The localization receiver operating characteristic (LROC) curve is a standard method to quantify performance for the task of detecting and locating a signal. This curve is generalized to arbitrary detection/estimation tasks to give the estimation ROC (EROC) curve. For a two-alternative forced-choice study, where the observer must decide which of a pair of images has the signal and then estimate parameters pertaining to the signal, it is shown that the average value of the utility on those image pairs where the observer chooses the correct image is an estimate of the area under the EROC curve (AEROC). The ideal LROC observer is generalized to the ideal EROC observer, whose EROC curve lies above those of all other observers for the given detection/estimation task. When the utility function is nonnegative, the ideal EROC observer is shown to share many mathematical properties with the ideal observer for the pure detection task. When the utility function is concave, the ideal EROC observer makes use of the posterior mean estimator. Other estimators that arise as special cases include maximum a posteriori estimators and maximum-likelihood estimators.
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