**We examined how well human observers can discriminate the density of surfaces in two halves of a rotating three-dimensional cluttered sphere. The observer's task was to compare the density of the front versus back half or the left versus right half. We measured how the bias and sensitivity in judging the denser half depended on the level of occlusion and on the area and density of the surfaces in the clutter. When occlusion level was low, observers in the front-back task were biased to judge the back as denser, and when occlusion level was high they were biased to judge the front as denser. Weber fractions decreased as density increased for both the front-back and left-right tasks, consistent with previous findings for two-dimensional density discrimination. Weber fractions did not vary significantly with area for the front-back task, but increased with area for the left-right task, and we attribute this difference to occlusions that have different effects in the two tasks. We also ran model observers that compared the image occupancies of the two halves against a known expected difference. As the occlusion level increased, this expected difference followed a similar trend as the biases of the human observers, with a roughly constant offset between them. Weber fractions for human and model observers followed some similar trends, but there were discrepancies as well that can be partly explained by the information available to human versus model observers in carrying out their respective tasks.**

*η*, namely the number of surfaces per cm

^{3}, and the area

*A*of each surface. The total number of surfaces in each scene was

*N*=

*ηV*, where

*V*is the fixed volume of the sphere. The value

*N*was rounded to the nearest integer. See Table 1 for the values of

*η*,

*A*, and

*N*that were used in the different conditions of the experiments.

*η*and

*A*in Table 1 were chosen such that their product

*λ*is called the

*occlusion factor*(Langer & Mannan, 2012). It is the expected total area of the surfaces in the clutter per cm

^{3}. The greater the value of

*λ*, the more occlusions tend to occur. Note that the area

*A*decreases and mean density

*η*increases as one goes down each column of Table 1, and these variations exactly cancel to keep

*λ*constant in each column. Also, the mean density

*η*increases and area

*A*is constant on the main diagonal (top left to bottom right), and area increases and mean density is constant on the cross-diagonal (bottom left to top right). These variations in

*λ*,

*η,*and

*A*will be indicated by three arrows in subsequent figures.

^{1}

*η*in the two halves were chosen separately for each mean density value

*η*, namely we defined nine density difference levels:

*Z*

_{0}= 58 and

*Z*= 82 cm from the virtual subject's position, that is, the center of projection. This depth range corresponds to the diameter of the sphere. The projection plane or display screen was defined at

_{max}*Z*= 70 cm, which was the center depth of the sphere. Perspective effects were present, but were relatively weak since many surfaces were partly occluded and so there was large variation in image sizes of visible portions of the surfaces.

_{mid}*X*-axis with an amplitude of 10° and a rotational velocity of ±10°/s. Each stimulus was presented for 4 seconds, namely two periods of motion. The motion gave a strong kinetic depth effect. Moreover, dynamic occlusions between surfaces typically specified their ordinal depth, and so there was less of a tendency for depth reversals than what one typically has in 3D cluttered scenes, namely if one uses small random dots. The motion may also help segment the front and back halves because the rotation yields opposite motion directions in the front versus back halves.

*λ*= 0.08], Supplementary File S2 [medium occlusion

*λ*= 0.04], and Supplementary File S3 [low occlusion

*λ*= 0.02]). The different density and area combinations in the figure correspond to the entries in Table 1. Figure 2a shows examples of the Experiment 1 stimuli for different densities in the two halves. Specifically, the four rows illustrate density differences

*Z*. For negative depth-luminance covariance (DLC−), the luminance was chosen to be proportional to

*Z*was the depth (cm) of the center of the square,

*Z*

_{0}and

*Z*were the near and far limits of the clutter, as defined earlier. We chose this power law based on a simple

_{max}*Y*value as the gray background. See the Appendix in Scaccia and Langer (2018) for more details on display calibration and linearization.

*A*and density

*η*and their product λ, and the sign of luminance variation across the volume (WB/BW or DLC−/DLC+).

