5,490 research outputs found
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
Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type
We study the Cauchy problem for the Korteweg de Vries (KdV) equation with
small dispersion and with monotonically increasing initial data using the
Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero
dispersion limit, is obtained using the steepest descent method for oscillatory
Riemann-Hilbert problems. The asymptotic solution is completely described by a
scalar function \g that satisfies a scalar RH problem and a set of algebraic
equations constrained by algebraic inequalities. The scalar function \g is
equivalent to the solution of the Lax-Levermore maximization problem. The
solution of the set of algebraic equations satisfies the Whitham equations. We
show that the scalar function \g and the Lax-Levermore maximizer can be
expressed as the solution of a linear overdetermined system of equations of
Euler-Poisson-Darboux type. We also show that the set of algebraic equations
and algebraic inequalities can be expressed in terms of the solution of a
different set of linear overdetermined systems of equations of
Euler-Poisson-Darboux type. Furthermore we show that the set of algebraic
equations is equivalent to the classical solution of the Whitham equations
expressed by the hodograph transformation.Comment: 32 pages, 1 figure, latex2
Illiquid Assets and Optimal Portfolio Choice
The presence of illiquid assets, such as human wealth or a family owned business, complicates the problem of portfolio choice. This paper is concerned with the problem of optimal asset allocation and consumption in a continuous time model when one asset cannot be traded. This illiquid asset, which depends on an uninsurable source of risk, provides a liquid dividend. In the case of human capital we can think about this dividend as labor income. The agent is endowed with a given amount of the illiquid asset and with some liquid wealth which can be allocated in a market where there is a risky and a riskless asset. The main point of the paper is that the optimal allocations to the two liquid assets and consumption will critically depend on the endowment and characteristics of the illiquid asset, in addition to the preferences and to the liquid holdings held by the agent. We provide what we believe to be the first analytical solution to this problem when the agent has power utility of consumption and terminal wealth. We also derive the value that the agent assigns to the illiquid asset. The risk adjusted valuation procedure we develop can be used to value both liquid and illiquid assets, as well as contingent claims on those assets.
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