Hyperspectral Data Processing: Algorithm Design and Analysis

15.2 Kernel-Based LSMA (KLSMA)

This section revisits and extends the three least squares-based techniques, OSP/LSOSP, NCLS, and FCLS, that have shown success in hyperspectral unmixing (Chang, 2003a) to their corresponding kernel-based counterparts, each of which represents three categories of techniques implementing LSMA. One is the category of abundance-unconstrained methods that include the well-known Gaussian maximum likelihood (GML) estimation and OSP, both of which have been shown essentially the same technique in Chapter 12. Another is the category of partially abundance-constrained methods, of which the NCLS method developed for constrained signal detection by Chang and Heinz (2000) is a representative. A third category of fully abundance-constrained methods among which the fully constrained least squares (FCLS) method developed by Heinz and Chang (2001) is the most widely used method for this purpose. So, when it comes to extend LSMA to kernel-based LSMA (KLSMA), it is natural to consider these three methods for kernel extension. The works in Kwon and Nasrabadi (2005b), Broadwater et al. (2007), and Liu and Chang (2009) are early attempts to accomplish this goal.

In what follows, we follow the work in Liu et al. (2012) to develop a unified kernel theory for extending LSMA to KLSMA by first developing the kernel-based LSOSP (KLSOSP) that is derived directly from the structure of the OSP. This KLSOSP is then used to derive a KNCLS that is in turn to be used to further derive ...