NMF in =-=[38]-=-. Unfortunately, we were not able to produce an auxiliary function for the new objective function in (21) – in particular because it can no longer be easily decomposed as a convex part and concave par... |

50 | Semi-supervised linear spectral unmixing using a hierarchical Bayesian model for hyperspectral imagery
- Dobigeon, Tourneret, et al.
- 2008
(Show Context)
Citation Context ...nal bandwidth of 10nm. The scene of interest, of size of 50×50 pixels, consists of a part of lake and a coastal area composed of soil and vegetation. This dataset has been previously studied in [13], =-=[47]-=- and, thus, the unmixing results obtained in the current work can be compared to those reported in these later references. This dataset will be referred to as the “Moffett” image in the following. The... |

41 | Unmixing hyperspectral images using a generalized bilinear model
- Halimi, Altmann, et al.
- 2011
(Show Context)
Citation Context ... 2 rue Camichel, BP 7122, 31071 Toulouse cedex 7, France. (e-mail: nicolas.dobigeon@enseeiht.fr). using bilinear models [8], [9]. As explained in [10], several bilinear models have been proposed [11]–=-=[13]-=-, and they mainly differ by the constraints imposed on the nonlinearity term. The linear-quadratic model advocated in [14] also incorporates pairwise interactions between the endmembers components. Co... |

40 |
A generalized divergence measure for nonnegative matrix factorization. Torun, Poland. Neuroinformatics workshop
- Kompass
- 2005
(Show Context)
Citation Context ...|bij), (7) where d(x|y) is the β-divergence between positive scalars x and y. The β-divergence was introduced in various forms in [28]–[30] and has become a standard measure of fit in NMF, see, e.g., =-=[31]-=-–[34]. In this paper we use the following definition: dβ(x|y) def= xβ β (β−1) + yβ β − x y β−1 β−1 β ∈ R\{0, 1} x log xy − x+ y β = 1 x y − log xy − 1 β = 0 . (8) The limiting cases β = 0 and β ... |

35 |
Algorithms for nonnegative matrix factorization with the βdivergence
- Févotte, Idier
(Show Context)
Citation Context ..., (7) where d(x|y) is the β-divergence between positive scalars x and y. The β-divergence was introduced in various forms in [28]–[30] and has become a standard measure of fit in NMF, see, e.g., [31]–=-=[34]-=-. In this paper we use the following definition: dβ(x|y) def= xβ β (β−1) + yβ β − x y β−1 β−1 β ∈ R\{0, 1} x log xy − x+ y β = 1 x y − log xy − 1 β = 0 . (8) The limiting cases β = 0 and β = 1 c... |

34 | Automatic relevance determination in nonnegative matrix factorization with the/spl beta/-divergence,”
- Tan, Fevotte
- 2013
(Show Context)
Citation Context ...d can either be fixed or learnt from training data by cross-validation. Under certain assumptions, the β-divergence can be mapped to a log-likelihood function for the Tweedie distribution [28], [35], =-=[36]-=-, parametrized with respect to its mean. In our setting, this translates into E[Y|M,A,R] = MA+R. In particular, the values β = 0, 1, 2 underlie multiplicative Gamma noise, Poisson noise and Gaussian a... |

33 |
Spectral unmixing of vegetation, soil and dry carbon cover in arid regions: Comparing multispectral and hyperspectral observations
- Asner, Heidebrecht
(Show Context)
Citation Context ...e of prime interest when analyzing hyperspectral data since it provides a comprehensive and meaningful description of the collected measurements in various application fields including remote sensing =-=[1]-=-, planetology [2], food monitoring [3] or spectromicroscopy [4]. SU consists in decomposing P multi-band observations Y = [y1, . . . ,yP ] into a collection of K individual spectra M = [m1, . . . ,mK ... |

