Vertex component analysis: a fast algorithm to extract endmembers spectra from hyperspectral data

Chapter in Book Proceedings with Peer Review First Iberian Conference, IbPRIA 2003, Puerto de Andratx, Mallorca, Spain, JUne 4-6, 2003. Proceedings

Vertex component analysis: a fast algorithm to unmix hyperspectral 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.

Parallel implementation of vertex component analysis for hyperspectral endmember extraction

International Conference with Peer Review 2012 IEEE International Conference in Geoscience and Remote Sensing Symposium (IGARSS), 22-27 July 2012, Munich, Germany

Analysis and recognition of similar environmental sounds

Dissertação apresentada na Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa para obtenção do grau de Mestre em Engenharia Informática

2D PCA-Based localization for mobile robots in unstructured environments

In this paper a new PCA-based positioning sensor and localization system for mobile robots to operate in unstructured environments (e. g. industry, services, domestic ...) is proposed and experimentally validated. The inexpensive positioning system resorts to principal component analysis (PCA) of images acquired by a video camera installed onboard, looking upwards to the ceiling. This solution has the advantage of avoiding the need of selecting and extracting features. The principal components of the acquired images are compared with previously registered images, stored in a reduced onboard image database, and the position measured is fused with odometry data. The optimal estimates of position and slippage are provided by Kalman filters, with global stable error dynamics. The experimental validation reported in this work focuses on the results of a set of experiments carried out in a real environment, where the robot travels along a lawn-mower trajectory. A small position error estimate with bounded co-variance was always observed, for arbitrarily long experiments, and slippage was estimated accurately in real time.

Forest soil short term response to prescribed fire: an approach by Principal Components Analysis

In the current context of serious climate changes, where the increase of the frequency of some extreme events occurrence can enhance the rate of periods prone to high intensity forest fires, the National Forest Authority often implements, in several Portuguese forest areas, a regular set of measures in order to control the amount of fuel mass availability (PNDFCI, 2008). In the present work we’ll present a preliminary analysis concerning the assessment of the consequences given by the implementation of prescribed fire measures to control the amount of fuel mass in soil recovery, in particular in terms of its water retention capacity, its organic matter content, pH and content of iron. This work is included in a larger study (Meira-Castro, 2009(a); Meira-Castro, 2009(b)). According to the established praxis on the data collection, embodied in multidimensional matrices of n columns (variables in analysis) by p lines (sampled areas at different depths), and also considering the quantitative data nature present in this study, we’ve chosen a methodological approach that considers the multivariate statistical analysis, in particular, the Principal Component Analysis (PCA ) (Góis, 2004). The experiments were carried out in a soil cover over a natural site of Andaluzitic schist, in Gramelas, Caminha, NW Portugal, who was able to maintain itself intact from prescribed burnings from four years and was submit to prescribed fire in March 2008. The soils samples were collected from five different plots at six different time periods. The methodological option that was adopted have allowed us to identify the most relevant relational structures inside the n variables, the p samples and in two sets at the same time (Garcia-Pereira, 1990). Consequently, and in addition to the traditional outputs produced from the PCA, we have analyzed the influence of both sampling depths and geomorphological environments in the behavior of all variables involved.

Optimization of River Water Quality Surveys by Multivariate Analysis of Physicochemical, Bacteriological and Ecotoxicological Data

This study aims to optimize the water quality monitoring of a polluted watercourse (Leça River, Portugal) through the principal component analysis (PCA) and cluster analysis (CA). These statistical methodologies were applied to physicochemical, bacteriological and ecotoxicological data (with the marine bacterium Vibrio fischeri and the green alga Chlorella vulgaris) obtained with the analysis of water samples monthly collected at seven monitoring sites and during five campaigns (February, May, June, August, and September 2006). The results of some variables were assigned to water quality classes according to national guidelines. Chemical and bacteriological quality data led to classify Leça River water quality as “bad” or “very bad”. PCA and CA identified monitoring sites with similar pollution pattern, giving to site 1 (located in the upstream stretch of the river) a distinct feature from all other sampling sites downstream. Ecotoxicity results corroborated this classification thus revealing differences in space and time. The present study includes not only physical, chemical and bacteriological but also ecotoxicological parameters, which broadens new perspectives in river water characterization. Moreover, the application of PCA and CA is very useful to optimize water quality monitoring networks, defining the minimum number of sites and their location. Thus, these tools can support appropriate management decisions.

Vertex component analysis GPU-based implementation for hyperspectral unmixing

Endmember extraction (EE) is a fundamental and crucial task in hyperspectral unmixing. Among other methods vertex component analysis ( VCA) has become a very popular and useful tool to unmix hyperspectral data. VCA is a geometrical based method that extracts endmember signatures from large hyperspectral datasets without the use of any a priori knowledge about the constituent spectra. Many Hyperspectral imagery applications require a response in real time or near-real time. Thus, to met this requirement this paper proposes a parallel implementation of VCA developed for graphics processing units. The impact on the complexity and on the accuracy of the proposed parallel implementation of VCA is examined using both simulated and real hyperspectral datasets.

