7.1 Introduction

What makes endmember extraction unique in hyperspectral data exploitation is that an endmember represents the purity of a spectral signature that can be used to specify a spectral class. Interestingly, endmember extraction has not received much attention in multispectral data analysis in the last decades because low spectral and spatial resolutions of multispectral imaging sensors result in collected data sample vectors, which are more likely to be mixed instead of being pure. Consequently, the likelihood of finding endmembers is rather small. Under such circumstances, there is no reason to perform endmember extraction in multispectral data other than conducting mixed data analysis. However, with the use of high spatial and spectral resolution bands many subtle material substances that cannot be resolved by multispectral sensors can be now revealed by hyperspectral imaging sensors as pure signatures. Such substances generally provide vital information in image analysis. One of such substances is endmembers that may appear as mixed sample vectors in multispectral data but turn out to be pure signature vectors in hyperspectral data. Accordingly, finding endmembers has emerged as a fundamental and critical data preprocessing that offers basic understanding of hyperspectral data. On the other hand, finding these substances is a big challenge since their existence generally cannot be known by prior knowledge or visualized by inspection from their spatial presence.

Despite the fact that many EEAs have been developed, several critical issues arising in endmember extraction have been either overlooked or not appropriately addressed. The first and foremost is the issue in determining how many endmembers are assumed to be present in the image data, denoted by p. This prior knowledge is usually not available and cannot be known a priori. It must be obtained from the data itself by unsupervised means. Another issue that needs to be addressed is how to find them once the value of p is determined. Over the past few years, algorithms developed for endmember extraction have focused on the second issue while avoiding the first issue by simply selecting p on an empirical basis or assuming that it is provided a priori by inspection or given by prior knowledge. Interestingly, the first issue of determining p in endmember extraction has not been tackled until recently a series of publications (Chaudhry et al., 2006; Chang and Plaza, 2007; Plaza and Chang, 2007; Chang et al., 2006) were reported on using a new concept called virtual dimensionality (VD), which was first introduced in Chapter 17 in Chang (2003a) as the number of spectrally distinct signatures and later published in Chang and Du (2004). Due to the fact that endmembers are always spectrally distinct signatures, VD provides a reasonable and good estimate for p even if it may not be accurate. Nevertheless, experiments show that VD is probably as close as we can get when it comes to estimation of p.

Two main streams have been developed to design an EEA. One is derived from the principle of orthogonal projection (OP). The well-known PPI (Boardman, 1994) was the first to materialize this concept to find endmembers. It assumes that endmembers are more likely to be those whose orthogonal projections on a set of randomly generated unit vectors, referred to as skewers, are either minimal or maximal. As a result, these endmembers should appear at extreme points of skewers due to convexity. In other words, when data sample vectors are orthogonally projected on a set of skewers, the desired endmembers usually have either maximal or minimal projections occurring at either end of skewers. The second mainstream for designing EEAs is to use minimal /maximal simplex volume as a criterion to find a simplex that embraces all data samples with the minimal volume such as minimal-volume transform (MVT) developed by Craig (1994) or a simplex that is embedded in the data space with the maximal volume such as N-finder algorithm (N-FINDR) developed by Winter (1999).

In addition to OP and simplex volume there is a third criterion, least-squares error (LSE), that is also derived from the notion of convexity geometry and can also be used to design EEAs. It is derived from linear spectral mixture analysis (LSMA) and assumes that a set of endmembers used for spectral unmixing produces the smallest LSE compared to the same number of data sample vectors that are used for spectral unmixing. One representative is a fully constrained least-squares-based EEA (FCLS-EEA) derived from the FCLS developed by Heinz and Chang (2001). There are also several EEAs in this same category that are very close to FCLS-EEA, for example, iterative error analysis (IEA) developed by Neville et al. (1999) and iterated constrained endmember (ICE) by Berman et al. (2004).

A fourth criterion is to use sample spectral statistics for designing EEAs, which can be categorized into second-order statistics and high-order statistics (HOS). Of particular interest in second-order statistics are second-order component analysis (CA)-based criteria, which assume that a set of endmembers produce the least possible spectral correlation among the same number of data sample vectors. One of such EEAs is referred to as standardized PCA (SPCA)-EEA developed by Ji and Chang (2006), which uses SPCA to find a set of endmembers that yield the least statistical spectral correlation among a given number of data sample vectors. This type of criterion does not satisfy the abundance constraints as FCLS does. As for HOS-based EEAs an independent component analysis (ICA)-based EEA, recently developed by Wang and Chang (2006b), represents one example in this category.

Most recently, a fifth criterion has also received interest where it uses sample spatial/spectral correlation for searching endmembers, such as automated morphological endmember extraction (AMEE) proposed by Plaza et al. (2002) and the one developed by Roggea et al. (2007).

While an EEA performs endmember extraction dimensionality reduction (DR) is required to reduce data dimensionality to cope with the problem of the so-called curse of dimensionality. Therefore, in addition to the number of endmembers, p, to be generated, another relevant issue is the number of components or dimensions to be retained after DR, denoted by q. Interestingly, the concept of VD presented in Chapter 5 has been also shown to be effective to estimate p (Chang and Plaza, 2006; Plaza and Chang, 2006) as well as q (Wang and Chang, 2006).