16.3 Correlation-Weighted Hyperspectral Measures for Target Discrimanition and Identification

The signature vector-based spectral measures described in Section 16.2 calculate the spectral similarity value between a pair of two signature vectors using only the spectral information provided by L bands within these two signature vectors. So, if a material signature vector is mixed by other substances, the spectral characteristics of the signature vector to be processed do not necessarily characterize the spectral properties of the material signature vector it represents. This often occurs in real applications when a material signature vector is either mixed with other signature vectors such as background signatures or embedded in a single signature vector as a subsample target. In both cases, using a signature vector-based spectral measure to measure material similarity is generally not effective. In order to resolve this dilemma, signature vector-based hyeprspectral measures are extended to correlation-weighted hyperspectral measures that can be categorized into two classes. One comprises of hyperspectral measures that introduce a priori sample spectral correlation into signature vector-based spectral measure so as to improve discrimination performance in spectral similarity. The other is made up of hyperspectral measures weighted by a posteriori sample spectral correlation to do what a priori sample spectral correlation does.

16.3.1 Hyperspectral Measures Weighted by A Priori Correlation

A hyperspectral measure weighted by a priori correlation assumes that there is known correlation available prior to material discrimination and identification. A good example of best utilization of such prior correlation is the OSP approach recently developed by Harsanyi and Chang (1994) that separates desired target signature vectors from undesired target signature vectors to achieve better hyperspectral image classification. This OSP concept can be used to design new hyperspectral measures as follows.

Assume that there are p target signature vectors which are known a priori and s_i is a signature vector of interest. We can define a matrix, U to be a matrix formed by all a priori known target signature vectors except s_i. By taking advantage of the following orthogonal projector defined by Harsanyi and Chang (1994) or (2.86) in Chapter 2:

(2.86)

we can de-correlate the signature vector s_i via orthogonal projection from all other known target signatures in U by projecting s_i to the space orthogonal to the space linearly spanned by undesired target signature vectors in U, and derive a family of several OSP-based hyperspectral measures. Depending on how to use , these OSP-based hyperspectral measures can be either used for discrimination or identification.

16.3.1.1 OSP-Based Hyperspectral Measures for Discrimination

Two separate problems, discrimination, and identification are considered. When discrimination is performed, it only needs to discriminate one signature vector from another. When identification is performed, we generally assume that there is a database, available for signature identification.

Two OSP-based hyperspectral measures can be designed for discrimination between two signature vectors s_i and s_j. One includes as a weighting matrix into ED in (16.2), called ED_OSP, defined by

(16.10)

where U is an undesired signature matrix formed by all known signatures excluding s_i and s_j.

The other is derived from the concept of OPD in (16.3), called OSP-based divergence, D_OSP, defined by

(16.11)

with the same U defined in (16.10).

If in (16.10) is assumed to be the identity matrix, then (16.10) is reduced to ED in (16.1). On the other hand, (16.11) is an extension of the OPD by including to eliminate the interference caused by undesired target signature vectors. However, it should be noted that unlike the ED_OSP, which is a nonnegative measure, the in (16.11) can take positive or negative values depending upon whether or not s_i and s_j point to the same direction. To avoid this problem, the absolute value is used in (16.11).

16.3.1.2 OSP-Based Hyperspectral Measures for Identification

In the previous subsection, the signature vectors s_i and s_j are assumed to be signature vectors in real data and (16.10) and (16.11) are used for signature discrimination. In this subsection, we assume that there is a database or spectral library available to be used to identify a signature vector in real data s_i. In this case, D_OSP can be modified to perform signature identification in two different ways depending on how to use the matched signature either from the database Δ, denoted by ID_OSP,Δ or itself, denoted by ID_OSP as follows:

(16.12)

(16.13)

where U_k is a matrix formed by all signature vectors in Δ excluding signature vector d_k.

Using (16.12) and (16.13) a signature vector s_i can be identified via a database Δ by the one in Δ that yields the ID_OSP,Δ or ID_OSP.

16.3.2 Hyperspectral Measures Weighted by A Posteriori Correlation

In many applications, obtaining a priori correlation information is very difficult, if not impossible. Therefore, it is highly desirable if we can generate necessary information directly from the image data without relying on prior knowledge. It has been demonstrated in Chapter 12 that the a priori correlation provided by can be approximated by the inverse of the sample spectral correlation matrix, R⁻¹, which can be used to account for a posteriori correlation. In light of this interpretation, the sample spectral correlation/covariance is used to derive new a posteriori correlation-weighted hyperspectral measures.

16.3.2.1 Covariance Matrix-Weighted Hyperspectral Measures

As noted in Section 12.5, the RX detector (RXD) defined by (12.76) was developed by Reed and Yu (1990) for anomaly detection. If the r and μ in (12.76) are replaced with s_i and s_j, respectively, then the RX anomaly detector becomes

(16.14)

Since (16.14) is a Mahalanobis distance (MD)-like measure, we can define a new hyperspectral MD-like measure via (16.14), called MD_RX as follows:

(16.15)

Two comments on (16.15) are noteworthy.

