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An NCM-based Bayesian algorithm for hyperspectral unmixing
Abstract
This paper studies a new Bayesian algorithm to unmix hyperspectral images. The algorithm is based on the recent normal compositional model introduced by Eismann. Contrary to the standard linear mixing model, the endmember spectra are assumed to be random signatures with know mean vectors. Appropriate prior distributions are assigned to the abundance coefficients to ensure the usual positivity and sum-to-one constraints. However, the resulting posterior distribution is too complex to obtain a closed form expression for the Bayesian estimators. A Markov chain Monte Carlo algorithm is then proposed to generate samples distributed according to the full posterior distribution. These samples are used to estimate the unknown model parameters. Several simulations are conducted on synthetic and real data to illustrate the performance of the proposed method.
... A common research problem in hyperspectral image (HSI) analysis is that of estimating endmembers from a given set of pixels, often referred to as endmember extraction. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] Often endmembers are extracted simultaneously with estimating of their respective proportions, this is known as pixel unmixing. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] A vast majority of approaches to this problem rely on a linear model to describe the mixing relationships between endmembers. ...
... [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] Often endmembers are extracted simultaneously with estimating of their respective proportions, this is known as pixel unmixing. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] A vast majority of approaches to this problem rely on a linear model to describe the mixing relationships between endmembers. 17 An example is the linear mixture model (LMM) shown in Equation (1), with the corresponding constraints in Equation (2), where is the number of endmembers in the scene, is an error term, and and are the endmembers and proportions, respectively, of a given pixel . ...
A method of incorporating the multi-mixture pixel model into hyperspectral endmember extraction is presented and
discussed. A vast majority of hyperspectral endmember extraction methods rely on the linear mixture model to describe
pixel spectra resulting from mixtures of endmembers. Methods exist to unmix hyperspectral pixels using nonlinear
models, but rely on severely limiting assumptions or estimations of the nonlinearity. This paper will present a
hyperspectral pixel endmember extraction method that utilizes the bidirectional reflectance distribution function to
model microscopic mixtures. Using this model, along with the linear mixture model to incorporate macroscopic
mixtures, this method is able to accurately unmix hyperspectral images composed of both macroscopic and microscopic
mixtures. The mixtures are estimated directly from the hyperspectral data without the need for a priori knowledge of the
mixture types. Results are presented using synthetic datasets, of multi-mixture pixels, to demonstrate the increased
accuracy in unmixing using this new physics-based method over linear methods. In addition, results are presented using
a well-known laboratory dataset.
... A common research problem in hyperspectral image (HSI) analysis is that of estimating endmember proportions from a given set of pixels, often referred to as pixel unmixing. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] A vast majority of approaches to this problem rely on a linear model to describe the mixing relationships between endmembers. 1 An example is the linear mixture model (LMM) shown in Equation (1), with the corresponding constraints in Equation (2), where is the number of endmembers in the scene, is an error term, and and are the endmembers and proportions, respectively, of a given pixel . 2,3 Reliance on the linear model is a result of the mathematical amenability of linear models and the prevalence of macroscopic mixtures, also known as areal or checkerboard mixtures. ...
... LMM proportions were estimated using quadratic programming to minimize the RSS term given in Equation (16), derived from the LMM shown in Equation (1). Similarly, the AD-LMM proportions were estimated using quadratic programming to minimize the RSS term given in Equation (14), derived from Equation (13). Thus, allowing a comparison between MPE and proportions estimated with the LMM in both reflectance and albedo-domains. ...
A method of incorporating macroscopic and microscopic reflectance models into hyperspectral pixel unmixing is
presented and discussed. A vast majority of hyperspectral unmixing methods rely on the linear mixture model to
describe pixel spectra resulting from mixtures of endmembers. Methods exist to unmix hyperspectral pixels using
nonlinear models, but rely on severely limiting assumptions or estimations of the nonlinearity. This paper will present a
hyperspectral pixel unmixing method that utilizes the bidirectional reflectance distribution function to model
microscopic mixtures. Using this model, along with the linear mixture model to incorporate macroscopic mixtures, this
method is able to accurately unmix hyperspectral images composed of both macroscopic and microscopic mixtures. The
mixtures are estimated directly from the hyperspectral data without the need for a priori knowledge of the mixture types.
