StarTrek Illusion—General object constancy phenomenon?

Abstract

We report a new powerful type of contrast and size illusion caused by apparent motion in depth, which we called the StarTrek illusion. We found that an optic flow pattern consistent with objects moving in depth strongly modulates their apparent contrast. Disks that appeared to move away from the observer appeared to grow higher in contrast and larger, while their retinal size and contrast remained constant. We explain this illusion of contrast by the term contrast constancy: Normally, objects lose their contrast when viewed from far away, but when this expected loss does not happen, the brain infers that the physical contrast of the object increases as the object moves away. This is perceived as the illusory increase of the object's contrast. The contrast constancy is largely analogous to the well-known size constancy phenomenon. We discovered that the two phenomena are related. By adjusting the size of the disks during the optic flow motion, the illusory contrast increase could be easily canceled or even reversed. On the other hand, the illusory size increase could not be manipulated the same way by contrast modulation. Our results suggest that the brain may use the same scaling factor to account for the size and contrast change with distance and that the estimated object size affects the contrast calculation.

Introduction

One of the amazing abilities of our visual system is to perceive unchanging objects as they are: unchanging, no matter how extensively their environment varies or our point of view changes. A well-known phenomenon that manifests the perceived stability of the visual world is size constancy. Size constancy can be observed in three-dimensional scenes or in photographs and drawings (Boring, 1964) as long as depth in the scene is perceived. Leibowitz, Brislin, Perlmutrer, and Hennessy (1969) found that different depth cues could affect the magnitude of size constancy. They measured the overestimation of the object size at varying distances by measuring the magnitude of the Ponzo illusion, and the results showed that the illusory effect was about 45% for three-dimensional actual scenes compared to 30% for two-dimensional photographs of the same scene. It is not surprising that stronger depth cues produced a stronger illusion. There is evidence that size constancy is already present in early infancy (Slater, Mattock, & Brown, 1990). This ability to compensate the effect of distance is not only found in the visual system but also in another sensory modality: Tactile size constancy was found recently (Jackson & Shaw, 2000; Taylor-Clarke, Jacobsen, & Haggard, 2004). 

These phenomena demonstrate one goal: to convey the actual physical dimensions of the world to our perception. Because we have such a prominent ability to perceive the true size at various distances, we could be using similar mechanisms for other visual features as well. In fact, Georgeson and Sullivan (1975) had found that our contrast perception mechanisms compensate for the variation of contrast sensitivity with spatial frequency. They termed this phenomenon “contrast constancy.” While Georgeson's study did not explicitly point out the relationship between the perceptions of contrast and depth, its demonstration of the relation between apparent contrast and spatial frequency might imply such relationship, because spatial frequency varies with distance. Although the contrast constancy has not been profoundly explored the same way as the size constancy, the phenomenon itself gives an example of other aspects of constancy. 

It has long been known that the perception of surface lightness is depth dependent (Gilchrist, 1977; Logvinenko & Maloney, 2006; Pereverzeva & Murray, 2009). Some researchers used a wide range of manipulation of depth cues (Kitazaki, Kobiki, & Maloney, 2008; Landy, Maloney, Johnston, & Young, 1995) to investigate how various combinations of depth cues affect lightness perception in three-dimensional scenes. Since the results of all these studies suggest that depth cues contributes to lightness perception, one might expect that depth perception can also affect contrast perception. In fact, recent studies (Aslin, Battaglia, & Jacobs, 2004) show that contrast adaptation or contrast gain control is depth dependent. 

Recently, we observed a new powerful type of contrast and size illusion caused by apparent motion in depth, which we called the StarTrek illusion. We found that an optic flow pattern that consists of disks moving in depth strongly modulates their contrast. Objects appearing to move away from the observer grew higher in contrast and larger, while their retinal size and the actual contrast remained constant. This phenomenon can be observed with as few as 3–5 objects and for different kinds of objects: Dots, light, dark, or Difference-of-Gaussian (DoG) disks were among the objects tested. The size of the disks per se did not significantly affect the illusion strength. However, by adjusting the size of disks during the optic flow motion, the illusory contrast increase could be canceled completely or reversed. 

