From 6 to 8 different undergraduates, all with 20/20 vision or better, participated in each condition. Subjects gave informed consent before the experiment and were permitted to leave at any time, although none did. They were told that they would evaluate the number of randomly-shown disks that were presented to them under various conditions of brightness and contrast. The protocol was approved by the institutional review board (IRB) of Northeastern University.

Stimuli were generated using Matlab in conjunction with Psychtoolbox (Brainard, 1997) and presented on a Dell LCD monitor viewed at 60 cm. The stimuli consisted of disks (0.5° across) distributed pseudo-randomly over a square patch of 10° on each side (as in Figure 1A). The patch was divided into a 12 by 12 grid of imaginary cells, and the disks were placed in a subset of these cells with jittering. In intermingled displays, two sets of disks with different contrasts were intermingled in a single patch presented at the center of the screen. In segregated displays, only one set of disks was presented, also in the same 10° square patch. Since the disks were small relative to the field, disk contrasts were defined in Weberian rather than in Michelson terms (Peli, 1996) as δL/L, where L is the background luminance and δL is the luminance difference between the disk and L. White (74.0 cd/m²) and gray (26.1 cd/m²) disks were presented on a dark gray field of 18.2 cd/m². Thus, both sets of disks had positive contrasts, these being 3.11 (white) and 0.43 (gray).

The set size (N) is the number of disks in the target (comparison) set, that is, the set to be reported. N was always randomized over trials, and varied from 30 to 70 when blocked or from 20 to 80 otherwise. Thus, the total number of disks in intermingled displays was N + 50 disks, as the standard set contained 50 disks.

There were two main conditions, segregated and intermingled. In the segregated case, the disk display in each trial comprised either gray or white disks. The set size, N, varied randomly from N = 20–80 in steps of 10. Set size and target color (gray or white) were both randomized across trials, so subjects did not know which to expect. In the intermingled blocked condition, gray and white disks were intermingled. Trials with gray disks as targets and trials with white disks as targets were run in separate blocks of 45 trials. Four of the 8 subjects completed two gray trial blocks first and then two white blocks, while the other four were run in the opposite order. Set size, N, was (untypically) randomized from N = 30–70 in steps of 5.

In the intermingled randomized condition, subjects were pre-cued (8 subjects) or post-cued (7 other subjects) to judge the number of either the gray disks or the white disks in each intermingled display. The target set size (the number of disks to be judged) varied from N = 20–80 in steps of 10. There were always 50 disks of the other color to be ignored, so if the targets were gray, then N gray disks were intermingled with 50 white ones, and if the targets were white, then N white disks were intermingled with 50 gray ones. Whether the target set was gray or white was randomized across trials.

A post-cue intermingled display with white target is illustrated in Figure 3. In a pre-cue trial, the fixation cross would have turned white (or gray) for 1 s before the display. Timing was otherwise the same in all conditions; a fixation cross was presented for 1 s, followed by a disk display for 1.5 s. Subjects input their numerosity estimate after each trial by using the keys on a standard number pad. The inter-trial interval was 3 s.

Subjects were told to evaluate the numerosity of the target set while ignoring the non-target set. They were not told the other (non-target) set in the intermingled display had a fixed numerosity (50).

Set size and target disk color were within-subject variables. Each set size (N) was repeated 10 times for each combination of N and two disk colors (white or gray), for each subject. Thus, subjects each ran 140 trials when the step size was 10 and 180 trials when the step size was 5. The trials were divided into four blocks, with 35 or 45 trials in each block. Subjects could rest between blocks. They completed 10 practice trials without feedback for each disk color, to become familiar with the procedure. Subjects were then run for about 20 min. in the main experiment, with short breaks between blocks. Each trial lasted 6–7 s.

A 2 (disk color) × 7 (disk number) two-way repeated-measures ANOVA was performed on the mean numerosity estimates from 6 subjects. Judged numerosity increased significantly [F_{(6, 30)} = 68.59, p < 0.01] over the 7 set sizes (N), as shown in Figure 4. As predicted by contrast constancy, the gray disks were no more numerous than white ones, the effect of disk color being insignificant [F_(1,5) = 3.73, p = 0.11]. There is a hint in Figure 4 that 80 gray disks appeared more numerous than 80 white ones, but the color-number interaction was not significant [F_(6,30) = 1.18, p = 0.34].

One of the 8 subjects run with a pre-cue was an obvious outlier and was excluded. The mean numerosity judgements of the remaining 7 subjects is plotted in Figure 5, top. Numerosity again increased with set size [F_(6,36) = 45.88, p < 0.01]. The numerosity of gray disks was overestimated relative to the white disks [F_(1,6) = 30.88, p < 0.01] equally over the whole range of set size (N), there being no interaction [F_(6,36) = 0.71]. (The outlier subject also overestimated the numerosity of gray disks, but to a much-exaggerated degree). Thus, the numerosity illusion persisted even when a pre-cue permitted subjects to attend fully to the target disks.

