Linear Spectral Unmixing With Three Criteria, Least Squares Error, Simplex Volume and Orthogonal Projection

Abstract

Although Linear Spectral Mixture Analysis (LSMA) and endmember finding are discussed in Chaps. 2 and 3 respectively as separate topics, both subjects are actually closely related. As a matter of fact, many Endmember-Finding Algorithms (EFAs) are indeed designed from the concept of Linear Spectral Unmixing (LSU) carried out by LSMA. Nonetheless, it does not imply that LSU is an endmember finding technique or vice versa. The link between these two is the geometric structure resulting from simplex that is shared by LSU and endmember finding. When Fully Constrained Least Squares (FLCS) performs LSU, referred to as Fully Abundance-Constrained LSU (FAC-LSU), it imposes two physical constraints, Abundance Sum-to-one Constraint (ASC) and Abundance Non-negativity Constraint (ANC), and it assumes that all data sample vectors are embraced by a simplex with vertices formed by a set of signatures of interest, known as endmembers. The vertices used to form a simplex are exactly the same signatures used to form a Linear Mixing Model (LMM) for LSMA. Basically, endmember finding does not perform LSU, but it can use FAC-LSU to produce the best possible endmembers by finding maximal Simplex Volume (SV) as does N-FINDR (Winter 1999a, b) or minimal SV as does Minimum Volume Transform (MVT) (Craig 1994). Such use of simplex may lead to a belief that endmember finding can perform LSU and vice versa. Unfortunately, it is not necessarily true as will be shown in Chap. 9. This chapter is devoted to the investigation and exploration of their relationships.