To quantify the location of this most effective noise frequency range, a log Gaussian function was fitted to the data with the
lmfit package in Python. The function was defined as
where
b determined the upper asymptote of the function,
m the minimum value at the dip,
σ the width of the dip, and
μ the location of the dip. The parameter of interest was
μ, the location of the dip on the spatial frequency axis (i.e., the noise frequency that most strongly reduced White's illusion). This value was computed for all observers and all grating frequencies, and is plotted in
Figure 6. Two observers (observeres n2 and e1,
Supplemental Figures S2 and S8) did not show a clear effect of the noise, and one was extremely variable in her responses (observer n7,
Supplemental Figure S7), so they were excluded from this analysis. An additional observer (n4,
Supplemental Figure S4) only showed a clear noise effect at the two higher grating frequencies, so no function was fit to her low-grating frequency data. It is clear from these results that all observers for which a clear effect of the noise was measurable were most affected by noise in the range between 1 and 5 cpd. Furthermore, the most effective noise frequency increased with increasing grating frequency, but not proportionally. The mean slope across observers in
Figure 6 is 0.63 for both line segments. The 95% confidence intervals (bootstrapped with 10,000 trials with replacement) are [0.51, 0.75] for the lower segment and [0.52, 0.76] for the upper segment.