Results are presented using synthetic datasets, of macroscopic and microscopic mixtures, to demonstrate the increased
accuracy in unmixing using this new physics-based method over linear methods. In addition, results are presented using
a well-known laboratory dataset. Using these results, and other published results from this dataset, increased accuracy in
unmixing over other nonlinear methods is shown.
... Hyperspectral image unmixing involves estimating the endmembers present in the scene and/or the fractional abundance of each endmember in every pixel [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Most unmixing algorithms are based on the linear mixing model (LMM) [1] shown in Eqs. ...
Much work in the study of hyperspectral imagery has focused on macroscopic mixtures and unmixing via the linear mixing model. A substantially different approach seeks to model hyperspectral data non-linearly in order to accurately describe intimate or microscopic relationships of materials within the image. In this paper we present and discuss a new model (MacMicDEM) that seeks to unify both approaches by representing a pixel as both linearly and non-linearly mixed, with the condition that the endmembers for both mixture types need not be related. Using this model, we develop a method to accurately and quickly unmix data which is both macroscopically and microscopically mixed. Subsequently, this method is then validated on synthetic and real datasets.
... Therefore design of the tree species classifier was based on forming the representative design (training) set of trees for each of the species. A similar approach is developed in various publications [7], [17], [18]. To start the classification, classes of i k trees from species i ...
This work describes the task of inventorying Baltic mixed forests at an individual tree level. The development of a practicable methodology for semi-automated identification of tree species was targeted. Data acquisition equipment and preprocessing software, explored forest area, processing approaches, obtained classification results as well as newly developed software are described. To resolve the core problem – tree species identification – a classification approach is proposed for processing multi-spectral imagery data from the vicinity of tree tops. A multi-class classifier is designed from multi-spectral data of interactively selected trees included in initial design (training) sets for two conifer and three deciduous species of interest. An approach for the stabilization of the classification results is proposed, based on improving the representativeness of the design sets by selection of trees from different locations, dismissing trees with overlapping crowns and anomalies, followed by the calculation of a spectral dissimilarity parameter of the design sets and dismissing the sets of trees of any species which are too similar. The best classification results were obtained using a two-stage procedure. In the first stage, species clusters were created by adding randomly selected trees from the whole analyzed forest area. Final classification of all trees was done by using a Bayes classifier designed on the basis of cluster properties. A procedure for increasing robustness of the clustering stage is proposed, based on performing multiple clustering attempts, each using a randomly sampled subset of a chosen design set for the classifier design, and making a decision about the class of each tree by the majority vote from the results of these attempts. This classification algorithm was tested against the set of trees, for which information was available from field work. It is shown that a mean classification error below 3% can be achieved and the maximum error rate was decreased substantially by applying the proposed approach for selection of representative design sets.
... These prior distributions ensure the positivity and sum-to-one constraints of the abundance coefficients. They are based on a reparametrization of the abundance vectors and are much more flexible than the priors previously studied in [7], [21] or [11]. Of course, the accuracy of the abundance estimation procedure drastically depends on the hyperparameters associated to these priors. ...
This paper describes a new algorithm for hyperspectral image unmixing. Most
of the unmixing algorithms proposed in the literature do not take into account
the possible spatial correlations between the pixels. In this work, a Bayesian
model is introduced to exploit these correlations. The image to be unmixed is
assumed to be partitioned into regions (or classes) where the statistical
properties of the abundance coefficients are homogeneous. A Markov random field
is then proposed to model the spatial dependency of the pixels within any
class. Conditionally upon a given class, each pixel is modeled by using the
classical linear mixing model with additive white Gaussian noise. This strategy
is investigated the well known linear mixing model. For this model, the
posterior distributions of the unknown parameters and hyperparameters allow
ones to infer the parameters of interest. These parameters include the
abundances for each pixel, the means and variances of the abundances for each
class, as well as a classification map indicating the classes of all pixels in
the image. To overcome the complexity of the posterior distribution of
interest, we consider Markov chain Monte Carlo methods that generate samples
distributed according to the posterior of interest. The generated samples are
then used for parameter and hyperparameter estimation. The accuracy of the
proposed algorithms is illustrated on synthetic and real data.