This newly discovered phenomenon is interesting not only because it is the first clear demonstration that the percept of depth has a rather strong effect on contrast perception, irrespective of the target's context, but it also suggests that the brain applies a regulating rule for contrast variation with distance, the same way as it does for size, which we discuss in detail later. Furthermore, we argue that the modulation process that the brain applies for size happens before that for contrast and that the apparent size strongly contributes to the apparent contrast. 

Methods

Stimuli

The stimuli were displayed on a gray background and viewed on a linearized 21″ ViewSonic G225f monitor, except for Experiment 4, in which the stimuli were viewed through a Wheatstone stereoscope on a pair of G225f monitors. The display resolution was set to 1600 × 1200 pixels, and for the typical viewing distance of 70 cm, a pixel subtended 1 arcmin. 

The target was a set of high-contrast randomly located disks moving on a gray background (Figure 1). Peripheral random disks on a gray background formed a static stencil mask. The mask had a 10° circular aperture positioned in the center of the screen, through which the moving disks could be seen. Their motion created an optic flow consistent with the disks being positioned on a fronto-parallel plane moving back and forth with constant speed, i.e., in a triangle-wave fashion. From the point of view of the observer, the density of the disks became higher when they appeared to move away from the observer. Thus, we refer to this motion phase as “stimulus contraction” and refer to the motion phase when disks move toward the observer as “stimulus expansion” correspondingly. In Experiment 2, the simulated depth modulation was 100%, i.e., from the viewing distance to twice the viewing distance. The nearby disks were allowed to overlap in the course of their inward motion; however, given the low density of the disks, this happened rarely. As the disks moved inward, new disks appeared along the boundary of the aperture from behind the occluding pattern of the stencil mask consistent with the pattern of the optic flow. We tested disks of various diameters from 0.05° to 0.3° and DoG profile disks of the same diameters, the DoG diameter defined as two standard deviations for the broader of the two constituent Gaussians. Depending on the disk size, 100–1000 disks were displayed in each trial, which included one stimulus contraction–expansion motion cycle and lasted for 2 s. Various disk sizes, positive and negative luminance polarity, and DoG profile disks were studied in separate experimental blocks. Subjects carried out 450 trials for each experimental block. 

Figure 1

 

An example of the stimulus used. The white bars illustrate the radial optic flow created by the moving random disks visible through the circular aperture in the center of the screen.

Subjects

Fifteen observers with normal or corrected visual acuity were tested. Thirteen of the observers were naive to the purpose of the study; only two were experienced psychophysical observers. Observers were trained for a short time (2–5 min) to get acquainted with the stimuli and the task. 

Psychometric procedure

Observers had to indicate whether the disks appeared to be getting higher contrast or lower contrast as they moved away by pushing left and right mouse buttons, respectively. A nulling paradigm was used to measure the illusory contrast modulation. In order to null the illusory effect, a gradual luminance decrement or increment, depending on the target's luminance polarity, was applied to all the moving disks as they moved away and an equal gradual increment was applied as they returned. We found that the contrast modulation given by the following formula produced a fairly constant contrast-change percept in the course of the optic flow and was suitable for the nulling paradigm: 

$C\left(d\right)=C\left(d_0\right)/(1+A\frac{\Delta d}{d_0}),$

(1)

where d ₀ stands for the actual viewing distance, Δd = d − d ₀ stands for the modulation of the distance from observer d as simulated by the optic flow, and C(d ₀) stands for the contrast of the stimulus for d = d ₀. The nulling amplitude, A > 0, of the contrast modulation was calculated by a modified version of the Bayesian adaptive algorithm, devised by Kontsevich and Tyler (1999). Note that because A was always positive, contrast always decreased as the simulated distance d increased. We termed

$\frac{\Delta d}{d_0}$

the motion amplitude factor. The illusion strength was measured as the nulling value of ΔC = C(d) − C(d ₀). 

