Fully Abundance-Constrained Sequential Endmember Finding: Linear Spectral Mixture Analysis

Abstract

The Fully Constrained Least Squares (FCLS) method discussed in Chap. 2 is a Linear Spectral Mixture Analysis (LSMA) technique which has been widely used for data unmixing. It assumes that data sample vectors can be represented by a set of signatures via a Linear Mixing Model (LMM) by which data sample vectors can then be unmixed by FCLS in terms of the abundance fractions of these signatures present in the data sample vectors subject to two physical constraints, Abundance Sum-to-one Constraint (ASC) and Abundance Non-negativity Constraint (ANC) imposed on LMM. Because of its use of ASC and ANC, FCLS has also been used to find endmembers in a similar manner to N-FINDR because ASC and ANC can be interpreted as the two abundance constraints used to impose on a simplex. In this case, FCLS is considered as SiMultaneous FCLS Endmember-Finding Algorithm (SM FCLS-EFA) in a similar manner to SiMultaneous N-FINDR (SM N-FINDR). So, in parallel to development of N-FINDR in Chang (2013), this chapter also develops a similar theory for SM FCLS-EFA for finding endmembers. Specifically, two sequential versions of SM FCLS, to be called SeQuential FCLS-EFA (SQ-FCLS-EFA) and SuCcessive FCLS (SC-FCLS-EFA), can also be derived as Endmember-Finding Algorithms (EFAs), both of which can be considered as the counterparts of N-FINDR, SeQuential N-FINDR (SQ N-FINDR), and SuCcessive N-FINDR (SC N-FINDR) in Chap. 6, respectively. However, unlike N-FINDR, which is designed to find pure signatures as endmembers, FCLS-EFA is specifically designed to find signatures that are not necessarily pure but rather spectrally distinct signatures to represent data best in terms of an LMM. So, to reflect its nature more accurately in finding signatures for LMM, the endmembers found by FCLS-EFA are indeed Virtual Signatures (VSs) as defined in Chang et al. (2010). To deal further with random issues in the use of initial conditions, three versions of SM FCLS-EFA—Initialization Driven FCLS-EFA (ID-FCLS-EFA), Iterative FCLS-EFA (IFCLS-EFA) , and Random FCLS-EFA (RFCLS-EFA)—are also developed corresponding to their respective counterparts developed for N-FINDR in Chang (2013), which are Initialization Driven N-FINDR (ID-N-FINDR), Iterative N-FINDR (IN-FINDR), and Random N-FINDR (RN-FINDR).