Weighted least squares error approaches to abundance-constrained linear spectral mixture analysis

Abstract

Linear spectral mixture analysis (LSMA) is a widely used technique for subpxiel detection and mixed pixel classification. Due to mathematical tractability it is generally implemented without constraints. However, it has been shown that the constrained LSMA can improve the unconstrained LSMA, specifically in quantification when accurate estimates of abundance fractions are required. When the constrained LSMA is considered, two constraints are generally imposed on abundance fractions, abundance sum-to-one constraint (ASC) and abundance nonnegativity constraint (ANC), referred to as abundance-constrained LSMA (AC-LSMA). A general and common approach to solving the AC-LSMA is to estimate abundance fractions in the sense of least-squares error (LSE) subject to the imposed constraints. Since the LSE is not weighted in accordance with significance of bands, the effect caused by the LSE is assumed to be uniform over all the bands, which is generally not necessarily true. This paper extends the commonly used LSE-based AC-LSMA to weighted LSE-based AC-LSMA with the weighting matrix that is derived from various approaches such as parameter estimation, pattern classification and orthogonal subspace projection (OSP). As demonstrated by experiments, the weighted LSE-based AC-LSMA generally performs better than the commonly used LSE-based AC-LSMA where the latter can be considered a special case of the former with the weighting matrix reduced to the identity matrix.