Untypical Contrast Normalization Explains the "Weak Outnumber Strong" Numerosity Illusion

Abstract

Less salient, lower contrast disks appear to be more numerous than more salient, higher contrast disks when intermingled in equal numbers into the same display (Lei and Reeves, 2018), but they are equal in perceived numerosity when segregated into different displays. Comparative judgements indicate that the apparent numerosity of the lower contrast disks is unaffected by being intermingled with high contrast disks, whereas the high contrast disks are reduced in numerosity by being intermingled with the low contrast ones (Lei and Reeves, 2018). Here, we report that this illusion also occurs for absolute judgements of the numerosities of displays of from 20 to 80 disks. A model based on luminance-difference contrast normalization (LDCN) explains the illusory loss of high-contrast (salient) items along with veridical perception of the low-contrast ones. The model correctly predicts that perceived numerosity is linearly related to the square-root of the number of disks, with the extent of the illusion depending on an attentionally-weighted function of contrast and assimilation.

Keywords: contrast; contrast-dependent numerosity illusion; illusion; model; numerosity perception; segregation.

Figure 1

Example stimulus and psychometric function. (A) An example disk array in a discrimination task. The standard disk set (here, the white ones) had a fixed numerosity of 50 while the numerosity of the comparison disk set (here, gray) varied in a range of 30–70. In this example, both sets had a numerosity of 50. (B) An example psychometric function from one subject plots the probability of the gray disks being chosen as more numerous against their physical numerosity. The dots represent one subject's data and the curve the best-fitting Weibull function. The PSE, 45.7, indicates how many gray disks matched 50 white disks in perceived numerosity, implying that the white disks were under-estimated relative to the gray ones.

Figure 2

Latencies for reporting which set of intermingled disks, white or gray, were more numerous. Mean RTs were in the range of 0.8–1.2 s, typical for estimation and far faster than counting. The standard set contained 50 disks. Bars show ±1 SE calculated between subjects; confidence intervals are 28% larger than the SEs.

Figure 3

Post-cue intermingled display. A verbal prompt as to which disk set to enumerate was given after the display until the subject entered his or her response.

Figure 4

Mean numerosity of segregated displays of 20 to 80 all white or all gray disks. Bars show +1 between-subjects SEs for the gray disks and −1 for the white ones; confidence intervals are 28% larger than the SEs.

Figure 5

Mean judged numerosity vs. set size (N) when disk sets were intermingled. i.e., both white and gray disks appeared together on each trial. Top: the target set (i.e., the set to be judged) was randomized but pre-cued. Middle: the target set was randomized but post-cued. Bottom: target set was blocked. Error bars: +1SE for gray disks, −1SE for white disks.

Figure 6

Mean judged numerosity (J) vs. √N when disk sets were segregated; i.e., only white disks (Jw) or only gray disks (Jg) appeared on each trial.

Figure 7

Mean judged numerosity vs. √N when disk sets were intermingled. Top: the set to be judged was randomized and pre-cued on each trial. Middle: the set to be judged was randomized and post-cued. Bottom: the set to be judged was blocked. Lines show linear best-fits to √N. These fits are empirical.

Figure 8

White numerosity when segregated (“seg”), as predicted by Equation 2, and when intermingled and either pre-cued or blocked, as predicted by Equation 3. Data are shown by dots and predictions by straight lines. Parameters A = 7.00, B = 2.02 were first fit to judgements of the segregated W disks. Parameters w = 0.43 and w = 0.81 were then best fit to judgements of the pre-cued and blocked intermingled W disks, with A and B unchanged. The different weights in different conditions of attention illustrate how attention can be captured in the model expressed by Equation 3. However, direct evidence that attention can alter the effective contrast energy that underpins the numerosity illusion is lacking, so this finding remains suggestive rather than conclusive; further evidence would be required to confirm it.