Fully Geometric-Constrained Sequential Endmember Finding: Simplex Volume Analysis-Based N-FINDR

Abstract

An endmember is considered as an idealistic, pure spectral signature according to the definition given in Schwengerdt (Schowengerdt 1997). Because of the purity of their spectral signatures, endmembers can be used to specify distinct spectral classes. As a result, endmember finding is one of most fundamental tasks in hyperspectral data exploitation because endmembers provide crucial information in identifying material substances. One general approach is to use Simplex Volume Analysis (SVA) (Chang 2013b) which assumes endmembers as vertices of a simplex, and endmembers can be then identified by finding a simplex with maximal volume that is fully embedded in the data space or a simplex with minimal volume that embraces the entire data space. Over the past few years, SVA has become a major trend in finding endmembers. The most notable is the N-finder algorithm (N-FINDR) developed by Winter (1999a, b). Since N-FINDR was introduced, many SVA-based endmember-finding algorithms currently available in the literature are either derived from N-FINDR or modified as its variants. However, when N-FINDR comes to practical implementation, four major obstacles need to be overcome. One is the number of endmembers which must be known a priori. Second, the use of random initial endmembers to initialize N-FINDR generally results in different sets of final found endmembers. Consequently, the results are inconsistent and not reproducible. Third, the requirement of dimensionality reduction (DR) produces different results because of using different DR techniques. Last, but not least is the exceedingly high computational cost caused by an exhaustive search for endmembers all together and simultaneously. This chapter develops a theory of SVA and re-visits its major player, N-FINDR, from a practical implementation point of view to cope with the above-mentioned issues. Three sequential versions of N-FINDR—SeQuential N-FINDR (SQ N-FINDR) discussed in Chang (2013a, b), Circular N-FINDR (CN-FINDR), and SuCcessive N-FINDR (SC N-FINDR) —along with their real time processing counterparts are presented (Chang 2013a). In particular, in order to address the issue caused by using random initial endmembers, two new versions of N-FINDR—Iterative N-FINDR (IN-FINDR) and Random N-FINDR (RN-FINDR)—are also developed. To expand the real time capability of these two algorithms further, a new concept of multiple-pass N-FINDR is also introduced to implement IN-FINDR and RN-FINDR in multiple passes so that in each pass N-FINDR can be carried out in real time.