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We present the application of a task-based image quality evaluation procedure that combines an automatic signal detection technique with a novel nonparametric free-response data analysis method. The procedure delivers an index measuring the detectability of signals at unknown locations, in a manner that requires only a small number of image samples, thus making it convenient for practical applications. We are using the free-response methodology, which allows for multiple marks to be retrieved from an image. An overall signal detectability metric is obtained by using an exponential transformation of the free-response operating characteristic (EFROC) curve. A nonparametric estimator with expressions for its variance calculations are presented. The procedure allows for scaling of the detectability index to any given image size, a property that makes it advantageous for application to phantom experiments in which multiple signals can be inserted, or have relatively large background volumes that can be scanned for false signals. Such experiments can provide substantial signal detectability information with only a few images. We apply this technique to evaluate an energy-based scatter estimation method for PET, which is used in conjunction with an image reconstruction algorithm that incorporates energy-dependent corrections for scatter and random coincidences [Phys. Med. Biol. 51 (2006) 2919-2937]. A comparison of PET time-of-flight (TOF) performance versus no-TOF is also presented.