1. The covariance matrix-weighted hyperspectral measure, MD_RX defined by (16.15) is essentially derived from the widely used maximum likelihood classifier (MLC) by replacing s_i and s_j in (16.15) with a data sample vector r to be classifier and the mean of the jth class, μ_j, (16.15), respectively, as follows:

(16.16)

where MLC assigns to the data sample vector a class that yields the minimum of over all the p classes, .

2. Since the MLC described in (16.16) is supervised in the sense that the knowledge of , means of p classes, must be available a priori, the MLC in (16.16) can be further improved by including an undesired signature annihilator defined by (2.86) operating on the data sample vector in (16.16) in a similar manner that the OSP is defined by (2.85) as follows:

(16.17)

where U_j is a matrix made up of all signatures excluding s_j, that is, .

3. In (16.16) and (16.17) the knowledge of the number of classes, p and is assumed to be known a priori. If such knowledge is not available, there is impossible to perform classification. The class means μ_j in (16.16) must be replaced by the global mean μ. As a result, the MLC specified by (16.16) becomes the well-known anomaly detector, RXD defined by (12.76):

(12.76)

16.3.2.2 Correlation Matrix-Weighted Hyperspectral Measures

As also noted in Section 12.4.1.2, replacing the used in the OSP in (12.22) with the inverse of spectral correlation matrix, R⁻¹ yields the constrained energy minimization (CEM) in (12.52). Using the same token we can also define a new hyperspectral measure weighted by a posteriori correlation from the ED_OSP, called MD_CEM defined by

(16.18)

Interestingly, comparing (16.18) to (16.15) MD_RX and MD_CEM have the same identical structure except the a posteriori correlation in (16.15) provided by K⁻¹ as opposed to R⁻¹ used to account for a posteriori correlation in (16.18).

As special cases where the sample spectral covariance matrix K and the sample spectral correlation matrix R are whitened, that is, and , both (16.15) and (16.18) are reduced to ED

(16.19)

16.3.2.3 Covariance Matrix-Weighted Matched Filter Distance

According to RXD specified by (12.76), we can also derive a new hyperspectral measure, denoted by MFD_RX by replacing the both r's in (12.76) with s_j and s_i, respectively, as follows:

(16.20)

where μ is the sample mean of the data to be processed. Since s_j and s_i are not the same signature vectors, the values produced by in (16.20) are not necessarily non-negative. In this case, the absolute value is used in (16.20).

16.3.2.4 Correlation Matrix-Weighted Matched Filter Distance

As an alternative, we can also take advantage of CEM in another way to account for a posteriori correlation by designing a new hyperspectral measure, called CEM-matched filter distance, MFD_CEM defined by

(16.21)

Comparing (16.21) with (16.20), (16.20) can be also obtained by replacing s_i, R⁻¹ and s_j with s_i − μ, K⁻¹ and s_j − μ, respectively.

In analogy with (16.17), if K and R are whitened, that is, and , and μ = 0, (16.20) and (16.21) are reduced to SAM

(16.22)

where SAM(s_i,s_j) is defined by (16.2). Interestingly, the four proposed a posteriori correlation-based hyperspectral measures, MD_RX, MD_CEM, MFD_RX, and MFD_CEM turn out to be the same four target discrimination measures developed in Chang and Chiang (2002).

Recently, an adaptive coherence estimator (ACE) developed by Kraut et al. (1999) and

(16.23)

where d is the target signal to be detected and r is a data sample vector to be processed. By virtue of (16.23) we can further define a new discrimination measure, denoted by MFD_ACE by replacing d with s_i and r with s_j as

(16.24)

If we further use K^−1/2 as a whitening matrix as discussed in Section 6.3.1 of Chapter 6 (i.e., making K an identity matrix I by K=I) and define , , and , then (16.23) and (16.24) can be re-expressed respectively as follows

(16.25)

which is a square of an inner product of two unit vectors, and , and

(16.26)

which also becomes the SAM measure of two unit signature vectors and similar to (16.22) with signature vector lengths being unit length.

In addition to discrimination specified by (16.15) and (16.18) and (16.21), (16.22), (16.24) and (16.26) these equations can be also used to perform identification in the same way that (16.12) and (16.13) do by comparing a real signature vector against a database Δ. However, such identification is rather straightforward because the used a posteriori correlation is provided by either sample covariance matrix K or sample correlation matrix R compared to the a priori correlation which must be determined by the undesired signature matrix U used for annihilation.

Since all the spectral measures defined by (16.10) and (16.11), (16.15) and (16.18), and (16.20), (16.21), (16.24) involve the calculation of correlation either provided by or the inverse of the sample spectral correlation matrix, R⁻¹ or the inverse of the sample covariance matrix K⁻¹, they can be referred to as second-order hyperspectral measures. On the other hand, the sample spectral correlation is not included in any pure signature vector-based spectral similarity measure. So, SAM and SID can be considered as the first-order spectral measures. Figure 16.1 summarizes relationships among various first-order and second-order hyperspectral measures.

Figure 16.1 Block diagram of various hyperpsectral measures.