Efficient performance in visual detection tasks requires excluding signals from irrelevant spatial locations. Indeed, researchers have found that detection performance in many tasks involving multiple potential target locations can be explained by the uncertainty the added locations contribute to the task. A similar type of Location Uncertainty may arise within the visual system itself. Converging evidence from hyperacuity and crowding studies suggests that feature localization declines rapidly in peripheral vision. This decline should add inherent position uncertainty to detection tasks. The current study used a modified detection task to measure how intrinsic position uncertainty changes with eccentricity. Subjects judged whether a Gabor target appeared within a cued region of a noisy display. The eccentricity and size of the region varied across blocks. When subjects detected the target, they used a mouse to indicate its location. This allowed measurement of localization as well as detection errors. An ideal observer degraded with internal response noise and position noise (uncertainty) accounted for both the detection and localization performance of the subjects. The results suggest that position uncertainty grows linearly with visual eccentricity and is independent of target contrast. Intrinsic position uncertainty appears to be a critical factor limiting search and detection performance.

*extrinsic*and

*intrinsic*. Extrinsic position uncertainty occurs when the location of the signal is randomly sampled from a set of potential locations or is otherwise imprecisely specified within the task. Experiments that directly manipulate extrinsic position uncertainty are sometimes called

*uncertainty*experiments (e.g., Cohn & Lasley, 1974; Pelli, 1985) and sometimes single-fixation

*visual search*experiments (e.g., Cameron, Tai, Eckstein, & Carrasco, 2004; Eckstein, 1998; Palmer, 1994; Palmer, Verghese, & Pavel, 2000).

*certain*condition, the observer is told in advance that the target, if present, will always appear within the leftmost patch. In the

*uncertain*condition, the target can appear in either (but not both) of the noise patches. Because the observer can rule out one of the locations in the first condition, we should expect to see better performance in this condition compared with the second condition. More generally, as we increase the number of noise patches and locations in which the target can appear, the observer's performance will tend to decrease because the observer is forced to consider an increasing number of noisy feature responses. However, imagine that the two patches in our sample task are spatially adjacent and that the observer now has intrinsic position uncertainty such that he cannot discern whether a particular feature response came from the first or the second display location. In this case, his uncertainty about the provenance of the responses requires that he consider both responses regardless of the condition. That is, if the observer has intrinsic uncertainty that confuses the spatial sources of the signals, then we should not expect any difference in performance between the certain and uncertain conditions. More generally, intrinsic uncertainty will reduce the effect of added extrinsic uncertainty.

*f*noise). The area surrounding the circular region was set to the mean luminance of 20 cd/m

^{2}. The 1/

*f*noise was created by filtering Gaussian white noise, truncating the waveform at ±2

*SD*, scaling to obtain the desired root-mean-square (rms) amplitude and then adding a constant to obtain the mean luminance. The rms contrast of the noise region was fixed at 10% across all trials.

*Location-Uncertain*(LU) condition, the target appeared at a random location within the cued region; however, to keep any part of the target from appearing outside of the cued target region, the target center was not allowed to fall within 0.3 degree of the edge of the cued region. The size of the target region for the LU condition changed with eccentricity such that the diameter of the cued region was equal to the eccentricity of the region center (e.g., a target region centered at 5° was bounded by a circle whose diameter was 5°). In the

*Location-Specified*(LS) condition, the target (when present) always appeared in the center of the cued region that was 0.8° in diameter. Observers were tested at 5 different eccentricities along the upper vertical meridian (

*ɛ*= {0.0°, 1.5°, 2.5°, 3.5°, 5.0°}), over a variety of target contrasts, and the trials were blocked by cue condition, target region eccentricity, and target contrast.

*intrinsic position uncertainty*reflects factors intrinsic to the observer that contribute to uncertainty in the spatial source of the template or feature response associated with an image patch, while

*intrinsic response uncertainty*reflects factors intrinsic to the observer that contribute to uncertainty in the magnitude of the response.

