Recursive Hyperspectral Sample Processing of Orthogonal Projection-Based Simplex Growing Algorithm

Abstract

The simplex growing algorithm (SGA) developed by Chang et al. (A growing method for simplex-based endmember extraction algorithms. IEEE Transactions on Geoscience and Remote Sensing 44(10): 2804–2819, 2006b) has been used for finding endmembers and was studied in Chap. 10 of Chang (Real time progressive hyperspectral image processing: endmember finding and anomaly detection, Springer, New York, 2016). It can be considered a sequential version of the well-known N-finder endmember finding algorithm (N-FINDR) developed by Winter (Proceedings of 13th international conference on applied geologic remote sensing, Vancouver, BC, Canada, pp. 337–344, 1999a; Image spectrometry V, Proceedings of SPIE 3753, pp. 266–277, 1999b) to find endmembers one after another by growing simplexes one vertex at a time. However, one of the major hurdles for N-FINDR and SGA is the calculation of a simplex volume (SV), as discussed in Chap. 2, which poses a great challenge in designing any algorithms using a SV to find endmembers. This chapter develops an orthogonal projection (OP)-based approach to SGA, called OPSGA, which essentially resolves this computational issue. The idea is based on a geometric SV (GSV) from structures of simplexes. If we consider a j-vertex simplex S j specified by j previously found endmembers as a base and the next endmember mj+1 to be found as a new vertex to be added to S j to form a new (j + 1)-vertex simplex, Sj+1, then calculating the GSV of this new (j + 1)-vertex simplex, Sj+1, is equivalent to multiplying the GSV of S j , which is considered a base with the OP on the base S j from mj+1, which is considered its height. As a result, finding mj+1 to yield a Sj+1 with the maximal GSV is equivalent to finding mj+1 with the maximal OP on S j . On this interpretation, OPSGA converts the issue of calculating a determinant-based SV (DSV) to finding OP without actually computing matrix determinants. Accordingly, OPSGA can be considered a technique for calculating a GSV by OP (GSV-OP). To further reduce the computational complexity, OPSGA is also extended to a recursive Kalman filtering-like recursive hyperspectral sample processing of OPSGA (RHSP-OPSGA), which has several advantages and benefits in terms of computational savings and hardware implementation over N-FINDR and SGA.