Observers carried out three blocks of 150 trials per block for each condition. Uncertainties for the measurement of ΔC were taken as the maximum of the two: (i) variation of the ΔC estimate calculated from the results of the adaptive algorithm and (ii) variation of the ΔC estimates in between the three experimental blocks. The resulting uncertainties (one SEM) are represented by error bars in the figures. 

Results

Experiment 1: Effect of the disk's luminance polarity, luminance profile, and size

In Experiment 1, we tested light disks (white), dark disks (black), and the difference-of-Gaussian (DoG) luminance profile disks of two sizes. Small-scale disks were 0.05 degree in diameter; large-scale disks were 0.1 degree in diameter. Figure 2 shows the illusory contrast increase as a function of the luminance polarity, luminance profile, and size. The illusory effect was strong for all three subjects tested. On average, the illusory contrast increase was about 30%. The strength of the illusion varied little among the four conditions [F _condition(3,7) = 0.026, p > 0.5, two-way ANOVA] but varied significantly among the subjects [F _subject(2,7) = 13.221, p < 0.005, two-way ANOVA]. Here and in the other figures, individual subject's data are shown by different colors. This shows that the nature and size of the objects creating the optic flow was of little significance and the illusion could be induced by stimuli of various luminance polarities, luminance profiles, and sizes. 

Figure 2

 

The effect of the disks' luminance polarity (light/dark), the disks' luminance profile (flat/DoG), and the DoG spatial scale (small/large). The contrast decrease required to null the illusory contrast increase is plotted along the y-axis. Different colors and symbols indicate the three experimental subjects tested.

Experiment 2: Effect of the motion amplitude

In order to test the effect of motion amplitude on this illusion, we modulated the amplitude of the optic flow by adjusting the motion amplitude factor

$\frac{\Delta d}{d_0}$

. Motion amplitude factor 1 corresponds to the in-depth motion to twice the viewing distance. The trial duration was adjusted so as to keep the speed of the optic flow constant among different motion amplitudes. For example, if the traveled distance changed from 70 cm to 140 cm, the trial duration changed from 2 s to 4 s. The results (Figure 3) show the illusory contrast increase as a function of the motion amplitude. Five subjects participated in this experiment. The illusion's strength first increased linearly for most subjects. Then, for those subjects, which experienced the strongest illusion it decelerated as the motion amplitudes increased. 

Figure 3

 

The effect of the motion amplitude. Motion amplitude factors 0.5, 1, and 2 correspond to the in-depth motion from 70 cm to 105 cm, 140 cm and 210 cm, respectively. The different colors and symbols indicate the different experimental subjects.

Experiment 3: Effect of the disparity modulation

To study the effect of other depth cues, we varied binocular disparity using the Wheatstone stereoscope. The disparity of the disks varied in accord with the depth change indicated by the optic flow pattern in this experiment. Five subjects participated in this experiment. After a training section, subjects were able to merge the two depth cues. We ran two conditions: one with the disparity modulation alone and the other with the disparity modulation added to the original optic flow motion. 

Figure 4 compares the two new conditions with the original optic flow condition. The results differed significantly among subjects [F _subject(4,11) = 17.251, p < 0.005, two-way ANOVA]. The effect of the condition was also significant [F _condition(2,11) = 8.302, p < 0.01, two-way ANOVA]. For four of the five subjects tested, the disparity alone produced the weakest illusion, which is only 1/2 to 3/4 of the effect of the optic flow. For the condition where the two depth cues were combined, the illusion strength was not significantly different than that for the optic flow alone, which shows that the binocular disparity was a weaker depth cue and added very little or nothing to the effect of the optic flow. 

Figure 4

 

The effect of the disparity modulation. No disparity modulation was present for the optic flow condition. Disparity modulation alone was present for the “disparity” condition. Both types of modulations were present for the “both” conditions. The different colors and symbols indicate the different experimental subjects.

Experiment 4: Effect of the optic flow direction

Previous experiments comprised the full stimulus contraction–expansion motion cycle. However, some observers noticed that the illusion was more obvious for the stimulus expansion phase. In order to test the effect of the optic flow direction, we blocked the stimulus contraction and expansion phases. Otherwise, the methods and stimuli were the same as in the previous experiments. 

