## Alan Gilchrist

Print publication date: 2006

Print ISBN-13: 9780195187168

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780195187168.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2019. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 15 February 2019

# An Anchoring Model of Errors

Chapter:
(p.294) 11 An Anchoring Model of Errors
Source:
Seeing Black and White
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780195187168.003.0011

# Abstract and Keywords

This chapter offers a specific error-driven theory of lightness framed in terms of the rules of anchoring. The lightness model described in Gilchrist et al. (1999) is presented in more detail in this chapter. Its principal strength is the wide range of lightness errors it can accommodate. Central to the model is its construction of the relationship between simple and complex images. The discussion starts with a consideration of lightness in simple images. At the heart of the anchoring model lies a critical assumption about the relationship between simple and complex images: the rules of anchoring found in simple images are applicable to frames of reference embedded within complex images. The same anchoring rules are held to apply to both.

The lightness model described in Gilchrist et al. (1999) is presented here in more detail. Its principal strength is the wide range of lightness errors it can accommodate. Central to the model is its construction of the relationship between simple and complex images. But first we will consider lightness in simple images.

# ANCHORING IN SIMPLE IMAGES

Let us begin with a definition of simple images. I will define the simplest image as one that contains only two shades of gray, and these fill the entire visual field. There has not been complete agreement on what constitutes the simplest image. Evans (1974, p. 84) has argued that a spot seen against darkness constitutes the simplest image. Heinemann (1972, p. 146) has nominated the disk/annulus display. Koffka (1935, p. 111) has argued that a completely homogeneous visual field (ganzfeld) is simpler than a point of light in a dark field, but he did not claim that such an image produces lightness. For that, at least one edge is necessary. As Wallach noted (1963, p. 112), “Opaque colors which deserve to be called white or gray, in other words ‘surface colors,’ will make their appearance only when two regions of different light intensity are in contact with each other.”

In practice, this can be achieved by placing the observer's head inside a large dome, as described in Chapter 9. Under these simple conditions, the computation of lightness can be exhaustively described by three rules.

1. 1. Highest luminance rule. The highest luminance is defined as white (reflectance 90%), and the darker region is seen relative (p.295) to this standard, using the Wallach ratio principle. The formula, which applies to both regions, is:

$Display mathematics$
where Lt is the luminance of the target, Lh is the highest luminance, and lightness is defined as perceived reflectance.

2. 2. Area rule. Area is held to influence lightness only when the darker region has the larger area. For a two-region dome, the combined formula for highest luminance and area is:

$Display mathematics$
where Ad is the area of the darker region, as a percentage of the total area in the field. Note that as Ad increases, lightness also increases. When Ad is 50 or less, this formula reduces to formula 1 above.

3. 3. Scale Normalization Rule. The range of perceived grays in the dome is expanded in proportion to the degree to which the physical luminance range is truncated relative to the canonical range of 30:1. If the physical range exceeds 30:1, the perceived range is proportionately compressed. The formula is:

$Display mathematics$
where R is the actual range. The value .56 represents the slope of the regression line obtained by plotting perceived range against actual range in a disk/ganzfeld experiment reported by Gilchrist and Bonato (1995) and shown in Figure 9.24. Expansion always projects away from the anchor, just as compression always collapses toward the anchor. Perceived reflectance of a specific patch is predicted by combining this formula with formula 1 above, as follows:
$Display mathematics$

Domes that contain more than two regions will still be considered simple as long as they contain no segmentation factors, which are described later.

# SIMPLE VERSUS COMPLEX IMAGES

The fact that we could achieve such a command of the rules of anchoring in simple images is gratifying. But the significance of this is very limited unless these findings can be applied to complex images. The question of how simple images are related to complex images has been addressed sporadically, but it has never gotten the attention it deserves. The induction experiments of the contrast period were based on the tacit assumption that the behavior of isolated points of light can be directly extrapolated to all the points that compose a complex (p.296) image. Why else would one care so much about the brightness of spots of light in a dark room? But with the computational period came experiment after experiment contradicting this assumption of direct extrapolation.

Arend (Arend & Goldstein, 1987; Arend & Spehar, 1993a) has criticized disk/annulus experiments as fatally flawed, suggesting that such stimulus conditions are too ambiguous to teach us anything about lightness in complex scenes. Clearly the results obtained under simple conditions cannot be mindlessly applied to complex images. And yet, it would be surprising if there were no systematic relationship between lightness in simple images and lightness in complex images.

# THE APPLICABILITY ASSUMPTION

At the heart of the anchoring model lies a critical assumption about the relationship between simple and complex images: the rules of anchoring found in simple images are applicable to frames of reference embedded within complex images. The same anchoring rules are held to apply to both. The values so computed in local frameworks do not represent the final perceived values, however, until they are combined with values computed globally. This is the Kardos principle of co-determination. But the anchoring model holds that the formulae describing the lightness computation within a given framework are the same whether a simple image inside a dome, a local framework such as a shadow, or (presumably) the entire visual field. These rules are three: the highest luminance rule, the area rule, and the scale normalization rule.

## Area and the Applicability Assumption

Ana Radonjić and I recently completed an extensive experiment that tested the role of area in simple images but in a larger sense was really a test of the applicability assumption. Nine domes were created. Each one had the same two shades of gray, but in different proportions, using radial sectors, as seen in Figure 11.1. We had several reasons to revisit the question of area. The predicted area rule graphed in Figure 9.10 was based on only a few data points. And the radial sector method removed the confound between relative area and retinal eccentricity; the proportion of light and dark was always the same in the fovea and in the periphery, as long as the subject fixated the central vertex.

The results presented in Figure 11.1 show that lightness was significantly influenced by relative area only when the darker region was greater than 180°. This confirms the shape of the curves predicted in Figure 9.10. More importantly, however, it implies that the area rule (p.297)

Figure 11.1. Perceived lightness of light and dark regions within a simple image that filled the observer's entire visual field, as relative area was varied, with luminance values constant.

obeys the applicability assumption, given that the area rule has already been demonstrated several times (Diamond, 1955; Newson, 1958; Stevens, 1967; Stewart, 1959) in complex images.

It should be noted, however, that these results did not show the gamut expansion predicted by the scale normalization rule—that is, in those domes to which the area rule does not apply, the range of perceived reflectances was slightly less than the actual range (approximately 5:1), casting some doubt on the status of the scale normalization rule.

## Frameworks in Complex Images

A complex image will be defined as an image that contains at least one segmentation factor or more than one framework. A framework will be defined by a group of patches in the retinal image that are segregated, or a group of patches that belong together or are grouped together.

Frameworks of illumination can be seen clearly in some of the wonderful photographs by Cartier-Bresson, one of which is shown in Figure 11.2. In Figure 11.2 I have added a set of probe disks. Though the disks are equal in luminance, they appear as different shades of gray within the different frameworks. Adding probe disks to Adelson's (p.298)

Figure 11.2. Probe disks of identical luminance appear different within different frames of reference. (Henri Cartier-Bresson, Trastevere, Rome, Italy, 1959. Photograph courtesy of Magnum Photos.)

checkered shadow image (Fig. 11.3) shows that it is not the local contrast ratio that determines the lightness of these disks, but rather the framework within which each falls. The disks on squares A and B have identical local contrast (their backgrounds are also identical), and yet they appear different in lightness. The disks flanking the letter A have very different local contrast and yet they appear roughly equal in lightness, and the same holds for the disks flanking the letter B. The weak simultaneous contrast effect seen there is small relative to the difference between disks inside the shadow and disks outside.

