Unmixing hyperspectral data: independent and dependent component analysis

The development of high spatial resolution airborne and spaceborne sensors has improved the capability of ground-based data collection in the fields of agriculture, geography, geology, mineral identification, detection [2, 3], and classification [4–8]. The signal read by the sensor from a given spatial element of resolution and at a given spectral band is a mixing of components originated by the constituent substances, termed endmembers, located at that element of resolution. This chapter addresses hyperspectral unmixing, which is the decomposition of the pixel spectra into a collection of constituent spectra, or spectral signatures, and their corresponding fractional abundances indicating the proportion of each endmember present in the pixel [9, 10]. Depending on the mixing scales at each pixel, the observed mixture is either linear or nonlinear [11, 12]. The linear mixing model holds when the mixing scale is macroscopic [13]. The nonlinear model holds when the mixing scale is microscopic (i.e., intimate mixtures) [14, 15]. The linear model assumes negligible interaction among distinct endmembers [16, 17]. The nonlinear model assumes that incident solar radiation is scattered by the scene through multiple bounces involving several endmembers [18]. Under the linear mixing model and assuming that the number of endmembers and their spectral signatures are known, hyperspectral unmixing is a linear problem, which can be addressed, for example, under the maximum likelihood setup [19], the constrained least-squares approach [20], the spectral signature matching [21], the spectral angle mapper [22], and the subspace projection methods [20, 23, 24]. Orthogonal subspace projection [23] reduces the data dimensionality, suppresses undesired spectral signatures, and detects the presence of a spectral signature of interest. The basic concept is to project each pixel onto a subspace that is orthogonal to the undesired signatures. As shown in Settle [19], the orthogonal subspace projection technique is equivalent to the maximum likelihood estimator. This projection technique was extended by three unconstrained least-squares approaches [24] (signature space orthogonal projection, oblique subspace projection, target signature space orthogonal projection). Other works using maximum a posteriori probability (MAP) framework [25] and projection pursuit [26, 27] have also been applied to hyperspectral data. In most cases the number of endmembers and their signatures are not known. Independent component analysis (ICA) is an unsupervised source separation process that has been applied with success to blind source separation, to feature extraction, and to unsupervised recognition [28, 29]. ICA consists in finding a linear decomposition of observed data yielding statistically independent components. Given that hyperspectral data are, in given circumstances, linear mixtures, ICA comes to mind as a possible tool to unmix this class of data. In fact, the application of ICA to hyperspectral data has been proposed in reference 30, where endmember signatures are treated as sources and the mixing matrix is composed by the abundance fractions, and in references 9, 25, and 31–38, where sources are the abundance fractions of each endmember. In the first approach, we face two problems: (1) The number of samples are limited to the number of channels and (2) the process of pixel selection, playing the role of mixed sources, is not straightforward. In the second approach, ICA is based on the assumption of mutually independent sources, which is not the case of hyperspectral data, since the sum of the abundance fractions is constant, implying dependence among abundances. This dependence compromises ICA applicability to hyperspectral images. In addition, hyperspectral data are immersed in noise, which degrades the ICA performance. IFA [39] was introduced as a method for recovering independent hidden sources from their observed noisy mixtures. IFA implements two steps. First, source densities and noise covariance are estimated from the observed data by maximum likelihood. Second, sources are reconstructed by an optimal nonlinear estimator. Although IFA is a well-suited technique to unmix independent sources under noisy observations, the dependence among abundance fractions in hyperspectral imagery compromises, as in the ICA case, the IFA performance. Considering the linear mixing model, hyperspectral observations are in a simplex whose vertices correspond to the endmembers. Several approaches [40–43] have exploited this geometric feature of hyperspectral mixtures [42]. Minimum volume transform (MVT) algorithm [43] determines the simplex of minimum volume containing the data. The MVT-type approaches are complex from the computational point of view. Usually, these algorithms first find the convex hull defined by the observed data and then fit a minimum volume simplex to it. Aiming at a lower computational complexity, some algorithms such as the vertex component analysis (VCA) [44], the pixel purity index (PPI) [42], and the N-FINDR [45] still find the minimum volume simplex containing the data cloud, but they assume the presence in the data of at least one pure pixel of each endmember. This is a strong requisite that may not hold in some data sets. In any case, these algorithms find the set of most pure pixels in the data. Hyperspectral sensors collects spatial images over many narrow contiguous bands, yielding large amounts of data. For this reason, very often, the processing of hyperspectral data, included unmixing, is preceded by a dimensionality reduction step to reduce computational complexity and to improve the signal-to-noise ratio (SNR). Principal component analysis (PCA) [46], maximum noise fraction (MNF) [47], and singular value decomposition (SVD) [48] are three well-known projection techniques widely used in remote sensing in general and in unmixing in particular. The newly introduced method [49] exploits the structure of hyperspectral mixtures, namely the fact that spectral vectors are nonnegative. The computational complexity associated with these techniques is an obstacle to real-time implementations. To overcome this problem, band selection [50] and non-statistical [51] algorithms have been introduced. This chapter addresses hyperspectral data source dependence and its impact on ICA and IFA performances. The study consider simulated and real data and is based on mutual information minimization. Hyperspectral observations are described by a generative model. This model takes into account the degradation mechanisms normally found in hyperspectral applications—namely, signature variability [52–54], abundance constraints, topography modulation, and system noise. The computation of mutual information is based on fitting mixtures of Gaussians (MOG) to data. The MOG parameters (number of components, means, covariances, and weights) are inferred using the minimum description length (MDL) based algorithm [55]. We study the behavior of the mutual information as a function of the unmixing matrix. The conclusion is that the unmixing matrix minimizing the mutual information might be very far from the true one. Nevertheless, some abundance fractions might be well separated, mainly in the presence of strong signature variability, a large number of endmembers, and high SNR. We end this chapter by sketching a new methodology to blindly unmix hyperspectral data, where abundance fractions are modeled as a mixture of Dirichlet sources. This model enforces positivity and constant sum sources (full additivity) constraints. The mixing matrix is inferred by an expectation-maximization (EM)-type algorithm. This approach is in the vein of references 39 and 56, replacing independent sources represented by MOG with mixture of Dirichlet sources. Compared with the geometric-based approaches, the advantage of this model is that there is no need to have pure pixels in the observations. The chapter is organized as follows. Section 6.2 presents a spectral radiance model and formulates the spectral unmixing as a linear problem accounting for abundance constraints, signature variability, topography modulation, and system noise. Section 6.3 presents a brief resume of ICA and IFA algorithms. Section 6.4 illustrates the performance of IFA and of some well-known ICA algorithms with experimental data. Section 6.5 studies the ICA and IFA limitations in unmixing hyperspectral data. Section 6.6 presents results of ICA based on real data. Section 6.7 describes the new blind unmixing scheme and some illustrative examples. Section 6.8 concludes with some remarks.

