A Least-Squares Approach to Fully Constrained Linear Spectral Mixture Analysis Using Linear Inequality Constraints

Fully constrained linear spectral mixture analysis (FCLSMA) has been used for material quantification in remotely sensed imagery. In order to implement FCLSMA, two constraints are imposed on abundance fractions, referred to as Abundance Sum-to-one Constraint (ASC) and Abundance Nonnegativity Constraint (ANC). While the ASC is linear equality constraint, the ANC is a linear inequality constraint. A direct approach to imposing the ASC and ANC has been recently investigated and is called fully constrained least-squares (FCLS) method. Since there is no analytical solution resulting from the ANC, a modified fully constrained least-squares method (MFCLS) which replaces the ANC with an Absolute Abundance Sum-to-one Constraint (AASC) was proposed to convert a set of inequality constraints to a quality constraint. The results produced by these two approaches have been shown to be very close. In this paper, we take an opposite approach to the MFCLS method, called least-squares with linear inequality constraints (LSLIC) method which also solves FCLSMA, but replaces the ASC with two linear inequalities. The proposed LSLIC transforms the FCLSMA to a linear distance programming problem which can be solved easily by a numerical algorithm. In order to demonstrate its utility in solving FCLSMA, the LSLIC method is compared to the FCLS and MFCLS methods. The experimental results show that these three methods perform very similarly with only subtle differences resulting from their problem formations.