Hyperspectral Data Processing: Algorithm Design and Analysis

13.5 Relationship Between FVC-FLSMA and LCDA

Recently, a constrained linear discriminant analysis approach, called linearly constrained discriminant analysis (LCDA), was developed by Du and Chang (2001a) where the within-class and between-class scatter matrices were replaced by intradistance and interdistance, respectively, and the class means were also aligned with orthogonal directions. As shown in Du and Chang (2001a), LCDA solution has the same equation specified by (13.21). So, LCDA is essentially FVC-FLSMA. Furthermore, the total scatter matrix S_T is the sum of within-class scatter matrix S_W and between-class scatter matrix S_B in (2.35) and is a constant matrix. The problem specified by (13.7) can be further shown to be equivalent to finding w_l that satisfies

(13.25)

The solution to (13.25) can be obtained by that turns out to be the same as (13.12). As shown in Chang (2003a), the total scatter matrix S_T is related to the data training sample covariance matrix K_t by where N is total number of training samples. Using this fact, the problem specified by (13.25) is also equivalent to the following problem:

(13.26)

The solution to (13.26) is also that is also similar to (13.12) ...