Cortical magnification is the area of cortex that represents a unit area of retina; the variation of magnification with eccentricity is expressed by the magnification function, but each visual representation area could have a different magnification function. The parameters of the magnification function for a particular task are therefore potentially diagnostic of the cortical area whose properties define the limits of the performance. This concept provides a bridge between anatomy and psychophysical performance. If two tasks exhibit different magnification properties (other than simple scaling), the implication is that their limits are set by different cortical projection geometries (after control of optical factors). Perhaps other explanations are possible, but the presence of magnification properties would generate a differential hypothesis to motivate corresponding anatomical or physiological studies.

The defining property of the magnification function for a particular task is the change in minimum angle of resolution for a particular task with distance from the fovea. To a first approximation, many magnification functions can be fitted with a linear function of resolution angle A versus eccentricity E of the form

in degrees, where m is the slope of the line (which is always positive) and E2 is the intercept, which is always negative for real resolution and positive m. Levi, Klein, and Aitsebaomo (1985) pointed out that the characteristic value for such linear magnification is the constant E2 because m just plays the role of a scaling parameter. For our purposes, note that E2 is so called because it has the property that angle A doubles when the stimulus reaches that eccentricity, i.e., when E = E2.

However, magnification functions often deviate from linearity, and can be well-fitted by an exponential function of the form

which may be seen to be of similar form to the linear approximation by taking the logarithm of both sides:

In these functions, μ is the logarithmic slope and Elog 2 is the eccentricity at which log A increases by 0.3.

Reflection symmetry is a visual property of relevance to humans because other humans and most animals have pronounced bilateral symmetry, while inanimate objects in the natural environment typically do not have obvious symmetry (Julesz, 1971; Barlow & Reeves, 1979; Tyler, 1996a). There has been extensive interest in the mechanisms of human symmetry processing (Wagemans, 1995; Tyler, 1996b). In previous studies, we have investigated the detectability of static and dynamic symmetry around a central axis across varying distances of separation of the symmetric regions by a noise mask. The function of separation represents the summation range over which symmetry can be perceived, and its width provides a measure of the symmetry summation width. Tyler, Hardage, and Miller (1995) found that detection of static targets was possible for widths up to 6 arcmin for unscaled targets and with up to 64º of separation by a blank mask for eccentricity-scaled targets (Tyler & Hardage, 1996). Symmetry detection in dynamic noise showed a much narrower summation width with a noise mask (Tyler et al., 1995). In a further exploration of unscaled noise, it was found that symmetry detection did not show the expected decline with eccentricity, either in sensitivity or summation width (Tyler, 1999). In the current study, we provide more detailed analysis of the eccentricity properties and compare the eccentricity functions for static and dynamic reflection symmetry, again using unscaled noise.

Previous studies have reported reduced detectability for static symmetry when the axis is placed in eccentric vision (Julesz, 1971; Corballis & Roldan, 1974; Saarinen, 1988; Herbert & Humphrey, 1993; Saarinen, Rovamo, & Virsu, 1989; Barrett, Whitaker, McGraw, & Herbert, 1999). Most of these studies found, by various methods, that sensitivity declined with eccentricity by various methods, but the declines reported were surprisingly gradual if symmetry detection is considered as a position-based task. Corballis and Roldan (1974), for example, found little difference between symmetry detectability at 0º and at about 6º eccentric. We therefore wished to determine the peripheral scaling properties for symmetry detection in terms of both peak sensitivity and spatial tuning range. Rather than presupposing an eccentricity scaling for the noise elements, as did previous authors, the noise here consisted of unscaled binary random elements of fixed size, allowing us to determine the scaling empirically.

The stimuli were generated on a Macintosh IIfx with a Motorola 68040 CPU and were presented on a noninterlaced monochrome display with a frame rate of 67 Hz, subtending 23.5° wide by 17.6° high at a viewing distance of 57 cm. Patterns consisted of 307,200 black and white pixels at a dot density of 100%, with a random placement bilaterally reflected about a vertical axis. The dots subtended 2.2 arcmin in diameter. In the symmetric test patterns (Figure 1a and 1b), all points greater than a specified distance about the designated axis were mirror-reflected, providing for a gap of random dots in the central strip separating the halves of the symmetry pattern (no gap in Figure 1a; 40 pixels in Figure 1b). In the asymmetric null patterns (Figure 1c), every pixel was colored either black or white at random throughout the image. Lighting was ambient fluorescent, with the screen hooded to reduce glare.

