
@article{chang_c.-i._fast_2006,
	title = {A fast iterative algorithm for implementation of pixel purity index},
	volume = {3},
	issn = {{1545-598X}},
	doi = {10.1109/LGRS.2005.856701},
	abstract = {The pixel purity index {(PPI)} has been widely used in hyperspectral image analysis for endmember extraction due to its publicity and availability in the Environment for Visualizing Images {(ENVI)} software. Unfortunately, its detailed implementation has never been made available in the literature. This paper investigates the {PPI} based on limited published results and proposes a fast iterative algorithm to implement the {PPI,} referred to as fast iterative {PPI} {(FIPPI).} It improves the {PPI} in several aspects. Instead of using randomly generated vectors as initial endmembers, the {FIPPI} produces an appropriate initial set of endmembers to speed up its process. Additionally, it estimates the number of endmembers required to be generated by a recently developed concept, virtual dimensionality {(VD)} which is one of the most crucial issues in the implementation of {PPI.} Furthermore, it is an iterative algorithm, where an iterative rule is developed to improve each of the iterations until it reaches a final set of endmembers. Most importantly, it is an unsupervised algorithm as opposed to the {PPI,} which requires human intervention to manually select a final set of endmembers. The experiments show that both the {FIPPI} and the {PPI} produce very close results, but the {FIPPI} converges very rapidly with significant savings in computation.},
	number = {1},
	journal = {Geoscience and Remote Sensing Letters, {IEEE}},
	author = {Chang, {C.-I.} and Plaza, A.},
	year = {2006},
	keywords = {Automatic target generation process {(ATGP),} endmember extraction algorithm, endmember pixel, fast iterative pixel purity index, feature extraction, remote sensing, unsupervised algorithm, virtual dimensionality, virtual dimensionality {(VD)}},
	pages = {63--67}
},

@article{grana_two_2009,
	title = {Two lattice computing approaches for the unsupervised segmentation of hyperspectral images},
	volume = {72},
	url = {http://portal.acm.org/citation.cfm?id=1539945},
	abstract = {Endmembers for the spectral unmixing analysis of hyperspectral images are sets of affinely independent vectors, which define a convex polytope covering the data points that represent the pixel image spectra. Strong lattice independence {(SLI)} is a property defined in the context of lattice associative memories convergence analysis. Recent results show that {SLI} implies affine independence, confirming the value of lattice associative memories for the study of endmember induction algorithms. In fact, {SLI} vector sets can be easily deduced from the vectors composing the lattice auto-associative memories {(LAM).} However, the number of candidate endmembers found by this algorithm is very large, so that some selection algorithm is needed to obtain the full benefits of the approach. In this paper we explore the unsupervised segmentation of hyperspectral images based on the abundance images computed, first, by an endmember selection algorithm and, second, by a previously proposed heuristically defined algorithm. We find their results comparable on a qualitative basis.},
	number = {10-12},
	journal = {Neurocomput.},
	author = {Manuel Grana and Ivan Villaverde and Jose O. Maldonado and Carmen Hernandez},
	year = {2009},
	keywords = {auto-associative morphological memories, endmember induction, hyperspectral images, lattice associative memories, lattice computing, lattice independence},
	pages = {2111--2120}
},

