Receiver operating characteristic (ROC) analysis is performed by a curve, called ROC curve, plotted based on detection probability, <inline-formula> <tex-math notation="LaTeX">P_{\text {D}} ...</tex-math></inline-formula>, versus false alarm probability, <inline-formula> <tex-math notation="LaTeX">P_{\text {F}} </tex-math></inline-formula>, and has been widely used as an evaluation tool for signal detection. Specifically, the area under an ROC curve (AUC) is calculated and used as a detection measure. Unfortunately, finding distributions of <inline-formula> <tex-math notation="LaTeX">P_{\text {D}} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">P_{\text {F}} </tex-math></inline-formula> to generate a continuous ROC curve is practically infeasible. This article investigates approaches to generating a discrete 2D ROC curve of (<inline-formula> <tex-math notation="LaTeX">P_{\text {D}} </tex-math></inline-formula>,<inline-formula> <tex-math notation="LaTeX">P_{\text {F}} </tex-math></inline-formula>) without appealing for probability distributions. Since <inline-formula> <tex-math notation="LaTeX">P_{\text {D}} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">P_{\text {F}} </tex-math></inline-formula> are determined by the same threshold <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula> to specify a detector, an ROC curve of (<inline-formula> <tex-math notation="LaTeX">P_{\text {D}} </tex-math></inline-formula>,<inline-formula> <tex-math notation="LaTeX">P_{\text {F}} </tex-math></inline-formula>) can only be used to evaluate the effectiveness of a detector but not target detectability (TD) and also not background suppressibility (BS). To address this issue, a 3D ROC curve is generated as a function of (<inline-formula> <tex-math notation="LaTeX">P_{\text {D}} </tex-math></inline-formula>,<inline-formula> <tex-math notation="LaTeX">P_{\text {F}} </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>) by introducing a specific threshold parameter <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula> as a third independent variable. As a result, a 3D ROC curve along with its derived three 2D ROC curves of (<inline-formula> <tex-math notation="LaTeX">P_{\text {D}} </tex-math></inline-formula>,<inline-formula> <tex-math notation="LaTeX">P_{\text {F}} </tex-math></inline-formula>), (<inline-formula> <tex-math notation="LaTeX">P_{\text {D}} </tex-math></inline-formula>,<inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>), and (<inline-formula> <tex-math notation="LaTeX">P_{\text {F}} </tex-math></inline-formula>,<inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>) can further be used to design new quantitative measures to evaluate the effectiveness of a detector and its TD and BS. To demonstrate the full utility of 3D ROC analysis in target detection, extensive experiments are performed on two types of targets, prior target detection and anomaly detection, to conduct a comprehensive analysis on 3D ROC curves using new designed detection measures to evaluate target/anomaly detection performance.
Hyperspectral target detection (HTD) and hyperspectral anomaly detection (HAD) are designed by completely different functionalities in terms of how to carry out target detection. Specifically, HTD is ...a reconnaissance technique looking for known targets as opposed to HAD which is a surveillance technique seeking unknown targets of interest. So, HTD is generally designed by the hypothesis testing theory to derive likelihood ratio test (LRT)-based detectors. However, such hypothesis testing theory-based HTD requires the targets under the alternative hypothesis to be known. In addition, it also requires knowledge of the probability distribution under each hypothesis such as Gaussian distributions. Accordingly, the LRT-based HTD cannot be directly applied to HAD. This article develops a dual theory of LRT-based HTD for HAD, which converts HTD to HAD by making LRT-based detectors anomaly detectors. In addition, by virtue of this dual theory a new signal-to-noise ratio (SNR)-based theory can be also developed for HAD. Interestingly, the commonly used hyperspectral anomaly detector, referred to as Reed and Xiaoli detector (RXD), which is derived from the generalized LRT (GLRT), can be also rederived by this dual theory as well as the new developed SNR-based HAD theory.
