Consider the ideal detector for a signal-known-exactly in a simple detection task. Each stimulus presentation consists either of noise alone or of noise plus a signal. The ideal detector computes the cross-correlation of the display with a prewhitened matched template of the target to obtain a scalar-valued template response
R, which is then compared to a criterion. If
R exceeds the criterion, the detector responds “target present”; otherwise, it responds “target absent.” Ideal performance in this task varies monotonically with the signal-to-noise ratio
d′, which, in the current task, is manipulated by varying the contrast
c of the target signal:
where Φ
−1[·] represents the inverse of the standard normal integral and PC
max(
c) is a psychometric function representing the proportion-correct detection performance for an observer whose goal is to minimize detection errors (i.e., false alarms + misses). The psychometric function is modeled as a cumulative Weibull function with two parameters: a contrast threshold parameter
c T and a steepness parameter
s:
In human observers, due largely to inhomogeneities in retinal sampling and cortical representation, the contrast threshold varies as a function of the target's position in the retinal image. We assume that the threshold changes as an exponential function of retinal eccentricity (Peli et al.,
1991):
Combining
Equations A1 and
A2, we describe the effective signal-to-noise ratio for a human observer detecting a signal-known-exactly:
We compute the effective response noise
σ r (
c,
ɛ) by assuming that the difference between the signal and noise response means is fixed to 1. In this case,
σ r (
c,
ɛ) = 1/
d′(
c,
ɛ), which leads to the result in
Equation 2 of the main text:
However, even when the signal is specified exactly, human observers behave as though some of the stimulus parameters are uncertain (Tanner,
1961). One of the parameters for which humans demonstrate intrinsic uncertainty is stimulus position. We represent this uncertainty by modeling an intrinsic position noise whose standard deviation
σ p varies as a linear function of eccentricity:
We estimated the linear coefficient
p = 0.09 from the asymptotic localization performance of the human observers. Moreover, we found that this intrinsic noise is anisotropic, with greater variance along the radial axis
ρ (i.e., the direction of increasing retinal eccentricity) than along the tangential axis
τ (perpendicular to the radial axis). The ratio of noise along each axis appears constant across eccentricities with
and
Note that the resulting covariance matrix Σ
p (
ɛ,
θ) now varies as a function of direction. After estimating the intrinsic uncertainty parameter, we estimated the parameters describing the effective response noise (i.e.,
c T(0),
m T,
s) from the detection performance in the Location-Specified condition using a maximum likelihood procedure described in the
11 section. Note that our procedure for estimating the response noise parameters took into account the effect of the intrinsic position uncertainty on detection performance.