Anchoring of lightness values by relative luminance and relative area

Abstract

Surface lightness is widely thought to depend on the relative luminance coming from neighboring surfaces. But relative luminance can produce only relative lightness values. Specific lightness values can be derived only with an anchoring rule that specifies how relative luminance values in the retinal image are mapped onto the lightness scale. We explored the anchoring rules governing very simple images consisting of two adjacent surfaces that fill the entire visual field. These were painted onto the interior of a large hemisphere that surrounded the observer's head. Lighter and darker radial sectors of the same two shades of gray were painted onto nine such hemispheres, but with different relative areas. The region of highest luminance was always seen as white. The lightness of the darker sector depended on relative area, appearing lighter as the darker sector became larger, but this effect was stronger when the darker sector was larger than the lighter, a pattern of results shown to be consistent with over a dozen prior studies of relative area and lightness.

Introduction

Human visual perception of the black, white, and gray shade of surfaces, known as lightness perception, is very good in the real world, and largely independent of the illumination level in any part of the scene. But this ability has not been fully explained scientifically. We know that the lightness of a surface does not depend on its luminance, i.e., the intensity of light it reflects, because luminance depends on the illumination falling on the surface even more than it does on the reflectance of the surface. Thus, any luminance value can be seen as any shade of gray. We also know that lightness depends on relative luminance values. For example, a disk of constant luminance can be made to appear as any shade of gray from white to black merely by changing the luminance of the region surrounding it. The apparent shade of gray of the disk depends simply on the ratio between disk luminance and surround luminance. This is the celebrated ratio principle proposed by Wallach (1948). 

But relative luminance is also ambiguous. Relative luminance can produce only relative lightness values (Gilchrist et al., 1999). The perception of specific (or absolute) lightness values requires the application of an anchoring rule—a rule by which relative luminance values in the retinal image are pegged to specific values on the gray scale. Two different anchoring rules can be found in the literature. We will call these the highest luminance rule and the average luminance rule. According to the highest luminance rule (Evans, 1974; Land & McCann, 1971; Marr, 1982; Wallach, 1976) the highest luminance in the visual field is automatically perceived as white, and lower values are evaluated relative to this white standard. According to the average luminance rule (Buchsbaum, 1980; Helson, 1964) the average luminance in a scene is treated as middle gray, while higher and lower values are evaluated relative to this middle gray standard. 

Li and Gilchrist (1999) tested these rules against each other in the simplest image capable of producing the experience of a surface: two adjacent regions of different luminance that fill the entire visual field. If the interior of the dome were completely homogeneous, no surface would be seen. The observer would experience an infinite fog (Metzger, 1930, p. 13). Light/dark borders are crucial for seeing a surface and at least one such border must be present in the visual field before a surface will be seen. 

Li and Gilchrist placed the observer's head inside an opaque hemisphere, the interior of which was painted with two shades of matte gray. The results unequivocally supported the highest luminance rule. But they also found evidence that relative area plays a role in anchoring. Their stimuli included three domes in which the area of the darker region was (1) much smaller, (2) equal to, and (3) much larger than the lighter region. While the darker region appeared as the same lightness in the first two of these domes, it appeared substantially lighter in the third dome. A survey of the literature on lightness and area (reviewed here in the discussion) showed the same pattern of results. Li and Gilchrist suggested that strong effects of area on lightness occur only when the darker region is larger than the lighter region. 

Based on the fragmentary data reported by Li and Gilchrist (1999) and a survey of the literature described below, Gilchrist et al. (1999) proposed the theoretical area function shown in Figure 1. 

Figure 1

 

Area function suggested by Gilchrist et al. (1999).

We report a more extensive set of experiments using the hemisphere method. We spray-painted two shades of gray on the interiors of nine large opaque hemispheres and we varied relative area using a pattern of radial sectors. The observer's head was located within this dome so that the two shades filled the observer's entire visual field. Apart from providing a convenient fixation point (the apex of the smaller sector), the use of radial sectors also guaranteed that the relative proportion of light and dark was the same in the fovea and in the periphery. 