Consistent with these effects, Rock et al. (1992) showed that achromatic targets are phenomenally grouped based on their equal perceived lightness, not on equal luminance values or on equal local luminance ratios.

(p.299)

Figure 11.3. All of the probe disks added to this Ted Adelson display are identical, but those inside the shadow appear light gray while those outside the shadow appear dark gray. Within each of these regions it matters little whether the disk is on a light square or a dark square (http://web.mit.edu/persci/people/adelson/checkershadow_illusion.html). Reprinted with permission.

## Segmenting Frameworks

Clearly an important question is how such frameworks are identified within the retinal image. The anchoring model postulates two strong factors and series of weaker ones. The strong segmentation factors are the factors identified by Kardos: penumbra and depth boundaries (i.e., corners and occlusion edges). The Gestalt grouping factors create weaker frameworks as well. The black background in a simultaneous contrast display, for example, would be a weak framework. To these grouping factors we can add T-junctions and X-junctions. Grouping factors, of course, are just the flip side of segmentation factors. When frameworks interpenetrate one another, like the set of black squares in checkerboard contrast, it is more appropriate to speak of grouping factors than segmentation factors.

## Co-determination

Because frameworks in complex images are not completely isolated from each other functionally, the rules of anchoring must be applied to them using the Kardos principle of co-determination. In other words, each target surface in a complex image is a member, virtually by definition, of more than one framework, each of which exerts an influence on its lightness. In the model, a separate lightness value is computed for a given target within each of the frameworks to which (p.300)

Figure 11.4. Applicability assumption: The three rules of anchoring that govern simple frameworks (domes) can be seen at work in frameworks that are part of complex images. (Top) Highest luminance rule. (Middle) Area rule. (Bottom) Scale normalization rule.

it belongs, including the global framework (the whole image) and at least one local framework. Then a weighted average is taken to determine perceived lightness.

Co-determination in complex images can be seen in the three displays shown in Figure 11.1. Each display is composed of two side-by-side frameworks. I will treat each pair of frameworks as a global framework, ignoring the larger context.

Figure 11.4, top, illustrates the highest luminance rule. Consider the two targets marked as equal in luminance. The target on the shadowed right side appears approximately white, because it is the highest (p.301) luminance in its framework. It would be seen as pure white if the image on the right were painted onto the inside of a dome so that it filled the whole visual field. However, in the context of the adjacent lighted Mondrian, that target appears light gray rather than white. This illustrates the co-determination. Both parts of the compromise are phenomenally available here. If there were local anchoring but no global anchoring, the right-hand target would appear white. If there were no local anchoring, it would appear the same as the left-hand target. Clearly the percept lies between these values.

Figure 11.4, middle, shows the area rule at work. The small dark disk on the left appears darker than the larger disk on the right (even though they are physically equal), and the small light disk on the right looks lighter than the large disk on the left (also physically equal). If anchoring were totally global, these perceived differences wouldn't exist. But if anchoring were totally local, the differences would be much greater. If each of the two patterns were painted onto the inside of a dome, for example, the small disk on the left would appear slightly darker than it does here and the large disk on the right would appear much lighter, almost white, due to the area rule. Again, the differences that we see in these iso-luminant disks show the co-determination because they lie between zero difference and the extreme difference of the two-dome case.

The Wolff illusion (Fig. 9.22) shows the same thing for perceived area.

Contrast-contrast (Chubb et al., 1989), seen at the bottom, can be viewed as an application of the scale normalization effect to complex images. The left half of the display contains a full luminance range but the right half does not. Thus, in the right half the perceived range is expanded, and this is seen as greater contrast. Again, the illusion seen here is much weaker than if each half were presented separately within a dome, due to the role of global anchoring that is induced by placing them side by side.

## Weighting the Co-Determination

The lightness of any given target is a weighted average of the values computed for that target in both the global framework and any local frameworks. The weighting depends primarily on the strength of each local framework. For a target belonging to the global framework and one local framework, the formula is:

$Display mathematics$
where W1 is the weight of the local framework and 1-W1 is the weight of the global framework. Ltl and Lhl represent target luminance and highest luminance in the local framework; Ltg and Lhg represent these values in the global framework.

(p.302) The strength of a local framework depends mainly on its size, its degree of articulation, and the strength of its segregation. Greater articulation and larger size give the framework greater weight. Quantifying the strength of the weights will require more work, but a rough idea can be gotten from existing data. For example, in the textbook example of simultaneous contrast, if the gray targets appear to differ by about one Munsell step, a typical result, this implies a local framework weight of about 12%. Cataliotti and Gilchrist (1995) found that when a group of five squares is placed on a group of five squares (the staircase Gelb effect display, to be described soon), the black square appears as a Munsell 5.5. This implies a local framework weight of about 40%. The five squares have greater articulation than a local framework in the simultaneous contrast display (5 versus 2), and they are more strongly segregated (by a depth boundary).

### Coplanarity as a Graded Variable

In my experiments that led to the coplanar ratio principle (Gilchrist, 1980) I treated belongingness due to coplanarity as an all-or-none affair, even though Kardos (and Koffka with less clarity) had presented it as a graded factor, as required by the principle of co-determination. Other work (Gogel & Mershon, 1969; Wishart et al., 1997) suggests that the Gestaltists were correct.

# LIGHTNESS COMPUTATION IN COMPLEX IMAGES: A SUMMARY

1. 1. Complex images are segmented into frameworks.

2. 2. Within a given framework each target is assigned a value according to three rules of anchoring for simple images: highest luminance, area, and scale normalization.

3. 3. Local values are weighted according to the area and articulation of the local framework, and these values are combined with the global values.

## Modification of Katz's Rules

Katz's law of field size and his principle of articulation have been incorporated into the model, but with some modification. First, the model reflects the empirical findings that the law of field size works for perceived size but not for retinal size. More important, Katz claimed that greater field size and articulation produce greater constancy within a field. In the model, however, these factors produce stronger anchoring within the framework (or field). In many cases this would lead to greater constancy, but not in all cases. When the backgrounds in simultaneous lightness constancy are articulated, as shown in Figure 10.2d, the result is less constancy—that is, a greater illusion.

# (p.303) ROOTS OF THE MODEL

This anchoring model emerged somewhat by accident in an experiment designed to pit the anchoring notion against a contrast notion.

## Anchoring versus Contrast

In our work on the anchoring problem, we had repeatedly found that results that had been reported earlier from the perspective of contrast or induction could be understood as well or better from the perspective of anchoring. For example, the highest luminance rule accounts for the empirical finding that contrast effects primarily involve influences from the brighter region to the darker. In addition, the role of relative area in anchoring seems to account for those situations in which upward induction is obtained. These findings led to the obvious but radical possibility that everything that had previously been attributed to contrast (or induction) is really a manifestation of anchoring. Thus, Joe Cataliotti and I (Cataliotti & Gilchrist, 1995) conducted a series of experiments designed to tease apart contrast and anchoring interpretations of phenomena typically attributed to contrast.

