2.1.

Dual-Window RX Detector

Consider a three-dimensional hyperspectral cube with resized samples $\mathbf{X}={\{{\mathbf{x}}_i\}}_{i=1}^n$ in ${\mathbb{R}}^d$ ($d$ is the number of spectral bands) and $n$ is the total number of samples. For each pixel $\mathbf{y}$ (of size $d\times 1$), surrounding data are collected inside the outer window (of size $w_{\mathrm{out}}\times w_{\mathrm{out}}$) while outside the inner window (of size $w_{\mathrm{in}}\times w_{\mathrm{in}}$), centered at the pixel $\mathbf{y}$. The selected data are resized into a two-dimensional matrix ${\mathbf{X}}_s=\{{\mathbf{x}}_i{\}}_{i=1}^s$ ($s$ is the number of chosen samples, $s=w_{\mathrm{out}}\times w_{\mathrm{out}}-w_{\mathrm{in}}\times w_{\mathrm{in}}$). Hence, the matrix ${\mathbf{X}}_s$ (of size $d\times s$) is obtained for every pixel $\mathbf{y}$ on its own local window.

A single pixel form of the RX algorithm is often approximated by the following equation:^6,13,28

In Ref. 21, KRX has been investigated via projecting data into a high-dimensional feature space in which the data become more separable. In the kernel-induced feature space, the mapping function $\mathrm{\Phi }$ maps the pixel $\mathbf{y}\to \mathrm{\Phi }(\mathbf{y})\in {\mathbb{R}}^{d^{\prime }\times 1}$ ($d^{\prime }\gg d$ is the dimension of the kernel feature space) and $\mathbf{\Phi }=\mathrm{\Phi }({\mathbf{x}}_1)$, $\mathrm{\Phi }({\mathbf{x}}_2),\cdots ,\mathrm{\Phi }({\mathbf{x}}_s)\in {\mathbb{R}}^{d^{\prime }\times s}$. The corresponding output of KRX is represented as

2.2.

Proposed Decision-Fusion Detector

Adaptive anomaly detection is used to detect anomalies whose spectral signatures are different from the local background; depending upon the definition of local, the resulting anomaly detection performance will be different. In the setting of dual-window implementation, the pixels between the inner and outer windows are considered as local background; of course, the change of dual-window sizes will end up with different anomaly detection performances. Note that the purpose of the inner window is to prevent the background from being contaminated by the central pixel when it is a target; thus, the size of the inner window should be slightly larger than the target size; under a complete unknown environment, this information is unknown as well. Inspired by multiclassifier fusion,²⁹ such difficulty in appropriate window setting may be mitigated by detector fusion.

In the proposed decision-fusion approach, detection outputs for a pixel $\mathbf{y}$ using $m$ detectors with $m$ different windows are expressed as $\{r_i(\mathbf{y}),i=\mathrm{1,2},\cdots ,m\}$, where $r_i(\mathbf{y})$ represents the $i$’th output using the $i$’th pair $(w_{\mathrm{in}},w_{\mathrm{out}})$ via Eq. (1) or Eq. (2). The outputs of an entire image are normalized to have a range of [0, 1] and compared with a prescribed threshold $\eta$. A pixel will be claimed to an anomaly if the output is larger than $\eta$. The number of times that the pixel $\mathbf{y}$ is assigned to be an anomaly will be counted:

The final class-label decision follows a voting process expressed as

In MW-RX, for a pixel $\mathbf{y}$, after obtaining RX outputs with multiple dual windows, the maximum value will be taken²⁷

3.1.

Hyperspectral Data

The first experimental data we employed are the hyperspectral digital imagery collection experiment (HYDICE) image³⁰ This scene consists of $80\times 100\text{pixels}$ for an urban area. The spatial resolution is approximately 1 m. 175 bands of spectral coverage 0.4 to $2.5\mu \mathrm{m}$ remain after removal of water vapor absorption bands. There are approximately 21 anomalous pixels, representing cars and roof. The scene and the ground-truth map of anomalies are shown in Fig. 1.

Fig. 1

(a) Pseudocolor image of hyperspectral digital imagery collection experiment (HYDICE) urban scene, (b) ground-truth map of 21 anomalous pixels.

The second dataset was acquired by the HyMap airborne hyperspectral imaging sensor,³¹ which provides 126 spectral bands spanning the wavelength interval 0.4 to $2.5\mu \mathrm{m}$. The image dataset, covering one area of Cooke City, Montana, was collected on July 4, 2006, with the spatial size $200\times 800\text{pixels}$. Each pixel has approximately 3 m of ground resolution. Seven types of targets, including four fabric panel targets, and three vehicle targets, were deployed in the region of interest. In our experiment, we crop a subimage of size $100\times 300\text{pixels}$, including all these targets (anomalies) as depicted in Fig. 2. Figure 3 further illustrates the spectral signatures of the seven targets, which are significantly different from the mean of background.