As in the precursor study (Tyler, 1999), the independent variables were width of the random gap and distance of the axis of symmetry from fixation. The vertical axis of symmetry was located in the center of the monitor, and a red fixation point was placed at 2º, 5º or 10º to the right of fixation. Measurements for a range of durations, from 30 ms to 1800 ms, were taken by the method of constant stimuli with duration increments of about 0.25 octaves to the nearest 15 ms chosen to span a range of performance levels from chance (50% correct) to 100% correct. Each trial consisted of a brief (<300 ms) period of dynamic random fields with frame durations of 15 ms, followed by the symmetric or asymmetric test period, followed by another similar period of random fields. The noise in the test period was either invariant for the test duration (static) or continually refreshed (dynamic) at the same rate as the pre- and posttest periods. The pre- and posttest noise presentations were designed to eliminate masking by luminance onset and offset effects, but to mask any luminance afterimages following the presentation of static patterns. Tests consisted of blocks of 50 trials for each duration and eccentricity condition, and observers were instructed to identify any symmetry in the pattern with a “yes” response and asymmetry (random appearance) with a “no” response. Tyler et al., (1995) showed that criterion bias effects were minimal in this paradigm.

Four observers participated in the experiments. Observer L.H. was experienced in psychophysical testing; A.I. and C.A. were naive observers. L.H. was functionally monocular (with essentially no vision in the right eye); the other two observers used uncorrected (normal) binocular vision. All were given practice to become familiar with the detection task and the position and orientation of the symmetry axis to be detected in each block of trials.

Duration psychometric functions of percent correct as a function of test duration were measured for a range of different widths of occluding noise separating the symmetric halves of the patterns, constraining the symmetry to separations varying from 0 to 3° away from its reflection axis. This procedure was followed at eccentricities of 2, 5 and 10º of the symmetry axis relative to fixation. Sample psychometric functions are provided in Figure 2 to illustrate the importance of the criterion percent correct level. As described by Tyler et al. (1995), duration psychometric functions for symmetry detection do not have a uniform slope but often exhibit a steep portion for low detection levels, followed by a sharp corner into an extended plateau region, where increasing duration affords no improvement in detection probability. Beyond this plateau region, performance again rises rapidly (although this rise is not strongly evident in the data of Figure 2). This figure is provided to illustrate the justification for the separate analyses at high and low criterion levels for the peripheral data in the present study.

A quantitative evaluation of the properties of such psychometric functions was provided in Tyler et al. (1995). It should be emphasized that this plateau behavior does not reflect simply the presence of fast and slow mechanisms of symmetry processing. If the stimuli were homogeneous, the fast process would always be able to carry performance to 100% before the slow process took effect. Instead, the behavior must reflect an inhomogeneity in the population of random symmetry patterns, such that some (about one half of the population of samples where the plateau is at 75%) are inaccessible to the fast process and must wait for the slow process to operate before being detectable at all. At all eccentricities and noise gap widths, some of the patterns had symmetry detectable in less than 100 ms. For noise gap widths greater than a few pixels, most of the remaining patterns required durations of 2 sec or more for the symmetry to be detectable. The jump from ~50 ms to ~2 sec is puzzling in relation to known pattern processing mechanisms, because one might expect that the bilateral symmetry patterns could be processed with two attentional fixations. The typical attentional processing rate is similar to the saccadic refixation rate of 3 per sec (Loftus, Duncan, & Gehrig, 1992; Duncan, Ward, & Shapiro, 1994), implying that one should be able to detect the symmetry in a presentation time of about 800 ms. The need for a further second or two to reach 100% performance remains hard to explain.

Duration psychometric functions such as those in Figure 2 were analyzed at the criterion level of 60% correct in Tyler (1999) to reveal a reverse eccentricity scaling property for the width of the early detection region. This result is seen in the replot of Figure 3a, where the foveal noise-masking function shows a broad skirt at the lowest sensitivity levels, while the peripheral functions do not have this feature.