It is well known that, in general, a target surface will appear darker when a brighter region is placed next to it. We chose a dramatic example of this phenomenon, one frequently invoked to illustrate contrast: the Gelb effect. We used the spatial function of lateral inhibition to distinguish between anchoring and contrast accounts of this effect. It is generally agreed that contrast effects due to lateral inhibition decrease rapidly with lateral distance between the inducing region and the target. Shapley and Reid (1985) had also proposed that edge integration weakens with distance in a similar way. But anchoring involves no such fall-off. As long as the target and inducer lie within the same framework (or field of illumination), the distance between them should not matter. We were able to create several experiments that exploited this difference.

## Staircase Gelb Effect

Our basic experiment involved a stepwise version of the Gelb effect.1 First we suspended a black square (reflectance 3%) in midair and illuminated it with a bright projector spotlight so that it appeared completely white. The illumination intensity at the target location was 30 times that of the ambient room illumination, adjusted to give a black surface in the spotlight the same luminance as white in ambient illumination. Then we added a dark-gray square (reflectance 12%) next to the black square. Then, in like fashion, a middle-gray square (30%) was placed in the next position, then a light-gray square (60%), and finally a white square (90%). This left a horizontal row of five squares, arranged in ascending steps, roughly equal on the Munsell scale,2 from black at (p.304)

Figure 11.5. Perceived lightness of black square in a spotlight as lighter squares are added. The amount of darkening depends only on the amount by which each new square raises the highest luminance in the group, but not on its proximity to the black square.

one end to white at the other. The projector beam formed a horizontal rectangle to accommodate all five squares. Observers matched each target using a 16-step Munsell chart under separate illumination.

Our test would be the amount of darkening produced in the black target square when each brighter square was added to the group. According to the spatial function of lateral inhibition (and to Shapley and Reid's claim), the darkening of the target should decrease as new, brighter squares are added in positions successively more distant from the target. But according to anchoring, the darkening should depend merely on the degree to which each new square increases the maximum luminance in the group.

Our results are shown in Figure 11.5. If the darkening effect of the highest luminance were weakened by distance, we would have obtained a negative but positively accelerated curve. But we obtained a straight line, which strongly supports the anchoring prediction. We found no evidence for a spatial function, despite the fact that the brightest square, the white square, was separated from the target square by 4.5° of visual angle.

In another condition we switched the position of the dark-gray square and the white square so that the white square was now adjacent to the black target square. Ten additional observers matched the target square under these conditions, producing a mean Munsell of 6.0, exactly the same mean shown in Figure 11.5 when the white square was far from the target.

(p.305)

Figure 11.6. The Mondrian world (Annan & Gilchrist, 2004). The lightest squares, physically dark gray, appear white. Adapted with permission from Gilchrist et al., 1999.

In a further experiment we explored the spatial function by testing 10 additional observers at one-third the viewing distance. This multiplied all the visual angles within the display by a factor of three, producing a separation of 13.5° of visual angle between the target square and the white square. The spatial function of lateral inhibition clearly refers to retinal distance, not perceived distance. Despite the greater retinal separation, not one of the squares appeared significantly darker in the far condition compared to the near.3

We conducted a modified version of our basic five-squares experiment using a 4.6° square Mondrian composed of 15 squares ranging in reflectance from black to middle gray (Munsell 6.0). This Mondrian was presented in a special beam of light just as with the five squares. Ten naive observers matched the perceived lightness of all 15 squares in the Mondrian. Then a white square was added to the Mondrian and a separate group of 10 observers again matched all 15 squares. All 15 squares appeared darker when the white square was present. However, we found that the darkening of squares adjacent to the white square was not significantly greater than the darkening of remote squares.4

And finally, to eliminate the possibility of uncontrolled influences from the larger surround, we replicated the Mondrian experiment in what we called a Mondrian world (Fig. 11.6). Observers' heads were (p.306) placed within a trapezoidal chamber, all of the interior surfaces of which were covered with a Mondrian pattern composed of dark-gray shades ranging from black to dark gray (Munsell 4.0). One group of observers matched a subset of these squares and a separate group of observers matched the same squares after a real white square had been introduced into the display. Again we found no difference between the darkening effect on adjacent squares and that on remote squares.

## Uniform Influence of Anchor within a Framework

We concluded that when the highest luminance is increased within a group of surfaces that belong together so as to constitute a frame of reference, the standard of whiteness used to compute gray shades of lower values is changed for all elements within the framework, regardless of distance from the highest luminance. This implies that concepts like contrast and induction are better understood in terms of anchoring.

It should be noted that the lack of spatial function applies only to members of a single framework. When two isolated surfaces are presented in midair at some distance from each other, and one of the two is the highest luminance in the field, the lightness of the darker square depends in a systematic way on the separation between them. This is a solidly established empirical result (Cole & Diamond, 1971; Dunn & Leibowitz, 1961; Fry & Alpern, 1953; Leibowitz, Mote & Thurlow, 1953; Newson, 1958; Stewart, 1959) that also makes sense from an anchoring perspective. The greater the proximity between the two targets, the more they form a group and the more the lightness of the darker target is anchored by its relationship to the brighter target.

## New Finding: Strong Compression of Gray Scale

The straight line shown in Figure 11.5 shows that the anchoring power of the highest luminance is roughly homogeneous across the members of the coplanar group. But the shallow slope of this line reveals a new finding, a strong compression on the lightness scale (the y-axis) that caught us completely by surprise. The darkening effect of each higher luminance was much less than one would expect based on Wallach's ratio principle supported by edge integration. Although the five squares physically spanned the entire gray scale from black to white, they were perceptually compressed into the upper half of the gray scale, and the darkest square appeared no darker than Munsell 6.0.

This is a substantial and surprising failure of lightness constancy, and the failure has a very specific pattern. It is not the case that each of the five squares shows the same degree of erroneous increase in its perceived lightness. There is a gradient of constancy failure, with black showing a whopping failure of about half of the entire gray scale, white showing no failure at all, and the other three squares showing (p.307) failures proportionately spaced between these. This amounts to a compression of the gamut of perceived gray shades, and we will refer to this as gamut compression after Brown and MacLeod's (1997) term gamut expansion.

## Why Is the Gamut Compression Surprising?

We expected that once the true white had been added to the group of squares, the black square, with a luminance 30 times less than white, would finally appear black. Certainly a disk surrounded by an annulus 30 times brighter appears black. Nor can the compression be explained by the separation between the white and black squares, because the compression remains when the two squares are adjacent. This compression is a prime example of an illegitimate error, described in Chapter 10.

The Gelb effect is one of the most celebrated phenomena in lightness. In virtually all descriptions of the illusion, even by authorities like Katz (1935), Koffka (1935), Woodworth (1938), Wallach (1963), Evans (1948), and Helson (1964), the black target is said to appear black when a white paper is placed next to it. However, the empirical results tell a different story, one that is consistent with the compression we obtained. Even when a white target and a black target are presented side by side within the spotlight, the black target does not appear black; it appears lighter than middle gray. Stewart (1959) who conducted the most systematic study of the Gelb effect, reported Munsell values of 6.9 to 8.5, depending on conditions. Newson (1958) reported values of 5.5, 4.9, and 5.3. Gogel and Mershon (1969) reported a value of about 4.75.5

The Gelb target appears black in the spotlight only when it is completely surrounded by white (Horeman, 1963; McCann & Savoy, 1991).

## Source of the Compression

What can account for such a dramatic compression in lightness values? Three possible hypotheses can be suggested: (1) adaptation, (2) a lightness ceiling effect, and (3) co-determination.

