Hyperspectral Data Processing: Algorithm Design and Analysis

13.6 Abundance-Constrained Least Squares FLDA (ACLS-FLDA)

It should be noted that the FVC-FLDA solution (13.12) is not abundance-constrained in the sense that there is no constraint imposed on the abundance vector α. Therefore, the FVC-FLDA solution does not guarantee that , that is, for all . In order to obtain an abundance-constrained FLDA, we first consider the following unconstrained LSE problem with a weighting matrix given by the within-class scatter matrix S_W that minimizes the LSE:

(13.27)

Using this square-root form, the LSE in (13.27) can be further expressed as

(13.28)

Now, if we let and , (13.28) can be further reduced to one that minimizes the following unconstrained LSE:

(13.29)

which is exactly the same least squares mixing problem considered in LSMA. By virtue of (13.29) ...