The present paper provides a comparative analysis of the scaling properties of the upper regions of the static symmetry detection functions with those for symmetry in dynamic noise (Jenkins, 1983). Figure 3b shows the performance of the same observer who was shown in Figure 3a at criterion levels of 80% and 90% correct. Each datum point represents the outcome of 100 to 150 trials, which translates into a mean variation of about 0.1 of any given sensitivity level. The focus of the present paper is the variation of peak duration sensitivity for symmetry with eccentricity. It can be seen in Figure 3a that the peak sensitivity declines gradually with eccentricity, but that the decline is only about a factor of 2 by 10º, in contrast to the factor of about 14 expected for other position-discrimination tasks such as Vernier acuity (Levi et al., 1985). This behavior is considered further in “Quantitative Analysis.”

As expected from (Tyler, 1999), sensitivities were highest at each eccentricity when symmetry pattern halves were not separated by a random strip. At the high criterion levels of Figure 3b, the measured tunings widths for symmetry detection are narrower than they are at the low criterion level of Figure 3a. With increasing noise separations, duration sensitivities decreased rapidly toward the measurement limit of 0.056 (1.8 sec). The summation widths at half height of these functions (Figure 4) were also relatively stable for all observers across eccentricity. The mean summation width for the three observers at the eccentricities where all three were tested did not vary significantly (p > 0.1). Thus, the implication is that the symmetry structure in the most detectable patterns often extends a substantial distance from the symmetry axis. At the more demanding 90% level, the defining property is restricted to elements within about ±0.5º of the symmetry axis.

For symmetry detection in dynamic fields (Figure 5), the width tunings across eccentricity at a threshold criterion of 60% resemble those for static symmetry at this high percent correct (Figure 3) except that they tend to increase in width. Duration sensitivities show little variation in peak sensitivity for all three observers from 0º to 10º in the periphery. Half-height summation widths tended to broaden with increasing eccentricity (Figure 6), doubling from the very tight value of 0.15º with central fixation to about 0.3º at 10º eccentricity (increase significant at p < 0.05). There was no evidence for a secondary skirt to the function similar to that observed for static symmetry detection at 0º eccentricity and >1.5º noise gap (as found at the 60% level in Tyler et al., 1995).

The main point of interest in this study was to evaluate the change in sensitivity for the two types of symmetry detection task as a function of eccentricity. For conventional linear plots of duration thresholds against eccentricity, the data fell on curved trajectories, but switching to log duration versus linear eccentricity captured the curvature in the data within experimental error. These plots for sensitivity as a function of eccentricity derived from data such as those in Figures 3 and 5 are shown in Figure 7 (upper panels).

A second set of analyses was conducted for the criterion of 90% correct on the duration psychometric functions (Figure 7, lower panels). It may be seen that they fall on straight lines similar to those of the upper panels in Figure 7 but with higher duration thresholds and steeper log slopes in the eccentricity functions. The equation defining this behavior is

The data imply that duration sensitivities bear a logarithmic relationship to effective contrast sensitivity (as would be expected if they were governed by an exponential decay process) and perform the eccentricity analysis in these log-linear coordinates. In symbolic form, such a process would be defined by

and

Although this approach to the eccentricity functions captures the eccentricity variation in a simple manner, it complicates the issue of the doubling eccentricity for these tasks, because the power functions are accelerating with eccentricity. Nevertheless, we can readily derive the eccentricity for which the power fits would generate a doubling of sensitivity (Elog2), just as for the linear fits. The negative intercepts are very shallow in comparison to those for grating acuity.

Note that the error terms on the Elog2 values are given only in the decreasing direction because in the increasing direction the error could exceed infinity for the shallowest slopes. The decreasing direction is the one needed to distinguish the values from those for acuity.

Summary statistics for the mean behavior of the three observers are provided in Table 1and plotted in Figure 7. These fits reveal that there is a consistent decrease of about a factor of two in the detection threshold (that is, at 0º noise gap) for static symmetry detection relative to dynamic symmetry. Summation widths for static symmetry detection are larger than those for dynamic symmetry detection for each observer out to 5º, but by 10º, the respective summation widths converge. The slopes for the 60% cases are not significantly different from zero, while those for 90% case are significantly steeper (p < 0.05).

The Elog2 values of eccentricity at which duration sensitivity is doubled range from a high of 42º for the static symmetry at low sensitivity (60% correct) to a low of 8º for static symmetry at high sensitivity (90% correct), with the dynamic symmetry values falling between these extremes.