*intrinsic uncertainty observer*. To determine the performance of the intrinsic uncertainty observer, we derived the optimal detection and localization strategies for a task in which a known target is located randomly within a cued region in a field of random spatial noise. Figure 2 illustrates the important aspects of the model. In each trial, the observer attempts to determine whether or not a target signal is present within a cued region of the display and where the target signal, if present, is located. We assume that the observer receives independent feature responses from each of

*n*

_{ D }discrete nonoverlapping spatial locations within the display. A subset comprising

*n*

_{ C }of the

*n*

_{ D }locations falls within the cue circle representing the possible target locations. Each of the feature responses is corrupted with intrinsic response noise and their perceived locations are perturbed with intrinsic position noise. The addition of intrinsic position noise implies that responses originating within the cue region may be perceptually displaced. Therefore, in deciding whether the target is present in the display, the observer must integrate across the entire display (rather than just the cued region), taking into account the probability that each response represents a patch from the cued region. We assume that, except for the intrinsic noise, the observer is ideal. That is, it integrates information optimally across spatial locations. The details of the intrinsic uncertainty observer are described in 1. Here, we define the parameters that govern the two types of intrinsic uncertainty.

*σ*

_{ r }(

*c*,

*ɛ*) of target contrast

*c*and eccentricity

*ɛ*:

^{−1}(·) represents the standard normal integral, and

*c*

_{T}and

*s*, respectively, represent the contrast threshold and steepness parameters for a hypothetical localized psychometric function corrected for the observer's response bias

*β*and intrinsic position noise

*σ*

_{ p }(see 1 for derivation). We constrained the contrast thresholds to rise as an exponential function of eccentricity:

*m*

_{ p },

*m*

_{T},

*c*

_{T}(0),

*s*,

*β*}. We estimated the intrinsic uncertainty coefficient

*m*

_{ p }directly from the observers' localization errors (see Estimating intrinsic position uncertainty section). The remaining parameters were estimated via a maximum likelihood procedure from the psychometric functions measured in the conditions without extrinsic uncertainty (i.e., the Location-Specified conditions).

*c*/

*c*

_{T}) as the measure of signal strength. Localizations from hit trials were aggregated across subjects and binned by normalized target contrast into 20 quantiles. The dots represent the standard deviation (

*σ*

_{loc}) of the localization error in each bin, while each of the solid curves represents the least squares fit to the data for a reversed logistic function forced to pass through the standard deviation of the target location distribution (i.e., a uniform disk distribution with radius

*r*=

*ɛ*/2).

^{1}The localization errors do not go to zero but asymptote.

*m*

_{ p }=

*σ*

_{ p }/

*ɛ*. To estimate this parameter across eccentricities, we normalized the localization data by calculating the

*normalized localization error*(

*σ*

_{loc}/

*ɛ*) from the data at each eccentricity.

*σ*

_{ext}across trials also increases linearly with eccentricity and we can define an extrinsic position uncertainty coefficient

*m*

_{ext}=

*σ*

_{ext}/

*ɛ*analogous to

*m*

_{ p }that remains constant across eccentricity conditions. At low contrasts, the normalized localization error approaches

*m*

_{ext}, at high contrasts, the normalized error asymptotes at

*m*

_{ p }. Thus, we expected that the normalized location errors for each of the eccentricities should overlap, at least at the asymptotes. Figure 5 plots these normalized localization errors across eccentricity conditions. The overlap of the data—particularly the high-contrast asymptotes—across eccentricities supports our assumption in Equation 1 that the intrinsic position uncertainty changes linearly with eccentricity (i.e., that

*m*

_{ p }is approximately constant across eccentricity). A Chow (1960) test found no significant difference between the individual fits,

*F*(4, 64) = 1.67,

*p*= 0.17. On average across eccentricities,

_{ p }≅ 0.09.