33 |
Bioucas-Dias, “Nonlinear mixture model for hyperspectral unmixing
- Nascimento, M
- 2009
(Show Context)
Citation Context ...EIHT, 2 rue Camichel, BP 7122, 31071 Toulouse cedex 7, France. (e-mail: nicolas.dobigeon@enseeiht.fr). using bilinear models [8], [9]. As explained in [10], several bilinear models have been proposed =-=[11]-=-–[13], and they mainly differ by the constraints imposed on the nonlinearity term. The linear-quadratic model advocated in [14] also incorporates pairwise interactions between the endmembers component... |

31 |
Comparative study between a new nonlinear model and common linear model for analysing laboratory simulated-forest hyperspectral data,” Remote Sensing of Environment
- Fan, Hu, et al.
- 2009
(Show Context)
Citation Context ...erated according to: • the Nascimento model (NM) [11] yp = K∑ k=1 akpmk + K−1∑ i=1 K∑ j=i+1 bipmismj + np, with [ ap bp ] ∈ S2K−1 (28) and bp = [b1p, . . . , b(K−1)p]T , • the Fan bilinear model (FM) =-=[12]-=- yp = K∑ k=1 akpmk + K−1∑ i=1 K∑ j=i+1 aipajpmismj + np, with ap ∈ SK , • the generalized bilinear model (GBM) [15] yp = K∑ k=1 akpmk + K−1∑ i=1 K∑ j=i+1 γijp aipajpmimj +np, 6with ap ∈ SK and where ... |

31 | Families of alpha- beta- and gamma- divergences: Flexible and robust measures of similarities. Entropy 2010
- Cichocki, Amari
(Show Context)
Citation Context ...he measure of fit in Eq. (5) to be such that D(A|B) = ∑ ij d(aij |bij), (7) where d(x|y) is the β-divergence between positive scalars x and y. The β-divergence was introduced in various forms in [28]–=-=[30]-=- and has become a standard measure of fit in NMF, see, e.g., [31]–[34]. In this paper we use the following definition: dβ(x|y) def= xβ β (β−1) + yβ β − x y β−1 β−1 β ∈ R\{0, 1} x log xy − x+ y β... |

29 |
A convex model for nonnegative matrix factorization and dimensionality reduction on physical space
- Esser, Moller, et al.
- 2012
(Show Context)
Citation Context ...ean distance, for the computation of a regular NMF Y ≈MA. Note finally that other articles that have addressed hyperspectral unmixing with regular NMF (i.e., in the standard linear model), e.g., [24]–=-=[27]-=-. To the best of our knowledge, the formulation of robust NMF described by Eq. (5), where R is nonnegative and penalized by the `2,1 norm (and where the abundances sum to one), is entirely novel. Furt... |

28 |
Systems Inc.), ENVI User’s guide Version 4.0
- RSI
- 2003
(Show Context)
Citation Context ...re spectral components have been generated according to four different linear and nonlinear models. The endmember spectra have been extracted from the spectral library provided with the ENVI software =-=[41]-=-. The first image, denoted as ILMM, is composed of pixels following the standard LMM (no nonlinear component) yp = K∑ k=1 akpmk + np, (27) with ap ∈ SK . The three other images, denoted INM, IFM and I... |

24 | Supervised nonlinear spectral unmixing using a post-nonlinear mixing model for hyperspectral images,” Univ
- Altmann, Halimi, et al.
- 2011
(Show Context)
Citation Context ...ar-quadratic model advocated in [14] also incorporates pairwise interactions between the endmembers components. Conversely, to approximate a large range of second-order nonlinearities, Altmann et al. =-=[15]-=- introduce a polynomial post-nonlinear model that has demonstrated its ability to describe most of the nonlinear effects, in particular in vegetated areas [16]. A common feature of these models is tha... |