Five subjects participated in this experiment. The results are plotted in Figure 5. Across these subjects, there was a significantly stronger effect for stimulus expansion than for stimulus contraction [t(4) = 5.62, p < 0.01]. On average, the perceived illusory contrast increase for the stimulus expansion condition is slightly less than 30%, which is largely equivalent as the full motion cycle (see Experiment 1), and is greater than the stimulus contraction condition by almost 10%. 

Figure 5

 

The effect of the optic flow direction. The “contraction” phase appeared as disks moving away from the observers; the “expansion” phase appeared as disks moving toward the observers. The different colors and symbols indicate 5 experimental subjects.

Experiment 5: Correlations with the density of disks

Because the density of the disks was constantly changing as they moved back and forth in the course of the optic flow, we wanted to know by how much the density change contributes to the illusion. In the first section of Experiment 5, only the first (low-density) and the last (high-density) frames of the optic flow were presented on the screen for a quarter of the normal trial duration in each trial. Two sizes of the disks (0.1° and 0.3°) were used for different subjects. The task remained the same as in the previous experiments. 

The mere density modulation (y-axis) and the full optic flow (x-axis) effects on contrast are compared in Figure 6a. Each datum represents a different subject; each subject was tested with either the 0.1° (black) or 0.3° (red) disk size. As before, there was a strong variation of the strength of the illusion among subjects. On average, the density modulation accounted for only about 1/3 of the illusion's magnitude for each subject. This is indicated by the linear fits shown with the solid lines: black line for the 0.1° disk size and red line for the 0.3° disk size. The slope of the black line is 0.41 and that of the red line is 0.34. The results demonstrate that the dynamics of the optic flow is an essential component of the illusion. In the second part of Experiment 5, we asked whether the density change contributes to the illusory effect of the optic flow at all. 

Figure 6

 

(a) Comparison between the effects of a mere density modulation (y-axis) and the full optic flow (x-axis). Each datum represents a different subject; color marks the 0.1° and 0.3° disk sizes used for different subjects. Solid lines show linear fits to the data. (b) The effects of constant density, dynamic density, and the original optic flow. Note that only the stimulus expansion phase was shown for the three conditions. The different colors and symbols indicate the different experimental subjects.

Methods for this part of the experiment were largely identical to those for the previous experiments, except that the number of disks visible within the circular aperture remained constant through the whole trial. To this end, disks randomly appeared and disappeared as they moved along their usual trajectories in such a way that the total number of the disks within the aperture remained constant. Each disk was present on the screen for five video frames only, just enough to create an impression of optic flow in a given direction. To make sure that the disks shown in this way were able to produce the illusory effect, we added a control condition (referred to as “dynamic density” condition) where the disks randomly appeared and disappeared the same way as for the constant density condition, but their density changed in the normal fashion, the same as for the original optic flow condition. Because the illusion strength for the stimulus expansion was nearly identical to that for the full stimulus contraction–expansion motion cycle (see Figures 2 and 5), only the stimulus expansion phase was shown in this experiment to save time. 

Four of the five subjects from Experiment 4 participated in this experiment and their individual results were plotted in Figure 6b. Because only the expansion phase was shown, the results of “constant density” condition and the control “dynamic density” condition were compared with the results of “stimulus expansion” condition in Experiment 4. One can see that the “dynamic density” condition produced almost the same illusory effect as the optic flow. On average, the illusory contrast increase was 25%. Thus, the fact that disks appeared and disappeared in the course of the optic flow did not significantly affect the illusion. However, for the “constant density” condition, the illusory effect was less than 15%. The difference was significant [F _condition(2,35) = 10.193, p < 0.005, two-way ANOVA], and Tukey HSD test indicates that the effect of “constant density” condition was significantly different from the other two at the 0.005 level. The same as in the previous experiments the overall strength of the illusion varied significantly among subjects [F _subject(3,35) = 5.902, p < 0.005, two-way ANOVA]. However, considering that only one out of four subjects experienced a major drop in perceiving the illusion, we conclude that the density factor is important but not decisive for the illusion. Altogether, this experiment demonstrates that both the optic flow motion pattern and the accompanying density modulation were important factors contributing to the strong illusion of contrast. We hypothesize that this was because both factors provided strong cues for depth modulation. 

