Human observers are able to localize the relative position of objects defined by Gaussian variations in either luminance (1st order) or contrast (2nd order). However, positional sensitivity is significantly poorer for 2nd- than for 1st-order stimuli. These judgments require the visual system to construct a representation of pertinent variation from a number of individual retinal samples—a process known as interpolation. We compared 1st- and 2nd-order interpolation mechanisms to examine whether differences in this process underlie differences in positional sensitivity. Observers were required to judge the relative position of two vertically separated 1st- or 2nd-order Gaussian distributions. The distributions were discretely sampled, and both sample number and separation were systematically varied. Results showed that with fixed sample separation (1.9 or 7.7 arcmin), optimum localization was obtained with a minimum of 4–6 samples, for both 1st- and 2nd-order stimuli. When sample number is maintained above this critical value, marked changes in sample separation (0 to 9 arcmin) had relatively little impact on thresholds for both 1st-order and 2nd-order stimuli. These results suggest that both 1st-order and 2nd-order interpolation mechanisms are limited by sample number rather than separation, and require a similar number of samples to mediate positional judgments.

^{−2}, with a frame rate of 120 Hz and CIE 1932 xy chromaticity coordinates of x = 0.273 and y = 0.283. The host computer was an Edsys (Pentium II) PC housing a CRS VSG2/4 graphics card. A chin and forehead rest was used to maintain the viewing distance at 2 meters.

^{−2}) within a rectangular vertical aperture (2.74° × 2.74° with upper and lower segments separated by 6.42 arcmin), and were windowed by a Gaussian function in the horizontal direction (see Figures 1a– 1d). The 1st-order (luminance-defined) stimuli were created by adding random noise (N(x)) to the Gaussian luminance profile (G(x)). The mathematical description of these stimuli in the horizontal direction is given by:

_{m}= mean luminance

_{2}(x, y), which is a random noise mask (m = .2) that is laid down to cover any residual systematic changes in local mean luminance, i.e. to remove any local second order signals.

^{2})/2

*σ*

^{2}); N(x) = 2(round(rand(x)) − 1. x denotes the horizontal pixel location,

*σ*is the standard deviation of the Gaussian envelope, round transforms a rational number to an integer, rand generates a random number between 0 and 1, and b and m are constants assigned the same values as in Equation 1. The contrast of the Gaussian was 0.5 in both Equations 1 and 2 and this value controls modulation depth in both cases.

*μ*is the offset corresponding to the 50% level on the psychometric function (offset corresponding to perceived alignment) and

*θ*provides an estimate of alignment threshold (half the offset between the 27% and 73% levels on the psychometric function approximately). The window containing both distributions was randomly jittered from trial-to-trial (0–0.14 deg.) to prevent the subjects using the edges of the screen as an alternative positional reference. Similarly, for the sampled distributions, the location of the samples relative to the underlying distribution was randomly jittered from trial-to-trial over a range equal to ±1 sample separation. This avoids a constant cue, such as the peak of the distribution, being used to make the localization judgement.

*t*(3) = 3.5,

*p*= .04, Cohen's

*d*= 1.05). This increased rate of threshold elevation may be indicative of additional noise sensitivity either at the initial stage, which is tuned to higher spatial frequencies, or at the rectification step in the 2nd-order processing cascade. The intercepts of the fitted lines are also higher for 2nd-order stimuli (

*t*(3) = 5.86,

*p*= .01, Cohen's

*d*= 3.15), the combination producing alignment thresholds for 2nd-order stimuli that were significantly poorer than their 1st-order counterparts across the range of Gaussian widths tested (Volz & Zanker, 1996).

*σ*) of approximately 13 and 37 arcmin respectively. These values seem unlikely to pertain to our localization task, since at our largest sample separation of 8.7 arcmin only two or three samples would fall into a ±1

*σ*summation zone for 1st-order stimuli while 2nd-order detectors should be able to use approximately 8. This should result in a larger decline in performance for 1st- than 2nd-order stimuli at the largest sample separation. That result is not obtained here. However, estimates of the size of Ricco's area, the region in which stimulus intensity and area are inversely proportional at threshold, do vary across observers and with eccentricity (Sukumar & Waugh, 2007), even over the relatively small extent of our current stimuli. The area also varies with luminance (Cornsweet & Yellott, 1985) although Sukumar and Waugh (2007) used a mean luminance of 52 cd.m

^{−2}(Waugh, 2008, personal communication) which is very similar to the 54.2 cd.m

^{−2}used here. In future the summation zones for contrast detection thresholds and localization should be determined within the same observers and using the same stimuli.

_{mk}. The variance of the linear regression thresholds, Th(Gw), is