How the window of visibility varies around polar angle

Abstract

Contrast sensitivity, the amount of contrast required to detect or discriminate an object, depends on spatial frequency (SF): The Contrast Sensitivity Function (CSF) peaks at intermediate SFs and drops at lower and higher SFs and is the basis of computational models of visual object recognition. The CSF varies from foveal to peripheral vision, but only a couple studies have assessed changes around polar angle of the visual field. Sensitivity is generally better along the horizontal than the vertical meridian, and better at the lower vertical than the upper vertical meridian, yielding polar angle asymmetries. Here, we investigate CSF attributes at polar angle locations at both group and individual levels, using Hierarchical Bayesian Modeling. This method enables precise estimation of CSF parameters by decomposing the variability of the dataset into multiple levels and analyzing covariance across observers. At the group level, peak contrast sensitivity and corresponding spatial frequency with the highest sensitivity are higher at the horizontal than vertical meridian, and at the lower than upper vertical meridian. At an individual level, CSF attributes (e.g., maximum sensitivity, the most preferred SF) across locations are highly correlated, indicating that although the CSFs differ across locations, the CSF at one location is predictive of the CSF at another location. Within each location, the CSF attributes co-vary, indicating that CSFs across individuals vary in a consistent manner (e.g., as maximum sensitivity increases, so does the SF at which sensitivity peaks), but more so at the horizontal than the vertical meridian locations. These results show similarities and uncover some critical polar angle differences across locations and individuals, suggesting that the CSF should not be generalized across iso-eccentric locations around the visual field. Our window of visibility varies with polar angle: It is enhanced and more consistent at the horizontal meridian.

Figure 1.. Experimental schematics and procedure.

(a) CSF (Contrast Sensitivity Function) analysis. The CSF depicts how contrast sensitivity depends on spatial frequency (SF). The inverse of the contrast threshold (c⁻¹) is contrast sensitivity (S) at a particular SF. Bayesian inference was used to estimate the CSF parameters — peak contrast sensitivity (peak-CS: maximum sensitivity), peak spatial frequency (peak-SF: the most preferred SF), and bandwidth (closely related with CSF shape). Bandwidth was fixed across locations (see text and Methods), and results of the cutoff spatial frequency (cutoff-SF: the highest discernable SF) and area under the log CSF (AULCSF: total sensitivity) are reported in Supporting Information. Higher values of these attributes are related to higher sensitivity, better performance, and/or a wider range of visible SFs and contrasts. (b) Polar angle asymmetries. In many visual dimensions, performance is higher at the horizontal than the vertical (HVA), and at the lower than the upper vertical meridian (VMA). (c) Task. A test Gabor stimulus was presented briefly at one of the four polar angle locations, and participants judged its orientation (CW or CCW). SF and contrast of the stimulus varied on each trial. (d) Schematics of covariance analysis. The covariance of the three CSF attributes (peak-CS, peak-SF, bandwidth) for all three locations is a 9 × 9 matrix. The covariation of locations for each attribute (blue cells), and the covariation of attributes within each location (pink cells) were assessed. Covariance for bandwidth is shaded in gray because the bandwidth was identical across locations for each observer in the HBM and thus, the covariance is not informative in most cases (but see Figure 4). (e) Interpretation of covariance analysis. Covariance of locations for each attribute (d, blue cells) tests whether CSFs and their corresponding key attributes co-vary as a function of polar angle. For example, a perfect positive correlation (coefficient of 1) for peak-CS among locations X and Y, indicates that an observer with a higher peak-CS at location X than other individuals also has a higher peak-CS at location Y. (e, blue panel top row). This relation does not hold for a scenario where there is no correlation (coefficient of 0) for any CSF attributes between locations X and Y. Covariance within each location (d, pink cells) allows investigation of how individuals’ CSFs relate to one another within each location. A perfect positive covariation (correlation coefficient of 1) of all combinations of attributes within a location indicates that observers’ CSFs are organized diagonally in the SF-contrast space: the higher the peak-CS, the higher the peak-SF, and the wider the bandwidth (e, pink panel, top row). The fewer the correlations, the more variability in the pattern of individual differences (e, pink panel, bottom row is a scenario where no correlations are present among CSF attributes within a location).

Figure 2.. Polar angle sensitivity differences.

(a) CSFs. An example participant’s data are shown in Figure S2. (b) Peak contrast sensitivity. (c) Peak spatial frequency. Results for cutoff-SF and AULCSF are shown in Figure S3. ***p < 0.001, **p < 0.01, *p < 0.05. Error bars are ±1SEM. HVA and VMA denote Horizontal-Vertical Anisotropy and Vertical Meridian Asymmetries, respectively.

Figure 3.. Covariation of each CSF attribute across polar angle locations.

(a) Peak contrast sensitivity. (b) Peak spatial frequency. Each cell corresponds to a blue cell in Figure 1d. The covariation of cutoff-SF and AULCSF are shown in Figure S4, and confidence intervals are in Figure S7. For observers’ data examples, see Figure S5. ***p < 0.001, **p < 0.01.

Figure 4.. Covariance of CSF attributes within each location (pink cells in Figure 1d–e).

Correlation between pairs of attributes at the (a) Horizontal, (b) Upper vertical, and (c) Lower vertical meridian. Note that here, the bandwidth is informative because the bandwidth varies as a function of individual observer (it is fixed only across locations). The color of each cell corresponds to the strength of correlation. Inset demonstrates schematics of simulated individual CSFs at each location, assuming that the significant/non-significant correlations are perfect/null correlations (coefficient of 1/0) for simplicity. Each cell corresponds to a pink cell in Figure 1d. The covariation of all attributes including cutoff-SF and AULCSF are shown in Figure S6. Confidence intervals are in Figure S7. ***p < 0.001, **p < 0.01, *p < 0.05. Black and white letters are for visibility.