@article{grana_lattice_2011,
	title = {Lattice independent component analysis for functional magnetic resonance imaging},
	volume = {181},
	issn = {0020-0255},
	lccn = {0000},
	url = {http://www.sciencedirect.com/science/article/B6V0C-515SRJV-2/2/4511f04ffeea5febd80c9b61ff2d3b3d},
	doi = {10.1016/j.ins.2010.09.023},
	abstract = {{{\textless}p{\textgreater}{\textless}br/{\textgreater}We} introduce a lattice independent component analysis {(LICA)} unsupervised scheme to functional magnetic resonance imaging {(fMRI)} data analysis. {LICA} is a non-linear alternative to independent component analysis {(ICA),} such that {ICA's} statistical independent sources correspond to {LICA's} lattice independent sources. In this paper, {LICA} uses an incremental lattice source induction algorithm {(ILSIA)} to induce the lattice independent sources from the input dataset. The {ILSIA} computes a set of Strongly Lattice Independent vectors using properties of lattice associative memories regarding Lattice Independence and Chebyshev best approximation. The lattice independent sources constitute a set of Affine Independent vectors that define a simplex covering the input data. {LICA} carries out data linear unmixing based on the lattice independent sources basis. Therefore, {LICA} is a hybrid combination of a non-linear lattice based component and a linear unmixing component. The principal advantage over {ICA} is that {LICA} does not impose any probabilistic model assumptions on the data sources. We compare {LICA} with {ICA} in two case studies. Firstly, including simulated {fMRI} data, {LICA} discovers the spatial location of meaningful sources with less ambiguity than {ICA.} Secondly, including real data from an auditory stimulation experiment, {LICA} improves over some state of the art {ICA} variants discovering the activation patterns detected by Statistical Parametric Mapping {(SPM)} on the same data.{\textless}/p{\textgreater}},
	number = {10},
	journal = {Information Sciences},
	author = {Manuel Gra\~{n}a and Darya Chyzhyk and Maite {Garc\'{i}a-Sebasti\'{a}n} and Carmen Hern\'{a}ndez},
	month = may,
	year = {2011},
	keywords = {{fMRI,} independent component analysis, lattice associative memories, lattice independence},
	pages = {1910--1928}
},

@article{plaza_impact_2006,
	title = {Impact of Initialization on Design of Endmember Extraction Algorithms},
	volume = {44},
	issn = {0196-2892},
	doi = {10.1109/TGRS.2006.879538},
	abstract = {Many endmember extraction algorithms {(EEAs)} have been developed to find endmembers that are assumed to be pure signatures in hyperspectral data. However, two issues arising in {EEAs} have not been addressed: one is the knowledge of the number of endmembers that must be provided a priori, and the other is the initialization of {EEAs,} where most {EEAs} initialize their endmember-searching processes by using randomly generated endmembers, which generally result in inconsistent final selected endmembers. Unfortunately, there has been no previous work reported on how to address these two issues, i.e., how to select a set of appropriate initial endmembers and how to determine the number of endmembers p. This paper takes up these two issues and describes two-stage processes to improve {EEAs.} First, a recently developed concept of virtual dimensionality {(VD)} is used to determine how many endmembers are needed to be generated for an {EEA.} Experiments show that the {VD} is an adequate measure for estimating p. Second, since {EEAs} are sensitive to initial endmembers, a properly selected set of initial endmembers can make significant improvements on the searching process. In doing so, a new concept of endmember initialization algorithm {(EIA)} is thus proposed, and four different algorithms are suggested for this purpose. It is surprisingly found that many {EIA-generated} initial endmembers turn out to be the final desired endmembers. A further objective is to demonstrate that {EEAs} implemented in conjunction with {EIA-generated} initial endmembers can significantly reduce the number of endmember replacements as well as the computing time during endmember search},
	number = {11},
	journal = {Geoscience and Remote Sensing, {IEEE} Transactions on},
	author = {A. Plaza and {C.-I.} Chang},
	year = {2006},
	keywords = {Automatic target generation process {(ATGP),} endmember extraction algorithm, endmember initialization algorithm, endmember initialization algorithm {(EIA),} endmember search, feature extraction, geophysics computing, hyperspectral data, iterative error analysis, iterative error analysis {(IEA),} maximin-distance algorithm, multidimensional signal processing, remote sensing, unsupervised fully constrained least squares, unsupervised fully constrained least squares {(UFCLS)} algorithm, virtual dimensionality},
	pages = {3397--3407}
},

