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.

Towards Human-like Bimanual Movements in Anthropomorphic Robots: A Nonlinear Optimization Approach

Previously we have presented a model for generating human-like arm and hand movements on an unimanual anthropomorphic robot involved in human-robot collaboration tasks. The present paper aims to extend our model in order to address the generation of human-like bimanual movement sequences which are challenged by scenarios cluttered with obstacles. Movement planning involves large scale nonlinear constrained optimization problems which are solved using the IPOPT solver. Simulation studies show that the model generates feasible and realistic hand trajectories for action sequences involving the two hands. The computational costs involved in the planning allow for real-time human robot-interaction. A qualitative analysis reveals that the movements of the robot exhibit basic characteristics of human movements.

A filter inexact-restoration method for nonlinear programming

A new iterative algorithm based on the inexact-restoration (IR) approach combined with the filter strategy to solve nonlinear constrained optimization problems is presented. The high level algorithm is suggested by Gonzaga et al. (SIAM J. Optim. 14:646–669, 2003) but not yet implement—the internal algorithms are not proposed. The filter, a new concept introduced by Fletcher and Leyffer (Math. Program. Ser. A 91:239–269, 2002), replaces the merit function avoiding the penalty parameter estimation and the difficulties related to the nondifferentiability. In the IR approach two independent phases are performed in each iteration, the feasibility and the optimality phases. The line search filter is combined with the first one phase to generate a “more feasible” point, and then it is used in the optimality phase to reach an “optimal” point. Numerical experiences with a collection of AMPL problems and a performance comparison with IPOPT are provided.

Global vs. local nonlinear optimization techniques for human-like movement of an anthropomorphic robot

In this paper a comparison between using global and local optimization techniques for solving the problem of generating human-like arm and hand movements for an anthropomorphic dual arm robot is made. Although the objective function involved in each optimization problem is convex, there is no evidence that the admissible regions of these problems are convex sets. For the sequence of movements for which the numerical tests were done there were no significant differences between the optimal solutions obtained using the global and the local techniques. This suggests that the optimal solution obtained using the local solver is indeed a global solution.

Towards human-like bimanual movements in anthropomorphic robots: a nonlinear optimization approach

Previously we have presented a model for generating human-like arm and hand movements on an unimanual anthropomorphic robot involved in human-robot collaboration tasks. The present paper aims to extend our model in order to address the generation of human-like bimanual movement sequences which are challenged by scenarios cluttered with obstacles. Movement planning involves large scale nonlinear constrained optimization problems which are solved using the IPOPT solver. Simulation studies show that the model generates feasible and realistic hand trajectories for action sequences involving the two hands. The computational costs involved in the planning allow for real-time human robot-interaction. A qualitative analysis reveals that the movements of the robot exhibit basic characteristics of human movements.

Semidiscretization and long-time asymptotics of nonlinear diffusion equations

We review several results concerning the long time asymptotics of nonlinear diffusion models based on entropy and mass transport methods. Semidiscretization of these nonlinear diffusion models are proposed and their numerical properties analysed. We demonstrate the long time asymptotic results by numerical simulation and we discuss several open problems based on these numerical results. We show that for general nonlinear diffusion equations the long-time asymptotics can be characterized in terms of fixed points of certain maps which are contractions for the euclidean Wasserstein distance. In fact, we propose a new scaling for which we can prove that this family of fixed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities for values close to zero.

A mixed finite element method for nonlinear diffusion equations

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model.

Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation

This paper is concerned with the modeling and analysis of quantum dissipation phenomena in the Schrödinger picture. More precisely, we do investigate in detail a dissipative, nonlinear Schrödinger equation somehow accounting for quantum Fokker–Planck effects, and how it is drastically reduced to a simpler logarithmic equation via a nonlinear gauge transformation in such a way that the physics underlying both problems keeps unaltered. From a mathematical viewpoint, this allows for a more achievable analysis regarding the local wellposedness of the initial–boundary value problem. This simplification requires the performance of the polar (modulus–argument) decomposition of the wavefunction, which is rigorously attained (for the first time to the best of our knowledge) under quite reasonable assumptions.

