Humans are exquisitely sensitive to changes in relative position. A fundamental and long-standing question is how information for position acuity is integrated along the length of the target, and why visual performance deteriorates when the feature separation increases. To address this question, we used a target made of discrete samples, each subjected to binary positional noise, combined with reverse correlation to estimate the behavioral “receptive field” (template), and a novel 10-pass method to quantify the internal noise that limits position acuity. Our results show that human observers weigh individual parts of the stimulus differently and importantly, that the shape of the template changes markedly with feature separation. Compared to an ideal observer, human performance is limited by a template that becomes less efficient as feature separation increases and by an increase in random internal noise. Although systematic internal noise is thought to be one of the important components limiting detection thresholds, we found that systematic noise is negligible in our position task.

*λ*), 20 (3.33

*λ*), or 60 (10

*λ*) arcmin gap between the centers of the two innermost patches. Each segment consisted of five Gabor patches (carrier SF, 10 cpd), and the inter-patch separation was 5 arcmin (the actual stimulus can be seen by viewing Figure 1 from a distance of 1.3 m). The patches were constructed to have a 2/3 aspect ratio: the Gaussian envelope standard deviation was 2.33 and 3.50 arcmin for the horizontal and vertical orientations, respectively. Positional noise was produced by shifting the position of each Gabor patch in the vertical direction around the intended mean line position of the test (right) segment according to a discrete binary probability function. The binary noise amplitude was always 0.67 arcmin, either positive or negative. No noise was added to the reference (left) segment. The stimulus was briefly presented (200 ms) on a flat 21-in. Sony F520 monitor screen at 90 Hz refresh rate. Subjects were asked to maintain fixation at the center of the monitor screen. The mean luminance of the stimuli was 55 cd/m

^{2}, and the contrast of each Gabor patch was 84%. The monitor was viewed directly at a distance of 4 m.

*d*′) values from 0 to 2. The observers' task was to rate the position of the test line compared to the reference line by giving an integer number from −2 (above) to 2 (below), including 0 (aligned). The observers were instructed to attend to the whole of the jittered line, to determine the average location, and to compare this with the reference line. A rating scale signal detection paradigm was used to calculate

*d*′ for discriminating the direction of offset in noise (Levi et al., 2000). The position offset at which

*d*′ = 1 was taken as threshold. Trial-by-trial verbal feedback was provided.

*w*

_{i}is the weighting (classification image) of each Gabor patch,

*x*

_{i}is the patch position, and the subscript

*i*goes from 1 to 5 for the five Gabor patches (see Figure 1). The sum of the weightings is unity.

*σ*

_{temp}) can be calculated from Equation 1 by the following equation:

*σ*

_{x}is the standard deviation of

*x*

_{i}(external noise). For an ideal observer, this becomes

*σ*

_{x}/√

*N*for

*N*patches because

*w*

_{i}= 1/

*N*(the same weighting is given to each patch).

*r*(−2, −1, 0, +1, and +2) is given to stimulus

*s*(−2/3, 0, and +2/3 arcmin). The calculation of the human threshold (

*d*′ = 1) was computed based on signal detection theory (Levi et al., 2000). A nonlinear chi-square minimization was used to fit the data with six parameters: four criteria and two

*d*′s (one between zero and negative offset stimuli and the other one between zero and positive offset stimuli). Equations 3 and 4 for the case of the human threshold become:

*d*′

_{ave}is the average of the two

*d*′ values and stim is the stimulus offset.

^{5}= 32 possible external noise combinations in total (Figure 2; for stimulus numbering, see legend in Figure 3; U [up] and D [down] indicate the individual patch position). For a given 1 of the 96 stimuli, the data are randomly spread out (see Figures 2 and 3).

*r*(−2, −1, 0, +1, and +2) is given to each of the 96 stimuli. A nonlinear chi-square minimization was used to fit the data with 99 parameters: 4 criteria, and 95

*d*′s. Each of the 96 stimuli had 10 trials because of our “10-pass” methodology. Ten repeats were sufficient for pinning down

*d*′ for each stimulus with adequate sensitivity. The sequence of stimuli presented to observers was randomized in each of the 10 runs.