1. 1. Adaptation. The five squares constitute an island of bright light surrounded by a sea of dimmer light. Perhaps in its adaptation to the ambient illumination, the visual system cannot accommodate the entire range within the spotlight. But this hypothesis seems to be knocked out by a simple variation: if a narrow rectangular white border is placed around the perimeter of the five squares, but still within the spotlight, the compression is almost completely eliminated! Somehow the white border seems to insulate the five squares from influence by the global (or foreign) framework. I refer to this as the insulation effect but cannot explain it further. A black border has no such effect, (p.308)

Figure 11.7. Lightness matches for the five gray shades under three configurations. Compression is reduced when the black square is surrounded by a white border, regardless of whether the border makes contact with the black square. Adapted with permission from Gilchrist et al., 1999.

and a middle-gray border seems to provide half as much insulation as a white border (see Gilchrist et al., 1999, Fig. 12). Apparently such insulation cannot be reduced to a contrast effect. It is necessary merely that the five squares be surrounded by the white border; they need not make any contact with it, as can be seen in Figure 11.7 (from Gilchrist et al., 1999).

2. 2. Ceiling effect. When all five squares are presented simultaneously in the spotlight, the brightest one or two squares appear self-luminous. It might be argued that the compression we observe in the data line in Figure 11.5 is the result of a ceiling effect imposed on the data by the Munsell chart: no matches above white are possible. Would the data line revert to a slope of +1 if we allowed observers to make matches in the zone of self-luminosity? The answer seems to be no. Cataliotti and I created an extended Munsell chart with samples of luminosity in 7 steps beyond white. Data produced by this chart were difficult to interpret. Indeed, two arguments show that the basic idea is misguided. First, as has been amply shown (Jacobsen & Gilchrist, 1988a,b), the ratio principle (with its characteristic +1 slope) does not apply to the zone of self-luminosity, only to the zone of opaque surface grays. Luminous surfaces produce luminance matching, not ratio matching. Second, the compression seen in Figure 11.5 is seen in the three darkest-gray squares, none of which appears self-luminous. If (p.309) the compression were the result of a ceiling effect, there should be a knee in the curve, with opaque targets, such as those on the left, falling along a +1 slope, and the luminous targets, such as those on the right, falling along a horizontal line. No such knee is obtained.

3. 3. Co-determination. According to Kardos (1934), each of the squares will be seen both in relation to the neighboring group (the relevant framework) and in relation to the remainder of the laboratory (the foreign framework). This analysis is quite compelling when one views the display from the perspective of the observer. And it appears to account for the results, as we will see.

We seem to have here a dramatic example of what Kardos (1934) called co-determination by the relevant field of illumination and by the foreign field of illumination. This can be clarified by a thought experiment. Imagine we eliminate either the relevant field or the foreign field. To eliminate the foreign field, we would have to get rid of the laboratory scene. Merely turning off the lights would not be sufficient, however. We would have to enlarge the five squares so that they fill the entire visual field, say by painting them onto the interior of a dome. In that case they would be seen veridically On the other hand, we could eliminate the relevant field containing the five squares. We could present each square by itself. We know that each square would be seen as white under such conditions.6 The anchoring model described above is couched in terms of local and global frameworks rather than relevant and foreign, for reasons that are given later.

This analysis of the staircase Gelb phenomenon is illustrated in Figure 11.8. The x-axis gives the luminance of each square. The y-axis gives the predicted lightness value within each framework. Predictions from local anchoring are shown by the diagonal line (which coincides with veridicality). Predictions from global anchoring are shown by the horizontal line. The actual data obtained from observers who viewed the entire group of five squares are shown by the dashed line. One can readily see this line as a weighted average of the local and global lines. Even the error bars show a corresponding gradient, proportional in length to the discrepancy between local and global predictions for a given square.

If the compression is caused by local and global co-determination, it should be possible to move the data systematically between the local and global predictions by strengthening or weakening the competing frameworks.

### Determinants of Framework Strength

Testing the hypothesis that greater articulation within a framework produces stronger anchoring within that framework, Cataliotti and I replaced the five squares in the spotlight with a 15-patch Mondrian using the same five shades of gray. This produced a dramatic decrease (p.310)

Figure 11.8. Anchoring analysis of the staircase Gelb display. The perceived values of the five squares represent a compromise between their computed values in their local framework and the values computed in the global framework. Adapted with permission from Gilchrist et al., 1999.

in the compression. The black patches, for example, appeared almost black as opposed to Munsell 6.0 in the five-square row.

This implies that the 15-patch Mondrian constitutes a stronger framework than the five-square linear array. What makes it stronger? There are three obvious differences between the five-square condition and the 15-patch Mondrian. First, there are 15 patches in one case and five in the other. Second, the 15-patch Mondrian constitutes a larger framework than the five squares in a row. And third, the squares in the five-square case are arranged in a graduated line from darkest to lightest, unlike in the Mondrian. In short, we found all three of these factors to play some role.

1. 1. Testing the configurational factor, we compared the linear arrangement of the five squares to a five-patch Mondrian, holding both number of squares and total area constant. We also compared a 10-square linear array to a 10-patch Mondrian. The results are shown in Figure 11.9. In both comparisons, there is significantly less compression in the Mondrian configuration than in the linear configuration.

2. 2. One can also begin to see from this figure that the number of patches plays a role. This is seen more clearly in Figure 11.9, in which the data are presented for Mondrians containing 2, (p.311)

Figure 11.9. A collage of patches presented in a spotlight shows less compression than a line of patches. Adapted with permission from Gilchrist et al., 1999.

Figure 11.10. The greater the number of patches in a Mondrian under a spotlight, the less the compression. Adapted with permission from Gilchrist et al., 1999.

(p.312)

Figure 11.11. The larger the display under spotlight, the less the compression. Adapted with permission from Gilchrist et al., 1999.

5, or 10 patches. Here we see a strong effect of the number of patches.

3. 3. We also made a test of the effect of area. Figure 11.11 presents a comparison between the five-patch Mondrian already mentioned and another five-patch Mondrian identical in every respect except its size, which was five times larger along each dimension (the area being 25 times larger). Here we found a small but systematic effect, with less compression for the larger framework. Recall that when we increased the retinal size of the display by reducing the viewing distance, this did not reduce the compression. Again, Katz's law of field size seems to work for perceived size but not for retinal size.

In a further experiment, we made everything in the observer's field of view totally dark except for the five-square display.7 This produced a modest darkening of the five squares. This result, which is opposite to what would be predicted by contrast theories, might be explained by the reduced articulation in the surrounding framework.

# APPLYING THE MODEL: ILLUMINATION-DEPENDENT ERRORS

In applying the anchoring model to the illumination-dependent errors listed in Chapter 10, keep in mind this helpful rule: local anchoring produces constancy,8 while global anchoring produces failures of constancy. (p.313) When the local framework is a separate field of illumination and when it contains a white surface, a veridical lightness value will be given by Lt/Lh * 90%. Anchoring in the global framework is equivalent to luminance matching, which is the signature of zero constancy. The success of the model in accounting for illumination-dependent errors will hinge on the degree to which the strength of local anchoring correlates (negatively) with error size.

## The Fundamental Illumination-Dependent Error

Because the anchor in the brighter field is necessarily higher than the anchor in the darker field, the lightness of any target in the darker field will be lower when computed globally than when computed locally. Thus, global anchoring darkens surfaces in lower illumination. Global anchoring lightens surfaces in higher illumination due to scale normalization. The global range typically exceeds the standard black-white range. Thus, values in the global framework are compressed upward toward white.