Values of relative area were chosen so that the area of the smaller region was either doubled or halved on each step (equal intervals on a log scale), on the hypothesis that relative area might behave as relative luminance, with equal log differences perceived as equivalent. 

Experiment 1

Method

Stimuli

Nine opaque acrylic hemispheric domes 76 cm in diameter served as stimuli. Two radial sectors were spray-painted using matte paint onto the interior of each dome: one darker (reflectance 7.7%, equal to Munsell 3.3) and one lighter (reflectance of 36%, equal to Munsell 6.5). The domes varied in terms of the relative area of the lighter and darker sectors, with darker sectors of 11, 22, 45, 90, 180, 270, 315, 338 and 354 degrees. The exterior of each dome was painted matte white. 

Apparatus

As shown in Figure 2, each dome was suspended within a large rectangular diffusion chamber (122 cm wide, 122 cm high and 77 cm deep) the interior of which was painted matte white for maximum diffusion of the lighting. The illumination came from two 750 watt halogen light bulbs housed in a metal box attached to the center of the back of the diffusion chamber and opening into the chamber. The illumination intensity in the dome was controlled by a pair of sliding aluminum shutters that could occlude more or less of a rectangular aperture (50.5 cm × 20 cm) connecting the illumination box with the diffusion chamber. The light entering the diffusion chamber was reflected uniformly off the white walls, creating homogenous illumination that illuminated the interior of the dome by flowing through a 11 cm gap between the front rim of the dome and the front wall of the chamber. The observer's head was placed within a hockey mask recessed into the front wall of the diffusion chamber, guaranteeing that the observer's eyes were located within the suspended dome so that its interior filled the entire visual field. The distance between the observer's eyes and the center of the dome interior was 22 cm. 

Figure 2

 

Method of positioning the observer's head inside the dome. Diffuse light reflecting from the white interior walls of the apparatus entered the open front of the dome.

The observer was asked to fixate the apex of the smaller sector (except in the 180° dome, in which the observer was asked to fixate straight ahead). 

In Experiment 1, the luminance of the lighter sector was equated to the luminance of white paper on the outside front of the diffusion chamber under normal room illumination, at 25 cd/m ². To keep this value constant across domes, it was necessary to make small adjustments in the amount of light entering the chamber separately for each dome because the amount of ambient light in the dome was influenced (modestly) by the relative area of the sectors. 

Matching chart

The observer matched the lightness of the sectors in the dome using a 16 step Munsell chart on a white background, separately illuminated and housed in a metal chamber attached to the wall immediately to the left of the apparatus. The luminance of the white chip was 317 cd/m ². 

Procedure

The observer entered a darkened laboratory in which the only source of light was the illumination coming from the box with the Munsell chart and was given the following instructions: “In a minute you will place your head all the way into this hockey mask and once you feel comfortable we will turn the lights in this apparatus on. You will find that your head is in the middle of one of these domes (pointing to the other domes hanging on the laboratory walls) and you'll see it looks like this (pointing at a line diagram of the suspended dome attached to the front of the apparatus). You will see one surface of one shade and another surface of a different shade. You should look straight ahead, to the center and memorize the two shades. Once you are sure you remember them, tell us and we'll turn off the lights. Then you can pull out your head and look at this scale (pointing at the Munsell scale) and match the two shades that you saw.” 

Observers

One hundred and eighty observers participated in the study. All were undergraduate students at Rutgers University who volunteered in order to fulfill a course requirement. A separate group of 20 observers viewed each of the nine domes. 

Results

The results are shown by the dashed line in Figure 3. Consider first the dome with equal areas of light and dark (180°). The lighter half was seen as completely white (Munsell 8.9; reflectance 77%), even though its actual reflectance was 36%, equal to a light gray or Munsell 6.5. This provides further support for the highest luminance rule, not the average luminance rule. Were the average luminance rule correct, no whites or blacks would be seen and the gray shades perceived in the two halves would lie at equal distances from middle gray on the scale. The darker half was seen as middle gray (Munsell 5.5; reflectance 25%) which approximates the reflectance of the white anchor (90%) divided by the luminance ratio at the border (4.7:1), according to the ratio principle of Wallach (1948). 

