Template matching observers are a well-known family of models of detection and discrimination. In detection and discrimination tasks, a stimulus from one of two possible categories,
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):
\begin{equation}\tag{2}R = f \cdot I = \sum\limits_{x,y} {f\left( {x,y} \right)I\left( {x,y} \right)} \end{equation}
The observer then compares the template response
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).
If the observer's template has a different shape or if the template has the same shape and the background noise is not white, then performance will be suboptimal.
If the template matching observer is uncertain, the observer applies the template over the region of uncertainty Ω, computes the maximum of the template responses
\begin{equation}\tag{3}R = \mathop {\max }\limits_{{\bf{\upomega }} \in \Omega } {f_{\bf{\omega }}} \cdot I\end{equation}
and then compares that maximum response to a decision criterion
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.
In the signal detection theory framework (Green & Swets,
1974), the performance of an observer in a two-alternative detection or discrimination experiment is described by two numbers, the discriminability
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
\begin{equation}\tag{4}F\left( x \right) = \Phi \left[ {{{d^{\prime} \left( x \right)} \over 2}} \right]\end{equation}
Thus, from
Equation 1 we see that the UNI function expressed in
d′ units is
\begin{equation}\tag{5}d^{\prime} \left( x \right) = \ln \left( {{{{e^{\alpha x}} + \beta } \over {1 + \beta }}} \right){\rm{\ \ \ \ }}x \ge 0\end{equation}
The symbols in
Figure 1 show
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:
\begin{equation}\tag{6}d^{\prime} = {{\left[ {{\mathop{\rm mean}\nolimits} \left( {R\left| b \right.} \right) - {\mathop{\rm mean}\nolimits} \left( {R\left| a \right.} \right)} \right]\sqrt 2 } \over {\sqrt {{\mathop{\rm var}} \left( {R\left| b \right.} \right) + {\mathop{\rm var}} \left( {R\left| a \right.} \right)} }}\end{equation}
A slightly better approximation to the UTM observer's
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).
Each panel in
Figure 1 is for a different radius of position uncertainty around the center pixel of the target. For example, when the uncertainty radius is 32 pixels, the UMT observer computes the template response at each of
π32
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.
Gaussian white noise is popular as a background in psychophysical experiments. This popularity is largely due to the fact that it is spatially uncorrelated, which greatly simplifies mathematical modeling and analysis of the psychophysical data. However, the backgrounds that occur under natural conditions are spatially correlated. Gaussian 1/f noise has the power spectrum of natural images (Burton & Moorehead,
1987; Field,
1987), and hence contains some of the spatial correlations that occur under natural conditions. Gaussian 1/f noise is also popular because it is precisely specified, naturalistic, and relatively tractable for modeling and analysis. How do the spatial correlations in 1/f noise affect the performance of the UTM observers and the accuracy of the UNI function approximation? The circles in
Figure 3 show the simulated detectabilities of the UTM observer for the same windowed sinewave target, uncertainties, and background contrasts used with the white noise backgrounds. Because of the spatial correlations, the detectabilities in 1/f noise are lower than white noise (note change of axes); however, the solid curves show that the fit of the UNI function is just as accurate as in white noise.
The approximation also remains good for other targets. For example,
Figure 4 shows (for 1/f noise) simulations and fits for a raised-cosine target (full width = 17 pixels), which is a broadband target with nonzero mean. Although the fit is just as good as for the windowed sinewave target, the amplitude of the target must be higher to reach the same levels of detectability (note the different
x axis scales in
Figures 3 and
4).
The UNI function also provides a good approximation when the uncertainty is in orientation rather than position, although the effects of orientation uncertainty are smaller.
Figure 5 shows simulation and fits for detection of the windowed sinewave target for the cases of no uncertainty and uniform orientation uncertainty over the range ±89°.
In addition to simulations with noise backgrounds, simulations were run with background patches randomly sampled from natural images. The natural image patches were binned along the dimensions of mean luminance, RMS contrast, and cosine similarity between the amplitude spectrum of the background and a windowed sinewave target (see Sebastian et al.,
2017 for details). For this simulation, all the image patches were randomly drawn from the same bin (e.g., they all had the same contrast, mean luminance, and cosine similarity). The symbols in
Figure 6 plot the detectability of the UTM observer as function of target amplitude for several levels of position uncertainty. The solid curves are the best fitting UNI functions (
Equation 5). Thus, even for natural backgrounds the UNI function provides a good approximation.
The symbols in
Figure 7 plot the thresholds of the UTM observer in 1/f noise for a range of background contrasts and levels of position uncertainty. The straight lines in
Figure 7 are the predictions of Weber's law, which holds accurately for all levels of uncertainty. Setting
d′ = 1.0 in
Equation 5, we see that the threshold of the UTM observer is given by
\begin{equation}\tag{7}{a_t} = {{\ln \left[ {e + \left( {e - 1} \right)\beta }\, \right]} \mathord{\left/ {\vphantom {{\ln \left[ {e + \left( {e - 1} \right)\beta } \,\right]} \alpha }} \right. \kern-1.2pt} \alpha }\end{equation}
When there is no uncertainty this equation reduces to
\begin{equation}\tag{8}{a_t} = {1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-1.2pt} \alpha }\end{equation}
Weber's law implies that without uncertainty
\begin{equation}\tag{9}{a_t} = {k_b}{c_b}\end{equation}
where
cb is the background contrast and
kb is the slope of the masking function with 0 uncertainty (dark blue line in
Figure 8;
kb = 0.337). It follows that
\begin{equation}\tag{10}{k_b}{c_b} = {1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-1.2pt} \alpha }\end{equation}
and thus for arbitrary uncertainty that
\begin{equation}\tag{11}{a_t} = \ln \left[ {e + \left( {e - 1} \right)\beta } \,\right]k_b c_b \end{equation}
If we divide the slope for each uncertainty level by
kb, 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
β is proportional to the uncertainty radius
r \begin{equation}\tag{12}{{m\left( r \right)} \over {m\left( 0 \right)}} = \ln \left[ {e + \left( {e - 1} \right){k_u}r} \right]\end{equation}
where
m is the slope of the masking function,
ku = 0.39, and
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:
\begin{equation}\tag{13}d^{\prime} \left( x \right) = \ln \left( {{{{e^{{x \over {{k_b}{c_b}}}}} + {k_u}r} \over {1 + {k_u}r}}} \right){\rm{\ \ \ \ }}x \ge 0\end{equation}
This exercise demonstrates the potential usefulness of the UNI function for characterizing and understanding the predictions of UTM observer models.