Blind hyperspectral unmixing

Hyperspectral unmixing methods aim at the decomposition of a hyperspectral image into a collection endmember signatures, i.e., the radiance or reflectance of the materials present in the scene, and the correspondent abundance fractions at each pixel in the image. This paper introduces a new unmixing method termed dependent component analysis (DECA). This method is blind and fully automatic and it overcomes the limitations of unmixing methods based on Independent Component Analysis (ICA) and on geometrical based approaches. DECA is based on the linear mixture model, i.e., each pixel is a linear mixture of the endmembers signatures weighted by the correspondent abundance fractions. These abundances are modeled as mixtures of Dirichlet densities, thus enforcing the non-negativity and constant sum constraints, imposed by the acquisition process. The endmembers signatures are inferred by a generalized expectation-maximization (GEM) type algorithm. The paper illustrates the effectiveness of DECA on synthetic and real hyperspectral images.

Parallel sparse unmixing of hyperspectral data

In this paper, a new parallel method for sparse spectral unmixing of remotely sensed hyperspectral data on commodity graphics processing units (GPUs) is presented. A semi-supervised approach is adopted, which relies on the increasing availability of spectral libraries of materials measured on the ground instead of resorting to endmember extraction methods. This method is based on the spectral unmixing by splitting and augmented Lagrangian (SUNSAL) that estimates the material's abundance fractions. The parallel method is performed in a pixel-by-pixel fashion and its implementation properly exploits the GPU architecture at low level, thus taking full advantage of the computational power of GPUs. Experimental results obtained for simulated and real hyperspectral datasets reveal significant speedup factors, up to 1 64 times, with regards to optimized serial implementation.