Figure 3

 

Average lightness matches for the lighter and darker sectors in nine domes. Dashed line: Experiment 1; solid line: Experiment 2.

Overall the results are characterized by three main features. First, the lighter region always appears white, regardless of relative area. Second, the darker region becomes lighter as it grows larger. Third, the effect of relative area appears more pronounced when the darker region is larger than the lighter region. 

The lighter sector was seen as white and did not differ significantly across domes (Munsell 8.9–9.2). However, the darker sector was perceived as significantly lighter as its area increased, moving from Munsell 4.9 for the 11° dome, to Munsell 7.6 for the 354° dome, F(8, 171) = 24.95, p < 0.001. We compared each individual dome with all other domes using planned comparisons (Tukey HSD) and found that the dark region appeared significantly lighter in every dome in which it was larger than 180° (50% of area) compared to every dome in which it was smaller than or equal to 180° ( p level < 0.001 in all comparisons, except 90° dome compared to 270° dome, where p = 0.001, and 270° dome compared to 180° dome, where p < 0.01). 

To explore this further, we conducted two additional analyses of variance: one which included only the domes in which the darker sector was smaller than the lighter and another that included only the domes in which the darker sector was larger. We did not include the 180° dome in these on the assumption that the transition between the two hypothesized legs of the curve might be gradual rather than sharp. We wanted to compare only those domes that fall clearly within one of the two zones. These ANOVAs showed that when the darker sector was smaller than the lighter in area (from 11° to 90°), its lightness did not change significantly with an increase of area ( F < 1, ns, all pairwise comparisons using Tukey HSD p > 0.6). However, when the darker area was larger it became significantly lighter as its area increased from 270° to 354° F(3, 76) = 3.97, p < 0.05. The planned comparisons (Tukey HSD) showed that the darker sector appeared significantly lighter in the 338° dome (Munsell 7.5) or in the 354° dome (Munsell 7.6) compared to the 270° dome (Munsell 6.6), both at the level of significance p < 0.05. These tests are summarized in Table 1. 

Table 1

 

Results of pairwise comparisons (Tukey HSD) of darker sector lightness among domes in which it is smaller than (top) or greater than (bottom) 180 degrees, in Experiment 1 (1) and Experiment 2 (2).

	22°	45°	90°
11°	(1) ns (2) ns	(1) ns (2) ns	(1) ns (2) ns
22°		(1) ns (2) ns	(1) ns (2) ns
45°			(1) ns (2) ns
	315°	338°	355°
270°	(1) ns (2) ns	(1) p < 0.05 (2) p < 0.05	(1) p < 0.05 (2) p < 0.001
315°		(1) ns (2) p < 0.05	(1) ns (2) p < 0.001
338°			(1) ns (2) ns

To further explore the relation between the area and lightness of the darker region we ran a regression analysis for each of the two groups of domes and found that when the darker area was smaller than the lighter (11°–90°), there was no correlation between the area and lightness of the darker region i.e., the slope of the regression function did not differ from zero ( t < 1, ns). However, we found a significant correlation between area and lightness when the darker area was larger than the lighter (270°–355° domes), r = 0.36, F(1, 78) = 11.90, p < 0.01, i.e., the slope of the regression function significantly differed from zero, t(78) = 3.45, p < 0.01. However, when we directly compared the slopes of the two functions we failed to find a significant difference between them. 

This perceived increase in lightness of the darker sector was accompanied by a significant compression of the perceived range in every dome in which the darker sector was larger than 180° compared to every dome in which it is smaller than or equal to 180°, but because this compression is redundant with the lightness changes just reported, we will not describe the statistical details. 

Finally, these results do not support our speculation that area should be plotted on a log scale and thus we used a linear scale for the abscissa in Figure 3. 

Experiment 2

Because the variability of lightness matches for each dome was relatively high in Experiment 1, we decided to replicate it. In Experiment 2, we held the luminance of the dark sector constant across conditions at 6.3 cd/m ². In all other respects the method was the same as that of Experiment 1, including the number of observers. 