## Illumination Difference: Larger Error with Larger Difference

The greater is the illumination difference between two frameworks, the greater is the difference between the anchors in the two. This means that for any given target, its value when anchored globally will differ from its value when anchored locally by a greater amount. The local/global compromise will thus deviate more from the local computation alone, meaning it will deviate more from veridicality.

## Standard Gradient of Error: Error Size Depends on Target Reflectance

A white surface in the higher illumination will be computed as white in its local framework, but also white in the global framework. Thus, there will be no error for this surface. A black surface, however, will be black relative to its local framework but much lighter relative to the global framework. Thus, it will show a large error. Gray shades between these two poles will show a gradient of errors.

A similar logic applies to the darker field of illumination. A black will be computed as black in this field, but also as black relative to the global framework. A white surface, however, will be computed as white in this local field, but much darker gray relative to the global framework. Again, the intermediate shades show gradations of error.

## Background Reflectance: Larger Errors with Darker Backgrounds

This fact follows from the asymmetry in anchoring: white is special, black is not. According to the anchoring model, a target will be computed (p.314) as white in its local framework whenever it is the highest luminance, regardless of its actual shade of gray. Whenever both targets happen to be the highest luminance in their local frameworks, they will be computed as the same lightness value (white) even if they are actually very different. In this case local anchoring does not produce veridical perception.9 The darker the backgrounds on the lighted and shadowed sides, the more likely it is that each target will be the highest luminance in its field. This is especially true when the two fields are poorly articulated.

Strictly speaking, this analysis implies that the lack of constancy is associated with incremental targets, not necessarily with dark backgrounds. Thus, anchoring theory predicts that decremental targets on dark backgrounds would produce better constancy than incremental targets on relatively light backgrounds. Indeed, this is exactly what Kozaki (1963, 1965) has reported to be the case. This may also explain why better constancy has often been found with targets of lower reflectance (Evans, 1948; Hsia, 1943; Oyama, 1968): they are more likely to be decrements.

## Framework Size: Larger Errors with Smaller Frameworks

According to anchoring theory, the larger a local framework, the stronger the anchoring within that framework. This stronger local anchoring is held to be the reason that Katz obtained better constancy with larger field size. This is not merely a restatement of Katz's claim. Describing the result in terms of strength of local anchoring gives an account of why better constancy results from larger field size, an account that places the law of field size within a more comprehensive theory of errors.

## Framework Articulation: Less Error with Greater Articulation

According to the anchoring model, the greater the articulation within a framework, the stronger the anchoring within that framework. Again, this explains Katz's observation that greater articulation is associated with greater constancy.

Thus, the basic model of local and global anchoring appears consistent with all six of the main features of illumination-dependent failures that turned up in our survey of errors. At least this should establish the model as a strong candidate for a theory of illumination-dependent constancy failures. But to constitute a theory of errors in general, the model must work for background-dependent errors as well.

### Local and Global, or Relevant and Foreign?

The model given so far differs from the theory of Kardos in the definition of the interacting frameworks. While Kardos spoke of the relevant (p.315) framework and the foreign framework, our model speaks of local and global frameworks. The difference is simply that the global framework includes the local framework (as in a nested hierarchy), while relevant and foreign frameworks exclude one another.

For the illumination-dependent errors we have just reviewed, we could use either the local/global construction or the relevant/foreign construction, although the latter works a bit better. But for background-dependent errors, such as simultaneous contrast, the local/global construction seems to work distinctly better.10 Thus, to bridge the two classes of error, the model has been stated in terms of local and global frameworks.

This does, however, leave us with a problem in explaining the staircase Gelb display.

### The Problem of the Horizontal Global Line

According to the local/global analysis of lightness given above, each of the five squares in the staircase Gelb display is computed to be white in the global framework. This shows up as a horizontal G-line in Figure 11.8. That would make more sense if we were using the relevant/foreign construction of Kardos (1934), because each of the squares would indeed appear white11 relative to the foreign region outside the spotlight. But the model given here uses a local/global construction, which means that the local framework is part of the global.

This may seem puzzling. After all, the white square in the spotlight is the highest luminance in the global framework and each of the other four squares stands in a different ratio to the luminance of the white square. Why, then, is the G-line not sloping?

The short answer is that a horizontal G-line fits with plausible weighting values for the local and global framework. The empirical data shown in Figure 11.8 imply local and global weights that are roughly equal. This seems reasonable for a small field of only five elements. Adding some slope to the G-line can produce the slope of the empirical line only if global anchoring gets far more weight than local, and this doesn't seem intuitively reasonable.

In our prior paper (Gilchrist et al., 1999), the horizontal G-line was justified by the small area of the five squares relative to the area of the global framework. Thus, the five squares have little influence on the global anchor. This is consistent with the role of area we found in our anchoring experiments with domes. Each of the five squares is assigned a global value of white because each has a luminance equal to or higher than that of the global anchor (which is roughly that of a white surface in the normally illuminated laboratory). According to this analysis, if the group of five squares is made much larger, the G-line should acquire a slope and the compression should be reduced. (p.316)

Figure 11.12. This contrast display (due to Elias Economou) suggests that figure belongs to ground more than ground belongs to figure.

Cataliotti and I did find a reduction in compression when we increased the total area of the five squares by a factor of 25, but this result was predicted based on the law of field size, and we have no clear evidence that it produced a sloping G-line.

Alternatively, the horizontal G-line might be justified by a kind of figure/ground asymmetry. There are some reasons to believe that figure belongs to ground more than ground belongs to figure. This asymmetry can be seen in Figure 11.12. The fact that the left-hand gray target appears lighter than the right-hand target shows that each gray region is seen relative to its surround, not to the region it encloses. This is closely related to the Kardos principle of the next deeper depth plane. In terms of fields of illumination, the spotlight containing the five squares constitutes a figural region of illumination, while the remainder of the laboratory constitutes a background region. If so, targets in the spotlight would be influenced by the surrounding framework, but not vice versa.

## Depth and Lightness

The coplanarity principle of lightness fits happily within the anchoring model, but the idea of competing frameworks suggests that coplanarity should be viewed as a graded factor, as Koffka had suggested. Edge classification, by contrast, implies an all-or-none distinction, except in the case of compound edges.12 Empirical evidence that coplanarity effects are indeed graded can be found in reports by Wishart et al. (1997) and Gogel and Mershon (1969).

# APPLYING THE MODEL: BACKGROUND-DEPENDENT ERRORS

A crucial question now is how well the anchoring model performs on background-dependent failures, lightness errors produced by various (p.317) configurations of regions that appear to differ only in reflectance. The model has proved its worth in accounting for the variety of illumination-dependent errors. If the same model could account for background-dependent errors as well, this would constitute an important theoretical development. The place to begin, of course, is with the simultaneous lightness contrast display.

## McCann's Account of Simultaneous Contrast

Not only does the model work quite well for the simultaneous contrast display, but in fact just such an account has been already given by John McCann (1987, p. 280):

If global normalization of the entire field of view were complete, we would expect that observers would report the two gray squares with identical reflectances would have the identical appearance. If local mechanisms were the only consideration, then the gray square in the black surround should mimic the results found in the Gelb experiment, and should appear a white, since it is the maximum intensity in the local area. Observer results give important information about the relative importance of global and local interactions. The gray square in the black surround is one lightness unit out of nine lighter than the same gray in the white surround. If local spatial calculations were the only consideration, the gray in black should appear a 9.0. If global spatial considerations were the only consideration, the gray in black should appear a 5.0. The observer matched the gray in black to a 6.0. In other words, the spatial normalization mechanism is an imperfect global mechanism. Alternatively, it is a local mechanism that is significantly influenced by information from the entire image.