Experiment 6: Contrast illusion vs. size illusion

Most observers noticed that the moving disks appeared to grow and shrink as well as to change their contrast. The illusion appears to be a dynamic variant of a size–distance illusion, e.g., the well-known Ponzo illusion. In this experiment, we investigated how the size change correlated with the contrast change. 

In order to directly measure the size–distance illusory effect and compare it with the contrast–distance illusory effect, we decreased the diameter S of the disks for the stimulus contraction phase and asked the subjects to judge whether the disks became larger or smaller in the first part of this experiment. The variation of S with distance d was the same as the contrast variation in the previous experiments and is described by the analogous formula: 

$S\left(d\right)=S\left(d_0\right)/(1+A\frac{\Delta d}{d_0}).$

(2)

The amount of size change necessary to null the imaginary growth of the disks was determined by the same adaptive algorithm as before and served as a measure of the size–distance illusion. No contrast modulation was applied to the stimulus. In the course of this experiment, we discovered that the size manipulation strongly affected the perceived contrast of the disks. To pursue this further, we used exactly the same stimulus but asked the subjects to judge whether the dots became higher contrast or lower contrast in the second part of the experiment. 

The experimental results are shown in Figure 7. The left panel (Figure 7a) shows a comparison between the illusory size increase and the illusory contrast increase, both measured by varying the disk size to null the illusions. Each datum represents a different subject. The illusory contrast increase is plotted along the x-axis, and the illusory size increase is plotted along the y-axis. The black curve indicates the square root function fit (see 1 for details). The results show that on average the relative decrease of the disk size required to null the size illusion is about half of that to null the contrast illusion. 

Figure 7

 

(a) Comparison between the illusory size and contrast increase, both measured by varying the disk size to null the illusions. Each datum represents a different subject. The black curve shows a parameter-free model prediction y + 1 = x + 1 (see 1). (b) Comparison between the nulling contrast effects of the disk size and disk contrast. Solid lines show linear fits to the data.

The right panel (Figure 7b) compares how the illusion of contrast is nulled by the disks' size variation measured in this experiment and the disks' contrast variation measured in the previous experiment. The nulling contrast variation is plotted along the x-axis, and the nulling size variation is plotted along the y-axis. The best linear fits of both the 0.1° disk size (black) and the 0.3° disk size (red) are shown in the panel, with the slopes of 0.957 and 1.027, respectively. Strikingly, the results demonstrate that the illusion of contrast can be nulled by the relative decrease of the disk size exactly equal to the required relative decrease of their contrast. The effects of these two manipulations showed a very close correlation for all 15 observers (all data points in Figure 7 are close to the diagonal) even though the illusion's strength varied very significantly among observers. 

Experiment 7: Effect of contrast change on size perception

The previous experiment demonstrated that size modulation had a surprisingly strong effect on contrast perception for our experimental paradigm. In this experiment, we tested whether, similarly, the contrast change affected the size perception. 

We were not able to set up the experiment using methods similar to those for the last experiment: increasing or decreasing the contrast of the disks as they moved away and asking the subjects to judge whether the disks became larger or smaller. No matter how strong the change was, it was not enough to null the size illusion. Therefore, instead of varying contrast adaptively, we fixed the contrast modulation at 3 levels (none, two-fold increase, two-fold decrease) and measured the illusory size increase adaptively, the same as in Experiment 6. The idea was to test whether the strong contrast modulations affected size illusion at all. In the first condition, the contrast of the disks was set to 50% and remained constant during the optic flow. We used the same formula for the contrast variation as in Experiment 1, but the amplitude A was fixed here (A = 0). In the second condition, disks began to grow higher in contrast as they moved away and lower in contrast as they moved closer. The contrast gradually varied from 50% at the beginning of the optic flow to 100% at the longest distance from the observer (A = −0.5). In the last condition, disks decreased in contrast as they moved away and increased in contrast as they moved back. The contrast variation of the disks was from 50% to 25% (A = 1). Because it was difficult to judge the small disk size when the contrast was as low as 25%, we used the 0.3° disk size only. Fourteen subjects participated in this experiment. 