@article{ritter_lattice_2010,
	title = {A lattice matrix method for hyperspectral image unmixing},
	volume = {In Press, Corrected Proof},
	issn = {0020-0255},
	url = {http://www.sciencedirect.com/science/article/B6V0C-4YPT1TR-5/2/1a42da56378f4a7a81ea59c0ac72e5b5},
	doi = {10.1016/j.ins.2010.03.022},
	abstract = {In this manuscript we propose a method for the autonomous determination of endmembers in hyperspectral imagery based on recent theoretical advancements on lattice auto-associative memories. Given a hyperspectral image, the lattice algebra approach finds in a single-pass all possible candidate endmembers from which various affinely independent sets of final endmembers may be derived. In contrast to other endmember detection methods, the endmembers found using two dual canonical lattice matrices are geometrically linked to the data set spectra. The mathematical foundation of the proposed method is first described in some detail followed by application examples that illustrate the key steps of the proposed lattice based method.},
	journal = {Information Sciences},
	author = {Gerhard X. Ritter and Gonzalo Urcid},
	month = oct,
	year = {2010},
	keywords = {abundance maps, hyperspectral image analysis: endmember search, Lattice algebra: affine independence, lattice associative memories, lattice independence, Lattice matrices, neural networks: associative memories, spectral unmixing}
},

@inproceedings{winter_m._e._n-findr:_1999,
	address = {Denver, {CO,} {USA}},
	title = {{N-FINDR:} an algorithm for fast autonomous spectral end-member determination in hyperspectral data},
	volume = {3753},
	shorttitle = {{N-FINDR}},
	url = {http://link.aip.org/link/?PSI/3753/266/1},
	publisher = {{SPIE}},
	author = {Winter, M. E. and Descour, M. R. and Shen, S. S.},
	month = oct,
	year = {1999},
	pages = {266--275}
},


@article{chang_c.-i._estimation_2004,
	title = {Estimation of number of spectrally distinct signal sources in hyperspectral imagery},
	volume = {42},
	issn = {0196-2892},
	doi = {10.1109/TGRS.2003.819189},
	abstract = {With very high spectral resolution, hyperspectral sensors can now uncover many unknown signal sources which cannot be identified by visual inspection or a priori. In order to account for such unknown signal sources, we introduce a new definition, referred to as virtual dimensionality {(VD)} in this paper. It is defined as the minimum number of spectrally distinct signal sources that characterize the hyperspectral data from the perspective view of target detection and classification. It is different from the commonly used intrinsic dimensionality {(ID)} in the sense that the signal sources are determined by the proposed {VD} based only on their distinct spectral properties. These signal sources may include unknown interfering sources, which cannot be identified by prior knowledge. With this new definition, three {Neyman-Pearson} detection theory-based thresholding methods are developed to determine the {VD} of hyperspectral imagery, where eigenvalues are used to measure signal energies in a detection model. In order to evaluate the performance of the proposed methods, two information criteria, an information criterion {(AIC)} and minimum description length {(MDL),} and the factor analysis-based method proposed by Malinowski, are considered for comparative analysis. As demonstrated in computer simulations, all the methods and criteria studied in this paper may work effectively when noise is independent identically distributed. This is, unfortunately, not true when some of them are applied to real image data. Experiments show that all the three eigenthresholding based methods (i.e., the {Harsanyi-Farrand-Chang} {(HFC),} the noise-whitened {HFC} {(NWHFC),} and the noise subspace projection {(NSP)} methods) produce more reliable estimates of {VD} compared to the {AIC,} {MDL,} and Malinowski's empirical indicator function, which generally overestimate {VD} significantly. In summary, three contributions are made in this paper, 1) an introduction of the new definition of {VD,} 2) three {Neyman-Pearson} detection theory-based thresholding methods, {HFC,} {NWHFC,} and {NSP} derived for {VD} estimation, and 3) experiments that show the {AIC} and {MDL} commonly used in passive array processing and the second-order statistic-based Malinowski's method are not effective measures in {VD} estimation.},
	number = {3},
	journal = {Geoscience and Remote Sensing, {IEEE} Transactions on},
	author = {Chang, {C.-I.} and Du, Q.},
	year = {2004},
	keywords = {a priori inspection, eigenthresholding based methods, eigenvalues, eigenvalues and eigenfunctions, {Harsanyi-Farrand-Chang,} high spectral resolution, hyperspectral characterization, hyperspectral imagery, hyperspectral sensors, image sensors, intrinsic dimensionality, Malinowskis empirical indicator function, {Neyman-Pearson} detection theory-based thresholding methods, noise subspace projection, noise-whitened {HFC,} signal denoising, signal detection, signal energy measurement, signal sources, spectrally distinct signal sources estimation, target classification, target detection, unknown interfering sources, unknown signal sources, virtual dimensionality, visual inspection},
	pages = {608--619}
},