(On the Munsell scale, 9.0 is white and 5.0 is middle gray.)

The data obtained for the staircase Gelb display require a larger weight for local anchoring than do the data McCann reports for the simultaneous contrast display. This is to be expected, however, because Cataliotti and I already found in the staircase Gelb experiments that weighting of the local framework varies with the number of elements in it (articulation). In the case of simultaneous contrast, the local framework contains two elements, compared to five elements in the staircase Gelb display; thus, its weight should be lower for the contrast display.

## The Anchoring Model of Simultaneous Contrast

The application of the anchoring model is illustrated in Figure 11.13. Before applying the model, we must take a moment to reflect on the concept of the framework.

## Definition of Framework

When Katz and the Gestaltists used terms like field, or framework, they meant a region of common illumination. But if we maintain this usage (p.318)

Figure 11.13. The anchoring explanation of simultaneous lightness contrast.

we cannot apply the model to simultaneous contrast, because the two backgrounds are not perceived to represent two fields of illumination.

Intuitively, however, the simultaneous contrast display lends itself readily to a frameworks analysis. The display can be seen as a single global framework composed of two local frameworks, one defined by the black background and one defined by the white, even though these local frameworks are not fields of illumination. This kind of perceptual structure, according to Gestalt theory, appears in our visual experience due to the operation of the grouping principles. This line of thinking leads to an alternative definition of framework in terms of the Gestalt concept of belongingness. Specifically, a framework can be defined as a group of surfaces that belong together, more or less. In the simultaneous contrast display, one target belongs to the black background while the other belongs to the white, due to factors of proximity and surroundedness.

## Framework Segregation Factors

Of course, even in the domain of illumination-independent constancy, it is not sufficient to talk merely of fields of illumination. External fields of illumination must be represented internally, and this requires that fields be defined in proximal stimulus terms. I have, with Kardos, already emphasized the two strong factors by which fields are defined: penumbra and depth boundaries (corners and occlusions boundaries). I will assume that when these two factors are presented pictorially in (p.319) an image, without oculomotor and stereo cues, they produce only weak frameworks. I will also assume that the traditional grouping factors produce weak but functional frameworks. And I will suggest two additional grouping factors: edge junctions (especially T, X, and ?) and luminance gradients.

### Source of the Error: Local Anchoring

We noted earlier that for illumination-independent constancy, veridicality is associated with local anchoring while the errors come from global anchoring. For background-independent constancy these are reversed, at least when the display is uniformly illuminated. Global anchoring produces luminance matching (that is, veridicality), while the errors come from local anchoring. Thus, anything that strengthens the local frameworks will strengthen the contrast illusion.

Every theory of lightness accounts for the fundamental background-dependent error (dark backgrounds lighten, etc.) and the fact that increasing the background difference strengthens the illusion, and the anchoring theory is no exception. Thus I will not spend time on these points.

## Simultaneous Contrast and Perceptual Grouping

A central claim of the anchoring account should be emphasized. The simultaneous contrast illusion is held to be the product of grouping processes, and the manipulation of grouping factors should be able to modulate the strength of the illusion and even reverse its direction. We have already seen grouping factors at work in the Benary effect and the Koffka ring.

### Double Increments

The lack of a contrast effect in the double increments version of simultaneous contrast is a major clue to the source of the illusion. This fact is not obviously consistent with lateral inhibition accounts, but it flows directly from the highest luminance rule in the anchoring model. When both targets are increments, each will be the highest luminance in its local framework and receive a local assignment of white. Thus, the difference in local assignments that lies at the heart of the anchoring account is absent in the double increments version of the display, so no illusion is predicted.13

### Modulating the Strength of the Illusion

In anchoring terms, the simultaneous contrast illusion is weak because the local frameworks are poorly articulated and weakly segregated from each other. When the local frameworks are articulated, the illusion is strengthened (Adelson, 2000; Bressan & Actis-Grosso, 2004; (p.320) Gilchrist et al., 1999). When the white and black backgrounds are better segregated by introducing a luminance ramp between them, the illusion is also strengthened (Agostini & Galmonte, 1997; Shapley, 1986). Agostini & Galmonte (1999) have reported a series of experiments in which they varied spatial articulation in the Benary effect. All these results are strongly consistent with the anchoring model.

Kingdom (2003, p. 37) claims that the contrast illusion is enhanced when the target boundaries are blurred. If true, this would contradict the anchoring model. But Kingdom offers no data, and his claim is inconsistent with data showing the opposite (MacLeod, 1947; Thomas & Kovar, 1965).

### Grouping by Similarity

The Laurinen et al. variations shown in Figure 10.4 can be said to illustrate grouping by similarity. Laurinen et al. (1997) showed that when the target squares have texture of the same scale but different from the scale of texture on the backgrounds, as in the top and middle of Figure 10.4, the contrast illusion is almost eliminated. Bonato et al. (2003) also found this result varying type of texture rather than scale. Local frameworks are weakened because the grouping by similarity of each target and its background is weakened. In addition, the two targets tend to group with each other and the two backgrounds tend to group with each other. Inverting these grouping relationships, as I did in the bottom of Figure 10.4, strengthens the illusion. Bonato et al. (2003) also found this. Olkkonen et al. (2002) have used chromatic color to modulate illusion strength while holding relative luminances constant. When both targets share a common color and the two backgrounds share a different color, the illusion is reduced. These results follow directly from the anchoring model. And the model predicts that the illusion will be strengthened if the target and background on the left side share a common color and the target and background on the right side share a different color.

### Common Fate

Agostini and Proffitt (1993) have shown that a simultaneous contrast effect can be created even using the unlikely grouping principle of common fate. They distributed a flock of large white dots randomly across a blue field and set all the dots into motion in the same direction. A separate flock of black dots was distributed across the same blue field and set into motion in a different direction. A single gray dot moved with the white dots appeared slightly darker than another gray dot that moved with the black dots. This result cannot be explained by lateral inhibition because the immediate background of both dots was the blue field. When the movement stopped, of course, the two gray dots appeared identical.

(p.321) The finding by Agostini and Bruno (1996) that the illusion is twice as large when presented in a spotlight as when presented on paper is also consistent with the anchoring account. In the paper version, the global framework is very large and well articulated, including as it does much of the surrounding environment. The spotlight segregates a framework containing only the contrast pattern. Relative to this framework (which is weaker than the global framework) the two local frameworks have increased strength, causing a stronger illusion.

The same analysis applies to the contrast display when presented on a CRT screen. But Agostini and Bruno showed that even on a monitor, the strong illusion can be weakened by surrounding the contrast display with a Mondrian pattern that, in anchoring terms, strengthens the framework containing the whole SLC illusion.

## Reversing the Illusion

White's illusion is consistent with the anchoring model because it shows that the direction of the illusion depends on inducing stripes with which the targets are perceptually grouped rather than those with which the targets share a greater border. Todorović (1997) has created a variation (see Fig. 10.7) in which the aspect ratio between black and white adjacency to the target is pushed to a greater extreme.

