12.5 OSP Implemented Without Knowledge

As OSP was originally developed in Harsanyi and Chang (1994), it required complete endmember knowledge about the image data. Unfortunately, such requirement is seldom satisfied in reality. In order to resolve this issue, two approaches are developed previously. One is to generate desired complete knowledge directly from the image data in an unsupervised means so that the obtained unsupervised knowledge can be used as if it was provided a priori (Chang, 2003a, Chang et al., 2001) to make (d,U)-model applicable where the undesired target signature projector can be constructed from the generated U. Due to the fact that such generated unsupervised knowledge may not be accurate or reliable, an alternative approach is to implement OSP without appealing for unsupervised knowledge. One such approach is CEM described in Section 12.4.1 where only the desired target knowledge, d, is required. Instead of trying to find unknown signatures in U for annihilation, CEM suppresses all signatures other than the signature of interest. To accomplish that, CEM makes use of the inverse of the sample correlation matrix, to approximate the complete knowledge provided by in OSP. As a result, OSP and CEM can be related by (12.72)–(12.74) using TCIMF to bridge the gap. Another approach is to implement OSP without prior knowledge. As noted, (d,U)-model requires the knowledge of the desired target signature d and the undesired target signature matrix U. When both the d and the U are not available, OSP must be implemented whatever it can obtain directly from the data. According to (12.72) and (12.73), when the knowledge about the U is not available, the sample spectral correlation matrix can be used to approximate . Moreover, if the knowledge of the d is further not provided, the only available information that can be used for OSP is the image pixel vector r. In this case, the matched signatures used in OSP must be replaced by the d. So, substituting r and for d and in (12.9), respectively, results in a new filter that can be used for anomaly detection. Such a filter is referred to as OSP anomaly detector (OSPAD), denoted by δ^OSPAD(r) and given by

(12.75)

It should be noted that if we replace the d in CEM in (12.52) with the image pixel r, the resulting form would be the constant 1 for all the image pixel r, that is, . This is because CEM performs as an estimator rather than a detector as OSP does. Since no desired signature d needs to be estimated, the quantity of that is included in CEM to account for the estimation accuracy varies with the image pixel r. Therefore, CEM cannot be used to derive for anomaly detection as we did for OSP in (12.75). That also explains why OSP has better generalization properties than CEM and CEM can be considered as partial-knowledge version of OSP.

Interestingly, if we replace r and in δ^OSPAD(r) with r − μ and where μ and K are the sample mean and the sample covariance matrix, the resulting filter δ^OSPAD(r) turns out to be the well-known anomaly detector, referred to as RX detector (RXD), δ^RXD(r) developed by Reed and Yu (1990) and also known as Mahalanobis distance (Fukunaga, 1990):

(12.76)

If we once again replace the matched signature r^T in (12.75) and (12.76) with the L-dimensional unity vector , δ_OSPAD(r) becomes so-called low probability detection (LPD), δ^LPD(r) in (Harsanyi, 1993; Wang and Chang, 2004) given by

(12.77)

that was developed in Harsanyi's dissertation (Harsanyi, 1993) and uniform target detector δ^UTD(r):

(12.78)

that was derived in Chang (2003a). More details about δ^LPD(r) and δ^UTD(r) can be found in Chang (2003a) and Wang and Chang (2004).

As discussed in Section 12.4, we may sometimes have partial knowledge about target signatures that are not wanted, such as background. In this case, we may think that removing this knowledge prior to anomaly detection could improve anomaly detectability. As will be explained later, this is not necessarily true.

Following a similar treatment in Section 12.4, suppose that the knowledge about U is provided. We can implement δ^OSPAD(r) in conjunction with the undesired signature annihilator, in the same way that it is implemented in δ^OSP(r) to remove the undesired target signatures before anomaly detection. The resulting detector is called -OSP anomaly detector (-OSPAD), defined by

(12.79)

Similarly, RXD can be also implemented in conjunction with , called -RXD and denoted by as follows:

(12.80)

Surprisingly, according to the conducted experiments, δ^OSPAD(r) and δ^RXD(r) will be shown to perform very closely regardless of whether or not is included in detection. This is because anomaly detectors are generally designed to extract pixels whose signatures spectrally distinct from their surroundings rather than suppress signatures as OSP does. Another reason is that since can be viewed as an approximation of , an additional inclusion of does not improve the performance of δ^OSPAD(r) and δ^RXD(r), both of which already perform a similar task to that is carried out by in (12.79)–(12.80).

