Recursive Hyperspectral Sample Processing of Geometric Simplex Growing Algorithm

Abstract

Simplex volumes (SVs) have been used in the literature as a criterion for finding endmembers. A main issue that arises in finding SVs is inverting a nonsquare matrix, which involves excessive computing time in calculating the matrix determinant. This type of SV calculation is referred to as a determinant-based SV (DSV) calculation (Chap. 2). Therefore, several preprocessing steps are suggested for DSV calculation in Chap. 2 to ease computational complexity, such as reducing data dimensionality, easing computational complexity by manipulating determinant calculation, narrowing search regions via Pixel Purity Index (PPI), or developing algorithms such as the simplex growing algorithm (SGA) to grow simplexes one after another sequentially instead of finding all endmembers together simultaneously. However, all these DSV calculation techniques remain stuck with certain inherent drawbacks encountered in finding SVs using a matrix determinant calculation, in addition to another issue: the calculated SV may not be a true SV, as pointed out in Chap. 2. To resolve this dilemma, Chap. 11 developed an orthogonal projection (OP)-based growing simplex volume analysis (GSVA) approach, called orthogonal projection SGA (OPSGA), which calculates a geometric SV (GSV) from a simplex geometry point of view instead of resorting to the matrix determinant. It transforms finding a new endmember that yields the maximal DSV by growing previously simplexes into finding an endmember with the maximal OP onto a hyperplane that is linearly spanned by simplexes specified by previously found endmembers. Accordingly, OPSGA can be considered a technique for calculating GSV by OP (GSV-OP). As an alternative to OPSGA, this chapter presents another GSVA approach to finding GSV, referred to as the geometric SGA (GSGA), developed by Chang, Li, and Song (IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing 99:1–13, 2016c), which converts GSV calculation into a product of the height and base of a simplex. In parallel to OPSGA, GSGA can also be considered a technique for calculating the GSV by the simplex height (GSV-SH). In other words, finding the maximal SV is equivalent to finding the maximal height of a simplex. As a result, when simplexes grow one vertex at a time, GSV can be calculated by multiplying only successive heights where the heights can be found by the Gram–Schmidt orthogonalization process (GSOP). This is because the bases of previously grown simplexes are already known and need not be recalculated. Furthermore, in analogy with recursive hyperspectral sample processing of OPSGA (RHSP-OPSGA), a recursive hyperspectral sample processing of GSGA (RHSP-GSGA) can also be derived and implemented as a Kalman filter-like algorithm so as to achieve significant savings of computing time on a timely basis as it is implemented in real time. Interestingly, GSGA/RHSP-GSGA produces sets of endmembers identical to those of OPSGA/RHSP-OPSGA, even though both calculate SVs differently. In other words, the set of endmembers produced by GSGA through finding the maximal heights turns out to be the same set of endmembers found by OPSGA through maximal OPs. This is because finding the maximal OP by orthogonal subspace projection (OSP) can be shown to be equivalent to finding the maximal height by GSOP. However, there is a key difference between GSGA and OPSGA, where GSGA works on simplex edges, as opposed to OPSGA, which deals with vertices directly. Consequently, the initial conditions required by GSGA and OPSGA are also different, where GSGA must start with two vertices as an initial simplex edge, whereas OPSGA can start with any single vertex as its initial endmember in the same way SGA does. Another difference between GSGA and OPSGA is the computational complexity and computer processing time resulting from the use of OSP and GSOP. Despite the fact that OP and SVs are different criteria used to design endmember finding algorithms (EFAs), several recent studies showed that they were actually closely related (Chen 2014; Chang Real time progressive hyperspectral image processing: endmember finding and anomaly detection. Springer, 2016; IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens. (99):1–27, 2016b). This fact is further confirmed by OPSGA developed in Chap. 11 and GSGA developed in this chapter.