Using simulated annealing to obtain optimal linear end-member mixtures of hyperspectral data

Abstract

Current thinking in hyperspectral remote sensing suggests the existence of “pure” end-member components. Once identified these components become apices for an n-dimensional convex hull, or minimum encompassing hypervolume, containing all remaining data points. Data in the hypervolume are considered linear mixtures of the apex components. As such, each spectra can be constructed of linear combinations of end-member components. Deriving the relative proportions of end-member components is non-trivial and is often computed using variations of least-squares techniques. These solutions have not been shown to be optimal for combinatorial minimization problems. An exhaustive search of the domain space is not practical and the discrete nature of the data precludes the notion of search based on direction, i.e., hill climbing or steepest descent approaches. Simulated annealing (SA) is an optimization strategy easily applied to this domain. Loosely based on thermodynamic cooling models for crystalline materials, SA combines a “random walk” model with a Boltzmann's probability distribution based upon the temperature ( t) of a system. SA operates much like a phase cooling within a magma body, allowing the user to control the “cooling” schedule and preventing “quenching” to a non-optimal state or local minimum. For this study SA is applied to linear combinations of mineral spectra and reproduces the component mixtures within acceptable tolerances.