@article{chein-i_chang_linear_2010,
	title = {Linear Spectral Mixture Analysis Based Approaches to Estimation of Virtual Dimensionality in Hyperspectral Imagery},
	volume = {48},
	issn = {0196-2892},
	lccn = {0000},
	doi = {10.1109/TGRS.2010.2068552},
	abstract = {{{\textless}p{\textgreater}Virtual} dimensionality {(VD)} is a new concept which was originally developed for estimating the number of spectrally distinct signatures present in hyperspectral data. The effectiveness of the {VD} is determined by the technique used for {VD} estimation. This paper develops an orthogonal subspace projection {(OSP)} technique to estimate the {VD.} The idea is derived from linear spectral mixture analysis where a data sample vector is modeled as a linear mixture of a finite set of what is called as virtual endmembers in this paper. A similar idea was also previously investigated by the signal subspace estimate {(SSE)} and was later improved by hyperspectral signal subspace identification by minimum error {(HySime),} where the minimum mean squared error is used as a criterion to determine the {VD.} Interestingly, with an appropriate interpretation, the proposed {OSP} technique includes the {SSE/HySime} as its special case. In order to demonstrate its utility, experiments using synthetic images and real image data sets are conducted for performance analysis.{\textless}/p{\textgreater}},
	number = {11},
	journal = {Geoscience and Remote Sensing, {IEEE} Transactions on},
	author = {{Chein-I} Chang and Wei Xiong and Weimin Liu and {Mann-Li} Chang and {Chao-Cheng} Wu and {C.C.-C.} Chen},
	year = {2010},
	keywords = {geophysical image processing, {Harsanyi-Farrand-Chang} method, hyperspectral imagery, linear spectral mixing, linear spectral mixture analysis, mean square error methods, minimum mean squared error, orthogonal subspace projection, remote sensing, signal subspace estimate, spectrally distinct signatures, virtual dimensionality estimation, virtual endmembers},
	pages = {3960--3979}
},


@article{ritter_morphological_1999,
	title = {Morphological bidirectional associative memories},
	volume = {12},
	issn = {0893-6080},
	url = {http://www.sciencedirect.com/science/article/B6T08-3XMHNK7-6/2/b6b8330f98848c61ac8f5794c032c5b3},
	doi = {10.1016/S0893-6080(99)00033-7},
	abstract = {The theory of artificial neural networks has been successfully applied to a wide variety of pattern recognition problems. In this theory, the first step in computing the next state of a neuron or in performing the next layer neural network computation involves the linear operation of multiplying neural values by their synaptic strengths and adding the results. Thresholding usually follows the linear operation in order to provide for nonlinearity of the network. In this paper we discuss a novel class of artificial neural networks, called morphological neural networks, in which the operations of multiplication and addition are replaced by addition and maximum (or minimum), respectively. By taking the maximum (or minimum) of sums instead of the sum of products, morphological network computation is nonlinear before thresholding. As a consequence, the properties of morphological neural networks are drastically different from those of traditional neural network models. The main emphasis of the research presented here is on morphological bidirectional associative memories {(MBAMs).} In particular, we establish a mathematical theory for {MBAMs} and provide conditions that guarantee perfect bidirectional recall for corrupted patterns. Some examples that illustrate performance differences between the morphological model and the traditional semilinear model are also given.},
	number = {6},
	journal = {Neural Networks},
	author = {G. X. Ritter and J. L. {Diaz-de-Leon} and P. Sussner},
	month = jul,
	year = {1999},
	keywords = {Associative memories, Bidirectional associative memories, morphological associative memories, morphological neural networks},
	pages = {851--867}
},

