Recursive Geometric Simplex Growing Analysis for Finding Endmembers in Hyperspectral Imagery

Abstract

Simplex growing algorithm (SGA) is an endmember finding algorithm which searches for endmembers one after another by growing simplexes one vertex at a time via maximizing simplex volume (SV). Unfortunately, several issues arise in calculating SV. One is the use of dimensionality reduction (DR) because the dimensionality of a simplex is usually much smaller than data dimensionality. Second, calculating SV requires calculating the determinant of an ill-ranked matrix in which case singular value decomposition (SVD) is generally required to perform DR. Both approaches generally do not produce true SV. Finally, the computing time becomes excessive and numerically instable as the number of endmembers grows. This paper develops a new theory, called geometric simplex growing analysis (GSGA), to resolve these issues. It can be considered as an alternative to SGA from a rather different point of view. More specifically, GSGA looks into the geometric structures of a simplex whose volume can be actually calculated by multiplication of its base and height. As a result, it converts calculating maximal SV to finding maximal orthogonal projection as its maximal height becomes perpendicular to its base. To facilitate GSGA in practical applications, GSGA is further used to extend SGA to recursive geometric simplex growing algorithm (RGSGA) which allows GSGA to be implemented recursively in a similar manner that a Kalman filter does. Consequently, RGSGA can be very easily implemented with significant saving of computing time. Best of all, RGSGA is also shown to be most efficient and effective among all SGA-based variants.