Hyperspectral unmixing with simultaneous dimensionality estimation

This paper is an elaboration of the simplex identification via split augmented Lagrangian (SISAL) algorithm (Bioucas-Dias, 2009) to blindly unmix hyperspectral data. SISAL is a linear hyperspectral unmixing method of the minimum volume class. This method solve a non-convex problem by a sequence of augmented Lagrangian optimizations, where the positivity constraints, forcing the spectral vectors to belong to the convex hull of the endmember signatures, are replaced by soft constraints. With respect to SISAL, we introduce a dimensionality estimation method based on the minimum description length (MDL) principle. The effectiveness of the proposed algorithm is illustrated with simulated and real data.

Osmolality of preterm formulas suplemented with monprotein energy supplements

Background: Addition of energy supplements to preterm formulas is an optional strategy to increase the energy intake in infants requiring fluid restriction, in conditions like bronchopulmonary dysplasia. This strategy may lead to an undesirable increase in osmolality of feeds, the maximum recommended safe limit being 400 mOsm/kg. The aim of the study was to measure the changes in osmolality of several commercialized preterm formulas after addition of glucose polymers and medium-chain triglycerides. Methods: Osmolality was measured by the freezing point depression method. Six powdered formulas with concentrations of 14 g/100 ml and 16 g/100 ml, and five ready-to-feed liquid formulas were analyzed. All formulas, were supplemented with 10% (low supplementation) or 20% (high supplementation) of additional calories, respectively, in the form of glucose polymers and medium chain triglycerides, maintaining a 1:1 glucose:lipid calorie ratio. Inter-analysis and intra-analysis coefficients of variation of the measurements were always < 3.9%. Results: The mean osmolality (mOsm/kg) of the non-supplemented formulas varied between 268.5 and 315.3 mOsm/kg, increasing by 3–5% in low supplemented formulas, and by 6–10% in high supplemented formulas. None of the formulas analyzed exceeded 352.8 mOsm/kg. Conclusion: The supplementation of preterm formulas with nonprotein energy supplements with up to 20% additional calories did not exceed the maximum recommended osmolality for neonatal feedings.

Grid structure impact in sparse point representation of derivatives

In the Sparse Point Representation (SPR) method the principle is to retain the function data indicated by significant interpolatory wavelet coefficients, which are defined as interpolation errors by means of an interpolating subdivision scheme. Typically, a SPR grid is coarse in smooth regions, and refined close to irregularities. Furthermore, the computation of partial derivatives of a function from the information of its SPR content is performed in two steps. The first one is a refinement procedure to extend the SPR by the inclusion of new interpolated point values in a security zone. Then, for points in the refined grid, such derivatives are approximated by uniform finite differences, using a step size proportional to each point local scale. If required neighboring stencils are not present in the grid, the corresponding missing point values are approximated from coarser scales using the interpolating subdivision scheme. Using the cubic interpolation subdivision scheme, we demonstrate that such adaptive finite differences can be formulated in terms of a collocation scheme based on the wavelet expansion associated to the SPR. For this purpose, we prove some results concerning the local behavior of such wavelet reconstruction operators, which stand for SPR grids having appropriate structures. This statement implies that the adaptive finite difference scheme and the one using the step size of the finest level produce the same result at SPR grid points. Consequently, in addition to the refinement strategy, our analysis indicates that some care must be taken concerning the grid structure, in order to keep the truncation error under a certain accuracy limit. Illustrating results are presented for 2D Maxwell's equation numerical solutions.

Simulation of default events in a CDX and estimation of the spread

The portfolio generating the iTraxx EUR index is modeled by coupled Markov chains. Each of the industries of the portfolio evolves according to its own Markov transition matrix. Using a variant of the method of moments, the model parameters are estimated from a data set of Standard and Poor's. Swap spreads are evaluated by Monte-Carlo simulations. Along with an actuarially fair spread, at least squares spread is considered.

Logarithmically proportional objective function for planar surfaces recognition in 3D point cloud

3D laser scanning is becoming a standard technology to generate building models of a facility's as-is condition. Since most constructions are constructed upon planar surfaces, recognition of them paves the way for automation of generating building models. This paper introduces a new logarithmically proportional objective function that can be used in both heuristic and metaheuristic (MH) algorithms to discover planar surfaces in a point cloud without exploiting any prior knowledge about those surfaces. It can also adopt itself to the structural density of a scanned construction. In this paper, a metaheuristic method, genetic algorithm (GA), is used to test this introduced objective function on a synthetic point cloud. The results obtained show the proposed method is capable to find all plane configurations of planar surfaces (with a wide variety of sizes) in the point cloud with a minor distance to the actual configurations. © 2014 IEEE.