Fault diagnosis by refining the parameter uncertainty space of nonlinear dynamic systems

This paper deals with fault detection and isolation problems for nonlinear dynamic systems. Both problems are stated as constraint satisfaction problems (CSP) and solved using consistency techniques. The main contribution is the isolation method based on consistency techniques and uncertainty space refining of interval parameters. The major advantage of this method is that the isolation speed is fast even taking into account uncertainty in parameters, measurements, and model errors. Interval calculations bring independence from the assumption of monotony considered by several approaches for fault isolation which are based on observers. An application to a well known alcoholic fermentation process model is presented

Nonlinear inverse dynamic models of gas sensing systems based on chemical sensor arrays for quantitative measurements

Gas sensing systems based on low-cost chemical sensor arrays are gaining interest for the analysis of multicomponent gas mixtures. These sensors show different problems, e.g., nonlinearities and slow time-response, which can be partially solved by digital signal processing. Our approach is based on building a nonlinear inverse dynamic system. Results for different identification techniques, including artificial neural networks and Wiener series, are compared in terms of measurement accuracy.

Nonlinear Pile Behavior in Integral Abutment Bridges, HR-227, Part 1, 1982

The highway departments of all fifty states were contacted to find the extent of application of integral abutment bridges, to survey the different guidelines used for analysis and design of integral abutment bridges, and to assess the performance of such bridges through the years. The variation in design assumptions and length limitations among the various states in their approach to the use of integral abutments is discussed. The problems associated with lateral displacements at the abutment, and the solutions developed by the different states for most of the ill effects of abutment movements are summarized in the report. An algorithm based on a state-of-the-art nonlinear finite element procedure was developed and used to study piling stresses and pile-soil interaction in integral abutment bridges. The finite element idealization consists of beam-column elements with geometric and material nonlinearities for the pile and nonlinear springs for the soil. An idealized soil model (modified Ramberg-Osgood model) was introduced in this investigation to obtain the tangent stiffness of the nonlinear spring elements. Several numerical examples are presented in order to establish the reliability of the finite element model and the computer software developed. Three problems with analytical solutions were first solved and compared with theoretical solutions. A 40 ft H pile (HP 10 X 42) in six typical Iowa soils was then analyzed by first applying a horizontal displacement (to simulate bridge motion) and no rotation at the top and then applying a vertical load V incrementally until failure occurred. Based on the numerical results, the failure mechanisms were generalized to be of two types: (a) lateral type failure and (b) vertical type failure. It appears that most piles in Iowa soils (sand, soft clay and stiff clay) failed when the applied vertical load reached the ultimate soil frictional resistance (vertical type failure). In very stiff clays, however, the lateral type failure occurs before vertical type failure because the soil is sufficiently stiff to force a plastic hinge to form in the pile as the specified lateral displacement is applied. Preliminary results from this investigation showed that the vertical load-carrying capacity of H piles is not significantly affected by lateral displacements of 2 inches in soft clay, stiff clay, loose sand, medium sand and dense sand. However, in very stiff clay (average blow count of 50 from standard penetration tests), it was found that the vertical load carrying capacity of the H pile is reduced by about 50 percent for 2 inches of lateral displacement and by about 20 percent for lateral displacement of 1 inch. On the basis of the preliminary results of this investigation, the 265-feet length limitation in Iowa for integral abutment concrete bridges appears to be very conservative.

Nonlinear Finite Element Study of Piles in Integral Abutment Bridges, HR-227, Part 2, 1982