Hyperspectral target detection (HTD) can be generally categorized by its targets to be detected, <inline-formula> <tex-math notation="LaTeX">a </tex-math></inline-formula> priori targets with ...provided known target knowledge as <inline-formula> <tex-math notation="LaTeX">a </tex-math></inline-formula> priori target detection and <inline-formula> <tex-math notation="LaTeX">a </tex-math></inline-formula> posteriori targets with known target signatures (spectral shapes), but unknown abundance fractions needed to be estimated as <inline-formula> <tex-math notation="LaTeX">a </tex-math></inline-formula> posteriori target detection. As a result, target detection can be performed in three scenarios, full pure-pixel target detection corresponding to <inline-formula> <tex-math notation="LaTeX">a </tex-math></inline-formula> priori target detection, and subpixel and mixed-pixel target detection corresponding to <inline-formula> <tex-math notation="LaTeX">a </tex-math></inline-formula> posteriori target detection. To develop theories for these three types of target detection, this article develops three approaches. One is to rederive hypothesis testing-based detection theory using very basic statistical detection theory. Another two are new theories, signal-to-noise ratio (SNR)-based detection theory that uses SNR as a criterion to derive optimal detectors and spectral angle (SA)-based detection theory that calculates SA to perform HTD, both of which do not require prior probability distributions as hypothesis testing does. Specifically, it will be shown that many current hypothesis testing-derived likelihood ratio test (LRT)-based detectors can find their counterparts in the SNR-derived theory and the SA-derived detection theory. Finally, to evaluate the detection performance among the detectors developed from these three approaches, several effective detection measures resulting from 3-D receiver operating characteristic (ROC) analysis are used to conduct a comprehensive study and comparative analysis.
One fundamental task of hyperspectral imaging is spectral unmixing. In this case, the conventional pure pixel-based hyperspectral image classification (HSIC) may not work effectively for mixed ...pixels. This article presents a kernel-based approach to hyperspectral mixed pixel classification (HMPC) which includes two nonlinear mixed pixel classifiers, kernel constrained energy minimization (KCEM) and kernel linearly constrained minimum variance (KLCMV) to replace the widely used pure pixel-based support vector machine (SVM) classifier. Interestingly, what the binary-class and multiclass SVM classifiers are to pure pixel-based HSIC can be similarly derived for what a single-class KCEM detector and a multiclass KLCMV detector are to HMPC. In particular, the commonly used discrete classification map-based hard classification measures, average accuracy (AA) and overall accuracy (OA) for performance evaluation can be further generalized to real-valued mixed class abundance fractional map-based soft classification measures via 3-D receiver operating characteristic (3-D ROC) analysis-derived detection measures. Extensive experiments are conducted to demonstrate the utility of HMPC where KCEM/KLCMV not only significantly improve the classification performance of CEM/LCMV-based classifiers but also outperform many existing spectral-spatial classification methods.
This paper presents a statistical detection theory approach to hyperspectral image (HSI) classification which is quite different from many conventional approaches reported in the HSI classification ...literature. It translates a multi-target detection problem into a multi-class classification problem so that the well-established statistical detection theory can be readily applicable to solving classification problems. In particular, two types of classification, a priori classification and a posteriori classification, are developed in corresponding to Bayes detection and maximum a posteriori (MAP) detection, respectively, in detection theory. As a result, detection probability and false alarm probability can also be translated to classification rate and false classification rate derived from a confusion classification matrix used for classification. To evaluate the effectiveness of a posteriori classification, a new a posteriori classification measure, to be called precision rate (PR), is also introduced by MAP classification in contrast to overall accuracy (OA) that can be considered as a priori classification measure and has been used for Bayes classification. The experimental results provide evidence that a priori classifier as Bayes classifier which performs well in terms of OA does not necessarily perform well as a posteriori classifier in terms of PR. That is, PR is the only criterion that can be used as a posteriori classification measure to evaluate how well a classifier performs.
Virtual dimensionality (VD) is originally defined as the number of spectrally distinct signatures in hyperspectral data. Unfortunately, there is no provided specific definition of what "spectrally ...distinct signatures" are. As a result, many techniques developed to estimate VD have produced various values for VD with different interpretations. This paper revisits VD and interprets VD in the context of Neyman-Pearson detection theory where a VD estimation is formulated as a binary composite hypothesis testing problem with targets of interest considered as signal sources under the alternative hypothesis, and the null hypothesis representing the absence of targets. In particular, the signal sources under both hypotheses are specified by three aspects. One is signal sources completely characterized by data statistics via eigenanalysis, which yields Harsanyi-Farrand-Chang method and maximum orthogonal complement algorithm. Another one is signal sources obtained by a linear mixing model fitting error analysis. A third one is signal sources specified by inter-band spectral information statistics which derives a new concept, called target-specified VD. A comparative analysis among these three aspects is also conducted by synthetic and real image experiments.
