# Mirror Symmetry Is Subject to Crowding

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Participants

#### 2.2. Apparatus

#### 2.3. Stimuli

#### 2.4. Procedure

## 3. Results

^{2}= (sep − sep

_{min})(size − size

_{min})

_{size}) = separation at threshold (μ

_{sep}); i.e., μ

_{size}/μ

_{sep}= 1. In other words, the green circles in Figure 4 represent the sizes (or separations) corresponding to the grey dots in Figure 3 at eccentricities of 1 to 16° divided by the sizes (or separations) corresponding to the grey dots at fixation (0°); i.e., μ

_{E}/μ

_{0}. These limiting cases (${s}_{E}/{s}_{0}$ and μ

_{E}/μ

_{0}) show how the rectangular parabolas in Figure 3 shift up and to the right with eccentricity, respectively, within each panel.

_{E}/μ

_{0}. The y-axis in Figure 4 is labeled “Magnification Factor” because the curves represent the degree to which the functions shift up and to the right with eccentricity.

_{E}/μ

_{0}and the corresponding size at threshold by ${s}_{E}/{s}_{0}$. Once the data were shifted downward and leftward in this way, we found the best fitting rectangular parabola to the scaled data. Figure 5 shows the shifted data along with the best-fitting rectangular parabola. Also shown is the proportion of variability that the parabola explains. (Please note that our measure of fit is based on the diagonal distance from data point to curve in the log-log space shown [31]). Generally the fits were reasonably good and explained 72% of the variability in the data on average.

_{E}/μ

_{0}with $M=1+{k}_{H}E$ and ${s}_{E}/{s}_{0}$ with $M=1+{k}_{V}E$. Once the data were scaled we found the best fitting rectangular parabola. The results are not shown but they explained only 55% of the variability in the data on average, and are therefore substantially worse than the first analysis, which used μ

_{E}/μ

_{0}and ${s}_{E}/{s}_{0}$ to scale the data. Therefore, in spite of the seemingly nice fits of $M=1+{k}_{H}E$ to μ

_{E}/μ

_{0}and $M=1+{k}_{V}E$ to ${s}_{E}/{s}_{0}$ (r

^{2}≈ .97) these approximations leave a great deal of residual variability about the rectangular parabola fit to the scaled data.

^{αH}/β

_{H}and M = 1+E

^{αV}/β

_{V}respectively. We thought these non-linear functions might better fit μ

_{E}/μ

_{0}and ${s}_{E}/{s}_{0}$, respectively, and thus produce scaled data that are a better fit to a rectangular parabola. The obtained results explained 63% of the variability in the data and are thus substantially better than the second fit, yet still worse than for the original “raw fits”.

_{E}/μ

_{0}. It is disappointing that we are unable to find low-parameter fits to these scaling factors that are equally effective. It should be kept in mind, however, that the ${r}^{2}$ values reported above do not include variability attributable to eccentricity. That is, the ${r}^{2}$ values account only for the variability of the scaled data about the best fitting function. If the fits to the scaled data were “unscaled” by multiplying scaled size at threshold by ${s}_{E}/{s}_{0}$ and scaled separation at threshold by μ

_{E}/μ

_{0}, the resulting fits to the original data would have produced ${r}^{2}$ values that exceed .97, on average, in all cases.

## 4. Discussion

_{size}(or μ

_{sep}) for which μ

_{size}/μ

_{sep}= 1, reflect the changes in the maximum crowding effects with eccentricity; i.e., greatest extent of the crowding region or greatest elevation of target size at threshold. Figure 4 (green circles) shows that the magnitude of crowding increases with eccentricity at a faster rate than changes in size thresholds in the unflanked conditions. Using the best-fitting linear fits to these data gives a mean ${\overline{k}}_{H}$ = 2.21 (SEM = 0.16), which is about 2.8 times greater than ${\overline{k}}_{V}$ = 0.79. Therefore, in the case of symmetry discrimination we find that crowding effects increase with eccentricity at a greater rate than unflanked size-thresholds, as found in several previous studies employing different stimuli and tasks [22,23,25]. Clearly, symmetry is not exempt from crowding effects.

