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Kalman Filter-Based Estimation for Hyperspectral Signals

Most popular and widely used approaches in statistical signal estimation are mean squared error (MSE) based approaches among which Kalman filtering (KF) is the most powerful and effective technique that can be implemented in real time under a nonstationary environment. Recently, a Kalman filtering approach to linear spectral unmixing, called Kalman filter-based linear spectral unmixing (KFLU) was developed for mixed pixel classification by Chang and Brumbley (1999a, 1999b). However, its applicability to spectral characterization for spectral estimation, identification, and quantification has not been explored. This chapter presents new applications of KF in spectral estimation, identification, and abundance quantification for which three Kalman filter (KF)-based spectral characterization signal processing (KFSCSP) techniques are developed. These techniques are completely different from KFLU in the sense that the former performs a Kalman filter across a spectral coverage wavelength by wavelength (i.e., band-by-band) as opposed to the latter, which implements a Kalman filter pixel vector by pixel vector throughout an entire image cube. In addition, the proposed Kalman filter-based techniques do not require a linear mixture model as KFLU does. Accordingly, they are not linear spectral unmixing methods but rather spectral signature filters operating as if they are spectral measures. More specifically, the state equation implemented in KFLU is designed to keep track of pixel-to-pixel correlation present in a hyperspectral image cube, whereas the state equation used by KFSCSP techniques is designed to capture the band-to-band correlation within a single signature vector. As a result, KFSCSP techniques can be used for both laboratory data and non-image data analysis compared with KFLU, which is primarily developed for hyperspectral imagery. This is due to the fact that laboratory data do not usually have pixel-to-pixel correlation of which KFLU can take advantage, but they do have band-to-band correlation that can be taken into account by KFSCSP techniques. Also, KFSCSP techniques do not require a linear mixture model as does KFLU, which assumes that image pixels are linearly mixed by a number of image endmembers. Therefore, KFSCSP techniques do not need image endmembers to form a linear mixture model for their implementation. Accordingly, they would rather be considered as spectral signature filters.