*η*from trial to trial were chosen using a staircase procedure. We used a 1-up/1-down staircase with the nine stimulus levels Δ

*η,*which were described above in Equation 2. The staircases targeted a proportion of choosing each of the two halves in 50% of the trials, i.e., the point of subjective equality (PSE). For each of the four experiments, the staircases for the different conditions were randomly interleaved. Each staircase began at a randomly chosen Δ

*η*level from the set of nine levels and then terminated after 14 reversals.

*α*, slope

*β*and derived quantities JND and Weber fraction, which were defined as follows. Each staircase yielded the fraction of trials in which the subject chose the front half in Experiments 1 and 2 or right half in Experiments 3 and 4 as being more dense. We used the Palamedes toolbox (Prins & Kingdom, 2018) to fit a logistic function to these fractions

*x*is one of the nine density difference levels

*β*is the slope.

*x*−

*α*such that

*x*is the 75% threshold for choosing front or right, so

*Weber fraction*. For the front-back task, the denominator

*λ*, density

*η*, area

*A*, and the sign and type of luminance variation. For Experiments 3 and 4, we ran one-way ANOVAs for each of them rather than two-way ANOVAs that combined them, because these experiments had a different number and different set of subjects.

*p*values except if

*p*values are very small. A

*p*value smaller than 0.05 is considered to be significant.

*α*in Experiments 1 and 2 (front-back). There was a strong main effect for the occlusion factor

*λ*,

*F*(2, 174) = 59.11,

*p*< 10

^{−19}), namely a back bias (

*α*> 0) for small

*λ*, a near-zero bias for the middle

*λ*, and a front bias (

*α*< 0) for large

*λ*. There was also a main effect for the type of luminance variation,

*F*(1, 174) = 14.64,

*p*< 10

^{−3}, with a more negative mean bias for Experiment 2 than Experiment 1, i.e., a greater bias to see the front as denser for Experiment 2 than for Experiment 1. There was also an interaction between the factors of

*λ*and the type of luminance variation,

*F*(2, 174) = 4.28,

*p*= 0.015.

*λ*, we consider

*occupancy*fractions, that is, the fraction of the pixels that correspond to surfaces in each half. Figure 4a shows mean image occupancy fractions for the nine conditions of Table 1, specifically for the case of uniform density, i.e., Δ

*η*= 0. For each experiment and within each condition, there is little variation in these values from scene to scene. The error bars show the standard deviations multiplied by 10 to better illustrate their relative magnitudes.

*λ*, and the difference between the means increases as well. Another way to view this trend in the image occupancy of the two halves is to let Δ

*η*vary instead of just taking Δ

*η*= 0, and to ask the question: what would Δ

*η*need to be for the two halves to have the same image occupancy? The answer depends on the occlusion factor: for a larger occlusion factor

*λ*, the density difference Δ

*η*would need to be more negative. This effect is illustrated in Figure 3 by the black horizontal lines that show the level of density difference Δ

*η*that would yield equal image occupancy for the front and back halves for each of the conditions. Note how these Δ

*η*values decrease as the occlusion factor

*λ*increases.

*follow a similar trend. However, there is a crucial difference between these two trends, namely the observer biases are shifted relative to the density differences that give equal occupancy. In particular, the observer biases are closer to zero. This implies that observers are not merely judging the relative image occupancy of the two halves or the relative density of*

*α**visible*surfaces in the two halves, but rather they are judging the relative density of surfaces in the scene and taking account of the occlusion effects, albeit with a bias that depends on the amount of occlusion.

*λ*. Such effects are not surprising since the task in Experiment 2 is inherently more difficult than in Experiment 1, as discussed earlier in the Stimuli section, and the task is even more difficult when

*λ*is larger. Observers may change their bias when the task is more difficult, and possibly rely more on perceived image occupancy than on perceived density in that case.

*λ*effect was near significant,

*F*(1, 174) = 2.78,

*p*= 0.057, with Weber fractions decreasing as

*λ*increased. There was a strong main effect for the type of luminance variation,

*F*(2, 174) = 17.71,

*p*< 10

^{−4}), with greater Weber fractions for Experiment 2 (DLC) than Experiment 1 (WB). The greater Weber fractions in Experiment 2 is not surprising because the task in Experiment 2 is inherently more difficult.