Fast unsupervised extraction of endmembers spectra from hyperspectral data

Linear unmixing decomposes an hyperspectral image into a collection of re ectance spectra, called endmember signatures, and a set corresponding abundance fractions from the respective spatial coverage. This paper introduces vertex component analysis, an unsupervised algorithm to unmix linear mixtures of hyperpsectral data. VCA exploits the fact that endmembers occupy vertices of a simplex, and assumes the presence of pure pixels in data. VCA performance is illustrated using simulated and real data. VCA competes with state-of-the-art methods with much lower computational complexity.

Hyperspectral unmixing algorithm via dependent component analysis

This paper introduces a new method to blindly unmix hyperspectral data, termed dependent component analysis (DECA). This method decomposes a hyperspectral images into a collection of reflectance (or radiance) spectra of the materials present in the scene (endmember signatures) and the corresponding abundance fractions at each pixel. DECA assumes that each pixel is a linear mixture of the endmembers signatures weighted by the correspondent abundance fractions. These abudances are modeled as mixtures of Dirichlet densities, thus enforcing the constraints on abundance fractions imposed by the acquisition process, namely non-negativity and constant sum. The mixing matrix is inferred by a generalized expectation-maximization (GEM) type algorithm. This method overcomes the limitations of unmixing methods based on Independent Component Analysis (ICA) and on geometrical based approaches. The effectiveness of the proposed method is illustrated using simulated data based on U.S.G.S. laboratory spectra and real hyperspectral data collected by the AVIRIS sensor over Cuprite, Nevada.

Decision fusion for the classification of hyperspectral data: Outcome of the 2008 GRS-S data fusion contest

The 2008 Data Fusion Contest organized by the IEEE Geoscience and Remote Sensing Data Fusion Technical Committee deals with the classification of high-resolution hyperspectral data from an urban area. Unlike in the previous issues of the contest, the goal was not only to identify the best algorithm but also to provide a collaborative effort: The decision fusion of the best individual algorithms was aiming at further improving the classification performances, and the best algorithms were ranked according to their relative contribution to the decision fusion. This paper presents the five awarded algorithms and the conclusions of the contest, stressing the importance of decision fusion, dimension reduction, and supervised classification methods, such as neural networks and support vector machines.

Rockfall susceptibility assessment and remote geological mapping with LiDAR point clouds