Results

The lightness matches are shown by the solid line in Figure 3. The basic effect obtained in Experiment 1 was replicated. The lighter sector was perceived as white in all nine domes (Munsell 9.0–9.3; variations not significantly different), while the lightness of the darker sector significantly increased with the increase in its area, F(8, 171) = 16.40, p < 0.001. We compared each individual dome with all other domes using planned comparisons (Tukey HSD) and found that the darker sector was perceived as significantly lighter in all the domes in which its area was larger than the lighter compared to all domes in which it was smaller than or equal to the lighter (all Tukey HSD tests were significant at p level < 0.01); also pairwise comparisons revealed a marginally significant difference in lightness of the darker sector between the 22° and 180° dome ( p = 0.05). 

Two separate ANOVAs, one for a group of domes in which the darker sector was smaller than the lighter and another for a group of domes in which it was larger than the lighter, replicated the findings from Experiment 1. Area had a significant effect on the lightness of the darker sector only when the darker sector was larger than the lighter, F(3, 76) = 9.63, p < 0.001. Pairwise comparisons (Tukey HSD) showed that the darker sector was significantly lighter in the 338° dome (Munsell 7.0) or the 354° dome (Munsell 7.5) compared to the 270° dome (Munsell 6.2; p < 0.05 and p < 0.001, respectively) as well as in the 338° or the 355° dome when compared to the 315° dome (Munsell 6.2; p < 0.05 and p < 0.001, respectively). 

By contrast, area did not have an effect on lightness when the darker sector was smaller than the lighter ( F < 1, p > 0.6 for all pairwise comparisons using Tukey HSD). 

We conducted a regression analysis within each of the two groups of domes and obtained the same results as in Experiment 1. When the darker area was smaller than the lighter, there was no correlation between the area and lightness of the darker region (the slope did not differ significantly from 0, t < 1, ns). But, when the darker area was larger than the lighter, we found a significant correlation between the area and lightness of the darker sector, r = 0.46; F(1, 78) = 20.45, p < 0.001 (the slope was significantly different than zero, t(78) = 4.52, p < 0.001). However, as in Experiment 1, when we directly compared the slopes of the two functions we failed to find a significant difference between them. 

As in Experiment 1, the perceived lightness range went hand-in-hand with the lightness of the darker sector. 

General discussion

The stimuli we used in these experiments represent the simplest stimulus conditions for the perception of surface lightness. On one hand, a dome with a single homogeneous surface would not be seen as a surface. On the other hand, displays used in prior work, such as a disk/annulus surrounded by darkness, contain at least three different regions, not two. Under our minimal conditions of stimulation, our results establish that lightness is strongly influenced by relative area. The lightness of the darker sector varied by almost three Munsell units as a function of its relative area. 

Clearly anchoring in these simple images depends, not merely on photometric factors, but on geometric factors as well. In one sense this is not surprising. The image can be thought of as a pattern of photometric and geometric relationships. But it is not clear why a surface with the largest area tends to appear white. 

There is also an asymmetry between the two rules of anchoring that should be noted. The highest luminance rule is a firm rule, with almost no exceptions. But the largest area rule is not really a rule – it is more of a tendency. Presumably, the largest area appears completely white only in the limiting case – when its relative area is extremely large. Otherwise there is merely a tendency for the lightness of the largest area to move towards white. 

One leg or two?

Although our results are also consistent with a simple linear function, several statistics support the claim that strong effects of area on lightness occur only when the darker region is larger than the lighter region (which we will call the darker-larger zone), as proposed by Li and Gilchrist (1999): 

1.  
   Many significant area effects were found in the darker-larger zone; none were found in the darker-smaller zone (see Table 1).
2.  
   The correlation of area and lightness was significant (the slope of the regression line was significantly different from zero) in the darker-larger zone, but not in the darker-smaller zone.

Prior research on area and lightness

Approximately 15 published studies can be found in which lightness or brightness was measured as a function of relative area. All of these appear consistent with the suggestion that strong area effects occur only the darker-larger zone. 

In Wallach's classic disk/annulus experiments, the disk was darker than the annulus and its area was about 1/4 that of the annulus. When he reduced annulus area to be equal with disk area, the resulting disk lightness did not change, but when he reduced annulus area further to approximately 1/4 that of the disk, the disk became substantially lighter. These findings are consistent with the darker-larger rule. Indeed, Wallach himself (1948, p. 323) stated “It seems that, once the ring has an area equal to that of the disk, any further increase in its width does not affect the resulting color of the disk.” 