The results are shown in Figure 8. The illusory size increase is plotted along the y-axis, and the three conditions are plotted along the x-axis. Label “×2” indicates the contrast increase condition, where the contrast changed from 50% to 100%; label “×1” indicates the constant contrast condition, where the contrast remained 50%, and “×0.5” indicates the contrast decrease condition, where the contrast changed from 50% to 25%. There was very little variation of the size illusion among the three conditions, and altogether, the effect of the contrast modulation was insignificant [F _condition(2,77) = 0.966, p > 0.05, two-way ANOVA]. 

Figure 8

 

The effect of contrast modulation on the illusion of size. Three conditions are plotted along the x-axis. Label “×2” indicates the condition for which the contrast changed from 50% to 100%, label “×1” indicates the condition for which the contrast remained constant, and “×0.5” indicates the condition for which the contrast changed from 50% to 25%. The different colors and symbols indicate the different experimental subjects.

Discussion

The first three experiments explored the properties of the StarTrek illusion: Objects that appeared to move in depth also appeared to change their contrast. The illusory contrast change was very strong, 25%–30% on average, while for some subjects the illusion was much stronger. The illusory effect grew even stronger as the motion amplitude increased. Results show that the discovered illusory contrast modulation is a ubiquitous, powerful, and distinct phenomenon demonstrating a direct relationship between the perceived depth and the perceived contrast. We mainly used radial optic flow to create the perception of the depth change. The associated binocular disparity change produced a weaker illusion on its own and contributed little when combined with the optic flow. On the other hand, the density modulation present in a radial optic flow turned out to be a significant factor of the illusion's strength. The nature and size of the objects creating the optic flow was of little significance: Light and dark disks and DoGs of various sizes worked equally well. 

We suggest that the StarTrek phenomenon is a contrast-domain counterpart of a size–distance illusion, e.g., the well-known Ponzo illusion. The Ponzo illusion is a manifestation of the phenomenon of size constancy: Objects appear to maintain their physical size even when their angular size (size on the retinas) is changing. For example, human figures seen from far away may be tiny on the retinas, yet faraway humans do not appear as midgets, presumably, because the brain accounts for the angular size variation with distance. In addition, vice versa, if the object's angular size does not show the expected variation with distance, then the brain infers that its physical size must be varying. 

If size constancy is a strategy the brain uses to successfully recognize a certain object at different distances, it is possible that it uses a similar strategy for contrast perception. When an object moves farther away, its image on the retina gets smaller. As a result, the image loses high spatial frequency information and, consequently, some measure of contrast. Moreover, there is an overall shift of the image content to higher spatial frequencies, where contrast sensitivity of the human visual system is low (Georgeson & Sullivan, 1975). Our data suggest that the perception of an object's contrast remains relatively constant even though our sensitivity to contrast is reduced when viewing objects far away. If an object's contrast does not show the expected variation with distance, the brain comes up with a rationalization: The object's contrast must be varying. This evokes a compensatory response from the brain causing the StarTrek illusion. 

According to this hypothesis, the StarTrek illusion can be affected by how ecologically plausible the optic flow patterns are (Experiments 4 and 5). Normally, we move forward a lot more than we move backward. Correspondingly, the optic flow in the form of expansion is more common than the optic flow in the form of contraction. Besides, things moving toward us are more ecologically relevant (food, menace, etc.). These two factors may explain the stronger illusion for the expansion phase observed in Experiment 4: The visual system adapts stronger to the changing appearance of objects moving toward us than for objects moving away from us. Consequently, when these expected changes are not observed, we experience greater illusory effects for the expansion phase than for the contraction phase. 