*σ*

_{ p }, which represents the geometric mean of the standard deviations along the radial (fixation-to-target) and tangential axes. In fact, as shown in Figure 6, we found that human localization errors were anisotropic, showing greater variance along the radial axis

*ρ*than along the tangential axis

*τ*, with

*σ*

_{ ρ }:

*σ*

_{ τ }≈ 4:3 across all eccentricities. Preliminary data gathered along the horizontal meridian show a nearly identical relationship, confirming that these data indeed reflect the result of a radial/tangential anisotropy (i.e., rather than that of a vertical/horizontal anisotropy). Moreover, this radial/tangential anisotropy is consistent with results from position discrimination (Klein & Levi, 1987; White, Levi, & Aitsebaomo, 1992; Yap, Levi, & Klein, 1987) and crowding (Pelli et al., 2007; Petrov & Popple, 2007; Toet & Levi, 1992) studies, though somewhat less pronounced.

_{ p }= 0.09,

_{T}= 0.022,

_{T}(0) = 0.046,

*s*to vary as a function of eccentricity. However, in the intrinsic uncertainty observer we fit only a single steepness parameter across all eccentricities. The increase in steepness for the intrinsic uncertainty observer (solid curves) reflects solely the effect of the increasing intrinsic position uncertainty estimated from human localization performance (Figure 5). To illustrate the effect of the intrinsic position uncertainty, the dashed curves in Figure 7B show the performance of a simulated observer with the same intrinsic response noise as the intrinsic uncertainty observer, but with no intrinsic position noise. As the eccentricity increases, the model without intrinsic position uncertainty does an increasingly poorer job of accounting for the human data.

^{2}The important result shown here is that the effect of the intrinsic position uncertainty coefficient

*m*

_{ p }, estimated independently using localization data from the Location-Uncertain condition, is sufficient to account for the increase in the steepness of the psychometric functions with eccentricity.

Condition (model) | Observer | |||
---|---|---|---|---|

MMM | TUB | XNN | Combined | |

Location-Specified (IPU) | 0.9613 | 0.9542 | 0.8801 | 0.9323 |

Location-Uncertain (IPU) | 0.9040 | 0.8321 | 0.9081 | 0.8826 |

Location-Uncertain (NPU) | 0.6065 | 0.6883 | 0.7980 | 0.7071 |

*r*

^{2}(i.e., proportion of variance explained) values for each observer and condition. A nonparametric bootstrap test

^{3}for equivalence of these

*r*

^{2}values across the IPU and NPU models found that the differences were significant both for the combined subject data (

*p*< 0.001) and for the individual observers (MMM:

*p*< 0.001, TUB:

*p*= 0.03; XNN:

*p*= 0.044).

*σ*

_{loc}/

*ɛ*) as a function of normalized contrast (

*c*/

*c*

_{T}), and the data are collapsed across eccentricities. The solid curve represents the best fit of a reversed logistic function to the combined human data in Figure 5. Two features of these results are worth highlighting. The first is that the asymptotic localization performance of the intrinsic uncertainty observer closely matches that of the human observers. The second is that, as with the human observers, localization errors increase with decreasing signal strength. Both of these results are anticipated in the Estimating intrinsic position uncertainty section (see Figure 3). Nonetheless, these results are important because they show that an ideal observer model in which intrinsic position uncertainty is independent of target contrast can account for the observed increase in localization error with decreasing contrast.

*D*(1880, 34859) = 0.023,

*p*= 0.57. The left-hand plot in Figure 12 provides an additional view of the similarity of these distributions. However, the lower two panels in Figure 11 reveal differences in localization between the human and simulated observers,

*D*(1880, 34859) = 0.122,

*p*< 0.001. In particular, the human observers show systematically biased pointing behaviors. At small eccentricities, human observers tend to bias their pointing toward the fovea. This bias tapers off at about 3.0 degrees, after which subjects tend to bias their pointing behaviors away from the fovea. This modest bias may have been induced by the landmark cues provided by the fixation marker and the frame of the monitor (Diedrichsen, Werner, Schmidt, & Trommershauser, 2004; Werner & Diedrichsen, 2002), both of which were visible throughout each trial. Nonetheless, despite this bias, the asymptotic localization errors remain approximately equal between the human and simulated observers.