Perceived brightness was measured as a function of relative area in experiments by Diamond (1955) using adjacent rectangular patches and by Stevens (1967) using a disk/annulus display. As shown in Figure 4, both found strong effects on lightness when the darker region had the larger area, but little or no effect otherwise. 

Figure 4

 

Data from Stevens (left) and Diamond (right) with a measure of relative area on the abscissa and brightness on the ordinate. Darker-larger zone shown to left of vertical arrow.

Both Stewart (1959) and Newson (1958) obtained a pronounced effect on lightness in the Gelb effect while varying the size of an adjacent white inducing region. All of their stimuli fell within the darker-larger zone. Indeed, as can be seen in Figure 5, Newson's data curve reached an asymptote just where the area of his inducing region came to equal that of his Gelb surface, indicating that further increases in inducing region area would not have affected lightness. 

Figure 5

 

Effect of relative area on lightness (Newson, 1958). Darker-larger zone is to the left of the arrow marked Equal Areas.

Burgh and Grindley (1962) varied the overall retinal size of a simultaneous lightness contrast display and found no effect on the strength of the illusion, presumably because they did not vary relative size. 

Kozaki (1963) varied relative area in a haploscopically presented square center/surround stimulus surrounded by darkness. She obtained a strong effect on lightness when her stimuli fell in the darker-larger zone and a weak effect outside this zone. 

Yund and Armington (1975) found an effect of relative area on lightness when they tested center/surround stimuli that all fell within the darker-larger zone. 

Heinemann (1972) reported a disk/annulus study in which he varied the width of the annulus. For disks darker than the annulus, he obtained big effects only when annulus area was less than disk area, writing “…the form of the induction curves changes rapidly as the width of the annulus is increased from 1′ to 10.5′, but that further increase in the width of the annulus has little effect” (p. 154). Torii and Uemura (1965), in a very similar experiment, got effects of area only when annulus area was less than disk area. 

Bonato and Gilchrist (1994, 1999) increased the luminance of a target to find the threshold value at which it begins to appear self-luminous. All of their stimuli fell within the darker-larger zone because the target was always brighter and smaller than its background. Consistent with the area rule, they found that increasing the area of the target by a factor of 17 increased the self-luminosity threshold by a factor of 3. In pilot studies with domes, we found a parallel effect. When the lighter region becomes very small relative to the darker region, the lighter regions starts to appear self-luminous, even though the luminance ratio between regions is constant. 

In a further experiment Bonato and Gilchrist increased the perceived size of the target while keeping its retinal size constant and they increased the retinal size of the target while keeping its perceived size constant. Only the first of these produced an effect on luminosity threshold, suggesting that the darker-larger rule should be understood in terms of perceived area, not retinal area. 

Coren (1969) tested contrast effects using a reversible rabbit figure containing one lighter and one darker region. The relative area of his stimuli did not change physically, but because background is perceived to extend behind figure, the perceived areas of his lighter and darker regions did change with the perceptual reversal, in a manner consistent with the perceived size definition of the darker-larger rule. 

Bonato and Cataliotti (2000) found results consistent with the darker-larger rule in two luminosity threshold experiments using stimuli in which perceived area varied while retinal area was held constant. 

Finally, the darker-larger rule, when defined in terms of perceived area may explain the Wolff illusion (Gilchrist, 2006, Figure 9.22, p. 262; Wolff, 1934) in which dark gray appears darker when seen as figure than when seen as ground. The figure is constructed so that the aggregate figural area and the aggregate ground area are physically equal. But because the background appears to extend behind each of the small disks, the perceived area of the background is larger. 

The fact that all 15 of the prior studies of area and lightness show strong effects of area on lightness within the darker-larger zone, but not outside it supports the conclusion that the function we obtained in both experiments has two legs rather than one. Our failure to find a significant difference between the slopes of the two legs can be attributed to the relatively high variability in the data one obtains using two-region dome stimuli, presumably due to the relative simplicity of the stimulus. 

