16.2 Signature Vector-Based Hyperspectral Measures for Target Discrimanition and Identification

Spectral characteristics provide important and crucial features in material identification, discrimination, detection, and classification. Many spectral similarity measures have been developed and can be used for this purpose such as SAM (Schwengerdt, 1997), SID (Chang, 2000, 2003a, Chapter 2), Euclidean distance (ED), and many others (Chang, 2003a). When there is no prior target class information available, these measures are performed on a single signature vector basis to measure spectral variability between two signature vectors, in which case they are generally used for signature discrimination and identification, but not used for classification. Furthermore, they are effective only if the spectral signature vectors to be compared are true signatures of the materials that they really represent the signature vectors. However, this idealistic case is generally not true in many real applications where many factors may contaminate and corrupt spectral signature vectors to be identified. One scenario is mixed signature discrimination and identification where a spectral signature vector is mixed with a number of signature vectors resident in the signature vector. Another is subsample target discrimination and identification where the target to be identified is embedded in a single signature vector and its spectral signature vector is clearly mixed with other signature vectors that are also present in the signature vector. In either case, single signature vector-based spectral measures such as SAM may not work effectively and sometimes may even identify wrong targets. This section investigates the issue of discrimination and identification for mixed signature vectors and subsample targets and provides evidence that such examples indeed occur in real hyperspectral imagery where some commonly used spectral measures fail in discrimination and identification of mixed signature vectors and subsample targets. To remedy this problem it further develops new spectral measures for mixed signature vectors and subsample targets in identification and discrimination. Unlike single signature vector-based spectral measures described above, these measures take advantage of the sample spectral correlation to account for spectral variability of mixed signature vectors and subsample targets within the signature vectors. In particular, when a signature vector is an image pixel vector in an image data where mixed pixel vectors or subpixel targets may spatially and spectrally correlated with their neighboring pixel vectors, the inclusion of such sample spectral correlation in a spectral measure offers additional spectral information that any single signature vector-based spectral measure cannot provide. Two types of sample spectral correlation-based spectral measures are of interest. They are previously developed as target discrimination measures for anomaly classification in hyperspectral imagery and can be considered as candidates to be studied. One is Mahalanobis distance (MD)-based and the other is matched filter-based hyperspectral measures, both of which include the sample spectral covariance/correlation matrix to capture spectral mixed signature vectors and subsample target signature vectors more effectively. Due to the use of the sample spectral covariance/correlation matrix, these measures can be considered as second-order spectral measures as opposed to single signature vector-based spectral measures that can be thought of as first-order spectral measures which do not include the sample covariance/correlation matrix to account for sample correlation.

In what follows, we describe four signature vector-based spectral measures, ED SAM, OPD, and SID, all of which are discussed in Chang (2003a) and closely related to each other in one way or another. For example, when the angle between two signatures is small, ED and SAM are essentially the same measures as shown in Chang (2003a). On the other hand, the pair of SAM and OPD can be related by orthogonal projection. Additionally, while the OPD measures the divergence of one signature vector projection onto the other, SID can be considered as a stochastic version of OPD with orthogonal projection replaced with information divergence (Cover and Thomas, 1991). The definitions of these four measures are summarized as follows (Chang, 2003a).

Assume that and are two spectral signature vectors where L is the total number of spectral bands.

16.2.1 Euclidean Distance

The ED is one of most widely used metrics in mathematics to measure the distance between two spectral signatures, s_i and s_j, given by

(16.1)

16.2.2 Spectral Angle Mapper

The SAM measures spectral similarity by finding the angle between the spectral signatures s_i and s_j

(16.2)

where , and .

16.2.3 Orthogonal Projection Divergence

The concept of the OPD is first defined in Chang (2003a) which is originated from the orthogonal subspace projection (OSP) developed in Harsanyi and Chang (1994). It finds the residuals of orthogonal projections resulting from two pixel vectors, s_i and s_j given by

(16.3)

where for and I is the identity matrix.

16.2.4 Spectral Information Divergence

Let and be the two probability mass functions generated by s_i and s_j, respectively, with and . So, the self-information provided by s_i and s_j for band l is defined by

(16.4)

(16.5)

respectively. By virtue of (16.4) and (16.5) we can define the discrepancy of the self-information of band image B_l provided by s_j relative to the self-information of band image B_l provided by s_i, denoted by as

(16.6)

Averaging in (16.6) over all the band images results in

(16.7)

where is the average discrepancy in the self-information of s_j relative to the self- information of s_i. In context of information theory, in (16.7) is called the relative entropy of s_j with respect to s_i, which is also known as Kullback–Leibler information measure, directed divergence or cross entropy (Cover and Thomas, 1991). Similarly, we can also define the average discrepancy in the self-information of s_i relative to the self- information of s_j by

(16.8)

Summing (16.7) and (16.8) yields spectral information divergence (SID) defined by

(16.9)

which can be used to measure the discrepancy between two pixel vectors s_i and s_j in terms of their corresponding probability mass functions, p and q. It should be noted that while is symmetric, is not. This is because and . As a final remark on SID, it is worth noting that a recent work (Du et al., 2004) suggested a means of mixing SID and SAM as SID-SAM mixed measures specified by and . Their results have shown better discriminability in spectral similarity. Those who are interested in these measures can find more details in Du et al. (2004).