17.7 Conclusions

The performance of LSMA is completely determined by the number of signatures, p and signatures used to form a linear mixing model to unmix data sample vectors. Unfortunately, in real applications none of these two pieces of information is known accurately in advance. So, a key to success in LSMA is to find an appropriate signature matrix M to form a linear mixing model in (2.75) where r is an image pixel vector and n is a model correction term. In supervised LSMA (SLSMA), this matrix M is assumed to be known a priori. However, when it comes to ULSMA the knowledge of the signature matrix M is not available and must be obtained directly from the data. The two unsupervised approaches, LS-UVSFA in Section 17.2 and CA-UVSFA in Section 17.3, provide a means of finding such an unsupervised signature matrix M for ULSMA. Since the signatures found in the unsupervised signature matrix, M, are real data sample vectors and may not be pure as true endmembers assumed in SLSMA, they are called virtual signatures (VSs) for their distinction from true endmembers. So, the signature matrix M formed by VSs is also referred to as VS matrix. The performance of ULSMA is completely determined by two factors, the number of VSs, p, and the VSs used to form a linear mixing model, both of which are assumed to be known in SLSMA. Secondly, ULSMA generally outperforms SLSMA in cases that many unknown material signatures that cannot be identified by visual inspection or prior knowledge can now be found by LS-UVSFA/CA-UVSFA. Thirdly, the signatures used to form a linear mixing model for ULSMA are real data sample vectors. Accordingly, they are generally not real endmembers and can be in any form such as subsample vectors and mixed sample vectors. This is the reason that such data sample vectors are referred to as VSs for ULSMA to distinguish from true endmembers used for SLSMA. Last but not least, the performance of LSMA is generally evaluated by unmixed results of signatures that are used to form a linear mixing model either qualitatively or quantitatively. To this end, over the past years the signatures used for LSMA to perform unmixing as SLSMA are usually those in which users are interested. When it is extended to ULSMA the same logic is also applied where endmembers are assumed to be the signatures of interest. However, as demonstrated in the experiments conducted in Section 17.6 this may not be realistic or applicable for real-world applications where mixed BKG signatures that are generally not signatures of interest are also crucial for LSMA to perform background suppression. The ability of LSMA in background suppression has been often overlooked and it can be as important as signatures of interest such as endmembers. Unfortunately, finding appropriate BKG signatures is not a trivial matter. The LS-UVSFA and CA-UVSFA presented in this chapter are primarily designed for this purpose, both of which find a desired set of VSs that includes target and BKG signatures. Whether these VSs are pure or mixed is not of major concern for ULSMA.

As a final note, despite the fact that many efforts have been made to determine the number of endmembers (Kosaka et al., 2005; Nascimento and Dias, 2005; Eches et al., 2010; Cawse et al., 2010; Zare and Gader, 2007, Broadwater and Banerjee, 2009), there is no specific technique developed for determining and finding the number of true endmembers, both of which are actually two separate issues. The VD developed in Chapter 5 is particularly developed for the purpose of determining the number of spectrally distinct signatures not endmembers. Although, VD generally overestimates the number of endmembers as demonstrated in Chang et al. (2010) VD was shown indeed a good estimate for the number of signatures used to form a linear mixing model used by LSMA. For this reason VD has been used across the board to determine the number of signatures required by ULSMA.