Logic of the darker-larger rule

We have now found two anchoring rules for simple stimuli, one photometric and one geometric: (1) the highest luminance appears white and (2) the largest area tends to appear white. The darker-larger zone is precisely that zone in which these two rules conflict. When the darker region is less than 180° there is no contradiction between the highest luminance rule and the largest area tendency—the highest luminance also has the largest area. Lightness values are firmly anchored. But when the darker region is larger than 180° the photometric and geometric factors collide. The lighter region may have the highest luminance but the darker region has the larger area. As the darker region grows larger it presses to become white as the geometric factor asserts itself more strongly. Indeed when the darker area approaches 360°, its lightness approaches complete white. And in pilot work we found that when the lighter region becomes smaller than our smallest lighter sector, it comes to appear self-luminous, or glowing. 

Strange phenomena in the darker-larger zone

This collision of geometric and photometric factors when the darker region covers at least half the total area provides an explanation for a series of strange phenomena that occur only under these specific conditions. That is, in a two-part field, only when the darker region is larger than the lighter region: 

1.  
   lightness depends on relative area
2.  
   self-luminosity emerges
3.  
   gamut compression occurs—the range of perceived shades is compressed relative to the range of actual shades (Li & Gilchrist, 1999)
4.  
   brightness indention occurs (Schouten & Blommaert, 1995) – a dark halo is seen on the dark side of the boundary
5.  
   Heinemann's (1955) brightness enhancement effect occurs
6.  
   fluorence occurs (Evans, 1974, p. 100)

All six of these phenomena can be understood as attempts by the visual system to resolve the contradiction between the highest luminance principle and the largest area principle. 

Implications for theory

Theories of lightness can be conveniently grouped into low-level, mid-level, and high-level theories. The area effect would seem to make a challenge for both low-level and high-level theories. Clearly neither adaptation nor pupil size, as suggested by Hering (1874/1964) can explain the effect. As for spatial filtering models, typified by Blakeslee, Reetz, and McCourt (2009), it does not appear that these models in their current form could account either for the consistent appearance of the highest luminance as white or for the differential effect of area on lightness within the darker-larger zone as opposed to outside this zone. This is not to say that such models could not handle these effects in principle, with modifications. For example, Fred Kingdom (personal communication, April 30, 2009) notes that a higher contrast gain for “off-” compared to “on-” center filters however could easily handle the anisotropy in the present area data. 

Whittle (1994) showed that the function relating brightness to contrast for simple patch stimuli had a different shape for increments and decrements, at least when contrast was defined conventionally (e.g. Weber, Michelson). Increments showed a compressive nonlinearity while decrements showed a compressive, followed by accelerating, nonlinearity. Thus for medium-to-high contrasts, the brightness difference between a small dark region and its background is bigger than the brightness difference between a small light region and its background. These increment/decrement differences are readily explained in terms of the combined effects of local light adaptation and contrast gain (see Kingdom & Whittle, 1996; McIlhagga & Peterson, 2006). 

This effect of relative area on lightness seems inconsistent with an inverse optics approach to lightness (Gilchrist, Delman, & Jacobsen, 1983; Pizlo, 2001). According to the logic of inverse optics relatively veridical lightness values can be derived by systematically disentangling the various factors that combined to produce the image. So, for example, when a fog or veiling luminance is present in an image, the amount of contrast (difference in log luminance) at borders is reduced. When the visual system detects such a veiling luminance, it can, in effect, subtract a constant amount of light from every point in the image, thus recovering the original luminance ratios (Gilchrist & Jacobsen, 1983). The fog compresses the luminance ratios and the visual system expands them. But the area effect in lightness does not seem to correspond to any physical factor that affects the image. If the world worked in such a way that as a surface became larger, it would reflect more light (with no change in its reflectance), then the area rule would make the percept more veridical. But the world does not work that way. The fact that the area effect exists indicates that it must somehow be adaptive. But we are unable to suggest how. 

Simple images: A summary

In these experiments we explored the rules of anchoring under the minimum conditions for surface perception. An advantage of such simple conditions is that we can largely exhaust the rules of anchoring. 