The mean numerosity judgements of 7 subjects are plotted in Figure 5, middle. Numerosity again increased significantly with set size [F_(6,36) = 64.21, p < 0.01]. Gray disks were judged as more numerous than white ones [F_(1,6) = 63.31, p < 0.01], again equally over the whole range of set sizes, there being no interaction with N [F_(6,36) = 0.65]. This result indicates that the illusion remains even when the subject must retain both sets and is directed to attend to one or other set only in short-term memory.

The mean numerosity judgements of 8 subjects are plotted in Figure 5, bottom, against set size, N. Numerosity again increased over the 9 set sizes [F_(8,56) = 50.18, p < 0.01]. Gray disk numerosity was overestimated relative to the white disks [F_(1,7) = 6.80, p = 0.04], again with no interaction [F_(8,56) = 0.64].

The grand mean absolute numerosities in each condition are given in Table 1, along with the standard error of the difference between gray and white disks. The mean number presented, N, was 50 in all conditions. The grand mean numerosity reported was 31.3. The lower overall report in the post-cue condition (27.1, compared to 32.9 in the remaining conditions) may reflect a loss in short-term memory.

When segregated, the difference in numerosity (“Diff” in Table 1) of 1.15 in favor of gray over white disks was not significantly different from zero. When intermingled, the mean Diffs of 3.06, 3.49, and 2.71, for the blocked, pre-cue and post-cue condition, respectively, were all significant by t-tests (p < 0.05), their grand mean of 3.08 representing an illusion magnitude of 6.16%.

If contrast normalization worked in all cases, contrast constancy would occur in intermingled as well as segregated displays, and the numerosity illusion would not occur. However, it does occur. Lei (2015) and Lei and Reeves (2018) explained why the illusion does occur, such that the “weak conquer the strong,” by postulating that total contrast, C, depends on difference between the disk luminance and the next lower luminance. Such luminance differences may aid discrimination of stimuli processed by the same visual channel when no other cues exist to do so (Morgan et al., 2014).

For convenience, we refer to this luminance-difference contrast normalization hypothesis by an acronym, LDCN. Everything that follows is written for the normal polarity displays, to simplify symbols, but LDCN applies equally to reverse polarity displays except that the comparison is now to the next higher luminance.

According to LDCN, the contrasts δG/L of the weaker (gray) disks in intermingled displays (e.g., Figure 1A) are determined by the luminance difference between them and the field, L, that is, δG/L = (G-L)/L for gray disk luminance G, since the field has the next lower luminance. Thus, the LDCN contrast of the gray disks equals their Weber contrast both when the (weaker) gray disks are intermingled with (stronger) white disks and when they are segregated. Thus, gray-disk numerosity should be the same—predicting no illusion for gray disks in either normal or reversed polarity intermingled displays, as was found by Lei (2015) and Lei and Reeves (2018).

The contrasts of the white (or black) disks, when intermingled, however, are determined by the luminance difference between them and the gray ones, these having the next lower luminance. Thus, the LDCN total contrast of the white disks is less when intermingled than when segregated. However, the normalization factor, c = δW/L, does not change, as the peak-trough difference equals the luminance difference between the white disks and the field in both cases. Therefore, the normalized contrast of white intermingled disks is less than that of white segregated disks, and so numerosity constancy is violated in the observed direction (Lei and Reeves, 2018).

Denote the LDCN contrast difference by d = |δW/L - δG/L|, where δW/L is the Weber contrast (c) of the white disks and δG/L is the Weber contrast of the gray disks. Then

the factor d/c dictating the size of the illusion. Absent gray disks, d = c and Equation 3 reduces to Equation 2, the equation for segregated displays.

A prediction of LDCN is that the illusion should disappear for intermingled displays of opposite-polarity displays (white and black disks on a gray field), energy being the same as it would be in segregated displays. However, the illusion should return if the white and light gray disks on a dark field, which both have positive polarity, are replaced by black and dark gray disks on a light field, which both have negative polarity. Indeed the extent of the illusion should be the same for negative as for positive polarity displays, since E is unaffected by polarity. Both of these predictions of LDCN held up (Lei and Reeves, 2018).

A more subtle implication of the LDCN hypothesis is that reducing the luminance difference between the stronger and the weaker elements in an intermingled display should increase the numerosity illusion, since d is proportional to the luminance difference, but c is not. Indeed, Lei and Reeves (2018) reported that the illusion in a positive contrast display increased from a 16% reduction in the numerosity of the white elements to 27% by raising the luminance of the light gray elements toward that of the white ones – while keeping them visually distinct.

Since by hypothesis the numerosity of the stronger elements is reduced by the weaker elements in the same display, any manner of visually segregating the weaker from the stronger elements should suffice to eliminate the illusion by providing another cue to distinguish them (Morgan et al., 2014). Experiments were therefore run in Lei (2015) to test whether segregating the displays, not only spatially or by contrast polarity, as in Lei and Reeves (2018), but also by depth, shape, density, and duration, eliminates the numerosity illusion. In brief, segregating the elements always eliminated the illusion and returned numerosity constancy, as predicted.