The highway departments of the states which use integral abutments in bridge design were contacted in order to study the extent of integral abutment use in skewed bridges and to survey the different guidelines used for analysis and design of integral abutments in skewed bridges. The variation in design assumptions and pile orientations among the various states in their approach to the use of integral abutments on skewed bridges is discussed. The problems associated with the treatment of the approach slab, backfill, and pile cap, and the reason for using different pile orientations are summarized in the report. An algorithm based on a state-of-the-art nonlinear finite element procedure previously developed by the authors was modified and used to study the influence of different factors on behavior of piles in integral abutment bridges. An idealized integral abutment was introduced by assuming that the pile is rigidly cast into the pile cap and that the approach slab offers no resistance to lateral thermal expansion. Passive soil and shear resistance of the cap are neglected in design. A 40-foot H pile (HP 10 X 42) in six typical Iowa soils was analyzed for fully restrained pile head and pinned pile head. According to numerical results, the maximum safe length for fully restrained pile head is one-half the maximum safe length for pinned pile head. If the pile head is partially restrained, the maximum safe length will lie between the two limits. The numerical results from an investigation of the effect of predrilled oversized holes indicate that if the length of the predrilled oversized hole is at least 4 feet below the ground, the vertical load-carrying capacity of the H pile is only reduced by 10 percent for 4 inches of lateral displacement in very stiff clay. With no predrilled oversized hole, the pile failed before the 4-inch lateral displacement was reached. Thus, the maximum safe lengths for integral abutment bridges may be increased by predrilling. Four different typical Iowa layered soils were selected and used in this investigation. In certain situations, compacted soil (> 50 blow count in standard penetration tests) is used as fill on top of natural soil. The numerical results showed that the critical conditions will depend on the length of the compacted soil. If the length of the compacted soil exceeds 4 feet, the failure mechanism for the pile is similar to one in a layer of very stiff clay. That is, the vertical load-carrying capacity of the H pile will be greatly reduced as the specified lateral displacement increases.

Nonlinear dynamics and control of multibody systems

The dynamics of flexible systems, such as robot manipulators , mechanical chains or multibody systems in general, is becoming increasingly important in engineering. This article deals with some nonlinearities that arise in the study of dynamics and control of multibody systems in connection to large rotations. Specifically, a numerical scheme that adresses the conservation of fundamental constants is presented in order to analyse the control-structure interaction problems.

Covalidation of Integral Transforms and Method of Lines in Nonlinear Convection-Diffusion with Mathematica

The Mathematica system (version 4.0) is employed in the solution of nonlinear difusion and convection-difusion problems, formulated as transient one-dimensional partial diferential equations with potential dependent equation coefficients. The Generalized Integral Transform Technique (GITT) is first implemented for the hybrid numerical-analytical solution of such classes of problems, through the symbolic integral transformation and elimination of the space variable, followed by the utilization of the built-in Mathematica function NDSolve for handling the resulting transformed ODE system. This approach ofers an error-controlled final numerical solution, through the simultaneous control of local errors in this reliable ODE's solver and of the proposed eigenfunction expansion truncation order. For covalidation purposes, the same built-in function NDSolve is employed in the direct solution of these partial diferential equations, as made possible by the algorithms implemented in Mathematica (versions 3.0 and up), based on application of the method of lines. Various numerical experiments are performed and relative merits of each approach are critically pointed out.

Efficient Optimization Algorithms for Nonlinear Data Analysis

Identification of low-dimensional structures and main sources of variation from multivariate data are fundamental tasks in data analysis. Many methods aimed at these tasks involve solution of an optimization problem. Thus, the objective of this thesis is to develop computationally efficient and theoretically justified methods for solving such problems. Most of the thesis is based on a statistical model, where ridges of the density estimated from the data are considered as relevant features. Finding ridges, that are generalized maxima, necessitates development of advanced optimization methods. An efficient and convergent trust region Newton method for projecting a point onto a ridge of the underlying density is developed for this purpose. The method is utilized in a differential equation-based approach for tracing ridges and computing projection coordinates along them. The density estimation is done nonparametrically by using Gaussian kernels. This allows application of ridge-based methods with only mild assumptions on the underlying structure of the data. The statistical model and the ridge finding methods are adapted to two different applications. The first one is extraction of curvilinear structures from noisy data mixed with background clutter. The second one is a novel nonlinear generalization of principal component analysis (PCA) and its extension to time series data. The methods have a wide range of potential applications, where most of the earlier approaches are inadequate. Examples include identification of faults from seismic data and identification of filaments from cosmological data. Applicability of the nonlinear PCA to climate analysis and reconstruction of periodic patterns from noisy time series data are also demonstrated. Other contributions of the thesis include development of an efficient semidefinite optimization method for embedding graphs into the Euclidean space. The method produces structure-preserving embeddings that maximize interpoint distances. It is primarily developed for dimensionality reduction, but has also potential applications in graph theory and various areas of physics, chemistry and engineering. Asymptotic behaviour of ridges and maxima of Gaussian kernel densities is also investigated when the kernel bandwidth approaches infinity. The results are applied to the nonlinear PCA and to finding significant maxima of such densities, which is a typical problem in visual object tracking.