Characterizing the geological features and structures in three dimensions over inaccessible rock cliffs is needed to assess natural hazards such as rockfalls and rockslides and also to perform investigations aimed at mapping geological contacts and building stratigraphy and fold models. Indeed, the detailed 3D data, such as LiDAR point clouds, allow to study accurately the hazard processes and the structure of geologic features, in particular in vertical and overhanging rock slopes. Thus, 3D geological models have a great potential of being applied to a wide range of geological investigations both in research and applied geology projects, such as mines, tunnels and reservoirs. Recent development of ground-based remote sensing techniques (LiDAR, photogrammetry and multispectral / hyperspectral images) are revolutionizing the acquisition of morphological and geological information. As a consequence, there is a great potential for improving the modeling of geological bodies as well as failure mechanisms and stability conditions by integrating detailed remote data. During the past ten years several large rockfall events occurred along important transportation corridors where millions of people travel every year (Switzerland: Gotthard motorway and railway; Canada: Sea to sky highway between Vancouver and Whistler). These events show that there is still a lack of knowledge concerning the detection of potential rockfalls, making mountain residential settlements and roads highly risky. It is necessary to understand the main factors that destabilize rocky outcrops even if inventories are lacking and if no clear morphological evidences of rockfall activity are observed. In order to increase the possibilities of forecasting potential future landslides, it is crucial to understand the evolution of rock slope stability. Defining the areas theoretically most prone to rockfalls can be particularly useful to simulate trajectory profiles and to generate hazard maps, which are the basis for land use planning in mountainous regions. The most important questions to address in order to assess rockfall hazard are: Where are the most probable sources for future rockfalls located? What are the frequencies of occurrence of these rockfalls? I characterized the fracturing patterns in the field and with LiDAR point clouds. Afterwards, I developed a model to compute the failure mechanisms on terrestrial point clouds in order to assess the susceptibility to rockfalls at the cliff scale. Similar procedures were already available to evaluate the susceptibility to rockfalls based on aerial digital elevation models. This new model gives the possibility to detect the most susceptible rockfall sources with unprecented detail in the vertical and overhanging areas. The results of the computation of the most probable rockfall source areas in granitic cliffs of Yosemite Valley and Mont-Blanc massif were then compared to the inventoried rockfall events to validate the calculation methods. Yosemite Valley was chosen as a test area because it has a particularly strong rockfall activity (about one rockfall every week) which leads to a high rockfall hazard. The west face of the Dru was also chosen for the relevant rockfall activity and especially because it was affected by some of the largest rockfalls that occurred in the Alps during the last 10 years. Moreover, both areas were suitable because of their huge vertical and overhanging cliffs that are difficult to study with classical methods. Limit equilibrium models have been applied to several case studies to evaluate the effects of different parameters on the stability of rockslope areas. The impact of the degradation of rockbridges on the stability of large compartments in the west face of the Dru was assessed using finite element modeling. In particular I conducted a back-analysis of the large rockfall event of 2005 (265'000 m3) by integrating field observations of joint conditions, characteristics of fracturing pattern and results of geomechanical tests on the intact rock. These analyses improved our understanding of the factors that influence the stability of rock compartments and were used to define the most probable future rockfall volumes at the Dru. Terrestrial laser scanning point clouds were also successfully employed to perform geological mapping in 3D, using the intensity of the backscattered signal. Another technique to obtain vertical geological maps is combining triangulated TLS mesh with 2D geological maps. At El Capitan (Yosemite Valley) we built a georeferenced vertical map of the main plutonio rocks that was used to investigate the reasons for preferential rockwall retreat rate. Additional efforts to characterize the erosion rate were made at Monte Generoso (Ticino, southern Switzerland) where I attempted to improve the estimation of long term erosion by taking into account also the volumes of the unstable rock compartments. Eventually, the following points summarize the main out puts of my research: The new model to compute the failure mechanisms and the rockfall susceptibility with 3D point clouds allows to define accurately the most probable rockfall source areas at the cliff scale. The analysis of the rockbridges at the Dru shows the potential of integrating detailed measurements of the fractures in geomechanical models of rockmass stability. The correction of the LiDAR intensity signal gives the possibility to classify a point cloud according to the rock type and then use this information to model complex geologic structures. The integration of these results, on rockmass fracturing and composition, with existing methods can improve rockfall hazard assessments and enhance the interpretation of the evolution of steep rockslopes. -- La caractérisation de la géologie en 3D pour des parois rocheuses inaccessibles est une étape nécessaire pour évaluer les dangers naturels tels que chutes de blocs et glissements rocheux, mais aussi pour réaliser des modèles stratigraphiques ou de structures plissées. Les modèles géologiques 3D ont un grand potentiel pour être appliqués dans une vaste gamme de travaux géologiques dans le domaine de la recherche, mais aussi dans des projets appliqués comme les mines, les tunnels ou les réservoirs. Les développements récents des outils de télédétection terrestre (LiDAR, photogrammétrie et imagerie multispectrale / hyperspectrale) sont en train de révolutionner l'acquisition d'informations géomorphologiques et géologiques. Par conséquence, il y a un grand potentiel d'amélioration pour la modélisation d'objets géologiques, ainsi que des mécanismes de rupture et des conditions de stabilité, en intégrant des données détaillées acquises à distance. Pour augmenter les possibilités de prévoir les éboulements futurs, il est fondamental de comprendre l'évolution actuelle de la stabilité des parois rocheuses. Définir les zones qui sont théoriquement plus propices aux chutes de blocs peut être très utile pour simuler les trajectoires de propagation des blocs et pour réaliser des cartes de danger, qui constituent la base de l'aménagement du territoire dans les régions de montagne. Les questions plus importantes à résoudre pour estimer le danger de chutes de blocs sont : Où se situent les sources plus probables pour les chutes de blocs et éboulement futurs ? Avec quelle fréquence vont se produire ces événements ? Donc, j'ai caractérisé les réseaux de fractures sur le terrain et avec des nuages de points LiDAR. Ensuite, j'ai développé un modèle pour calculer les mécanismes de rupture directement sur les nuages de points pour pouvoir évaluer la susceptibilité au déclenchement de chutes de blocs à l'échelle de la paroi. Les zones sources de chutes de blocs les plus probables dans les parois granitiques de la vallée de Yosemite et du massif du Mont-Blanc ont été calculées et ensuite comparés aux inventaires des événements pour vérifier les méthodes. Des modèles d'équilibre limite ont été appliqués à plusieurs cas d'études pour évaluer les effets de différents paramètres sur la stabilité des parois. L'impact de la dégradation des ponts rocheux sur la stabilité de grands compartiments de roche dans la paroi ouest du Petit Dru a été évalué en utilisant la modélisation par éléments finis. En particulier j'ai analysé le grand éboulement de 2005 (265'000 m3), qui a emporté l'entier du pilier sud-ouest. Dans le modèle j'ai intégré des observations des conditions des joints, les caractéristiques du réseau de fractures et les résultats de tests géoméchaniques sur la roche intacte. Ces analyses ont amélioré l'estimation des paramètres qui influencent la stabilité des compartiments rocheux et ont servi pour définir des volumes probables pour des éboulements futurs. Les nuages de points obtenus avec le scanner laser terrestre ont été utilisés avec succès aussi pour produire des cartes géologiques en 3D, en utilisant l'intensité du signal réfléchi. Une autre technique pour obtenir des cartes géologiques des zones verticales consiste à combiner un maillage LiDAR avec une carte géologique en 2D. A El Capitan (Yosemite Valley) nous avons pu géoréferencer une carte verticale des principales roches plutoniques que j'ai utilisé ensuite pour étudier les raisons d'une érosion préférentielle de certaines zones de la paroi. D'autres efforts pour quantifier le taux d'érosion ont été effectués au Monte Generoso (Ticino, Suisse) où j'ai essayé d'améliorer l'estimation de l'érosion au long terme en prenant en compte les volumes des compartiments rocheux instables. L'intégration de ces résultats, sur la fracturation et la composition de l'amas rocheux, avec les méthodes existantes permet d'améliorer la prise en compte de l'aléa chute de pierres et éboulements et augmente les possibilités d'interprétation de l'évolution des parois rocheuses.