*M*, the number of possible signals, to vary as a function of eccentricity. The intrinsic position uncertainty model differs from these models in that it distinguishes the uncertainty due to intrinsic position uncertainty from that due to other forms of uncertainty. To be sure, other authors have suggested models of intrinsic inefficiency that use equivalent noise combined with intrinsic signal uncertainty (including position uncertainty) to account for nonlinearities in the psychometric function (e.g., Beutter, Eckstein, & Stone, 2003; Eckstein, Ahumada, & Watson, 1997; Zhang, Pham, & Eckstein, 2006) or to quantify the efficacy of various target location cues (Manjeshwar & Wilson, 2001). However, the intrinsic position uncertainty model is unique in that it represents the interaction between intrinsic and extrinsic position uncertainties as a function of peripheral eccentricity, anticipating the reduced impact of extrinsic position uncertainty on detection performance that results from increasing intrinsic position uncertainty. Figure 8B shows how an equivalent-noise model with steepness and threshold parameters fit to match detection performance in the Location-Specified condition fails to account for performance in the Location-Uncertain condition.

*per se*(Sullivan, Oatley, & Sutherland, 1972; Watt, 1984; Watt & Morgan, 1984; Wilson, 1986). Second, while the current study measures feature localization near threshold contrasts in spatial noise, position discrimination tasks use effectively zero-contrast backgrounds with elements whose contrasts can range from about 2 (Hess & Hayes, 1994; Levi & Tripathy, 1996) to as much as 50 (White et al., 1992) times the detection threshold. The exclusion of any sort of textured background in these studies is especially important considering that the introduction of just a single distracter element can substantially influence localization judgments in position discrimination tasks (Burbeck & Hadden, 1993; Hess & Badcock, 1995). Our goal in the current study was not, as in position discrimination experiments, to determine the minimum spatial offset that human observers can detect between elements under ideal conditions but rather their effective uncertainty with respect to the locations of features in a detection task.

*dual-response*detection task where subjects report not only when a target is detected but its apparent location. The large amount of additional information provided by this task strongly constrains models of detection performance, and it revealed clear evidence that position uncertainty is an important factor limiting detection performance in the periphery. This study also introduced a new ideal observer model for detection, the

*intrinsic uncertainty observer*, which may serve as the foundation for improved models of peripheral vision. This is potentially important because accurate models of peripheral vision are critical for understanding and predicting performance in natural visual tasks, which typically involve controlling the direction of gaze based on information detected in the periphery.

*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:

^{−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*:

*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:

*σ*

_{ p }varies as a linear function of eccentricity:

_{ 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

_{ 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.

*n*

_{ D }discrete spatial locations within the display. A subset of these, comprising

*n*

_{ C }locations, falls within the cue circle representing possible target locations. Let

*R*

_{ i }be the response obtained from display location (

*x*

_{ i },

*y*

_{ i }), where

*i*indexes the display locations and

*R*

_{ i }=

*r*

_{ i }+

*N*

_{ r }(

*i*) and

*N*

_{ r }(

*i*) is Gaussian noise with mean 0 and standard deviation

*σ*

_{ r }(

*i*). In addition, let

*J*be the target location chosen randomly from among the

*n*

_{ C }possible target locations. For mathematical convenience and without loss of generality, we assume that

*r*

_{ i }= 0.5 if

*i*=

*J*and

*r*

_{ i }= −0.5 otherwise.