Vertex component analysis: a geometric-based approach to unmix hyperspectral data

Hyperspectral remote sensing exploits the electromagnetic scattering patterns of the different materials at specific wavelengths [2, 3]. Hyperspectral sensors have been developed to sample the scattered portion of the electromagnetic spectrum extending from the visible region through the near-infrared and mid-infrared, in hundreds of narrow contiguous bands [4, 5]. The number and variety of potential civilian and military applications of hyperspectral remote sensing is enormous [6, 7]. Very often, the resolution cell corresponding to a single pixel in an image contains several substances (endmembers) [4]. In this situation, the scattered energy is a mixing of the endmember spectra. A challenging task underlying many hyperspectral imagery applications is then decomposing a mixed pixel into a collection of reflectance spectra, called endmember signatures, and the corresponding abundance fractions [8–10]. Depending on the mixing scales at each pixel, the observed mixture is either linear or nonlinear [11, 12]. Linear mixing model holds approximately when the mixing scale is macroscopic [13] and there is negligible interaction among distinct endmembers [3, 14]. If, however, the mixing scale is microscopic (or intimate mixtures) [15, 16] and the incident solar radiation is scattered by the scene through multiple bounces involving several endmembers [17], the linear model is no longer accurate. Linear spectral unmixing has been intensively researched in the last years [9, 10, 12, 18–21]. It considers that a mixed pixel is a linear combination of endmember signatures weighted by the correspondent abundance fractions. Under this model, and assuming that the number of substances and their reflectance spectra are known, hyperspectral unmixing is a linear problem for which many solutions have been proposed (e.g., maximum likelihood estimation [8], spectral signature matching [22], spectral angle mapper [23], subspace projection methods [24,25], and constrained least squares [26]). In most cases, the number of substances and their reflectances are not known and, then, hyperspectral unmixing falls into the class of blind source separation problems [27]. Independent component analysis (ICA) has recently been proposed as a tool to blindly unmix hyperspectral data [28–31]. ICA is based on the assumption of mutually independent sources (abundance fractions), which is not the case of hyperspectral data, since the sum of abundance fractions is constant, implying statistical dependence among them. This dependence compromises ICA applicability to hyperspectral images as shown in Refs. [21, 32]. In fact, ICA finds the endmember signatures by multiplying the spectral vectors with an unmixing matrix, which minimizes the mutual information among sources. If sources are independent, ICA provides the correct unmixing, since the minimum of the mutual information is obtained only when sources are independent. This is no longer true for dependent abundance fractions. Nevertheless, some endmembers may be approximately unmixed. These aspects are addressed in Ref. [33]. Under the linear mixing model, the observations from a scene are in a simplex whose vertices correspond to the endmembers. Several approaches [34–36] have exploited this geometric feature of hyperspectral mixtures [35]. Minimum volume transform (MVT) algorithm [36] determines the simplex of minimum volume containing the data. The method presented in Ref. [37] is also of MVT type but, by introducing the notion of bundles, it takes into account the endmember variability usually present in hyperspectral mixtures. The MVT type approaches are complex from the computational point of view. Usually, these algorithms find in the first place the convex hull defined by the observed data and then fit a minimum volume simplex to it. For example, the gift wrapping algorithm [38] computes the convex hull of n data points in a d-dimensional space with a computational complexity of O(nbd=2cþ1), where bxc is the highest integer lower or equal than x and n is the number of samples. The complexity of the method presented in Ref. [37] is even higher, since the temperature of the simulated annealing algorithm used shall follow a log( ) law [39] to assure convergence (in probability) to the desired solution. Aiming at a lower computational complexity, some algorithms such as the pixel purity index (PPI) [35] and the N-FINDR [40] still find the minimum volume simplex containing the data cloud, but they assume the presence of at least one pure pixel of each endmember in the data. 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. PPI algorithm uses the minimum noise fraction (MNF) [41] as a preprocessing step to reduce dimensionality and to improve the signal-to-noise ratio (SNR). The algorithm then projects every spectral vector onto skewers (large number of random vectors) [35, 42,43]. The points corresponding to extremes, for each skewer direction, are stored. A cumulative account records the number of times each pixel (i.e., a given spectral vector) is found to be an extreme. The pixels with the highest scores are the purest ones. N-FINDR algorithm [40] is based on the fact that in p spectral dimensions, the p-volume defined by a simplex formed by the purest pixels is larger than any other volume defined by any other combination of pixels. This algorithm finds the set of pixels defining the largest volume by inflating a simplex inside the data. ORA SIS [44, 45] is a hyperspectral framework developed by the U.S. Naval Research Laboratory consisting of several algorithms organized in six modules: exemplar selector, adaptative learner, demixer, knowledge base or spectral library, and spatial postrocessor. The first step consists in flat-fielding the spectra. Next, the exemplar selection module is used to select spectral vectors that best represent the smaller convex cone containing the data. The other pixels are rejected when the spectral angle distance (SAD) is less than a given thresh old. The procedure finds the basis for a subspace of a lower dimension using a modified Gram–Schmidt orthogonalizati on. The selected vectors are then projected onto this subspace and a simplex is found by an MV T pro cess. ORA SIS is oriented to real-time target detection from uncrewed air vehicles using hyperspectral data [46]. In this chapter we develop a new algorithm to unmix linear mixtures of endmember spectra. First, the algorithm determines the number of endmembers and the signal subspace using a newly developed concept [47, 48]. Second, the algorithm extracts the most pure pixels present in the data. Unlike other methods, this algorithm is completely automatic and unsupervised. To estimate the number of endmembers and the signal subspace in hyperspectral linear mixtures, the proposed scheme begins by estimating sign al and noise correlation matrices. The latter is based on multiple regression theory. The signal subspace is then identified by selectin g the set of signal eigenvalue s that best represents the data, in the least-square sense [48,49 ], we note, however, that VCA works with projected and with unprojected data. The extraction of the end members exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. As PPI and N-FIND R algorithms, VCA also assumes the presence of pure pixels in the data. The algorithm iteratively projects data on to a direction orthogonal to the subspace spanned by the endmembers already determined. The new end member signature corresponds to the extreme of the projection. The algorithm iterates until all end members are exhausted. VCA performs much better than PPI and better than or comparable to N-FI NDR; yet it has a computational complexity between on e and two orders of magnitude lower than N-FINDR. The chapter is structure d as follows. Section 19.2 describes the fundamentals of the proposed method. Section 19.3 and Section 19.4 evaluate the proposed algorithm using simulated and real data, respectively. Section 19.5 presents some concluding remarks.