The receiver operating characteristic (ROC) curve of detection probability (PD) versus the false alarm probability (PF), referred to as 2D ROC curve, has been widely used to evaluate hyperspectral ...anomaly detection (AD) performance. This article explores several fundamental and conceptual issues of a 2D ROC curve used for AD, which has been overlooked and never investigated in the past. How can a Neyman-Pearson (NP) detector work for AD? How is an ROC curve plotted without probability distributions? Why is a 2D ROC curve reported in the literature as a step function and later linearly interpolated as a linear piecewise function? How can an ROC curve be used to evaluate background suppression (BS)? To address all these issues, a mathematical theory of 2D ROC curve is rederived by a random Neyman-Pearson detector (RNPD) via a threshold parameter <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>, which actually determines PD and PF, and its detailed theoretical proofs along with comprehensive analysis are also provided. Specifically, a binary communication channel example is included to illustrate how RNPD works. This threshold <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>-driven approach, indeed, paves a way for deriving a 3D ROC curve as a function of three parameters: <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>, PD, and PF. By virtue of a 3D ROC curve, three 2D ROC curves of (PD,PF), (PD,<inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>), and (PF,<inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>) can be derived to perform AD performance analysis effectively in terms of anomaly detectability and BS. Experiments demonstrate that many anomaly detectors that claim to perform well on AD using 2D ROC curves are actually performed very poorly in BS.
Hyperspectral image classification (HSIC) methods based on convolutional neural network (CNN) continue to progress in recent years. However, high complexity, information redundancy, and inefficient ...description still are the main barriers to the current HSIC networks. To address the mentioned problems, we present a spatial-spectral dense CNN framework with a feedback attention mechanism called FADCNN for HSIC in this article. The proposed architecture assembles the spectral-spatial feature in a compact connection style to extract sufficient information independently with two separate dense CNN networks. Specifically, the feedback attention modules are developed for the first time to enhance the attention map with the semantic knowledge from the high-level layer of the dense model, and we strengthen the spatial attention module by considering multiscale spatial information. To further improve the computation efficiency and the discrimination of the feature representation, the band attention module is designed to emphasize the weight of the bands that participated in the classification training. Besides, the spatial-spectral features are integrated and mined intensely for better refinement in the feature mining network. The extensive experimental results on real hyperspectral images (HSI) demonstrate that the proposed FADCNN architecture has significant advantages compared with other state-of-the-art methods.
A known target detection assumes that the target to be detected is provided a priori, while anomaly detection is an unknown target detection without any prior knowledge. As a result, known target ...detection generally performs search-before-detect detection in an active mode, referred to as active target detection as opposed to anomaly detection, which performs throw-before-detect detection in a passive mode, referred to as passive target detection. Accordingly, techniques designed for these two types of detection are completely different. This article shows that there is indeed a mechanism, called target-to-anomaly conversion, which can convert hyperspectral target detection (HTD) to hyperspectral anomaly detection (HAD) via a novel idea, called dummy variable trick (DVT). By virtue of such target-to-anomaly conversion many well-known target detection techniques, such as likelihood ratio test (LRT), constrained energy minimization (CEM), and orthogonal subspace projection (OSP), the spectral angle mapper (SAM) and the adaptive cosine estimator (ACE) can be converted to their corresponding anomaly detectors, referred to as target-to-anomaly conversion-derived anomaly detectors (TAC-ADs). Since a target detector requires target knowledge while TAC-AD does not, a direct use of TAC-AD is not effective. To make TAC-AD work, a newly developed approach to effective anomaly space (EAS) is implemented in conjunction with TAC-AD so that anomalies can be retained in EAS and interference, and noise including background (BKG) can be removed from EAS. The experiments demonstrate that TAC-AD operating in EAS performs better than many existing anomaly detection approaches, including model-based methods.
