This theoretical note describes a simple equation that closely approximates the psychometric functions of template-matching observers with arbitrary levels of position and orientation uncertainty. We show that the approximation is accurate for detection of targets in white noise, 1/f noise, and natural backgrounds. In its simplest form, this equation, which we call the uncertain normal integral (UNI) function, has two parameters: one that varies only with the level of uncertainty and one that varies only with the other properties of the stimuli. The UNI function is useful for understanding and generating predictions of uncertain template matching (UTM) observers. For example, we use the UNI function to derive a closed-form expression for the detectability (d′) of UTM observers in 1/f noise, as a function of target amplitude, background contrast, and position uncertainty. As a descriptive function, the UNI function is just as flexible and simple as other common descriptive functions, such as the Weibull function, and it avoids some of their undesirable properties. In addition, the estimated parameters have a clear interpretation within the family of UTM observers. Thus, the UNI function may be the better default descriptive formula for psychometric functions in detection and discrimination tasks.
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