Hyperspectral Data Processing: Algorithm Design and Analysis

13.2 Feature Vector-Constrained FLSMA (FVC-FLSMA)

In this section, we extend FLDA discussed in Section 2.3.1.1 of Chapter 2 to an LSMA technique using Fisher's ratio as an unmixing criterion, referred to as FLSMA. One difficulty in doing so is that the FLDA-generated feature vectors are not endmembers to form a signature matrix M for LSMA Instead, they are discriminant vectors that are used to determine decision boundaries among classes. In particular, the number of such FLDA-generated discriminant feature vectors is one lower than the number of endmembers in M.

FLDA finds a set of feature vectors via Fisher's ratio or Rayleigh's quotient defined as

(2.35)

by solving a generalized eigenvalue problem specified by

(13.1)

where S_B and S_W are referred as between-class and within-class scatter matrices, respectively. Due to the fact that the rank of the between-class scatter matrix S_B is only p − 1, there are only p − 1 nonzero eigenvalues associated with (13.1). However, in order to implement LSMA, we need p feature vectors that can be used to form an endmember matrix M rather than discriminant vectors generated by (13.1). One way to mitigate this dilemma was proposed by Soltanian-Zadeh et al. (1996) and Du and Chang (2001a) who replaced Fisher's ratio with the ratio of interdistance ...