### The T-Junction as a Grouping Factor

The key to grouping in the Benary effect, White's illusion, and the Todorovć illusion appears to be the T-junction. Thus, I want to suggest that a T-junction increases the belongingness across the stem of the T, and/or decreases the belongingness across the top edge of the T, as Figure 11.14 shows.14 White's illusion is topologically equivalent to the

Figure 11.14. The role of T-junctions in anchoring.

(p.322) Benary illusion, but stronger. The greater strength of the contrast effect in White's illusion is probably due to the relatively high articulation level in each framework15 or, equivalently to the large number of T-junctions.

### Checkerboard Contrast

Checkerboard contrast can be attributed to the belongingness of one target square to the diagonal group of black squares and the belongingness of the other target to the group of white squares. Good continuation plays a role here. Now, of course, the targets might just as well be grouped with the horizontal rows or the vertical columns. These groupings would produce an equal appearance of the targets because each group would contain the same maximum luminance. Presumably grouping occurs in all of these directions, diagonal, vertical and horizontal, and the resulting compromise accounts for the weakness of the illusion. This analysis is illustrated in Figure 23 of Gilchrist et al. (1999).

### Reverse Contrast Illusions

The displays by Agostini and Galmonte, Economou and Gilchrist, and Bressan (see Figs. 10.10, 10.11, 10.12) show dramatically that grouping factors can totally reverse the direction of the illusion.

Elias Economou and I varied the strength of each of the grouping factors in his reverse contrast display, including good continuation of the bar ends, similarity of the bars, orientation alignment of the bars, and number of bars. In each case we merely asked observers to match the lightness of each of the target bars, and from that obtained a measure of the strength of the illusion. We found that lightness is a direct function of the strength of grouping, as can be seen in Figure 11.15. As the grouping factors are weakened, the reverse contrast illusion also weakens. To our surprise, scrambling the orientation of the bars, as shown in the bottom left graph in Figure 11.15, did not weaken the illusion,16 although rotating the target bars away from the orientation of the flanking bars did (bottom right graph in Fig. 11.15). These results provide strong evidence for the claim that simultaneous lightness contrast is fundamentally a phenomenon of perceptual grouping.

We also conducted stereo experiments that allowed us to place the target bars, flanking bars, and backgrounds in separate planes. As can be seen in Figure 11.16, the results are consistent with the anchoring model. The strength by which the target bars are anchored to either the flanking bars or the backgrounds is a direct function of the proximity in depth to those elements. This further confirms that the coplanar ratio principle is a graded function, not all or none, as I had originally believed (Gilchrist, 1980).

(p.323)

Figure 11.15. Reverse contrast illusion strength as various grouping factors are weakened.

## Economou/Gilchrist Experiments

Elias Economou and I conducted a series of experiments further testing the anchoring model of simultaneous contrast.

### Locus of Error

The anchoring model makes the very specific prediction that the bulk of the illusion is due to the lightening of the target on the black background caused by anchoring. A much smaller darkening of the other target is expected based on the scale normalization rule.17 We tested this in three separate experiments (Gilchrist et al., 1999). Each time we found a much larger deviation from veridicality for the target on the (p.324)

Figure 11.16. Reverse contrast illusion strength under various depth arrangements.

black background. Results consistent with ours have been reported by Adelson and Somers (2001), Bonato et al. (2001), and Logvinenko, Kane, and Ross (2002), who wrote, “the difference in lightness induction between Figs 1 and 2 arises from the dark surround.” This prediction can be verified merely by inspection in versions of simultaneous contrast that are especially strong. For example, in Figure 10.3, inspired by an Adelson figure, the lower ellipse approaches white much more than the upper ellipse approaches black, even though both ellipses are middle gray.

### Staircase Contrast

Staircase contrast displays presented by Hering (1874/1964, p. 125), Cornsweet (1970), and Shapley (1986, p. 51) are shown in Figure 11.17 (p.325)

Figure 11.17. Staircase contrast displays. (Left) Hering, 1874/1964, p. 125. (Right top) Shapley, 1986, p. 51. (Bottom right) Cornsweet, 1970, p. 279. Reprinted with permission.

(see also McArthur & Moulden, 1999, p. 1212). The implication is that all of the targets appear different from each other, whether decrements or increments. But the anchoring model predicts that the curve of target lightness across these panels should show a knee, with a horizontal section for the increments. Economou and I tested this prediction and got exactly that result (Fig. 11.18). Notice that the theoretical curve, shown in white, was derived directly from the anchoring model. The 4:1 global/local weights were derived from the typical size of simultaneous contrast: a Munsell difference of 0.7. Factoring in the scale normalization effect would steepen the slope.

### Target Luminance Variation

The anchoring model predicts a stronger illusion when darker targets are used. For the target on the black background (the main source of the illusion), the discrepancy between local and global values increases with darker targets. Testing this factor, we obtained just these results, as shown in Figure 11.19. Logvinenko and Ross (2005) obtained the same findings. It is not clear what a contrast theory would predict for this experiment.

(p.326)

Figure 11.18. Perceived lightness of targets in a staircase contrast display. No contrast effect was obtained for increments. The white line represents predictions from the anchoring model, assuming a 4:1 global:local weighting, but without taking into account scale normalization. Reprinted with permission from Gilchrist and Economou, 2003.

Figure 11.19. Simultaneous contrast is stronger with dark-gray targets, as predicted by the anchoring model.

## (p.327) Competing Groupings

The idea that a target is simultaneously part of more than one group and its lightness is a weighted average of its computed value in each of these is nicely illustrated in an experiment by Zaidi, Shy, and Spehar (1997). In their stimuli, shown in Figure 6.18, each gray target was grouped with one context region (either black or white) by coplanarity and with another context region by T-junctions. Mere inspection shows that for a restricted luminance range, the T-junction grouping is stronger than the coplanar grouping. However, by successively neutralizing each grouping, as shown in Figure 6.18b and 6.18c, Zaidi et al. also showed that both kinds of grouping separately influence target lightness, consistent with the principle of co-determination.

Adelson's illusions lend themselves readily to an anchoring analysis. The regions that appear as transparent are perceptually segregated as separate frameworks. The various junction types discussed by Adelson (1993), Anderson (1997), Todorović (1997), and Watanabe and Cavanagh (1993) can be considered segmentation factors. In the corrugated plaid, perceived depth planes serve as frameworks. Todorović (1997) has shown that when the display is modified to create a staircase pattern, as in Figure 6.15b, the illusion persists. This fact supports an anchoring analysis over an intrinsic-image analysis, because in the staircase figure, although the targets do not appear to be differently illuminated, they do lie in different frameworks as defined by planarity. Wishart et al. (1997) varied the perceived angle in the folds of the corrugated plaid. They found that the strength of the illusion varied with the angle. In anchoring terms, this implies that the more two adjacent surfaces depart from coplanarity, the less they are treated as belonging to each other for purposes of anchoring.

## The Luminance Gradient as a Grouping Factor

Luminance gradients are responsible for several delightful lightness illusions. It appears that these effects can be unified by the assumption that a luminance gradient functions to segregate (reduce their belongingness for anchoring) the luminance values on the two sides of the gradient. For instance, it has long been known that simultaneous lightness contrast is enhanced if the sharp border between the black and white backgrounds is replaced with a luminance gradient. If this gradient is thought to reduce the belongingness between the backgrounds, this would weaken the global framework, equivalent to strengthening the local frameworks. Because the model holds that the simultaneous contrast effect is caused by local anchoring, this should enhance the contrast effect.