23 |
Nonlinear hyperspectral mixture analysis for tree cover estimates in orchards,” Remote Sensing of Environment
- Somers, Cools, et al.
- 2009
(Show Context)
Citation Context ...evotte@unice.fr). N. Dobigeon is with University of Toulouse, IRIT/INP-ENSEEIHT, 2 rue Camichel, BP 7122, 31071 Toulouse cedex 7, France. (e-mail: nicolas.dobigeon@enseeiht.fr). using bilinear models =-=[8]-=-, [9]. As explained in [10], several bilinear models have been proposed [11]–[13], and they mainly differ by the constraints imposed on the nonlinearity term. The linear-quadratic model advocated in [... |

23 | Nonnegative matrix factorization - algorithms and applications
- Ho
- 2008
(Show Context)
Citation Context ...entifiable). The objective function (30) can be minimized using a minor modification of the MM algorithm described in Section III, similarly to the factorizations with missing data described in [34], =-=[49]-=-. After estimation, the missing pixels (l, p) belonging to the complement of O are reconstructed as [MA]lp and the aSAMs values between the original (complete) data Y and its reconstruction Ŷ are com... |

16 | Convergence-guaranteed multiplicative algorithms for non-negative matrix factorization with betadivergence
- Nakano, Kameoka, et al.
- 2010
(Show Context)
Citation Context ...problem min M C(M) = D(Y|MA+R) s.t. M ≥ 0. (9) When R = 0, this problem boils down to updating the dictionary matrix in NMF with the β-divergence. MM algorithms have been designed for that purpose in =-=[33]-=-, [34]. In this section, we extend the MM approach to the case where R ≥ 0. Denote by M̃ the estimate of M at current iteration. The first step of MM consists in building an upper bound G(M|M̃) of C(M... |

15 |
Non-linear spectral unmixing by geodesic simplex volume maximization
- Heylen, Burazerovic, et al.
- 2011
(Show Context)
Citation Context ...d least squares (FCLS) [43]. VCA and FCLS are standard endmember extraction and inversion methods designed for the LMM. Besides, we considered the nonlinear endmember extraction technique proposed in =-=[44]-=-, denoted as Heylen’s algorithm in what follows, coupled with four different inversion methods designed for various nonlinear models, namely the NM, FM, GBM and the very flexible polynomial post-nonli... |

14 | Blind spectral unmixing based on sparse nonnegative matrix factorization - Yang, Zhou, et al. |

14 | A hierarchical Bayesian framework for constructing sparsity-inducing priors
- Lee, Caron, et al.
- 2010
(Show Context)
Citation Context ...atistical model. As mentioned in Section II-C, the βdivergence underlies a Tweedie data distribution such that E[Y|MA+R] = MA+R. The term λ‖R‖2,1 can be seen a log-prior term. Using some results from =-=[40]-=-, the corresponding prior distribution p(rp) for each column of R can be obtained as a scale mixture of conditionally independent halfNormal distributions, with a Gamma distribution assigned to the sc... |

13 | Nonlinear unmixing of hyperspectral images: Models and algorithms
- Dobigeon, Tourneret, et al.
(Show Context)
Citation Context ...lying the observations and has resulted in interesting results for most applications. However, for several specific applications, LMM may be inaccurate and other nonlinear models need to be advocated =-=[7]-=-. For instance, in remotely sensed images composed of vegetation (e.g., trees), interactions of photons with multiple components of the scene lead to nonlinear effects that can be taken into account T... |

11 | Reconstruction of reflectance spectra using robust nonnegative matrix factorization
- Hamza, Brady
- 2006
(Show Context)
Citation Context ...vely. In [22], the `2,1 norm of (Y−MA) is minimized (noise free scenario). A so-called robust nonnegative matrix factorization approach was introduced for the reconstruction of reflectance spectra in =-=[23]-=-; however the term “robust” there refers to a different feature, namely the use of a data-fitting term (the hypersurface cost function) that is less sensitive to outlier observations than the traditio... |