The extended unsymmetric frontal solution for multiple-point constraints

The purpose of this paper is to discuss the linear solution of equality constrained problems by using the Frontal solution method without explicit assembling. Design/methodology/approach - Re-written frontal solution method with a priori pivot and front sequence. OpenMP parallelization, nearly linear (in elimination and substitution) up to 40 threads. Constraints enforced at the local assembling stage. Findings - When compared with both standard sparse solvers and classical frontal implementations, memory requirements and code size are significantly reduced. Research limitations/implications - Large, non-linear problems with constraints typically make use of the Newton method with Lagrange multipliers. In the context of the solution of problems with large number of constraints, the matrix transformation methods (MTM) are often more cost-effective. The paper presents a complete solution, with topological ordering, for this problem. Practical implications - A complete software package in Fortran 2003 is described. Examples of clique-based problems are shown with large systems solved in core. Social implications - More realistic non-linear problems can be solved with this Frontal code at the core of the Newton method. Originality/value - Use of topological ordering of constraints. A-priori pivot and front sequences. No need for symbolic assembling. Constraints treated at the core of the Frontal solver. Use of OpenMP in the main Frontal loop, now quantified. Availability of Software.

Parallel GPU architecture for hyperspectral unmixing based on augmented Lagrangian method

Hyperspectral imaging has become one of the main topics in remote sensing applications, which comprise hundreds of spectral bands at different (almost contiguous) wavelength channels over the same area generating large data volumes comprising several GBs per flight. This high spectral resolution can be used for object detection and for discriminate between different objects based on their spectral characteristics. One of the main problems involved in hyperspectral analysis is the presence of mixed pixels, which arise when the spacial resolution of the sensor is not able to separate spectrally distinct materials. Spectral unmixing is one of the most important task for hyperspectral data exploitation. However, the unmixing algorithms can be computationally very expensive, and even high power consuming, which compromises the use in applications under on-board constraints. In recent years, graphics processing units (GPUs) have evolved into highly parallel and programmable systems. Specifically, several hyperspectral imaging algorithms have shown to be able to benefit from this hardware taking advantage of the extremely high floating-point processing performance, compact size, huge memory bandwidth, and relatively low cost of these units, which make them appealing for onboard data processing. In this paper, we propose a parallel implementation of an augmented Lagragian based method for unsupervised hyperspectral linear unmixing on GPUs using CUDA. The method called simplex identification via split augmented Lagrangian (SISAL) aims to identify the endmembers of a scene, i.e., is able to unmix hyperspectral data sets in which the pure pixel assumption is violated. The efficient implementation of SISAL method presented in this work exploits the GPU architecture at low level, using shared memory and coalesced accesses to memory.

Estimation of signal subspace on hyperspectral data

Dimensionality reduction plays a crucial role in many hyperspectral data processing and analysis algorithms. This paper proposes a new mean squared error based approach to determine the signal subspace in hyperspectral imagery. The method first estimates the signal and noise correlations matrices, then it selects the subset of eigenvalues that best represents the signal subspace in the least square sense. The effectiveness of the proposed method is illustrated using simulated and real hyperspectral images.

Estimation of rotational frequency response functions

As it is widely known, in structural dynamic applications, ranging from structural coupling to model updating, the incompatibility between measured and simulated data is inevitable, due to the problem of coordinate incompleteness. Usually, the experimental data from conventional vibration testing is collected at a few translational degrees of freedom (DOF) due to applied forces, using hammer or shaker exciters, over a limited frequency range. Hence, one can only measure a portion of the receptance matrix, few columns, related to the forced DOFs, and rows, related to the measured DOFs. In contrast, by finite element modeling, one can obtain a full data set, both in terms of DOFs and identified modes. Over the years, several model reduction techniques have been proposed, as well as data expansion ones. However, the latter are significantly fewer and the demand for efficient techniques is still an issue. In this work, one proposes a technique for expanding measured frequency response functions (FRF) over the entire set of DOFs. This technique is based upon a modified Kidder's method and the principle of reciprocity, and it avoids the need for modal identification, as it uses the measured FRFs directly. In order to illustrate the performance of the proposed technique, a set of simulated experimental translational FRFs is taken as reference to estimate rotational FRFs, including those that are due to applied moments.

Hyperspectral signal subspace estimation

Given an hyperspectral image, the determination of the number of endmembers and the subspace where they live without any prior knowledge is crucial to the success of hyperspectral image analysis. This paper introduces a new minimum mean squared error based approach to infer the signal subspace in hyperspectral imagery. The method, termed hyperspectral signal identification by minimum error (HySime), is eigendecomposition based and it does not depend on any tuning parameters. It first estimates the signal and noise correlation matrices and then selects the subset of eigenvalues that best represents the signal subspace in the least squared error sense. 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.

Hyperspectral imagery framework for unmixing and dimensionality estimation

In hyperspectral imagery a pixel typically consists mixture of spectral signatures of reference substances, also called endmembers. Linear spectral mixture analysis, or linear unmixing, aims at estimating the number of endmembers, their spectral signatures, and their abundance fractions. This paper proposes a framework for hyperpsectral unmixing. A blind method (SISAL) is used for the estimation of the unknown endmember signature and their abundance fractions. This method solve a non-convex problem by a sequence of augmented Lagrangian optimizations, where the positivity constraints, forcing the spectral vectors to belong to the convex hull of the endmember signatures, are replaced by soft constraints. The proposed framework simultaneously estimates the number of endmembers present in the hyperspectral image by an algorithm based on the minimum description length (MDL) principle. Experimental results on both synthetic and real hyperspectral data demonstrate the effectiveness of the proposed algorithm.

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.