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.

On-line system for faults detection in induction motors based on PCA

Dissertation to obtain the degree of Master in Electrical and Computer Engineering

Principal component analysis of the yield curve

A Work Project, presented as part of the requirements for the Award of a Masters Degree in Finance from the NOVA – School of Business and Economics

Pair trading: Clustering based on principal component analysis

This study focuses on the implementation of several pair trading strategies across three emerging markets, with the objective of comparing the results obtained from the different strategies and assessing if pair trading benefits from a more volatile environment. The results show that, indeed, there are higher potential profits arising from emerging markets. However, the higher excess return will be partially offset by higher transaction costs, which will be a determinant factor to the profitability of pair trading strategies. Also, a new clustering approach based on the Principal Component Analysis was tested as an alternative to the more standard clustering by Industry Groups. The new clustering approach delivers promising results, consistently reducing volatility to a greater extent than the Industry Group approach, with no significant harm to the excess returns.

Analysis of metabolic flux distributions in relation to the extracellular environment in Avian cells

Continuous cell lines that proliferate in chemically defined and simple media have been highly regarded as suitable alternatives for vaccine production. One such cell line is the AG1.CR.pIX avian cell line developed by PROBIOGEN. This cell line can be cultivated in a fully scalable suspension culture and adapted to grow in chemically defined, calf serum free, medium [1]–[5]. The medium composition and cultivation strategy are important factors for reaching high virus titers. In this project, a series of computational methods was used to simulate the cell’s response to different environments. The study is based on the metabolic model of the central metabolism proposed in [1]. In a first step, Metabolic Flux Analysis (MFA) was used along with measured uptake and secretion fluxes to estimate intracellular flux values. The network and data were found to be consistent. In a second step, Flux Balance Analysis (FBA) was performed to access the cell’s biological objective. The objective that resulted in the best predicted results fit to the experimental data was the minimization of oxidative phosphorylation. Employing this objective, in the next step Flux Variability Analysis (FVA) was used to characterize the flux solution space. Furthermore, various scenarios, where a reaction deletion (elimination of the compound from the media) was simulated, were performed and the flux solution space for each scenario was calculated. Growth restrictions caused by essential and non-essential amino acids were accurately predicted. Fluxes related to the essential amino acids uptake and catabolism, the lipid synthesis and ATP production via TCA were found to be essential to exponential growth. Finally, the data gathered during the previous steps were analyzed using principal component analysis (PCA), in order to assess potential changes in the physiological state of the cell. Three metabolic states were found, which correspond to zero, partial and maximum biomass growth rate. Elimination of non-essential amino acids or pyruvate from the media showed no impact on the cell’s assumed normal metabolic state.

Analysis of functional magnetic resonance imaging time series by independent component analysis

Independent Component Analysis, Time Series Analysis, Functional Magnetic Resonance Imaging