Hyperspectral Data Processing: Algorithm Design and Analysis

3.4 3D ROC Analysis

From the hypothesis testing problem specified by (3.1), a detector makes a binary hard decision by thresholding a real-valued LRT, Λ(r), via a threshold τ (see (3.5)). Accordingly, the detector performance is determined by two parameters, Λ(r) and τ, both of which are real values. As a result, the detection rate, P_D, in (3.2) and (3.6) and the false alarm probability/rate P_F in (3.3) and (3.7) are indeed functions of Λ(r) and threshold τ. However, in the Neyman–Pearson detection theory the cost function and prior probabilities are assumed to be not known, nor is τ. In this case, the false alarm rate P_F is used as a cost function and the threshold τ becomes a dependent function of P_F via (3.7) by setting P_F = β in (3.4). This is contradictory to the original detection problem where P_F = β is actually obtained by a specific value of the threshold τ. Therefore, when an ROC curve is plotted in Figure 3.4 based on P_D versus P_F, the threshold τ is implicitly absorbed in P_F and there is no way to show how the threshold τ specifies P_F as the way it should be in Bayes detection theory in (3.1). To resolve this issue, this section develops a new approach to ROC analysis, referred as 3D ROC analysis, which extends the traditional 2D ROC analysis in Section 3.3 by including ...