13.9 Conclusions

This chapter presents a new approach to LSMA, referred to as FLSMA, which directly extends the well-known FLDA to LSMA in two different ways. One is called FVC-FLSMA that constrains the Fisher ratio-generated feature vectors to mutual orthogonal directions. Another is called AC-FLSMA that imposes the sum-to-one and non-negativity constraints on abundance fractions in the least squares sense. It has been shown that FVC-FLSMA operates in the same way as LCMV does, with the only difference that the data correlation matrix used in LCMV is replaced by the within-class scatter matrix in FLSMA. Because the within-class scatter matrix is a more effective measure than the data correlation matrix in pattern classification, FVC-FLSMA performs better than LCMV in mixed pixel classification. Additionally, it also shows that LCDA is essentially the same as FVC-FLSMA. There are also three types of AC-FLSMA that can be derived in parallel in the same fashion as three types of constrained least squares methods are developed for LSMA in Chang (2003a). They are called ASCLS-FLSMA, ANCLS-FLSMA, and AFCLS-FLSMA with their respective counterparts in the abundance-constrained least squares LSMA, SCLS, non-negativity constrained least squares (NCLS), and FCLS. Since the mixed pixel classification is performed by AC-FLSMA using Fisher's ratio as a classification measure and least squares error as an abundance estimation criterion, AC-FLSMA also performs better than abundance-constrained least squares-based LSMA (ACLS-LSMA) and abundance-unconstrained FVC-FLSMA. Two concluding remarks are noteworthy. FLSMA presented in this chapter can be extended to unsupervised FLSMA (UFLSMA) in a similar manner as unsupervised knowledge is generated to characterize unknown background in Ji et al. (2004), where UFLSMA performed as well as FLSMA if the unsupervised generated training sample pool provided sufficient representative samples for each of classes. Another remark is that the performance of FLSMA relies heavily on the provided training samples. If the image is ill-represented by a given sample pool, FLSMA may perform poorly.