13.1 Introduction

LSMA has been widely used in subpixel analysis and mixed pixel classification. Many algorithms have been developed for LSMA such as LS-LSMA, SNR-based OSP, and Mahalanobis distance-based Gaussian maximum likelihood estimation (GMLE). However, according to Juang and Katagiri (1992), LSE is not necessarily the best criterion to measure classification error and neither is SNR. Instead, FLDA is one of the major techniques widely used in pattern classification (Duda and Hart, 1973). It makes use of the so-called Fisher's ratio also known as Rayleigh quotient, which is the ratio of between-class scatter matrix to within-class scatter matrix, as a criterion to generate a set of feature vectors that constitute a feature space for better classification. A similar approach to FLDA was developed by Soltanian-Zadeh et al. (1996) who replaced Fisher's ratio with the ratio of interdistance to intradistance and aligned the generated feature vectors along mutual orthogonal directions. This approach has been shown to be successful in magnetic resonance (MR) image classification. Most recently, Soltanian-Zadeh et al.'s approach was further extended to linearly constrained discriminant analysis (LCDA) by Du and Chang for hyperspectral image classification to improve LSMA classification (Du and Chang, 2001a; Chang 2003b). Technically speaking, the feature vectors obtained by Soltanian-Zadeh et al. (1996) as well as those by Du and Chang (2001a) are not actually Fisher's feature vectors because Soltanian-Zadeh et al.'s interdistance to intradistance ratio is not Fisher's ratio.

FLDA is a traditional class membership-labeling technique. When it is used as an LSMA technique, it is implemented in a simple and straightforward manner on a pure pixel basis. Consequently, FLDA produces class maps different from fractional abundance maps generated by LSMA-based techniques that are gray-scale images. This chapter revisits FLDA and presents a new approach, to be called FLSMA. It directly extends pure pixel-based FLDA to a mixed pixel-based technique so as to perform subpixel detection and mixed pixel classification. It constrains FLDA in a way that the Fisher ratio-generated feature vectors are aligned along mutual orthogonal directions in the same way that both Soltanian-Zadeh et al.'s approach and LCDA align the feature vectors generated by the interdistance to intradistance ratio. Analogous to other mixed pixel-based techniques, FLSMA also generates fractional abundance maps with gray scales representing abundance fractions of classes to be classified. As discussed in Chang (2002b) and Chang (2003a), there are two types of constrained approaches, called TSCMPC and TACMPC, developed for LSMA. The TSCMPC constrains target signatures of interest along desired directions to derive a linearly constrained minimum variance (LCMV) approach (Chang, 2002b) that includes constrained energy minimization (CEM) as its special case, whereas TACMPC implements abundance sum-to-one constraint (ASC) and abundance non-negativity constraint (ANC) to derive three least squares abundance-constrained LSMA approaches: sum-to-one constrained least squares (SCLS), non-negativity constrained least squares (NCLS), and fully constrained least squares (FCLS). Interestingly, approaches similar to both TSCMPC and TACMPC can also be developed for FLSMA.

One approach is called FVC-FLSMA derived from TSMPC. It replaces the sample correlation matrix used in LCMV with the within-class scatter matrix. In particular, it can be shown that the classifiers derived by both FVC-FLSMA and LCDA are essentially the same. In addition, because FVC-FLSMA uses Fisher's ratio as a classification criterion as opposed to LCMV that uses LSE as a classification measure, FVC-FLSMA generally performs better than LCMV in classification as expected.

The other approach is abundance constrained least squares FLDA (ACLS-FLDA) derived from TACMPC. It is referred to as AC-FLSMA and implements Fisher's ratio to carry out mixed pixel classification while using the least squares error to perform abundance fraction estimation. Accordingly, in analogy with abundance-constrained LSMA (AC-LSMA), there are also three types of AC-FLSMA that can further be derived: abundance sum-to-one constrained least squares FLSMA (ASCLS-FLSMA), abundance non-negativity constrained least squares FLSMA (ANCLS-FLSMA), and abundance fully constrained least squares FLSMA (AFCLS-FLSMA). As will be demonstrated, AC-FLSMA generally performs better than its counterpart, AC-LSMA, to produce more accurate abundance fractions. It should be noted that the AC-FCLS is the same as FCLS developed in Heinz and Chang (2001) used in other chapters. The inclusion of “AC” in front of FCLS and FLSMA is simply to emphasize the constraints imposed on abundance fractions to distinguish from the FVC-FLSMA which imposes constraints on the feature vectors.