8 |
Hyperspectral imaging - an emerging process analytical tool for food quality and safety control
- Gowen
(Show Context)
Citation Context ...erspectral data since it provides a comprehensive and meaningful description of the collected measurements in various application fields including remote sensing [1], planetology [2], food monitoring =-=[3]-=- or spectromicroscopy [4]. SU consists in decomposing P multi-band observations Y = [y1, . . . ,yP ] into a collection of K individual spectra M = [m1, . . . ,mK ], called endmembers, and estimating t... |

8 |
Linear-quadratic mixing model for reflectances in urban environments
- Meganem, Déliot, et al.
(Show Context)
Citation Context ...], [9]. As explained in [10], several bilinear models have been proposed [11]–[13], and they mainly differ by the constraints imposed on the nonlinearity term. The linear-quadratic model advocated in =-=[14]-=- also incorporates pairwise interactions between the endmembers components. Conversely, to approximate a large range of second-order nonlinearities, Altmann et al. [15] introduce a polynomial post-non... |

8 | Real-time online singing voice separation from monaural recordings using robust low-rank modeling
- Sprechmann, Bronstein, et al.
- 2012
(Show Context)
Citation Context ... defined by ‖R‖2,1 = ‖e‖1 = P∑ p=1 ‖rp‖2 . (6) Eq. (5) defines a robust NMF problem. Robust NMF is a nonnegative variant of robust PCA [18] which has appeared in different forms in the literature. In =-=[19]-=-, the outlier term R is nonnegative and penalized by the `1 norm. In [20] and [21], R is real-valued and penalized by `1 and `1,2 norms, respectively. In [22], the `2,1 norm of (Y−MA) is minimized (no... |

8 |
Mapping ash tree colonization in an agricultural mountain landscape: Investigating the potential of hyperspectral imagery
- Sheeren, Fauvel, et al.
- 1987
(Show Context)
Citation Context ...68.20 39.56 41.66 30.93 28.94 IFM 98.16 171.10 145.41 198.64 11.21 28.55 56.15 30.30 28.64 9.28 10.20 IGBM 75.21 136.20 79.34 156.75 5.36 16.75 40.60 16.72 16.68 5.06 10.93 during the Madonna project =-=[48]-=-. The sub-image of interest, of size of 50×50 pixels, is known to be mainly composed of oak and chestnut trees, with an additional unknown non-plantedtree endmember (referred to as Endm. #3 in what fo... |

7 | Nonlinear spectral unmixing of hyperspectral images using Gaussian processes
- Altmann, Dobigeon, et al.
- 2013
(Show Context)
Citation Context ... For conciseness, only the abundance maps obtained with β = 1 are displayed as the results for β = 2 were visually very similar. The unmixing results are in good agreement with previous results [47], =-=[50]-=-. However, in addition to the standard description of the data by linearly mixed endmembers, the proposed model also provides information regarding the pixels that cannot be explained with the standar... |

6 | Bilinear models for nonlinear unmixing of hyperspectral images
- Altmann, Dobigeon, et al.
- 2011
(Show Context)
Citation Context ...eon is with University of Toulouse, IRIT/INP-ENSEEIHT, 2 rue Camichel, BP 7122, 31071 Toulouse cedex 7, France. (e-mail: nicolas.dobigeon@enseeiht.fr). using bilinear models [8], [9]. As explained in =-=[10]-=-, several bilinear models have been proposed [11]–[13], and they mainly differ by the constraints imposed on the nonlinearity term. The linear-quadratic model advocated in [14] also incorporates pairw... |

5 |
On the unmixing of MEx/OMEGA hyperspectral data
- Themelis, Rontogiannis, et al.
- 2010
(Show Context)
Citation Context ...st when analyzing hyperspectral data since it provides a comprehensive and meaningful description of the collected measurements in various application fields including remote sensing [1], planetology =-=[2]-=-, food monitoring [3] or spectromicroscopy [4]. SU consists in decomposing P multi-band observations Y = [y1, . . . ,yP ] into a collection of K individual spectra M = [m1, . . . ,mK ], called endmemb... |

