Decision Strategies Maximizing the Area Under the LROC Curve

For the 2-class detection problem (signal absent/present), the likelihood ratio is an ideal observer in that it minimizes Bayes risk for arbitrary costs and it maximizes AUC, the area under the ROC curve. The AUCoptimizing property makes it a valuable tool in imaging system optimization. If one considered a different task, namely, joint detection and localization of the signal, then it would be similarly valuable to have a decision strategy that optimized a relevant scalar figure of merit. We are interested in quantifying performance on decision tasks involving location uncertainty using the LROC methodology. We derive decision strategies that maximize the area under the LROC curve, ALROC . We show that these decision strategies minimize Bayes risk under certain reasonable cost constraints. We model the detection-localization task as a decision problem in three increasingly realistic ways. In the first two models, we treat location as a discrete parameter having finitely many values resulting in an (L+1) class classification problem. In our first simple model, we do not include search tolerance effects and in the second, more general, model, we do. In the third and most general model, we treat location as a continuous parameter and also include search tolerance effects. In all cases, the essential proof that the observer maximizes ALROC is obtained with a modified version of the Neyman-Pearson lemma using Lagrange multiplier methods. A separate form of proof is used to show that in all three cases, the decision strategy minimizes the Bayes risk under certain reasonable cost constraints.