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.

Tool for spatial and dynamic representation of artistic performances

This proposal aims to explore the use of available technologies for video representation of sets and performers in order to serve as support for composition processes and artistic performer rehearsals, while focusing in representing the performer’s body and its movements, and its relation with objects belonging to the three-dimensional space of their performances. This project’s main goal is to design and develop a system that can spatially represent the performer and its movements, by means of capturing processes and reconstruction using a camera device, as well as enhance the three-dimensional space where the performance occurs by allowing interaction with virtual objects and by adding a video component, either for documentary purposes, or for live performances effects (for example, using video mapping video techniques in captured video or projection during a performance).

Representing diffusion MRI in 5D for segmentation of white matter tracts with a level set method.

We present a method for segmenting white matter tracts from high angular resolution diffusion MR. images by representing the data in a 5 dimensional space of position and orientation. Whereas crossing fiber tracts cannot be separated in 3D position space, they clearly disentangle in 5D position-orientation space. The segmentation is done using a 5D level set method applied to hyper-surfaces evolving in 5D position-orientation space. In this paper we present a methodology for constructing the position-orientation space. We then show how to implement the standard level set method in such a non-Euclidean high dimensional space. The level set theory is basically defined for N-dimensions but there are several practical implementation details to consider, such as mean curvature. Finally, we will show results from a synthetic model and a few preliminary results on real data of a human brain acquired by high angular resolution diffusion MRI.

Parts, places, and perspectives : a theory of spatial relations based an mereotopology and convexity

This thesis suggests to carry on the philosophical work begun in Casati's and Varzi's seminal book Parts and Places, by extending their general reflections on the basic formal structure of spatial representation beyond mereotopology and absolute location to the question of perspectives and perspective-dependent spatial relations. We show how, on the basis of a conceptual analysis of such notions as perspective and direction, a mereotopological theory with convexity can express perspectival spatial relations in a strictly qualitative framework. We start by introducing a particular mereotopological theory, AKGEMT, and argue that it constitutes an adequate core for a theory of spatial relations. Two features of AKGEMT are of particular importance: AKGEMT is an extensional mereotopology, implying that sameness of proper parts is a sufficient and necessary condition for identity, and it allows for (lower- dimensional) boundary elements in its domain of quantification. We then discuss an extension of AKGEMT, AKGEMTS, which results from the addition of a binary segment operator whose interpretation is that of a straight line segment between mereotopological points. Based on existing axiom systems in standard point-set topology, we propose an axiomatic characterisation of the segment operator and show that it is strong enough to sustain complex properties of a convexity predicate and a convex hull operator. We compare our segment-based characterisation of the convex hull to Cohn et al.'s axioms for the convex hull operator, arguing that our notion of convexity is significantly stronger. The discussion of AKGEMTS defines the background theory of spatial representation on which the developments in the second part of this thesis are built. The second part deals with perspectival spatial relations in two-dimensional space, i.e., such relations as those expressed by 'in front of, 'behind', 'to the left/right of, etc., and develops a qualitative formalism for perspectival relations within the framework of AKGEMTS. Two main claims are defended in part 2: That perspectival relations in two-dimensional space are four- place relations of the kind R(x, y, z, w), to be read as x is i?-related to y as z looks at w; and that these four-place structures can be satisfactorily expressed within the qualitative theory AKGEMTS. To defend these two claims, we start by arguing for a unified account of perspectival relations, thus rejecting the traditional distinction between 'relative' and 'intrinsic' perspectival relations. We present a formal theory of perspectival relations in the framework of AKGEMTS, deploying the idea that perspectival relations in two-dimensional space are four-place relations, having a locational and a perspectival part and show how this four-place structure leads to a unified framework of perspectival relations. Finally, we present a philosophical motivation to the idea that perspectival relations are four-place, cashing out the thesis that perspectives are vectorial properties and argue that vectorial properties are relations between spatial entities. Using Fine's notion of "qua objects" for an analysis of points of view, we show at last how our four-place approach to perspectival relations compares to more traditional understandings.

