Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches

Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.Comment: This work has been accepted for publication in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensin

Development of iterative techniques for the solution of unsteady compressible viscous flows

The development of efficient iterative solution methods for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations is discussed. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. In this work, another approach based on the classical conjugate gradient method, known as the Generalized Minimum Residual (GMRES) algorithm is investigated. The GMRES algorithm has been used in the past by a number of researchers for solving steady viscous and inviscid flow problems. Here, we investigate the suitability of this algorithm for solving the system of non-linear equations that arise in unsteady Navier-Stokes solvers at each time step

An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary

Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. We do this in the usual way away from the boundaries, by replacing the incompressibility condition on the velocity by a Poisson equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy (in L-infinity), for both the velocity and the pressure. The scheme has a natural extension to 3-D.Comment: 50 pages, 14 figure

Institute for Computational Mechanics in Propulsion (ICOMP)

The Institute for Computational Mechanics in Propulsion (ICOMP) is a combined activity of Case Western Reserve University, Ohio Aerospace Institute (OAI) and NASA Lewis. The purpose of ICOMP is to develop techniques to improve problem solving capabilities in all aspects of computational mechanics related to propulsion. The activities at ICOMP during 1991 are described

An augmented lagrangian interior point method using diretions of negative curvature

We describe an efficient implementation of an interior-point algorithm for non-convex problems that uses directions of negative curvature. These directions should ensure convergence to second-order KKT points and improve the computational efficiency of the procedure. Some relevant aspects of the implementation are the strategy to combine a direction of negative curvature and a modified Newton direction, and the conditions to ensure feasibility of the iterates with respect to the simple bounds. The use of multivariate barrier and penalty parameters is also discussed, as well as the update rules for these parameters. Finally, numerical results on a set oftest problems are presented