Comparison between Statistical and Optimization Methods in Accessing Unmixing of Spectrally Similar Materials

This paper reports on the results from ordinary least squares and ridge regression as statistical methods, and is compared to numerical optimization methods such as the stochastic method for global optimization, simulated annealing, particle swarm optimization and limited memory Broyden-Fletcher-Goldfard-Sharon bound optimization method. We used each of the above mentioned methods in estimating the abundances of spectrally similar iron-bearing oxide/hydroxide/sulfate minerals in complex synthetic mixtures simulated from hyperspectral data. In evaluating the various methods, spectral mixtures were generated with varying linear proportions of individual spectra from the United States Geological Survey (USGS) spectral library. We conclude that ridge regression, simulated annealing and particle swarm optimization outperforms ordinary least squares method and the stochastic method for global optimization algorithms in estimating the partial abundance of each endmember. This result was independent of the error from either a uniform or gaussian distribution. For large remote sensing scenes, typically with millions of pixels and with many endmembers, we recommend using ridge regression. 1. BACKGROUND AND OBJECTIVE Remote sensors often record scenes in which the spectral signatures of various materials on the ground contribute to the spectrum measured from a single pixel. This can occur due to two reasons, (i) the spatial resolution of the image is low and adjacent objects can jointly occupy a single pixel and the resulting spectrum will be a composite of the individual objects, and (ii) distinct materials on the ground are combined into complex mixtures, for example, mixtures of minerals in the ground, which can occur regardless of the spatial resolution of the sensor (Keshava, 2003). Given such mixed pixels, the objective of unmixing (or sometimes known as “abundance estimation” or “fractional estimation”) is to identify the individual constituent materials present in the mixture, as well as the proportions in which they appear. Spectral unmixing is the procedure by which the measured spectrum of a mixed pixel is decomposed into a collection of constituent spectra, or more commonly known in the field of remote sensing as endmembers, and a set of corresponding fractions, or abundances, that indicate the proportion of each endmember present in the pixel. Spectral unmixing of hyperspectral remote sensing images is useful in determining abundances of different minerals. Most spectral unmixing techniques are variants of algorithms involving matrix inversion (Van der Meer & De Jong, 2000; Peddle & Smith, 2005; Miao et al., 2006; Settle, 2006). A major problem in spectral unmixing is the non-orthogonality of endmembers. The ability to estimate abundances in complex mixtures through spectral unmixing techniques is further complicated when considering very similar spectral signatures (Debba et al., 2006). It is known that iron-bearing oxide/hydroxide/sulfate minerals have similar spectral signatures and it is therefore difficult to estimate these abundances. Over the last two decades, several different unmixing models have been implemented, including least squares methods (Quarmby et al., 1992; Settle & Drake, 1993; Adams et al., 1995), neural networks (Atkinson et al., 1997; Liu et al., 2004), fuzzy classifiers (Foody, 1996), regression and decision trees (DeFries et al., 1999), support vector regression (Walton, 2008), gaussian mixture discriminant analysis (Ju et al., 2003) and maximum likelihood classifiers (Foody et al., 1992; Schowengerdt, Debba, P. (email [email protected]) is the corresponding and presenting author. Thanks to CSIR for funding. 1996). More recently, focus is on feature selection or feature extraction and using derivatives prior to spectral unmixing (Debba et al., 2006; Somers et al., 2009). This paper reports on the results from several statistical and optimization methods in estimating the abundances of spectrally similar iron-bearing oxide/hydroxide/sulfate minerals in complex synthetic mixtures using hyperspectral data. In using the various methods, spectral mixtures were generated with varying linear proportions of individual spectra of a set of iron-bearing oxide/hydroxide/sulfate minerals. The set of endmembers is commonly associated with sulphide-bearing mine wastes. Prior to unmixing, the mixed spectrum was first subjected to error from a uniform and gaussian distribution with signal-to-noise ratio of 500:1, which is typical for sensors like HyMap. Discussions on linear and non-linear mixtures and a literature survey on unmixing can be found in Keshava & Mustard (2002) and Keshava (2003). 2. METHODS OF SPECTRAL UNMIXING Spectral unmixing is a deconvolution process for estimating the contribution of individual endmembers. If we have K spectral bands, and we denote the ith endmember spectrum as si (K × 1) and the abundance of the ith endmember as ai, the observed spectrum x (K × 1) for any pixel in the scene can be expressed as x = a1s1 + a2s2 + · · ·+ aMsM + w