This work describes the task of inventorying Baltic mixed forests at an individual tree level. The development of a practicable methodology for semi-automated identification of tree species was targeted. Data acquisition equipment and preprocessing software, explored forest area, processing approaches, obtained classification results as well as newly developed software are described. To resolve the core problem - tree species identification - a classification approach is proposed for processing multi-spectral imagery data from the vicinity of tree tops. A multi-class classifier is designed from multi-spectral data of interactively selected trees included in initial design (training) sets for two conifer and three deciduous species of interest. An approach for the stabilization of the classification results is proposed, based on improving the representativeness of the design sets by selection of trees from different locations, dismissing trees with overlapping crowns and anomalies, followed by the calculation of a spectral dissimilarity parameter of the design sets and dismissing the sets of trees of any species which are too similar. The best classification results were obtained using a two-stage procedure. In the first stage, species clusters were created by adding randomly selected trees from the whole analyzed forest area. Final classification of all trees was done by using a Bayes classifier designed on the basis of cluster properties. A procedure for increasing robustness of the clustering stage is proposed, based on performing multiple clustering attempts, each using a randomly sampled subset of a chosen design set for the classifier design, and making a decision about the class of each tree by the majority vote from the results of these attempts. This classification algorithm was tested against the set of trees, for which information was available from field work. It is shown that a mean classification error below 3% can be achieved and the maximum error rat- was decreased substantially by applying the proposed approach for selection of representative design sets.
L’imagerie hyperspectrale est très largement employée en télédétection pour diverses applications, dans le domaine civil comme dans le domaine militaire. Une image hyperspectrale est le résultat de l’acquisition d’une seule scène observée dans plusieurs longueurs d’ondes. Par conséquent, chacun des pixels constituant cette image est représenté par un vecteur de mesures (généralement des réflectances) appelé spectre. Une étape majeure dans l’analyse des données hyperspectrales consiste à identifier les composants macroscopiques (signatures) présents dans la région observée et leurs proportions correspondantes (abondances). Les dernières techniques développées pour ces analyses ne modélisent pas correctement ces images. En effet, habituellement ces techniques supposent l’existence de pixels purs dans l’image, c’est-à-dire des pixels constitué d’un seul matériau pur. Or, un pixel est rarement constitué d’éléments purs distincts l’un de l’autre. Ainsi, les estimations basées sur ces modèles peuvent tout à fait s’avérer bien loin de la réalité. Le but de cette étude est de proposer de nouveaux algorithmes d’estimation à l’aide d’un modèle plus adapté aux propriétés intrinsèques des images hyperspectrales. Les paramètres inconnus du modèle sont ainsi déduits dans un cadre Bayésien. L’utilisation de méthodes de Monte Carlo par Chaînes de Markov (MCMC) permet de surmonter les difficultés liées aux calculs complexes de ces méthodes d’estimation. ABSTRACT : Hyperspectral imagery has been widely used in remote sensing for various civilian and military applications. A hyperspectral image is acquired when a same scene is observed at different wavelengths. Consequently, each pixel of such image is represented as a vector of measurements (reflectances) called spectrum. One major step in the analysis of hyperspectral data consists of identifying the macroscopic components (signatures) that are present in the sensored scene and the corresponding proportions (concentrations). The latest techniques developed for this analysis do not properly model these images. Indeed, these techniques usually assume the existence of pure pixels in the image, i.e. pixels containing a single pure material. However, a pixel is rarely composed of pure spectrally elements, distinct from each other. Thus, such models could lead to weak estimation performance. The aim of this thesis is to propose new estimation algorithms with the help of a model that is better suited to the intrinsic properties of hyperspectral images. The unknown model parameters are then infered within a Bayesian framework. The use of Markov Chain Monte Carlo (MCMC) methods allows one to overcome the difficulties related to the computational complexity of these inference methods.