5 | Minimum mean square distance estimation of a subspace
- Besson, Dobigeon, et al.
- 2011
(Show Context)
Citation Context ... outliers. These pixels probably correspond to areas where some interactions between several endmembers occur (e.g., water/vegetation, water/soil). Similar results have been already observed in [13], =-=[51]-=-, which confirms the relevance of the proposed method. For the Madonna image, the energy map exhibits residual terms that are mainly located in the area occupied by the oak trees and the unknown 3rd e... |

4 |
Quantifying nonlinear spectral mixing in vegetated areas: computer simulation model validation and first results
- Somers, Tits, et al.
(Show Context)
Citation Context ...e@unice.fr). N. Dobigeon is with University of Toulouse, IRIT/INP-ENSEEIHT, 2 rue Camichel, BP 7122, 31071 Toulouse cedex 7, France. (e-mail: nicolas.dobigeon@enseeiht.fr). using bilinear models [8], =-=[9]-=-. As explained in [10], several bilinear models have been proposed [11]–[13], and they mainly differ by the constraints imposed on the nonlinearity term. The linear-quadratic model advocated in [14] a... |

4 | Robust nonnegative matrix factorization for nonlinear unmixing of hyperspectral images
- Dobigeon, Févotte
- 2013
(Show Context)
Citation Context ...results obtained on synthetic data are reported in Section IV. Two real hyperspectral images are investigated in Section V. Section VI concludes. This article extends our preliminary conference paper =-=[17]-=- in a significant way. We here generalize the use of the squared Euclidean distance considered in [17] to the more general β-divergence. Additionally, we show how some of the multiplicative updates ar... |

3 |
Robust nonnegative matrix factorization using `2,1-norm
- Kong, Ding, et al.
- 2011
(Show Context)
Citation Context ... in different forms in the literature. In [19], the outlier term R is nonnegative and penalized by the `1 norm. In [20] and [21], R is real-valued and penalized by `1 and `1,2 norms, respectively. In =-=[22]-=-, the `2,1 norm of (Y−MA) is minimized (noise free scenario). A so-called robust nonnegative matrix factorization approach was introduced for the reconstruction of reflectance spectra in [23]; however... |

2 |
Spectral mixture analysis of EELS spectrumimages
- Dobigeon, Brun
- 2012
(Show Context)
Citation Context ...provides a comprehensive and meaningful description of the collected measurements in various application fields including remote sensing [1], planetology [2], food monitoring [3] or spectromicroscopy =-=[4]-=-. SU consists in decomposing P multi-band observations Y = [y1, . . . ,yP ] into a collection of K individual spectra M = [m1, . . . ,mK ], called endmembers, and estimating their relative proportions... |

2 | A comparison of nonlinear mixing models for vegetated areas using simulated and real hyperspectral data - Dobigeon, Tits, et al. |

2 |
Robust nonnegative matrix factorization,” Front
- Zhang, Chen, et al.
- 2011
(Show Context)
Citation Context ...MF problem. Robust NMF is a nonnegative variant of robust PCA [18] which has appeared in different forms in the literature. In [19], the outlier term R is nonnegative and penalized by the `1 norm. In =-=[20]-=- and [21], R is real-valued and penalized by `1 and `1,2 norms, respectively. In [22], the `2,1 norm of (Y−MA) is minimized (noise free scenario). A so-called robust nonnegative matrix factorization a... |

2 |
Robust nonnegative matrix factorization via `1 norm regularization,” ArXiv preprint
- Shen, Si, et al.
- 2012
(Show Context)
Citation Context ...m. Robust NMF is a nonnegative variant of robust PCA [18] which has appeared in different forms in the literature. In [19], the outlier term R is nonnegative and penalized by the `1 norm. In [20] and =-=[21]-=-, R is real-valued and penalized by `1 and `1,2 norms, respectively. In [22], the `2,1 norm of (Y−MA) is minimized (noise free scenario). A so-called robust nonnegative matrix factorization approach w... |