*λ*, we examined specific combinations of density

*η*and area

*A*. There was a main effect for changes in density along the main diagonal conditions,

*F*(2, 54) = 6.34,

*p*= 0.003, and again a main effect for the type of luminance (Experiment 1 vs. Experiment 2),

*F*(1, 54) = 7.18,

*p*= 0.009. On the cross diagonal, there was no main effect from changes in area,

*F*(2, 54) = 0.87,

*p*= 0.42, but there was again a main effect for the type of luminance,

*F*(2, 54) = 8.78,

*p*= 0.004. There was also a main effect within columns,

*F*(2, 174) = 8.1,

*p*< 10

^{−3}, with lower Weber fraction as we move down each column where density increases and area decreases, as well as type of luminance effect,

*F*(1, 174) = 18.8,

*p*< 10

^{−4}. This result is consistent with the result on the main diagonal where density had an effect. We conclude that the near-significant Weber fraction effect of the occlusion factor

*λ*in both Experiments 1 and 2 was primarily due to a strong density effect, not to occlusion per se. We will discuss these results further in the Discussion section.

*λ*, neither for Experiment 3,

*F*(2, 87) = 2.55,

*p*= 0.08, nor for Experiment 4,

*F*(2, 51) = 1.8,

*p*= 0.18. However, there was a strong main effect for density (on the main diagonal where area was fixed and density varied) for both Experiment 3,

*F*(2, 27) = 24.3,

*p*< 10

^{−6}, and for Experiment 4,

*F*(2, 15) = 10.9,

*p*= 0.001. There was a weaker but still significant effect of area (on the cross diagonal where density was fixed and area varied) both for Experiment 3,

*F*(2, 27) = 5.24,

*p*= 0.01, and Experiment 4,

*F*(2, 15) = 7.6,

*p*= 0.005. The density and area effects were in opposite direction, with Weber fractions decreasing as density increased, and Weber fractions increasing as area increased. These opposing effects might be the reason why there was no significant effect from the occlusion factor

*λ*, which is the product of density and area. There was also a main effect within columns, both for Experiment 3,

*F*(2, 87) = 14.62,

*p*< 10

^{−5}, and Experiment 4,

*F*(2, 51) = 35.5,

*p*< 10

^{−9}, namely Weber fractions decrease moving down each column of Table 1 as density increases and area decreases. This is consistent with the density and area results found on the main and cross diagonals, respectively, namely that Weber fractions increased as density increased and as area decreased.

*t*-test. Recall that Experiment 3 had black and white separated in the front and back and Experiment 4 had black and white separated in the left and right halves. We suspected that Experiment 4 would be easier since subjects would not need to disentangle white versus black in each half. However, a one-tailed

*t*-test on the signed differences of the Weber fractions showed no significant difference,

*t*= 0.81,

*p*= 0.22.

*η*= 0 level in the two experiments were the same. The crucial difference between the two is how observers deal with the more challenging occlusion effects in the front-back task versus the left-right task. In particular, we expected performance to be worse in Experiment 1 since observers need to account for the different image occupancies of the front versus back, which are due to occlusion, whereas in Experiment 3 (and Experiment 4) observers could perform the left versus right task in principle by just comparing the overall image occupancy in the two halves. A one-tailed

*t*-test indeed showed that Weber fractions were much higher for Experiment 1 than Experiment 3 (

*t*= 6.95,

*p*< 10

^{−6}).

*α*= 0), namely it knows the

*expected*difference in pixel counts in the two halves. We refer to this expected difference in each condition as

*τ*. We define the front-back model observers to respond “front” when the number of front pixels exceeds the number of back pixels by this threshold

*τ*. We computed

*τ*for each condition in advance, namely it was the mean difference in the number of pixels from visible front versus back surfaces over 10,000 scenes for that condition and for the Δ

*η*= 0 level. These

*τ*values were computed using the data in Figure 4a, and the density differences that correspond to the

*τ*values are plotted as black lines in Figure 3. Similarly, the left-right model observers compare the number of pixels corresponding to surfaces in the left and right halves. In this case the expected difference for the two halves is zero (

*τ*= 0).

