Optimal Joint Detection and Estimation That Maximizes ROC-Type Curves

Combined detection-estimation tasks are frequently encountered in medical imaging. Optimal methods for joint detection and estimation are of interest because they provide upper bounds on observer performance, and can potentially be utilized for imaging system optimization, evaluation of observer efficiency, and development of image formation algorithms. We present a unified Bayesian framework for decision rules that maximize receiver operating characteristic (ROC)-type summary curves, including ROC, localization ROC (LROC), estimation ROC (EROC), free-response ROC (FROC), alternative free-response ROC (AFROC), and exponentially-transformed FROC (EFROC) curves, succinctly summarizing previous results. The approach relies on an interpretation of ROC-type summary curves as plots of an expected utility versus an expected disutility (or penalty) for signal-present decisions. We propose a general utility structure that is flexible enough to encompass many ROC variants and yet sufficiently constrained to allow derivation of a linear expected utility equation that is similar to that for simple binary detection. We illustrate our theory with an example comparing decision strategies for joint detection-estimation of a known signal with unknown amplitude. In addition, building on insights from our utility framework, we propose new ROC-type summary curves and associated optimal decision rules for joint detection-estimation tasks with an unknown, potentially-multiple, number of signals in each observation.