The average inter-crossing number of equilateral random walks and polygons

In this paper, we study the average inter-crossing number between two random walks and two random polygons in the three-dimensional space. The random walks and polygons in this paper are the so-called equilateral random walks and polygons in which each segment of the walk or polygon is of unit length. We show that the mean average inter-crossing number ICN between two equilateral random walks of the same length n is approximately linear in terms of n and we were able to determine the prefactor of the linear term, which is a = (3 In 2)/(8) approximate to 0.2599. In the case of two random polygons of length n, the mean average inter-crossing number ICN is also linear, but the prefactor of the linear term is different from that of the random walks. These approximations apply when the starting points of the random walks and polygons are of a distance p apart and p is small compared to n. We propose a fitting model that would capture the theoretical asymptotic behaviour of the mean average ICN for large values of p. Our simulation result shows that the model in fact works very well for the entire range of p. We also study the mean ICN between two equilateral random walks and polygons of different lengths. An interesting result is that even if one random walk (polygon) has a fixed length, the mean average ICN between the two random walks (polygons) would still approach infinity if the length of the other random walk (polygon) approached infinity. The data provided by our simulations match our theoretical predictions very well.

Harmonic active contours.

We propose a segmentation method based on the geometric representation of images as 2-D manifolds embedded in a higher dimensional space. The segmentation is formulated as a minimization problem, where the contours are described by a level set function and the objective functional corresponds to the surface of the image manifold. In this geometric framework, both data-fidelity and regularity terms of the segmentation are represented by a single functional that intrinsically aligns the gradients of the level set function with the gradients of the image and results in a segmentation criterion that exploits the directional information of image gradients to overcome image inhomogeneities and fragmented contours. The proposed formulation combines this robust alignment of gradients with attractive properties of previous methods developed in the same geometric framework: 1) the natural coupling of image channels proposed for anisotropic diffusion and 2) the ability of subjective surfaces to detect weak edges and close fragmented boundaries. The potential of such a geometric approach lies in the general definition of Riemannian manifolds, which naturally generalizes existing segmentation methods (the geodesic active contours, the active contours without edges, and the robust edge integrator) to higher dimensional spaces, non-flat images, and feature spaces. Our experiments show that the proposed technique improves the segmentation of multi-channel images, images subject to inhomogeneities, and images characterized by geometric structures like ridges or valleys.

Measures of fit in multiple correspondence analysis of crisp and fuzzy coded data

When continuous data are coded to categorical variables, two types of coding are possible: crisp coding in the form of indicator, or dummy, variables with values either 0 or 1; or fuzzy coding where each observation is transformed to a set of "degrees of membership" between 0 and 1, using co-called membership functions. It is well known that the correspondence analysis of crisp coded data, namely multiple correspondence analysis, yields principal inertias (eigenvalues) that considerably underestimate the quality of the solution in a low-dimensional space. Since the crisp data only code the categories to which each individual case belongs, an alternative measure of fit is simply to count how well these categories are predicted by the solution. Another approach is to consider multiple correspondence analysis equivalently as the analysis of the Burt matrix (i.e., the matrix of all two-way cross-tabulations of the categorical variables), and then perform a joint correspondence analysis to fit just the off-diagonal tables of the Burt matrix - the measure of fit is then computed as the quality of explaining these tables only. The correspondence analysis of fuzzy coded data, called "fuzzy multiple correspondence analysis", suffers from the same problem, albeit attenuated. Again, one can count how many correct predictions are made of the categories which have highest degree of membership. But here one can also defuzzify the results of the analysis to obtain estimated values of the original data, and then calculate a measure of fit in the familiar percentage form, thanks to the resultant orthogonal decomposition of variance. Furthermore, if one thinks of fuzzy multiple correspondence analysis as explaining the two-way associations between variables, a fuzzy Burt matrix can be computed and the same strategy as in the crisp case can be applied to analyse the off-diagonal part of this matrix. In this paper these alternative measures of fit are defined and applied to a data set of continuous meteorological variables, which are coded crisply and fuzzily into three categories. Measuring the fit is further discussed when the data set consists of a mixture of discrete and continuous variables.

Detection of moving groups among early type stars

Our procedure to detect moving groups in the solar neighbourhood (Chen et al., 1997) in the four-dimensional space of the stellar velocity components and age has been improved. The method, which takes advantadge of non-parametric estimators of density distribution to avoid any a priori knowledge of the kinematic properties of these stellar groups, now includes the effect of observational errors on the process to select moving group stars, uses a better estimation of the density distribution of the total sample and field stars, and classifies moving group stars using all the available information. It is applied here to an accurately selected sample of early-type stars with known radial velocities and Strömgren photometry. Astrometric data are taken from the HIPPARCOS catalogue (ESA, 1997), which results in an important decrease in the observational errors with respect to ground-based data, and ensures the uniformity of the observed data. Both the improvement of our method and the use of precise astrometric data have allowed us not only to confirm the existence of classical moving groups, but also to detect finer structures that in several cases can be related to kinematic properties of nearby open clusters or associations.