La simulation est devenue dans la dernière décennie un outil essentiel du traitement statistique de modèles complexes et de la mise en oeuvre de techniques statistiques avancées, comme le bootstrap ou les méthodes d'inférence simulée. Ce livre présente les éléments de base de la simulation de lois de probabilité (génération de variables uniformes et de lois usuelles) et de leur utilisation en Statistique (intégration de Monte Carlo, optimisation stochastique). Après un bref rappel sur les chaînes de Markov, les techniques plus spécifiques de Monte Carlo par chaînes de Markov (MCMC) sont présentées en détail, à la fois du point de vue théorique (validité et convergence) et du point de vue de leur implémentation (accélération, choix de paramètres, limitations). Les algorithmes d'échantillonnage de Gibbs sont ainsi distingués des méthodes générales de Hastings-Metropolis par leur plus grande richesse théorique. Les derniers chapitres contiennent un exposé critique sur l'état de l'art en contrôle de convergence de ces algorithmes et une présentation unifiée des diverses applications des méthodes MCMC aux modèles à données manquantes. De nombreux exemples statistiques illustrent les méthodes présentées dans cet ouvrage destiné aux étudiants de deuxième et troisième cycles universitaires en Mathématiques Appliquées ainsi qu'aux chercheurs et praticiens désirant utiliser les méthodes MCMC. Monte Carlo statistical methods, particularly those based on Markov chains, are now an essential component of the standard set of techniques used by statisticians. This new edition has been revised towards a coherent and flowing coverage of these simulation techniques, with incorporation of the most recent developments in the field. In particular, the introductory coverage of random variable generation has been totally revised, with many concepts being unified through a fundamental theorem of simulation There are five completely new chapters that cover Monte Carlo control, reversible jump, slice sampling, sequential Monte Carlo, and perfect sampling. There is a more in-depth coverage of Gibbs sampling, which is now contained in three consecutive chapters. The development of Gibbs sampling starts with slice sampling and its connection with the fundamental theorem of simulation, and builds up to two-stage Gibbs sampling and its theoretical properties. A third chapter covers the multi-stage Gibbs sampler and its variety of applications. Lastly, chapters from the previous edition have been revised towards easier access, with the examples getting more detailed coverage. This textbook is intended for a second year graduate course, but will also be useful to someone who either wants to apply simulation techniques for the resolution of practical problems or wishes to grasp the fundamental principles behind those methods. The authors do not assume familiarity with Monte Carlo techniques (such as random variable generation), with computer programming, or with any Markov chain theory (the necessary concepts are developed in Chapter 6). A solutions manual, which covers approximately 40% of the problems, is available for instructors who require the book for a course. oui
Spectral unmixing using hyperspectral data represents a
significant step in the evolution of remote decompositional analysis
that began with multispectral sensing. It is a consequence of collecting
data in greater and greater quantities and the desire to extract more
detailed information about the material composition of surfaces. Linear
mixing is the key assumption that has permitted well-known algorithms to
be adapted to the unmixing problem. In fact, the resemblance of the
linear mixing model to system models in other areas has permitted a
significant legacy of algorithms from a wide range of applications to be
adapted to unmixing. However, it is still unclear whether the assumption
of linearity is sufficient to model the mixing process in every
application of interest. It is clear, however, that the applicability of
models and techniques is highly dependent on the variety of
circumstances and factors that give rise to mixed pixels. The outputs of
spectral unmixing, endmember, and abundance estimates are important for
identifying the material composition of mixtures
This paper is also the originator of the Markov Chain Monte Carlo methods developed in the following chapters. The potential of these two simultaneous innovations has been discovered much latter by statisticians (Hastings 1970; Geman and Geman 1984) than by of physicists (see also Kirkpatrick et al. 1983). 5.5.5 ] PROBLEMS 211
The analysis of hyperspectral data sets requires the determination of
certain basis spectra called 'end-members.' Once these spectra are