Fig. 2

Pseudocolor image of HyMap image; and ground-truth map of total seven types of targets (F1: 3 m red cotton target; F2: 3 m yellow nylon target; F3: 1 and 2 m blue cotton target; F4: 1 and 2 m red nylon target; V1: 1993 Chevy Blazer; V2: 1997 Toyota T100; V3: 1985 Subaru GL Wagon).

Fig. 3

Spectral signatures of the seven types of targets and the mean of the background in the HyMap data.

3.2.

Detection Performance

We investigate the effectiveness of the proposed RX-Fusion and KRX-Fusion. For KRX, a commonly used Gaussian radial basis function kernel is adopted.²¹ In this work, the kernel parameter is set to 50 for these two data according to our experimental study. As for windows $(w_{\text{in}},w_{\text{out}})$, since the size of anomalies is usually small, we set the general choices as listed in Table 1, which includes 12 pairs in total. Figure 4 first illustrates the performance with varying sizes of windows $(w_{\text{in}},w_{\text{out}})$ using the HYDICE urban data. The receiver-operating-characteristic (ROC) curve is employed to quantitatively evaluate the detection ability. The results clearly show that the performance of the detector changes significantly with different $(w_{\text{in}},w_{\text{out}})$ and indicate that it deteriorates if an inappropriate window is chosen, which motivates us to design a window-independent detector. The proposed RX-Fusion and KRX-Fusion, based on the decision-fusion strategy, simultaneously adopt multiple windows and produce the final decision map via a voting process.

Table 1

General choices for sizes of windows (win,wout).

Fig. 4

Detection performance with varying sizes of windows $(w_{\text{in}},w_{\text{out}})$ for the dual-window-based Reed-Xiaoli (RX) using the HYDICE urban data.

Figures 5 to 6 illustrate the area under ROC (AUC) performance of RX, KRX, RX-Fusion, and KRX-Fusion. In Fig. 5(a), the best $(w_{\text{in}},w_{\text{out}})$ for both RX and KRX is (7, 9); moreover, we observe that the AUC performance of RX and KRX is sensitive to the choice of sizes of windows, which is consistent with the performance in Fig. 4. In Fig. 5(b), the optimal $t$ values (out of 12) for RX-Fusion and KRX-Fusion are 5 and 4, respectively. Note that when $t=6$, the performance of RX-Fusion and KRX-Fusion is very similar to the best ones, which are also close to the case with the best window settings as shown in Fig. 5(b). In Fig. 6, for the HyMap data, the best $(w_{\text{in}},w_{\text{out}})$ for both RX and KRX is (7, 11), and the best $t$ values for RX-Fusion and KRX-Fusion are 9 and 8, respectively. In Fig. 6(b), if $t=6$, the performance of both RX-Fusion and KRX-Fusion is slightly worse, but much better than the cases with inappropriate window sizes as shown in Fig. 6(a).

Fig. 5

(a) AUC for RX and KRX with different windows; (b) AUC for RX-Fusion and KRX-Fusion with varying t.

Fig. 6

(a) AUC for RX and KRX with different windows; (b) AUC for RX-Fusion and KRX-Fusion with varying t.

Under the best parameters, Figs. 7 to 8 illustrate the ROC performance of the proposed RX-Fusion and KRX-Fusion compared with RX, KRX, MW-RX, and MW-KRX. For better visualization, we separate the cases of RX-Fusion and KRX-Fusion. From the results, it is obvious that the proposed RX-Fusion is always superior to RX and MW-RX, and the proposed KRX-Fusion outperforms KRX and MW-KRX. For the HYDICE urban data, MW-KRX exhibits a better performance than KRX; however, this is not true for the HyMap data. To further investigate the detection performance in the HYDICE urban data, Fig. 9 illustrates the detection maps when $P_f$ is fixed to a small value (e.g., 0.005) and $P_d$ is the maximum. The proposed RX-Fusion and KRX-Fusion still perform the best with the largest $P_d$, which is consistent with the results in Fig. 7.

Fig. 7

Receiver-operating-characteristic (ROC) performance of the proposed (a) RX-Fusion and (b) KRX-Fusion for the HYDICE urban data.

Fig. 8

ROC performance of the proposed (a) RX-Fusion and (b) KRX-Fusion for the HyMap data.

Fig. 9

Detection maps for HYDICE urban data when $P_f$ is fixed to 0.005. (a) RX: $P_d=0.7143$; (b) MW-RX: $P_d=0.6667$; (c) RX-Fusion: $P_d=0.8571$; (d) KRX: $P_d=0.8095$; (e) MW-KRX: $P_d=0.8571$.

Table 2 further summarizes the AUC performance. From the AUC values shown in Figs. 5 to 6, we can see that although the performances of suboptimal RX-Fusion and KRX-Fusion (i.e., $t=6$ when $m=12$) are slightly worse than the best RX and KRX (which are practically unknown), respectively, they are much better than their worst and average performances. This means, in reality, we can empirically choose $t$ to equal 50% of the total number of detectors; in other words, if half of detectors claim a pixel to be an anomaly, then it will be an anomaly in the final decision.

Table 2

Area under ROC (AUC) for several anomaly detectors using the two experimental data.