@article{ritter_g._x._morphological_1998,
	title = {Morphological associative memories},
	volume = {9},
	issn = {1045-9227},
	doi = {10.1109/72.661123},
	abstract = {The theory of artificial neural networks has been successfully applied to a wide variety of pattern recognition problems. In this theory, the first step in computing the next state of a neuron or in performing the next layer neural network computation involves the linear operation of multiplying neural values by their synaptic strengths and adding the results. A nonlinear activation function usually follows the linear operation in order to provide for nonlinearity of the network and set the next state of the neuron. In this paper we introduce a novel class of artificial neural networks, called morphological neural networks, in which the operations of multiplication and addition are replaced by addition and maximum (or minimum), respectively. By taking the maximum (or minimum) of sums instead of the sum of products, morphological network computation is nonlinear before possible application of a nonlinear activation function. As a consequence, the properties of morphological neural networks are drastically different than those of traditional neural network models. The main emphasis of the research presented here is on morphological associative memories. We examine the computing and storage capabilities of morphological associative memories and discuss differences between morphological models and traditional semilinear models such as the Hopfield net},
	number = {2},
	journal = {Neural Networks, {IEEE} Transactions on},
	author = {Ritter, G. X. and Sussner, P. and {Diaz-de-Leon,} J. L.},
	year = {1998},
	keywords = {addition, content-addressable storage, Hopfield net, linear operation, mathematical morphology, matrix algebra, maximum, minimum, morphological associative memories, morphological neural networks, neural nets, nonlinear activation function, Pattern recognition, pattern recognition problems, semilinear models, storage capabilities, synaptic strengths},
	pages = {281--293}
},


@article{ifarraguerri_a._multispectral_1999,
	title = {Multispectral and hyperspectral image analysis with convex cones},
	volume = {37},
	issn = {0196-2892},
	doi = {10.1109/36.752192},
	abstract = {A new approach to multispectral and hyperspectral image analysis is presented. This method, called convex cone analysis {(CCA),} is based on the bet that some physical quantities such as radiance are nonnegative. The vectors formed by discrete radiance spectra are linear combinations of nonnegative components, and they lie inside a nonnegative, convex region. The object of {CCA} is to find the boundary points of this region, which can be used as endmember spectra for unmixing or as target vectors for classification. To implement this concept, the authors find the eigenvectors of the sample spectral correlation matrix of the image. Given the number of endmembers or classes, they select as many eigenvectors corresponding to the largest eigenvalues. These eigenvectors are used as a basis to form linear combinations that have only nonnegative elements, and thus they lie inside a convex cone. The vertices of the convex cone will be those points whose spectral vector contains as many zero elements as the number of eigenvectors minus one. Accordingly, a mixed pixel can be decomposed by identifying the vertices that were used to form its spectrum. An algorithm for finding the convex cone boundaries is presented, and applications to unsupervised unmixing and classification are demonstrated with simulated data as well as experimental data from the hyperspectral digital imagery collection experiment {(HYDICE)}},
	number = {2},
	journal = {Geoscience and Remote Sensing, {IEEE} Transactions on},
	author = {Ifarraguerri, A. and Chang, {C.-I.}},
	year = {1999},
	keywords = {algorithm, convex cone analysis, convex cone boundary, discrete radiance spectra, geophysical measurement technique, image processing, linear combination, multidimensional signal processing, multispectral method, nonnegative component, terrain mapping, unsupervised unmixing},
	pages = {756--770}
}
