How the window of visibility varies around polar angle

Abstract

Contrast sensitivity, the amount of contrast required to discriminate an object, depends on spatial frequency (SF). The contrast sensitivity function (CSF) peaks at intermediate SFs and drops at other SFs. The CSF varies from foveal to peripheral vision, but only a couple of studies have assessed how the CSF changes with polar angle of the visual field. For many visual dimensions, sensitivity is better along the horizontal than the vertical meridian and at the lower than the upper vertical meridian, yielding polar angle asymmetries. Here, for the first time, to our knowledge, we investigate CSF attributes around polar angle at both group and individual levels and examine the relations in CSFs across locations and individual observers. To do so, we used hierarchical Bayesian modeling, which enables precise estimation of CSF parameters. At the group level, maximum contrast sensitivity and the SF at which the sensitivity peaks are higher at the horizontal than vertical meridian and at the lower than the upper vertical meridian. By analyzing the covariance across observers (n = 28), we found that, at the individual level, CSF attributes (e.g., maximum sensitivity) across locations are highly correlated. This correlation indicates that, although the CSFs differ across locations, the CSF at one location is predictive of that at another location. Within each location, the CSF attributes covary, indicating that CSFs across individuals vary in a consistent manner (e.g., as maximum sensitivity increases, so does the corresponding SF), 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 isoeccentric locations around the visual field.

Figure 1.

Experimental schematics and procedure. (A) CSF analysis. The CSF depicts how contrast sensitivity depends on 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-CS (maximum sensitivity), peak-SF (the most preferred SF), and bandwidth (closely related to CSF shape). Bandwidth was fixed across locations (see Methods). Results of the cutoff-SF (highest discernable SF) and AULCSF (total sensitivity) are reported in Supplementary Appendix. 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 meridian (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 (clockwise or counterclockwise). 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 was 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 covary 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). 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). The fewer the correlations, the greater the variability in the pattern of individual differences (E, pink panel, bottom, 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 Supplementary Figure A2. (B) Peak contrast sensitivity. (C) Peak spatial frequency. Results for cutoff-SF and AULCSF are shown in Supplementary Figure A3. *p < 0.05, **p < 0.01, ***p < 0.001. Error bars are ±1 SEM. 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 covariations of cutoff-SF and AULCSF are shown in Supplementary Figure A4, and confidence intervals are in Supplementary Figure A7. For observers’ data examples, see Supplementary Figure A5. **p < 0.01, ***p < 0.001.

Figure 4.

Covariance of CSF attributes within each location (pink cells in Figures 1D and 1E). 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. The insets demonstrate the 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 Supplementary Figure A6. Confidence intervals are shown in Supplementary Figure A7. *p < 0.05, **p < 0.01, ***p < 0.001. Black and white letters are for visibility.