Lei (2015) and Lei and Reeves (2018) were able to reject several other explanations of the illusion, including misclassification (counting white disks as gray ones more often than vice-versa), occlusion (by white disks of implicit gray ones), bounding contour (gray disks seeming to enclose more space than white ones, so appearing more numerous), and adaptation (the subject adapting faster to the more salient disks, and this lowering their numerosity). The absolute numerosity data shown above also reject the pure selective attention hypothesis.

Note that the LDCN contrast of the intermingled white disks, d = |δW/L - δG/L|, equals the sum of the (positive) white disk Weber contrast and the (negative) gray disk Weber contrast, that is, gray disk assimilation. Dresp-Langley and Reeves (2012) reported both contrast and assimilation occur for isolated gray patches on a lighter or darker gray field. In an important modeling exercise, Rudd (2010) showed that disk lightness in general depends on a weighted difference between disk contrast and assimilation due to a surrounding ring, δW/L - wδG/L, the weight (w) depending on attention. Our displays are not of the disk/ring variety, but yet may obey Rudd's rule. In the definition of LDCN used in Equation 3, w = 1, but below we discuss fits in which w <1, that, assimilation is weighted less than contrast.

We note here a limitation of Equation 3; there is no term for adaptation. However, Grasso et al. (2022) found that the numerosity of display of 29 colored items at constant luminance was reduced by adaptation to the same color, such that 34 items were matched to 29 items, a drop of 17%, while being uninfluenced by adaptation to a different color. Critically, color assimilation was used to separate perceived from physical color, and only the former adapted numerosity. We do not know if such adaptation would alter the extent of the numerosity illusion, and if so, how Equation 3 should be modified.

The absolute numerosity judgements in both segregated and intermingled displays can be fit by linear functions of √N with high precision (r > 0.98). The grand mean offset (B) applied to √N is 2.02, suggesting subitizing below N = 5, as reported before (e.g., Liss and Reeves, 1983). This reinforces LDCN but does not exclude other possible laws, such as Weber's law (Jevons, 1871; Dehaene, 2003) or a different compressive power law (Krueger, 1972, 1984), as the range of N is limited. Indeed, the current data fit the Weber law well (r = 0.987). However, the fits in Figures 6, 7, and a similar fit to the pre-cued absolute numerosity estimation data of Krueger (1984), for which J = 11.4(√N-2.74), 10 < N <200, r = 0.992, show that a √N law is a candidate.

LDCN contrast is an untypical measure but is analogous to an idea of Peli (1996), that the perceived contrasts in each spatial frequency band are determined by the amplitudes in that band compared to the amplitudes in the next lower band (only the very lowest band being contrasted against the field). Here the sizes, densities and distributions of the white and gray disks placed them all in the same spatial frequency band, so this is only an analogy, but we mention it as an alternative to defining contrast relative to the field.

The importance of contrast energy in determining numerosity is clear in our conditions in which disks were randomly located. Once displays are structured, however, additional factors come into play. For example, connecting pairs of elements by thin lines reduces numerosity as if some pairs are bound into single units (He et al., 2009). Grouping elements can alter perceived numerosity (Ginsburg, 1976, 1991; Im et al., 2016; Poom et al., 2019), though not always (Moscoso et al., 2021). It appears that judgements of numerosity rely not only on the low-level signal provided by total contrast energy, but also, like all other percepts, on higher-level Gestalt factors such as crowding (Anobile et al., 2015) grouping and connectivity, factors which we ignore here.

According to LDCN, the values of A and B obtained in the segregated displays should apply to the intermingled conditions with d/c multiplying √N. However, a fit of Equation 3 using the best-fit value of A, namely 7.00, when B was constrained to its grand mean of 2.02, badly over-predicted the illusion for pre-cued intermingled white disks when d/c was calculated based on the unweighted white and gray disk contrasts; the mean illusion, in fact 3.49, was predicted to be 6.30.

Therefore, Rudd's (2010) rule was applied in which the balance of contrast to assimilation is weighted by w, such that the contrast difference d, originally (W-G)/L, now equals (W-wG)/L = δW/L - wδG/L. The pre-cued intermingled white disks were least-squares best-fit with w = 0.43, so that d/c = 0.939, and the blocked intermingled white disks with w = 0.81, so that d/c = 0.885, with parameters A = 7.00 and B = 2.02 fixed. The outcome is shown in Figure 8. The fits are excellent (r > 0.995), but there is no independent evidence to determine the values of w, as might come from a separate assessment of the extent of assimilation. However, it is suggestive that assimilation in the blocked condition outweighed that in the pre-cue condition, as if the pre-cue permitted greater selection of the white disks and hence a reduced influence from the gray disks.