Because radial optic flow is normally associated with the density change of the flowing objects, this change was an important factor for the StarTrek illusion (Experiment 5). It is possible that the binocular disparity input to the optic flow did not have a similarly strong effect (Experiment 3) because at distances farther than several hundred meters, the binocular disparity becomes vanishingly small, while the optic flow cue is in effect at any distance. 

In Experiments 6 and 7, we compared the size and contrast illusions for the same stimulus. The results showed a surprising correlation between the object's size and its perceived contrast: 

1.  
   The strength of the size illusion was roughly half that of the contrast illusion.
2.  
   The relative amounts of size change and contrast change required to null the contrast illusion were about the same for any given observer. Note that this effect of size on the perceived contrast cannot be explained by the finite resolution of the visual system, because we used disks with angular diameter much larger than that of the Airy disk, and the effect was exactly the same for disks of the two different diameters we used.
3.  
   The size change affects the perceived contrast only when objects appear to move in depth. Simply changing the size of the disks without changing their apparent depth does not result in the perceived contrast change. This point is intuitively clear, but we also ran a control experiment, where subjects were required to match the contrast of a disk of varying size to a reference disk of a given constant size and contrast. The control experiment showed that the disk size had no effect on contrast judgments unless the disk diameter was comparable to the Airy disk diameter, which was an order of magnitude smaller than the smallest diameter used in our study.
4.  
   The contrast modulation did not affect the size illusion.

We propose a simple model of size and contrast perception, which explains the above 4 results. In particular, the model fits the data in the left panel of Figure 7 in a parameter-free fashion. The model suggests a global phenomenon that bridges size constancy and contrast constancy, which we termed general object constancy. Our hypothesis is that the brain uses a general object constancy mechanism that employs a single scaling function for both size constancy and contrast constancy, i.e., scales both retinal size and retinal contrast by the same amount as a function of distance. Additionally, the perceived contrast is scaled by the perceived size change. The model is illustrated by the diagram in Figure 9; mathematical details are discussed in 1. Because the size and contrast are both scaled as a function of distance, but the perceived size further contributes to scale the perceived contrast and not vice versa, the contrast illusion ends up about twice stronger than the size illusion (Figure 7a). The fact that the contrast illusion can be completely nulled by contrast modulation and size modulation (Experiment 6) but the size illusion can only be nulled by size modulation (Experiment 7) is also explained by the model. 

Figure 9

 

General object constancy mechanism. The brain scales both retinal size and retinal contrast of an object by a factor k as a function of distance. Additionally, the perceived size change contributes to the perceived contrast, which is indicated by the “×” symbol in the diagram.

Conclusions

The StarTrek illusion is one of the strongest illusions of contrast, which reveals intriguing new phenomena. It shows that size and contrast, apparently independent features, are directly linked: The contrast illusion nulled by a given amount of contrast change during the optic flow could also be nulled by the same amount of size change but not vice versa. This demonstrates that size calculation is done prior to the perceived contrast calculation and the resulting size is taken into account for the contrast calculation. Our results suggest that the StarTrek illusion demonstrates a “general object constancy” phenomenon uniting the well-known size constancy phenomenon and (the less known) contrast constancy phenomenon (Georgeson & Sullivan, 1975). According to our hypothesis, the brain applies a common scaling factor to the object's size and contrast to compensate for the changes in the object's appearance with viewing distance. 

Appendix A

General object constancy mechanism

Let s be the retinal size and c be the retinal contrast of an object. To explain our results, we suggest that when the in-depth motion happens, the brain first scales s and c by the same dimensionless factor k, which is some function of the relative depth d/d ₀: 

$S=s·k\left(\frac{d}{d_0}\right),$

(A1)

 

$C=c·k\left(\frac{d}{d_0}\right).$

(A2)

Here, d ₀ stands for the reference depth wherefrom the motion started, while S and C stand for the perceived size and contrast, respectively. Based on the data plotted in Figure 3, function k(

$\frac{d}{d_0}$

) is approximately linear for small motion amplitude factors and is decelerating for large amplitude factors. In addition, result (ii) in the Discussion section indicates that the retinal size, s, contributes to the perceived contrast, C, to the same degree as the retinal contrast, c. In addition, results (i) and (iv) indicate that the perceived size, S, and perceived contrast, C, are not symmetrical. Both these facts can be accommodated into the model by one simple modification of the contrast relationship: 

