**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.**

*x*is the value of stimulus property being varied,

*α*and

*β*are constants, and Φ(·) is the standard normal integral function. This function will be referred to here as the uncertain normal integral (UNI) function. Here, we demonstrate with simulations that the UNI function provides a close approximation to the psychometric functions of template-matching observers in white noise, 1/f noise, and natural backgrounds, for arbitrary levels of uncertainty about target position and orientation (and probably other dimensions of signal variation). The parameter

*β*varies only with the level of uncertainty. The parameter

*α*depends on the other properties of the stimuli, but does not vary with level of uncertainty. Thus, in the absence of observer uncertainty, the UNI function reduces to a normal integral function with a standard deviation of 2/α. We argue that the UNI function (a) is useful for characterizing and computing the predictions of template matching observers with uncertainty, (b) avoids practical and conceptual problems that sometimes arise with the other descriptive functions, and (c) may provide a more principled interpretation of psychometric data.

*a*and

*b*, is presented on each trial. In the absence of uncertainty, a template matching observer computes the dot product of a template

*f*(

*x, y*) with the input stimulus

*I*(

*x, y*):

*R*to a decision criterion

*γ*; if the response exceeds the criterion, then the observer responds that the stimulus was from category

*b*, otherwise that it was from category

*a*. In the case of detection in a background of Gaussian white noise, the stimulus from category

*a*is a sample of white noise, and the stimulus from category

*b*is a sample of white noise with an added target of some amplitude. If the observer's template has the same shape as the target, then the observer's accuracy will be the highest possible in this task (Peterson, Birdsall, & Fox, 1954; Green & Swets, 1974; Burgess, Wagner, Jennings, & Barlow, 1981; Geisler, 2011).

*γ*(e.g., Nolte & Jaarsma, 1967; Swensson & Judy, 1981; Shaw, 1982; Pelli, 1985). If the observer is uncertain about the position of the target, then Ω is the set of potential target locations,

**ω**= (

*x, y*); if uncertain about the orientation of the target, then Ω is the set of potential target orientations,

**ω**=

*θ*; and so on. Again, if the background is Gaussian white noise, the template has the shape of the target, and the uncertainty region corresponds to the actual range of variation of the target, then the uncertain template matching (UTM) observer will perform close to the optimal possible in the task (Peterson et al., 1954; Tanner, 1961; Nolte & Jaarsma, 1967; Shaw, 1982; Pelli, 1985; see later in this article). Of course, more biologically plausible UTM observers may have nonoptimal templates and intrinsic uncertainty. However, our primary concern here is with the performance of UTM observers independent of whether they are optimal or not. In other words, the aim is to demonstrate that Equation 1 accurately describes the psychometric functions of UTM observers.

*d*′ and the decision criterion

*γ*. The discriminability represents the intrinsic ability of the observer to perform the task, and the decision criterion any potential bias. For now we assume that the two stimulus categories are presented with equal probability, and that the UTM observer's criterion is always set to the optimal value (no bias). The effect of bias is easy to include. For equal presentation probabilities and no bias, an observer's accuracy is given by

*d*′ units is

*d*′ values of the UTM observer for detection of a raised-cosine-widowed sinewave target in Gaussian white noise, in a single interval identification (“Yes-No”) task. The target was 17 pixels in diameter and had a frequency of 1/8 cy/pixel. Each point is based on 2000 simulated trials. In computing the UTM observer's

*d*′ values, we applied the following formula to the sampled means and variances:

*d*′ values is given in the Appendix; however, using that approximation does not affect the quality of the UNI function predictions or any of the conclusions (see later).

*π*32

^{2}= 3,217 pixel locations and then selects the maximum. The different colored symbols represent different root-mean-squared (RMS) contrast levels for the noise background. The curves in the plot show the minimum squared-error fit of Equation 5. For each value of background contrast (colors) the parameter

*α*is fixed, and for each uncertainty radius (panels) the parameter

*β*is fixed (the

*α*values and single

*β*value in a panel were estimated simultaneously). Clearly, the UNI function provides a close approximation to the performance of the simulated UTM observer. The approximation remains close at other values of uncertainty, noise contrast, and target amplitude. Not surprisingly, the fits look equally good when plotted in units of proportion correct (Equation 1). For example, Figure 2 is a replot of the simulations and fits in Figure 1C.