Simple and complex images

Our clarification of the rules of anchoring in simple images represents an advance, but one that would be of limited interest unless it could be related to the problem of lightness computation in complex, so-called real world, images. The real challenge for a theory of lightness is to explain the computation of lightness in the kind of complex images we confront everyday in the world. Arend (1994) has argued that simple images are too simple to tell us anything about lightness under complex stimulation. But if lightness computation in complex images is different from that in simple images, neither can it be wholly unrelated. There must be some systematic relationship between these. 

But let us first define some terms. What is the essential difference between simple and complex images? It cannot be simply the number of patches in the image. Imagine a mondrian pattern that filled your whole visual field. It would not matter for lightness computation how many patches the mondrian contains. Each new patch would take its lightness from its luminance ratio with the highest luminance in the mondrian. There would be no qualitative change in this computation as we increase the number of patches. Qualitative changes in lightness computation occur only when organizational factors are added to the image, creating new frameworks. Thus, we will define a complex image as one that contains more than one perceptual framework, such as a framework of illumination. 

Applicability assumption

We endorse the claim, central to anchoring theory, that the rules of anchoring found for simple two-part images also apply to perceptual frameworks embedded in complex images (Gilchrist, 2006). This claim constitutes a key component of anchoring theory called the applicability assumption. There is one additional constraint in this application, however. The values of lightness computed within each of these local frameworks are subject to additional influences from outside the framework. This is known as global anchoring, and the combination of local and global values is called co-determination (Gilchrist et al., 1999; Kardos, 1934). 

Consider the highest luminance rule. If every surface in a complex image were anchored by the single highest luminance in the entire visual field, lightness perception would be wildly in error and lightness constancy would not exist. If the illumination level in sunlight were 30 times brighter than that within a shadow, for example, a black surface in the sunlight and a white surface in the shadow would be computed to have the same lightness value. The impressive degree to which perceived lightness values are independent of illumination level, called lightness constancy (Katz, 1935) implies that the highest luminance rule is applied separately within each region of illumination. At the same time, the relatively small failures of such independence suggest that these illumination frames of reference are not completely insulated from each other. There is some crosstalk, known as co-determination (Kardos, 1934) between frameworks. 

There is much evidence that the area function we obtained in our domes applies also to separate frameworks of illumination embedded in complex images. By the definition given above, the stimuli used in these previous studies of area and lightness can be considered complex images. These images often contained two regions of interest, one lighter and one darker, often called test and inducing regions, surrounded by a dark background. The two regions of interest are perceptually segregated from the dark surround. Thus, while the images used in all the previous studies contained at least three regions (and typically more) the important thing is that they all contained at least two perceptual frameworks. 

Perhaps these images are not as complex as those we encounter daily. But the transition from one to two frameworks in the image critically reveals how the nature of the computation shifts as multiple frameworks are introduced. Note that we are not saying that the computation within simple and complex images is the same, but merely that the computation within a framework is the same. The computation is different in that there is crosstalk (Sedgwick, 2003) between frameworks (Kardos, 1934). In both our domes experiment and in the earlier dark room experiments the darker region appeared lighter as it became larger. But the effect of surrounding the two regions of interest with darkness is to reduce the lightness difference between them, as seen in the staircase Gelb demonstration (Cataliotti & Gilchrist, 1995). Thus, even outside the zone of the darker-larger rule (when the darker region is the smaller region) the lightness difference between two adjacent regions with a given luminance ratio will be less when the two regions are surrounded by darkness than when the two regions fill the entire visual field. This is the concept of co-determination of Kardos (1934). In anchoring theory it is the weighted average of local and global anchoring (Gilchrist et al., 1999). 

Acknowledgments

This research was supported by grants from the National Science Foundation (BCS-0643827) and the National Institute of Health (BM 60826-02). The authors would like to thank Michael Rudd and Fred Kingdom for constructive comments on an earlier draft, and Jennifer Faasse, Simone Whyte, Camilo Marmolejo and Oscar Escobar for their invaluable help in data collection. 

Commercial relationships: none. 

Corresponding author: Alan Gilchrist. 

Email: [e-mail]. 

Address: Psychology Dept., Rutgers Univ., Newark, NJ 07102, USA.