*d*′. The

*d*′ values are free to float. The multi-pass nature of the stimuli is critical for being able to obtain an accurate estimate of the internal noise by this procedure.

*r*= 0.94).

*λ*) to widely spaced (10

*λ*) and found that position threshold in noise increased linearly with separation when plotted on a log–log scale (Figure 4a). For comparison, we also measured the threshold for a stimulus with no noise (red solid squares in Figures 4a and b) in one observer, RL. As expected, thresholds for the noisy stimulus are much higher (by almost a factor of 2) than those for the stimulus with no noise. The thresholds rise with increasing separation. However, the increase in threshold is much shallower than predicted by Weber's law (slope = 1). A power function was used to fit the threshold data;

*y*= 0.41

*λ*

^{0.5 ±}

^{0.06}(blue line) and

*y*= 0.27

*λ*

^{0.5 ±}

^{0.04}(red line) for the stimuli with and without noise, respectively. Adding noise to the stimulus shifts the threshold line upwards, with no change in the power constant, or slope of the regression line. The slope is about the same magnitude as reported previously, using zero noise Gabor stimuli (Whitaker, Bradley, Barrett, & McGraw, 2002).

*λ*of Figure 5a).

*λ*. Thus, the increase in random noise is an important factor in accounting for the change in threshold with increasing separation. The effect of external noise was to produce an upward shift of the power function (Figure 4b), increasing the thresholds by the same proportion for a wide range of separations. This kind of effect is typically attributed to changes in either template tuning or multiplicative internal noise, consistent with our results. Moreover, it implies that the change in random noise with separation is not additive in nature.

- Alignment performance could be limited by an orientation cue (Sullivan, Oatley, & Sutherland, 1972; Watt, 1984). For widely separated stimuli, an increase in offset displacement is necessary to maintain a constant detectable orientation cue. The constant orientation hypothesis predicts a slope of 1; however, as noted in the introduction, with narrowband stimuli (like those used here) the slope is considerably lower than 1.
- With increasing separation, there is a shift in the spatial scale of analysis such that larger filters are engaged as separation increases (Hess & Hayes, 1993; Whitaker et al., 2002). For narrowband stimuli such as Gabor patches, at small separations thresholds are inversely proportional to the carrier spatial frequency. At large separation, thresholds are determined by spatial scale characteristics of the envelope.

*λ*(1 deg).

*σ*) for separations up to ≈15

*σ*. For our stimuli, the Gaussian envelope standard deviation of 3.50 arcmin (for the vertical orientation) implies a floor of ≈0.52 min. The Whitaker et al. carrier floor for a very closely separated stimulus is approximately 0.3 min (for DW with an 8 c/deg carrier—their Figure 5) similar to RL's “no-noise” thresholds at the smallest separation (our Figure 4). Thus, we cannot completely exclude either mechanism; however, we discuss a more parsimonious alternative below.

- As the stimuli are separated, their eccentricity covaries with their separation, and because the cortical filters at large eccentricities are not as closely and regularly packed as in the fovea (Levi et al., 1988; Waugh & Levi, 1995), the position of stimulus features becomes more uncertain when separation/eccentricity increases. Indeed, when position stimuli are presented on an isoeccentric arc, at a given eccentricity the threshold remains constant over a range of separations. Thus, there is no effect of separation (Levi & Klein, 1990; Levi et al., 1988), and Weber's law (for non-isoeccentric stimuli) can be attributed to the effects of eccentricity. Consistent with this notion, our 10-pass results show that random positional noise increases with separation.

- Human observers weigh individual parts of the stimulus differently and importantly, the shape of the template changes with feature separation.
- Compared to an ideal observer, human performance is limited by a template that becomes less efficient as feature separation increases and by an increase in random internal noise.
- Systematic noise plays a negligible role in our position task.