*X*

_{ i },

*Y*

_{ i }) = (

*x*

_{ i },

*y*

_{ i }) + (

*N*

_{ x }(

*i*),

*N*

_{ y }(

*i*)) be the encoded location of the response

*R*

_{ i }, where (

*N*

_{ x }(

*i*),

*N*

_{ y }(

*i*)) is Gaussian noise with mean 0 and covariance matrix Σ

_{ p }(

*i*). Finally, to simplify notation, let

**W**

_{ i }= [

*R*

_{ i },

*X*

_{ i },

*Y*

_{ i }]. Because randomly changing the position of a noise patch has no effect on the decision or performance, we can assume that

**W**

_{ i }= [

*R*

_{ i },

*x*

_{ i },

*y*

_{ i }] for all

*i*≠

*J*and

**W**

_{ J }= [

*R*

_{ J },

*X*

_{ J },

*Y*

_{ J }]. In simulations, we represented the display as a triangular array of nonoverlapping patches whose centers were spaced 0.6° apart (the diameter of the target signal).

*β*(Green & Swets, 1966).

*j*indexes over possible target patches,

*k*indexes over encoded responses,

*p*(

*j*) = 1/

*n*

_{ C }represents the prior probability that the target was located in patch

*j*, and

*p*(

*k*∣

*j*) represents the probability that the encoded location (

*X*

_{ k },

*Y*

_{ k }) of

*R*

_{ k }was actually generated by patch

*j*in the display. Note that because the encoded locations in the image are binned into discrete spatial regions, this probability is computed as the integral over the region centered on (

*X*

_{ k },

*Y*

_{ k }) for a Gaussian centered on the true location (

*x*

_{ j },

*y*

_{ j }) with covariance Σ

_{ p }(

*j*).

*p*(

**W**

_{ i }∣signal,

*j*,

*k*) =

*p*(

**W**

_{ i }∣noise) for

*i*≠

*k*,

*l*(

**W**

_{1}, …,

*β*, the observer detects a target and must estimate its location. We assume that the observer's goal in the localization task is to minimize the average squared error between the true and estimated target locations. Therefore, the estimated target location (

**W**

_{1}, …,

*E*[·] represents the expectation operator.

*x*

_{ j },

*y*

_{ j }) term in the numerator, both the numerator and denominator of Equation A24 are equivalent to the likelihood term in Equations A16–A18. Thus,

- For target present trials, we set,
*r*_{ J }= 0.5 and*r*_{ i }= −0.5 for*i*≠*J*. For target absent trials, all*r*_{ i }were set to −0.5. - Gaussian noise samples
*R*_{ i }were generated for each of the*n*_{ D }locations in the display as described in the 2 section. - A location in the grid (
*X*_{ J },*Y*_{ J }) was selected randomly as the encoded target location according to*p*(*X*_{ J },*Y*_{ J }∣*x*_{ J },*y*_{ J }, Σ_{ p }(*J*)). - The observer decided whether the target was present in the display using Equation A18 and comparing it to the criterion
*β*. If*l*(**W**_{1}, …,$ W n D $) >*β*, the observer responded “target present”; otherwise, the observer responded “target absent.” - Finally, if the target signal was detected, the ideal observer calculated its expected location using Equation A25.

*m*

_{T},

*c*

_{T}(0),

*s*} along with the criterion

*β*were estimated by repeatedly simulating the intrinsic uncertainty observer with different parameter values along a four-dimensional grid and choosing the set of parameters (i.e., the point in the grid) that maximized the likelihood. The likelihoods were calculated as the probability of observing the empirical numbers of hits and false alarms given the hit and false alarm rates predicted by the intrinsic uncertainty observer. To obtain stable estimates, the predicted hit and false alarm rates were calculated by averaging over 100 runs of each block.

*r*

_{IPU}

^{2}=

*r*

_{NPU}

^{2}) by merging the data from the IPU and NPU predictions and repeatedly resampling appropriately sized samples to compute

*r*

_{IPU}*

^{2}and

*r*

_{NPU}*

^{2}, the coefficients of determination for the samples. After repeating this process for 50,000 iterations, we determined the proportion of samples

*α*for which

*r*

_{IPU}*

^{2}−

*r*

_{NPU}*

^{2}≥

*r*

_{IPU}

^{2}−

*r*

_{NPU}

^{2}. Finally, we used a two-tailed test for significance by setting

*p*= 2

*α*.