*λ*, which is constant within each column. However, the standard deviations of image occupancy are not constant within each column but rather they decrease from the top to bottom, typically by over 30%. The reason is that having a larger number of smaller surfaces drives the pixel counts of front and back surfaces to be closer to their expected values. (A similar effect was described in Figure 3 of Langer & Mannan, 2012.) It follows that the mean difference in the number of pixels of the front and back also will be constant within each column, and the standard deviations of the front-back difference will decrease from top to bottom within each column as well. Similarly, for Figure 4b, the means for left versus right are the same within each column, but again the standard deviation decreases moving down each column, and so the standard deviations of the difference will decrease moving down each column. Based on these observations about the standard deviations, we predicted that the model observer should be more sensitive to density differences moving down each column, both for the front-back task and the left-right task. We will see as follows that this prediction holds.

*X*-axis. Using a similar method as in Figure 4, we found that the standard deviations for the image occupancies for each half of the sphere and for each condition were reduced by approximately 40% in each condition for three-frame observers relative to the one-frame observers. We predicted that these reduced standard deviations in the image occupancy relative to the fixed differences in the mean image occupancies would decrease the model observer's Weber fraction.

*η*range for the model observer experiments. We defined density difference levels for the model observer, by dividing each of the human observer's stimulus levels by 10, so the model observer's levels were

*η*is larger and area

*A*is smaller, there is less variability in the number of front pixels and back pixels for each condition and so the observer can detect more reliably whether the difference in front versus back pixel counts is greater than the expected value

*τ*of that difference. Second, the three-frame model observer had lower Weber fractions than the one frame model observer. Thus, although the rotational motion itself (i.e., the image velocities of the surfaces) was not used by the model observer, the motion information did provide a significant benefit to the model observer, namely by providing more samples for comparing pixel counts in the two halves.

*λ*.

^{2}, an

*N*range of about 20 to 150 dots, and a fixed 10° diameter stimulus. Tsirlin et al. (2012) used a fixed density of about 20 dots/deg

^{2}, 3,000 dots, and a fixed stimulus size of about 13° × 13°. For our stimuli, the density range was approximately from 1.5 to 6 squares/deg

^{2}. Our smallest number of elements was 434 and our largest was 1,737, and the circle bounding our volume had a diameter of 20°. These values are thus in a similar range as studies already mentioned. This suggests that the back bias is robust to differences in densities, number of elements, and size of stimulus, as well as differences to the arrangement of the 3D stimulus (layers or volume), at least when the amount of occlusion is low.

^{2}, they found constant Weber fractions for lower densities and decreasing Weber fractions for higher densities, where the switch occurred at about 0.2 dots/deg

^{2}, depending on patch area. Anobile et al. (2015) subsequently showed that the switching point from constant to decreasing Weber fractions depends on eccentricity. For example, using centrally-presented patches of diameter 8° and presented in sequence, they found that the switch from constant to decreasing Weber fractions occurred at much higher densities, namely about 2 dots/deg

^{2}, and that the switching point decreased when the patches were presented more peripherally. Our stimuli had diameter of 20° and we did not control eye movements, and so they are a mix of central and peripheral presentation. Moreover, while our

*mean*2D densities varied from 1.5 to 6 elements/deg

^{2}, our 2D densities varied

*within*each stimulus, namely greatest at the center of the projected sphere and diminishing to zero at the circular edge. Overall though, our 2D densities and eccentricities were in the range that was similar to where Anobile and colleagues found decreasing Weber fractions as density increased, so our results on decreasing Weber fractions are consistent with their findings.

*λ*. The bias crossed over to the front when occlusion was higher. This dependence of bias on the level of occlusion has not been previously reported. We also found that these front and back biases did not depend on the density

*η*and the area

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^{1}Note the range of values of density and areas is larger in the central column, namely a factor of 4 instead of a factor of 2 range. A slightly cleaner design would have had (

*η*= 0.08,

*A*= 0.5,

*N*= 579) and (

*η*= 0.18,

*A*= 0.25,

*N*= 1,158) in the top and bottom elements of the middle column. This would have yielded a gradual change in all the parameters within the top and bottom rows as well.