_{E}/μ

_{0}) for grating and T orientation discrimination tasks reach levels of 54 and 58, respectively at 16° in the LowerVF, whereas for letter discrimination this ratio reaches only 33 at 16°. In the present tasks, averaged over observers and conditions we find that the μ

_{E}/μ

_{0}reaches 38 at 16°. Interestingly, we found ${\overline{k}}_{V}$ = 0.79 in the present symmetry task, 0.759 in the letter discrimination task and 0.54 and 0.57 in the grating and T orientation tasks. Therefore, in terms of these metrics the symmetry tasks studied here have more in common with the letter identification task than the grating and T orientation discrimination tasks.

## 5. Conclusions

## Acknowledgements

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**Figure 1.**Trigrams used for Experiment 1. The central element (target) was a symmetrical patch that could have a vertical (shown) or horizontal (not shown) axis of symmetry. The flankers could be above and below or to the left and right of the target. The entire configuration (trigram) could be parallel to or perpendicular to the line that connects the central target to the point of fixation. Note, the target and flankers were presented on a uniform grey background.

**Figure 2.**Size at threshold as a function of relative target/flanker separation at each eccentricity. Rows 1 and 2 plot results for two subjects and row 3 plots mean results. The four columns represent the four combinations of flanker positions with respect to the target (H and V) and position in the visual field (RightVF and LowerVF). Symbol colors represent eccentricities of 0 to 16°, as shown in the legend. Circles represent target/flanker separations of 1.25 to 8 times target size at each eccentricity. Triangles represent the unflanked condition at each eccentricity; these are depicted at 16 times target size for illustration.

**Figure 3.**Size at threshold as a function of the target/flanker separation at threshold. (Target/flanker separation at threshold = target size at threshold * relative separation.) Rows 1 and 2 plot results for two subjects and row 3 plots mean results. The four columns represent the four combinations of flanker positions with respect to the target (H and V) and position in the visual field (RightVF and LowerVF). Circles represent target/flanker separations of 1.25 to 8 times target size at each eccentricity. Triangles represent the unflanked condition at each eccentricity; these are depicted at 16 times target size for illustration. The continuous curves show the best fitting rectangular parabola; see text for details. The small gray point at the upper end of each curve represents the point on the curve for which size at threshold equals separation at threshold.

**Figure 4.**Horizontal and vertical magnification factors. The x-axes represent eccentricity in degrees visual angle. The blue circles represent target size at threshold in the unflanked conditions divided by target size at threshold at fixation (0°). These ratios show how the curves in Figure 3 shift upwards with eccentricity. The green circles represent sizes (and separations) for which size = separation at threshold divided by the same quantity obtained at fixation. These ratios show how the curves in Figure 3 shift rightwards with eccentricity. ${M}_{V}=1+{k}_{V}E$ and ${M}_{H}=1+{k}_{H}E$ are linear approximations to these vertical and horizontal shifts. In each panel ${k}_{V}$ and ${k}_{H}$ that best fit the data are shown along with the proportion of variability (${r}^{2}$) they explain in the data.

**Figure 5.**Scaled target size at threshold vs. scaled target/flanker separation at threshold. (See text for details.) The four columns represent the four combinations of flanker positions with respect to the target (H and V) and position in the visual field (RightVF and LowerVF). Symbol colors represent eccentricities of 0 to 16°. Circles represent target/flanker separations of 1.25 to 8 times target size at each eccentricity. Triangles represent the unflanked condition at each eccentricity; these are depicted at 16 times target size.

**Figure 6.**Effect target/flanker configuration in the RightVF and LowerVF. The ratio of size threshold for parallel flankers to perpendicular flankers is shown as a function of relative separation for the RightVF and LowerVF. The ratios have been formed from the average data (row three) from Figure 2.

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Roddy, G.; Gurnsey, R.
Mirror Symmetry Is Subject to Crowding. *Symmetry* **2011**, *3*, 457-471.
https://doi.org/10.3390/sym3030457

**AMA Style**

Roddy G, Gurnsey R.
Mirror Symmetry Is Subject to Crowding. *Symmetry*. 2011; 3(3):457-471.
https://doi.org/10.3390/sym3030457

**Chicago/Turabian Style**

Roddy, Gabrielle, and Rick Gurnsey.
2011. "Mirror Symmetry Is Subject to Crowding" *Symmetry* 3, no. 3: 457-471.
https://doi.org/10.3390/sym3030457