$S=s·k\left(\frac{d}{d_0}\right),$

(A3)

 

$C=c·k\left(\frac{d}{d_0}\right)\frac{S\left(d\right)}{S\left(d_0\right)},$

(A4)

where S(d ₀) is the perceived size at the starting depth d ₀ and S(d) is the perceived size at the current depth d. In other words, the perceived contrast is additionally scaled by the relative perceived size

$\frac{S\left(d\right)}{S\left(d_0\right)}$

. This model is illustrated by the diagram in Figure 9. Note that the perceived-size scaling factor equals 1 for the ecologically important case of size constancy (objects of constant size) where, presumably, S(d) = S(d ₀) for any d. Thus, in order to calculate the perceived size and contrast for the size constancy condition, the retinal contrast and size are scaled by the same depth factor k. 

The proposed model accounts for the results (i–iv) well. Without a loss of generality, we can assign k(1) = 1, and therefore, S(d ₀) = s(d ₀) and C(d ₀) = c(d ₀). When the retinal size and contrast remain constant, as is the case for the size–distance and contrast–distance illusions, we obtain 

$\frac{\Delta S}{S\left(d_0\right)}=\frac{S\left(d\right)}{S\left(d_0\right)}-1=k\left(\frac{d}{d_0}\right)-1,$

(A5)

 

$\frac{\Delta C}{C\left(d_0\right)}=\frac{C\left(d\right)}{C\left(d_0\right)}-1=k\left(\frac{d}{d_0}\right)·k\left(\frac{d}{d_0}\right)-1,$

(A6)

and therefore 

$\frac{\Delta S}{S\left(d_0\right)}+1=\sqrt{\frac{\Delta C}{C\left(d_0\right)}+1}.$

(A7)

 

This relationship fits the data in Figure 7a reasonably well given that the square root fit is parameter free. Thus, the model explains result (i). Result (ii) is explained as follows. To null the contrast illusion by means of the retinal contrast modulation (leaving the illusory size change unaffected), we require 

$\frac{c\left(d\right)}{c\left(d_0\right)}=\frac{1}{k^2\left(\frac{d}{d_0}\right)}.$

(A8)

Indeed, then 

$C=\frac{c\left(d_0\right)}{k\left(\frac{d}{d_0}\right)}·\frac{S\left(d\right)}{S\left(d_0\right)}=\frac{c\left(d_0\right)}{k\left(\frac{d}{d_0}\right)}·k\left(\frac{d}{d_0}\right)=c\left(d_0\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}$

(A9)

To null the contrast illusion by means of the retinal size modulation requires the same functional form: 

$\frac{s\left(d\right)}{s\left(d_0\right)}=\frac{1}{k^2\left(\frac{d}{d_0}\right)}.$

(A10)

Indeed, then 

$\frac{S\left(d\right)}{S\left(d_0\right)}=\frac{1}{k\left(\frac{d}{d_0}\right)}$

(A11)

and 

$C=\frac{c\left(d_0\right)}{k\left(\frac{d}{d_0}\right)}·\frac{k\left(d\right)}{k\left(d_0\right)}=c\left(d_0\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}$

(A12)

Thus, to null the contrast illusion 

$\frac{c\left(d\right)}{c\left(d_0\right)}=\frac{s\left(d\right)}{s\left(d_0\right)},$

(A13)

which explains result (ii). Result (iii) is explained by noting that for d = d ₀: k(

$\frac{d}{d_0}$

) = 1,

$\frac{S\left(d\right)}{S\left(d_0\right)}$

= 1, and therefore C = c(d ₀) = const. Finally, result (iv) is obvious because the retinal contrast c does not enter the formula for the perceived size S at all. 

Acknowledgments

Commercial relationships: none. 

Corresponding author: Jiehui Qian. 

Email: [e-mail]. 

Address: Northeastern University, 125 Nightingale Hall, 360 Huntington Ave, Boston, MA 02115, USA.