Nonideal particle distributions from kinetic freeze-out models

In fluid dynamical models the freeze-out of particles across a three-dimensional space-time hypersurface is discussed. The calculation of final momentum distribution of emitted particles is described for freeze-out surfaces, with both spacelike and timelike normals, taking into account conservation laws across the freeze-out discontinuity.

Radiative isotropic cosmologies with extra dimensions

We solve Einsteins equations in an n-dimensional vacuum with the simplest ansatz leading to a Friedmann-Robertson-Walker (FRW) four-dimensional space time. We show that the FRW model must be of radiation. For the open models the extra dimensions contract as a result of cosmological evolution. For flat and closed models they contract only when there is one extra dimension.

Particle and string fluid interpretation for the scalar sector of Kaluza-Klein theories

The scalar sector of the effective low-energy six-dimensional Kaluza-Klein theory is seen to represent an anisotropic fluid composed of two perfect fluids if the extra space metric has a Euclidean signature, or a perfect fluid of geometric strings if it has an indefinite signature. The Einstein field equations with such fluids can be explicitly integrated when the four-dimensional space-time has two commuting Killing vectors.

Particle creation due to cosmological contraction of extra dimensions

Particle production in a cosmological spacetime with extra dimensions is discussed. A five-dimensional cosmological model with a three-dimensional space expanding isotropically like in a radiative Friedmann-Robertson-Walker model and an internal space contracting to a constant small size is considered. The parameters of the model are adjusted so that time variations in internal space are compatible with present limits on time variations of the fundamental constants. By requiring that the energy density of the particles produced be less than the critical density at the radiation era we set restrictions on two more parameters: namely, the initial time of application of the semiclassical approach and the relative sizes between the internal space and the horizon of the ordinary Universe at this time. Whereas the production of massless particles allows a large range of variation to these parameters, the production of massive particles sets severe constraints on them, since, if they are overproduced, their energy density might very soon dominate the Universe and make cosmological dimensional reduction by extradimensional contraction unlikely.

Fractal dimension for Gaussian colored processes

The exact analytical expression for the Hausdorff dimension of free processes driven by Gaussian noise in n-dimensional space is obtained. The fractal dimension solely depends on the time behavior of the arbitrary correlation function of the noise, ranging from DX=1 for Orstein-Uhlenbeck input noise to any real number greater than 1 for fractional Brownian motions.

Measurement systems in wall production applications

Tässä tutkimuksessa kehitettiin prototyyppi betonielementin dimension mittaus järjestelmästä. Tämä järjestelmä mahdollistaa kolmiulotteisen kappaleen mittauksen. Tutkimuksessa kehitettiin myös stereonäköön perustuva kappaleen mittaus. Prototyyppiä testailin ja tulokset osoittautuivat luotettaviksi. Tutkimuksessa selvitetään ja vertaillaan myös muita lähestymistapoja ja olemassa olevia järjestelmiä kappaleen kolmiuloitteiseen mittaukseen, joita Suomalaiset yhtiöt käyttävät tällä alalla.

Multidimensional Deprivation in Contemporary Switzerland Across Social Groups and Time

We have investigated the phenomenon of deprivation in contemporary Switzerland through the adoption of a multidimensional, dynamic approach. By applying Self Organizing Maps (SOM) to a set of 33 non-monetary indicators from the 2009 wave of the Swiss Household Panel (SHP), we identified 13 prototypical forms (or clusters) of well-being, financial vulnerability, psycho-physiological fragility and deprivation within a topological dimensional space. Then new data from the previous waves (2003 to 2008) were classified by the SOM model, making it possible to estimate the weight of the different clusters in time and reconstruct the dynamics of stability and mobility of individuals within the map. Looking at the transition probabilities between year t and year t+1, we observed that the paths of mobility which catalyze the largest number of observations are those connecting clusters that are adjacent on the topological space.