Example 12.5

(Anomaly Detection)

In this example, several experiments were conducted to evaluate δ^OSPAD(r) specified by (12.75), δ^RXD(r) specified by (12.76) and δ^LPD(r) specified by (12.77). The same 401 simulated pixel vectors with added SNR 30:1 white Gaussian noise in Example 12.1 were used to detect the creosote leaves as an anomalous target. Figure 12.11(a)–(d) shows the detection results of δ^OSPAD(r), δ^RXD(r), and δ^LPD(r) when the pixels of the creosote leaves with abundance 10% were expanded from one pixel (pixel number 200), three pixels (pixel numbers 199, 200, 201), five pixels (pixel numbers, 198–202) to 11 pixels (pixel numbers 195–205). As we can see from Figure 12.11(a)–(d), the performance of anomaly detection in creosote leaves was deteriorated as the number of creosote leaves was increased.

As a matter of fact, according to our experiments, in order for the creosote leaves to qualify as an anomalous target, the number of pixels should not exceed 3. Figure 12.11 also shows that δ^LPD(r) could not be used to detect anomalies, but only for background detection. Table 12.5 also tabulates their respective detected abundance fractions where δ^OSPAD(r) performed slightly better than δ^RXD(r).

Figures 12.12 and 12.13 also show how SNR (10:1, 20:1, and 30:1) and abundance fractions (10%, 20%, and 30%) affected the performance of anomaly detection for δ^OSPAD(r) and δ^RXD(r), respectively.

As shown in Figures 12.12 and 12.13, the higher the SNR, the better the anomaly detection, and the more the abundance fractions of anomaly, the better the anomaly detection. Additionally, all these three figures (i.e., Figs. 12.11–12.13) demonstrated that both δ^OSPAD(r) and δ^RXD(r) performed comparably in terms of detected abundance fractions.

The next experiment was designed to see how many anomalies could be detected as distinct targets by δ^OSPAD(r) and δ^RXD(r) if the same 401 simulated pixels with added SNR 30:1 white Gaussian noise in Example 12.1 were also used. Figure 12.14 shows the detection results of δ^OSPAD(r) and δ^RXD(r) where two of three target signatures, blackbrush, creosote leaves, and sagebrush were selected with same abundance 10% at pixel number 100 and pixel number 300. Table 12.6 tabulates the detected abundance fractions of δ^OSPAD(r) and δ^RXD(r) in Figure 12.14 for three target signatures, blackbrush, creosote leaves, and sagebrush with same abundance 10% at pixel number 100 and pixel number 300.

As we can see from Table 12.6, the results were not as good as Figure 12.11(a), but the pixel number 300 was always detected. Interestingly, if the three target signatures, blackbrush, creosote leaves, and sagebrush were present at pixel numbers 100, 200, and 300 with same abundance 10%, Figure 12.15 shows that both δ^OSPAD(r) and δ^RXD(r) failed to detect these three target signatures but only the one at pixel number 200.

Table 12.7 tabulates the detection abundance fractions of δ^OSPAD(r) and δ^RXD(r) in Figure 12.15 for the three target signatures where the detected amounts of both creosote leaves and sagebrush were close.

However, from a visual inspection of Figure 12.15, only creosote leaves could be detected. This experiment provided evidence that an anomaly could not be simply determined by the detected amount of its abundance fraction. Rather, anomaly detection must be performed by comparing the detected abundance fraction of an anomaly against the abundance fractions detected in its surrounding neighborhood. The investigation of this issue was discussed in Hsueh and Chang (2004) and Hsueh (2004) and will be investigated in Chang (2013). Figure 12.15 and Table 12.7 demonstrate that anomaly detection could not be blindly implemented without some extra care.

Figure 12.11 Detection results of δ^OSPAD(r), δ^RXD(r), δ^LPD(r) with SNR 30:1 and abundance fraction 10%.

Table 12.5 Abundance fractions detected by δ^OSPAD(r) and δ^RXD(r) in Figure 12.11

Figure 12.12 Detection results of creosote leaves at pixel number 200 by δ^OSPAD(r) and δ^RXD(r) with various SNRs.

Figure 12.13 Detection results of creosote leaves at pixel number 200 by δ^OSPAD(r) and δ^RXD(r) with various abundance fractions.

Figure 12.14 Detection results of δ^OSPAD(r) and δ^RXD(r) with various signatures at different pixel numbers.

Table 12.6 Abundance fractions of two signatures detected by δ^OSPAD(r) and δ^RXD(r)

Figure 12.15 Detection results of blackbrush at pixel number 100, creosote leaves at pixel number 200, and sagebrush at pixel number 300 by δ^OSPAD(r) and δ^RXD(r).

Table 12.7 Abundance fractions of three signatures detected by δ^OSPAD(r) and δ^RXD(r) in Figure 12.15