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Weighted Abundance-Constrained Linear Spectral Mixture Analysis

Linear spectral mixture analysis (LSMA) has been used in a wide range of applications. It is, in general, implemented without constraints due to mathematical tractability. However, it has been shown that abundance-constrained LSMA (AC-LSMA) can improve abundance-unconstrained LSMA, specifically in quantification when accurate estimates of abundance fractions are necessary. As long as AC-LSMA is considered, two constraints are generally imposed: abundance sum-to-one constraint (ASC) and abundance nonnegativity constraint (ANC). A general and common approach to solving AC-LSMA is to estimate abundance fractions in the sense of least squares error (LSE) while satisfying desired imposed abundance constraints. Since the LSE resulting from each individual band in abundance estimation is not weighted in accordance with significance of each of full bands in the signatures used to unmix data sample vectors, the effect caused by LSE is assumed to be uniform over all bands, which is, in general, not necessarily true in practical applications. This chapter extends the commonly used AC-LSMA to three types of weighted AC-LSMA (WAC-LSMA) from three different perspectives: parameter estimation, pattern classification, and orthogonal subspace projection (OSP). As demonstrated by experiments, WAC-LSMA generally performs better than unweighted AC-LSMA where the latter can be considered a special case of WAC-LSMA with the weighting matrix chosen to be the identity matrix.