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.

Cochineal, a precious source of red: cochineal dyes characterization by high performance liquid chromatography with diode array detection and principal component analysis

Presented at Faculdade de Ciências e Tecnologias, Universidade de Lisboa, to obtain the Master Degree in Conservation and Restoration of Textiles

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 functional magnetic resonance imaging time series by independent component analysis

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

New insights on river water chemistry by using non-centred simplicial principal component analysis: a case study

The use of perturbation and power transformation operations permits the investigation of linear processes in the simplex as in a vectorial space. When the investigated geochemical processes can be constrained by the use of well-known starting point, the eigenvectors of the covariance matrix of a non-centred principalcomponent analysis allow to model compositional changes compared with a reference point.The results obtained for the chemistry of water collected in River Arno (central-northern Italy) have open new perspectives for considering relative changes of the analysed variables and to hypothesise the relative effect of different acting physical-chemical processes, thus posing the basis for a quantitative modelling

A novel method for automated classification of epileptiform activity in the human electroencephalogram-based on independent component analysis.

Diagnosis of several neurological disorders is based on the detection of typical pathological patterns in the electroencephalogram (EEG). This is a time-consuming task requiring significant training and experience. Automatic detection of these EEG patterns would greatly assist in quantitative analysis and interpretation. We present a method, which allows automatic detection of epileptiform events and discrimination of them from eye blinks, and is based on features derived using a novel application of independent component analysis. The algorithm was trained and cross validated using seven EEGs with epileptiform activity. For epileptiform events with compensation for eyeblinks, the sensitivity was 65 +/- 22% at a specificity of 86 +/- 7% (mean +/- SD). With feature extraction by PCA or classification of raw data, specificity reduced to 76 and 74%, respectively, for the same sensitivity. On exactly the same data, the commercially available software Reveal had a maximum sensitivity of 30% and concurrent specificity of 77%. Our algorithm performed well at detecting epileptiform events in this preliminary test and offers a flexible tool that is intended to be generalized to the simultaneous classification of many waveforms in the EEG.

On-line event detection by recursive dynamic principal component analysis and gas sensor arrays under drift conditions

Leakage detection is an important issue in many chemical sensing applications. Leakage detection hy thresholds suffers from important drawbacks when sensors have serious drifts or they are affected by cross-sensitivities. Here we present an adaptive method based in a Dynamic Principal Component Analysis that models the relationships between the sensors in the may. In normal conditions a certain variance distribution characterizes sensor signals. However, in the presence of a new source of variance the PCA decomposition changes drastically. In order to prevent the influence of sensor drifts the model is adaptive and it is calculated in a recursive manner with minimum computational effort. The behavior of this technique is studied with synthetic signals and with real signals arising by oil vapor leakages in an air compressor. Results clearly demonstrate the efficiency of the proposed method.

Drift compensation of gas sensor array data by common principal component analysis

A new drift compensation method based on Common Principal Component Analysis (CPCA) is proposed. The drift variance in data is found as the principal components computed by CPCA. This method finds components that are common for all gasses in feature space. The method is compared in classification task with respect to the other approaches published where the drift direction is estimated through a Principal Component Analysis (PCA) of a reference gas. The proposed new method ¿ employing no specific reference gas, but information from all gases ¿has shown the same performance as the traditional approach with the best-fitted reference gas. Results are shown with data lasting 7-months including three gases at different concentrations for an array of 17 polymeric sensors.

Characterization of soil chemical properties of strawberry fields using principal component analysis

One of the largest strawberry-producing municipalities of Rio Grande do Sul (RS) is Turuçu, in the South of the State. The strawberry production system adopted by farmers is similar to that used in other regions in Brazil and in the world. The main difference is related to the soil management, which can change the soil chemical properties during the strawberry cycle. This study had the objective of assessing the spatial and temporal distribution of soil fertility parameters using principal component analysis (PCA). Soil sampling was based on topography, dividing the field in three thirds: upper, middle and lower. From each of these thirds, five soil samples were randomly collected in the 0-0.20 m layer, to form a composite sample for each third. Four samples were taken during the strawberry cycle and the following properties were determined: soil organic matter (OM), soil total nitrogen (N), available phosphorus (P) and potassium (K), exchangeable calcium (Ca) and magnesium (Mg), soil pH (pH), cation exchange capacity (CEC) at pH 7.0, soil base (V%) and soil aluminum saturation(m%). No spatial variation was observed for any of the studied soil fertility parameters in the strawberry fields and temporal variation was only detected for available K. Phosphorus and K contents were always high or very high from the beginning of the strawberry cycle, while pH values ranged from very low to very high. Principal component analysis allowed the clustering of all strawberry fields based on variables related to soil acidity and organic matter content.

Colligation and Cross Moment-Based Approaches for Independent Component Analysis

Diplomityössä on käsitelty uudenlaisia menetelmiä riippumattomien komponenttien analyysiin(ICA): Menetelmät perustuvat colligaatioon ja cross-momenttiin. Colligaatio menetelmä perustuu painojen colligaatioon. Menetelmässä on käytetty kahden tyyppisiä todennäköisyysjakaumia yhden sijasta joka perustuu yleiseen itsenäisyyden kriteeriin. Työssä on käytetty colligaatio lähestymistapaa kahdella asymptoottisella esityksellä. Gram-Charlie ja Edgeworth laajennuksia käytetty arvioimaan todennäköisyyksiä näissä menetelmissä. Työssä on myös käytetty cross-momentti menetelmää joka perustuu neljännen asteen cross-momenttiin. Menetelmä on hyvin samankaltainen FastICA algoritmin kanssa. Molempia menetelmiä on tarkasteltu lineaarisella kahden itsenäisen muuttajan sekoituksella. Lähtö signaalit ja sekoitetut matriisit ovattuntemattomia signaali lähteiden määrää lukuunottamatta. Työssä on vertailtu colligaatio menetelmään ja sen modifikaatioita FastICA:an ja JADE:en. Työssä on myös tehty vertailu analyysi suorituskyvyn ja keskusprosessori ajan suhteen cross-momenttiin perustuvien menetelmien, FastICA:n ja JADE:n useiden sekoitettujen parien kanssa.