*x*axis scales in Figures 3 and 4).

*d′*= 1.0 in Equation 5, we see that the threshold of the UTM observer is given by

*c*is the background contrast and

_{b}*k*is the slope of the masking function with 0 uncertainty (dark blue line in Figure 8;

_{b}*k*= 0.337). It follows that

_{b}*k*, we can determine how the relative slope of the masking function depends on the uncertainty radius. Figure 8 plots the relative slopes. The solid line in Figure 8 plots the predicted relative slope under the assumption that the uncertainty parameter

_{b}*β*is proportional to the uncertainty radius

*r*

*m*is the slope of the masking function,

*k*= 0.39, and

_{u}*r*is the uncertainty radius. Substituting into Equation 5 gives a closed-form approximation for the performance of the UTM observer (in this specific task, with this specific target) as a function of target amplitude, background contrast, and position uncertainty:

*x*) in psychophysical experiments. Also, they provide an adequate fit to the performance of UTM observers over a limited range. However, as Pelli (1985) notes, they fail over a broader range. Also, in some situations they make rather nonsensical predictions. For example, measurements of psychometric functions for detection in 1/f noise show that the steepness parameter

*β*of the fitted functions tends to increase with retinal eccentricity (Najemnik & Geisler, 2005; Michel & Geisler, 2011). But, for both the Weibull and GNI function, this implies that detectability in the periphery will be better than in the fovea if the target amplitude is high enough (Figure 9B). In general, it seems very unlikely that supra-threshold objects in the periphery can have a bigger signal-to-noise ratio than in the fovea. With the UNI function it is possible to have steeper psychometric functions in the periphery without the detectability functions ever crossing (Figure 9A). Thus, there are cases (e.g., modeling visual search or differences between foveal or peripheral vision) where the UNI function may be more appropriate.

*d*′ psychometric functions, Lu and Dosher (1999, 2008) consider models with a nonlinear transducer plus sources of internal noise, rather than uncertainty. Such models can be designed so the

*d*′ psychometric functions do not cross, as in Figure 9A. However, these models have more parameters than the UNI function and do not include uncertainty. Below we note how the UNI function can be generalized to include all these factors.

*β*) of the Weibull function that are typically observed in psychophysical studies. The parameter

*α*simply scales the

*x*axis of both functions so the fits are equally good for other values of

*α*(Figure 10B). This figure demonstrates that the UNI function has similar flexibility for fitting psychometric function shapes.

*β*is only affected by the level of uncertainty and

*α*only by the other properties of the backgrounds and targets. There is evidence that position and phase uncertainty are important factors limiting peripheral vision (Bennett & Banks, 1987; Hess & Hayes, 1994; Levi, 2008; Michel & Geisler, 2011). To the extent that intrinsic (and extrinsic) uncertainty plays a role in real performance, fitting the UNI function would provide an estimate of the effect of uncertainty, separate from the other factors affecting performance. Whether the estimated parameters of the UNI function will provide new insight into the mechanisms of visual performance and individual differences remains to be seen. But regardless, it is a principled descriptive function that is about as flexible and simple the Weibull or GNI function, and it avoids some of their nonintuitive properties. Thus, it would seem to be a better default descriptive formula for psychometric functions in detection and discrimination tasks.

*β*and a different value of

*α*for each background contrast level. We conclude that the UNI function works equally well for the case of statistically independent template responses.

*x*can take values around some arbitrary value

*u*(e.g., discrimination of leftward and rightward motion):

*λ*is the lapse rate.

*β*captures all the effects of uncertainty, the formula can be generalized fairly easily to allow nonlinear and noisy template matching. This generalization involves additional parameters, so this may not be appropriate for simply summarizing psychometric functions and estimating thresholds, but it could be useful for developing and testing models. Specifically, let the model's

*d′*psychometric function for the case of no position or orientation uncertainty be

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*u*, and

_{a}, u_{b}, σ_{a}*σ*are the means and standard deviations of the two distributions and

_{b}*γ*is a decision criterion located at the cross point between the two distributions. Assuming

*u*>

_{b}*u*, the criterion is given by

_{a}