Expected E2s for contrast detection tasks are in the range of 3º to 5º (Levi et al., 1985), so those for the symmetry sensitivities are all significantly flatter than predicted (taking the criterion of 2 σ, p < 0.05) if this task was mediated by the local ganglion receptive field mechanisms implicated in contrast detection (Virsu & Rovamo, 1979).

Previous studies reviewed in the “Introduction” may be divided into two classes: those that assumed some eccentricity magnification function for their stimulus variation and found that symmetry detection was approximately equated by this manipulation, and those that assumed no magnification and found relatively minor changes in performance with eccentricity. Those that assumed a magnification do not provide strong support for this assumed scaling if symmetry processing is invariant with spatial frequency, because any assumed scaling would provide equal sensitivity. Taken together, these results suggest that symmetry processing is surprisingly robust to eccentric presentation, which provides the motivation for the finer-grained analysis of the present work. Given the apparent dissociation between processes mediating the low and high performance levels on the duration psychometric function (Tyler et al., 1995), eccentricity functions are evaluated for four diverse aspects of symmetry processing: 60% dynamic, 90% dynamic, 60% static, and 90% static performance.

The results show that observers were able to perform low-performance static and dynamic symmetry detection with approximately equal sensitivity from fixation to 10º in the periphery (a range that spans about half the extent of the cortical projection to area V1; Horton & Hoyt, 1991). This is a remarkable result considering that symmetry detection is essentially a position-mediated task in which the positions of the dots on either side of the symmetry axis must be compared for successful performance. Such position tasks typically have a very steep eccentricity scaling function (Levi et al., 1985) characterized by a negative intercept of about −0.6º. Similarly, Tyler and Hardage (1996) found that the duration sensitivity for symmetry detection in band-limited noise fell according to the -0.6º intercept. Other investigators using contrast or noise correlation thresholds, such as Saarinen (1988), Herbert and Humphrey (1993), and Barrett et al. (1999), have found the fall off with eccentricity to match the magnification function for luminance detection.

It may seem that the exponential fits of Figure 6 (straight lines on log-linear coordinates) would be liable to exaggerate the size of the negative intercepts due to the concave curvature of the analytic functions through the negative eccentricity range. The evident straightening of the curvature in the explicit curvature range is the chief argument for the exponential construct, but it has plausibility for the present temporal sensitivity analysis on the assumption that the signal effectiveness decays exponentially with increasing duration.

However, even if a linear extrapolation were assumed, the negative intercepts are still well outside the range of traditional acuity tasks. Linear extrapolations are generated by extrapolating the tangential slope at zero eccentricity to the zero sensitivity level (= infinite stimulus duration), the tangential slope being the lowest point at which the function is empirically defined. These linear-extrapolation negative intercepts range from −7.1º to −38.8º, evenly split between being larger and being smaller than the exponential extrapolations. So it is clear that the exponential assumption has not generated an overestimation of the negative intercepts. Even when ignoring the curved portion of the functions and focusing on the lowest levels by linear extroplation, the negative intercepts are truly much greater than those for grating acuity (and, a fortiori, for positional acuity tasks), regardless of the form of analysis.

It is therefore incontestable that different symmetry detection tasks evince different eccentricity scaling functions, ranging from almost flat to extremely steep. This diversity may be taken as evidence that the neural processing limiting symmetry perception in the various situations is operating at different levels of the cortical processing hierarchy. Tasks matching the luminance-detection scaling slope may be limited by the signal/noise ratio in the primary projection cortex. Tasks showing flat functions (as in the present study at the 60% correct level) or negative scaling (as in the width functions of Tyler, 1999) may be limited by the operation of a specialized symmetry detection mechanism that is constrained by factors other than the noise level of the input from primary cortex.

The present unscaled stimuli were made broadband in both energy spectrum and retinal location in order to allow the visual system to use any available mechanisms for symmetry detection. More colloquially, this study provides insight into the detectability of preferred symmetric designs such as Persian rugs. The data imply that duration sensitivities bear a logarithmic relationship to effective contrast sensitivity (as would be expected if they were governed by an exponential decay process) and validate the eccentricity analysis in these log-linear coordinates.