Decision strategies that maximize the area under the LROC curve

Abstract

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 the area under the receiver operating characteristic (ROC) curve [AUC]. The AUC-optimizing 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 localization ROC (LROC) methodology. Therefore, we derive decision strategies that maximize the area under the LROC curve, A(LROC). We show that these decision strategies minimize Bayes risk under certain reasonable cost constraints. The detection-localization task is modeled 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 A(LROC) is obtained with a modified version of the Neyman-Pearson lemma. 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.

Efficient visual-search model observers for PET

“…Under fairly basic conditions, this is accomplished with the Bayesian observer that computes the location-specific likelihood ratio: 16,17 …”

Application of the Hotelling and ideal observers to detection and localization of exoplanets

“…An optimal detector can be defined as the one that minimizes the expected cost associated to the decision. If the expected cost is to be minimized, the optimal observer takes the form [21,37] ( 19) which is called the generalized likelihood ratio [1]. If the observer concludes that a planet is present in the image, its estimated location r̂p is computed as (20) We still assume that the densities pr(g | H 0 ) and pr(g | H 1 , r p ) are as defined in Eqs.…”

Estimation receiver operating characteristic curve and ideal observers for combined detection/estimation tasks

“…The proof that these expressions give the ideal EROC observer is an easy adaptation of the proof presented by Khurd and Gindi for the ideal LROC observer [3,4]. For a given value P of the false-positive fraction, we have a constrained maximization problem for U TP (T 0 ), considered as a functional of the test statistic T(g) and the estimator θ(g), and as a function of the threshold T 0 .…”

“…Similar integrals in subsequent equations will also be over all parameter space unless otherwise specified. Using the angle bracket notation, the ordinate may be written as (4) A plot of U TP (T 0 ) versus P FP (T 0 ) as the threshold is varied generates the EROC curve [11]. Each point on the EROC curve gives the expected utility of our estimate of the parameter vector for the true-positive cases at a given false-positive fraction.…”

“…A useful figure of merit in this situation is the ALROC, the area under the localization receiver operating characteristic (LROC) curve [1,2]. Recently, Khurd and Gindi [3,4] determined the ideal LROC observer, whose LROC curve lies above those of all other observers for the given task. Of course, this also implies that the ideal LROC observer maximizes the ALROC for the given task.…”