found, the image cube can be 'unmixed' into the fractional abundance of
each material in each pixel. There exist several techniques for
accomplishing the determination of the end-members, most of which
involve the intervention of a trained geologist. Often these-end-members
are assumed to be present in the image, in the form of pure, or unmixed,
pixels. In this paper a method based upon the geometry of convex sets is
proposed to find a unique set of purest pixels in an image. The
technique is based on the fact that in N spectral dimensions, the
N-volume contained by a simplex formed of the purest pixels is larger
than any other volume formed from any other combination of pixels. The
algorithm works by 'inflating' a simplex inside the data, beginning with
a random set of pixels. For each pixel and each end-member, the
end-member is replaced with the spectrum of the pixel and the volume is
recalculated. If it increases, the spectrum of the new pixel replaces
that end-member. This procedure is repeated until no more replacements
are done. This algorithm successfully derives end-members in a synthetic
data set, and appears robust with less than perfect data. Spectral
end-members have been extracted for the AVIRIS Cuprite data set which
closely match reference spectra, and resulting abundance maps match
published mineral maps.
Given a set of mixed spectral (multispectral or hy- perspectral) vectors, linear spectral mixture analysis, or linear unmixing, aims at estimating the number of reference substances, also called endmembers, their spectral signatures, and their abundance fractions. This paper presents a new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis (VCA). The algorithm exploits two facts: 1) the endmembers are the vertices of a simplex and 2) the affine transformation of a simplex is also a simplex. In a series of experiments using simulated and real data, the VCA algorithm competes with state-of-the-art methods, with a computational complexity between one and two orders of magnitude lower than the best available method.
This paper studies a new Bayesian unmixing algorithm for hyperspectral images. Each pixel of the image is modeled as a linear combination of so-called endmembers. These endmembers are supposed to be random in order to model uncertainties regarding their knowledge. More precisely, we model endmembers as Gaussian vectors whose means have been determined using an endmember extraction algorithm such as the famous N-finder (N-FINDR) or Vertex Component Analysis (VCA) algorithms. This paper proposes to estimate the mixture coefficients (referred to as abundances) using a Bayesian algorithm. Suitable priors are assigned to the abundances in order to satisfy positivity and additivity constraints whereas conjugate priors are chosen for the remaining parameters. A hybrid Gibbs sampler is then constructed to generate abundance and variance samples distributed according to the joint posterior of the abundances and noise variances. The performance of the proposed methodology is evaluated by comparison with other unmixing algorithms on synthetic and real images.
This paper proposes a hierarchical Bayesian model that can be used for semi-supervised hyperspectral image unmixing. The model assumes that the pixel reflectances result from linear combinations of pure component spectra contaminated by an additive Gaussian noise. The abundance parameters appearing in this model satisfy positivity and additivity constraints. These constraints are naturally expressed in a Bayesian context by using appropriate abundance prior distributions. The posterior distributions of the unknown model parameters are then derived. A Gibbs sampler allows one to draw samples distributed according to the posteriors of interest and to estimate the unknown abundances. An extension of the algorithm is finally studied for mixtures with unknown numbers of spectral components belonging to a know library. The performance of the different unmixing strategies is evaluated via simulations conducted on synthetic and real data.
Given a set of mixed spectral (multispectral or hyperspectral) vectors, linear spectral mixture analysis, or linear unmixing, aims at estimating the number of reference substances, also called endmembers, their spectral signatures, and their abundance fractions. This paper presents a new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis (VCA). The algorithm exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. In a series of experiments using simulated and real data, the VCA algorithm competes with state-of-the-art methods, with a computational complexity between one and two orders of magnitude lower than the best available method.