Modelling of urban sensible heat flux at multiple spatial scales : a demonstration using airborne hyperspectral imagery of Shanghai and a temperature–emissivity separation approach

Urbanization related alterations to the surface energy balance impact urban warming (‘heat islands’), the growth of the boundary layer, and many other biophysical processes. Traditionally, in situ heat flux measures have been used to quantify such processes, but these typically represent only a small local-scale area within the heterogeneous urban environment. For this reason, remote sensing approaches are very attractive for elucidating more spatially representative information. Here we use hyperspectral imagery from a new airborne sensor, the Operative Modular Imaging Spectrometer (OMIS), along with a survey map and meteorological data, to derive the land cover information and surface parameters required to map spatial variations in turbulent sensible heat flux (QH). The results from two spatially-explicit flux retrieval methods which use contrasting approaches and, to a large degree, different input data are compared for a central urban area of Shanghai, China: (1) the Local-scale Urban Meteorological Parameterization Scheme (LUMPS) and (2) an Aerodynamic Resistance Method (ARM). Sensible heat fluxes are determined at the full 6 m spatial resolution of the OMIS sensor, and at lower resolutions via pixel aggregation and spatial averaging. At the 6 m spatial resolution, the sensible heat flux of rooftop dominated pixels exceeds that of roads, water and vegetated areas, with values peaking at ∼ 350 W m− 2, whilst the storage heat flux is greatest for road dominated pixels (peaking at around 420 W m− 2). We investigate the use of both OMIS-derived land surface temperatures made using a Temperature–Emissivity Separation (TES) approach, and land surface temperatures estimated from air temperature measures. Sensible heat flux differences from the two approaches over the entire 2 × 2 km study area are less than 30 W m− 2, suggesting that methods employing either strategy maybe practica1 when operated using low spatial resolution (e.g. 1 km) data. Due to the differing methodologies, direct comparisons between results obtained with the LUMPS and ARM methods are most sensibly made at reduced spatial scales. At 30 m spatial resolution, both approaches produce similar results, with the smallest difference being less than 15 W m− 2 in mean QH averaged over the entire study area. This is encouraging given the differing architecture and data requirements of the LUMPS and ARM methods. Furthermore, in terms of mean study QH, the results obtained by averaging the original 6 m spatial resolution LUMPS-derived QH values to 30 and 90 m spatial resolution are within ∼ 5 W m− 2 of those derived from averaging the original surface parameter maps prior to input into LUMPS, suggesting that that use of much lower spatial resolution spaceborne imagery data, for example from Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) is likely to be a practical solution for heat flux determination in urban areas.

Nature-Inspired Framework for Hyperspectral Band Selection

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)