(p.328) The same analysis can be applied to the dramatically enhanced contrast effect presented by Agostini and Galmonte (1997), shown in Figure 10.3. The shallow luminance gradient surrounding each gray target square can be thought of as a barrier to anchoring. Thus, each target belongs strongly to its immediate surround, but much less to more remote regions. For example, if the segregating effect were total, the gray square with the black immediate surround would be equivalent to a gray square in a black dome, which we know appears white (Li & Gilchrist, 1999). Of course the segregation is not total. Some global anchoring occurs, and thus that target appears very light gray, but not white.

# APPLYING THE MODEL: THE BRIGHTNESS INDUCTION LITERATURE

It might seem inappropriate to apply a model of lightness errors to the brightness induction experiments (described in Chapter 7). These experiments are about brightness, not lightness. However, lightness inevitably bleeds into brightness, presumably because the visual system struggles to interpret even such a highly reduced display as a set of surfaces. In any case, we have already found that the area rule, written for lightness, accounts very handily for the brightness data of those induction experiments in which area is varied. Thus, we will now proceed to apply the anchoring model to the other main brightness induction findings, to see whether anchoring or induction provides the better account.

## Basic Finding: Highest Luminance Is Crucial

The fundamental study in brightness induction is that of Heinemann (1955). We have already seen that his results can be explained by the highest luminance rule plus the area rule (see p. 243).

## The Role of Separation between Test and Inducing Fields

Experiments on the degree of separation, in the frontal plane, between inducing and test fields, reviewed in Chapter 5, have shown that the test field (of lower luminance) becomes darker as the separation is reduced. From the perspective of the lateral inhibition account, the closer the inducing field is to the test field, the greater the neural activation corresponding to the test field is inhibited by the inducing field.

From an anchoring standpoint, however, the perceived lightness of the test field will be some combination of its lightness relative to the dark surround and its lightness relative to the inducing field. Relative to the dark surround, the test field should appear white. Relative to the inducing field, the test field should appear as some shade of surface (p.329) gray, the specific shade determined by the test/inducing field luminance ratio, according to the formula given earlier:

$Display mathematics$
where Lt is the luminance of the test field, Lh is the highest luminance in the framework (the inducing field), and lightness is defined as perceived reflectance.18

Thus, target lightness should lie somewhere between white and the reflectance given by the above formula, depending on the degree to which the test field appears to belong to the inducing field. The greater the proximity between test and inducing field, the stronger the group formed by the two, and the more the lightness of the test field will be anchored by the inducing field. The greater the separation between test and inducing fields, the more the lightness of the test field is determined by its relationship to the dark surround.

Note that, according to anchoring theory, this separation effect applies only to test and inducing fields separated in space. When these fields are both part of an adjacent, coplanar group, the distance between them plays little or no role, as shown by Cataliotti and Gilchrist (1995), presumably because they are already strongly grouped together.

The anchoring account, unlike contrast/induction accounts, explains why the inducing field continues to appear white: because it is the highest luminance relative to both the test field and the dark surround. Thus, inducing field appearance becomes indifferent to local versus global weighting. This implies that separation between test and inducing fields should have no effect on inducing field appearance, and this is exactly what has been found.

## Anchoring versus Induction

For the test field, however, anchoring and lateral inhibition appear to make the same qualitative predictions when separation is varied. Is it possible to find conditions under which anchoring and lateral inhibition make different predictions? The answer lies in depth separation. The test field can be separated from the inducing field in depth even as the two fields remain retinally adjacent. According to anchoring theory, this separation should reduce the perceived belongingness between the two fields. But from the induction perspective it should have no effect.

Gogel and Mershon (1969) have conducted such experiments in depth separation, and the results decisively favor the anchoring interpretation. Increasing the depth separation makes the test field appear lighter.

## (p.330) Strong Role of Surroundedness

One of the clearest findings to emerge from the brightness induction literature is that the degree of induction is much greater when the inducing field surrounds the test field than when the inducing field and test field are merely adjacent to one another. This is entirely plausible from a belongingness perspective because surroundedness (closely related to figure/ground) is a grouping principle that increases the belongingness of the two regions over and above the grouping principle of proximity.19

Thus, for the results of brightness induction experiments, the anchoring account compares favorably to the induction account, even explaining a variety of results not explained by induction.

# SPECIAL PROBLEMS IN ANCHORING

## Temporal Anchoring Effects

Cataliotti and Bonato (2003) have demonstrated that anchoring influences occur over time as well as across space. They found that while a relatively dim disk presented within a dark room appears white, it will appear light gray if it is preceded by a much brighter disk. Although these two disks were presented in the same spatial location, they were viewed with separate eyes, ruling out adaptation as a cause of the effect. The strength of this darkening, of course, depended on the time interval between the two disks. They found a strong effect with a 1-second delay. The effect got weaker as the delay was increased, with no effect left using a 32-second delay.

Cataliotti and Bonato also measured articulation effects acting both spatially and temporally. In a variation on the Gelb effect conducted in a normally illuminated room, they showed that a black paper square that appears white (Munsell 9.5) in a spotlight perceptually darkens to Munsell 7.7 when a white square is placed next to it. But it darkens further, to a Munsell 5.8, when the white square is replaced by a Mondrian pattern of the same size composed of six patches ranging from white to black. The fact that the average luminance of the Mondrian was much lower than that of the white paper shows that this effect is an anchoring effect. Were it a contrast effect, the white paper would have a stronger darkening effect than the Mondrian. In a further experiment they showed that this articulation effect works over time as well, finding that the dim disk presented in darkness appears to darken more (to Munsell 6.0 rather than 7.2) when preceded by a Mondrian-patterned disk than when preceded by a homogeneous white disk, even though the white disk was much brighter.

### (p.331) A Lightness Hangover

Earlier (p. 305) I described an experiment conducted in what we called the Mondrian world. The interior walls of a small room were completely covered with a Mondrian pattern containing nothing but rectilinear patches of different shades of dark gray and black. An observer's head is placed within the room so that the entire visual field is covered with the Mondrian pattern. Patches of the highest luminance (actually dark gray) duly appear white, and no blacks are seen. When several real white patches are introduced, by sliding a panel that constitutes the far wall of the room, all the patches in the room appear to darken, as one would expect, but this darkening occurs very slowly. Initially the real whites appear self-luminous, and it takes up to 3 full minutes before the darkening is complete.

How can this hangover be explained? After all, a black paper that appears white in a spotlight (the Gelb effect) instantly appears to darken when a real white is placed next to it. Vidal Annan and I conducted a series of experiments to answer this question (Annan & Gilchrist, 2004). Our results ruled out adaptation as an explanation and established that it is the anchor itself that resists the change, not the lightness of individual surfaces. We discovered that the strength of the hangover is directly related to the number of patches that remain constant in luminance and are continually visible from before the whites are added until after. When each wall contains only a single dark-gray surface, the hangover is hardly found; the walls darken almost immediately when the true whites are added. But the greater the articulation of the walls, the longer the hangover persists.

We interpret our results as follows. When the illumination in a room is increased, it produces several effects. First the highest luminance is increased. But in addition, most visible surfaces also increase dynamically in luminance. These effects suggest that the illumination has changed and the anchor, which can be thought of as a surrogate for the illumination level, must be recalibrated. But in our Mondrian world, only the first of these effects occurs: the highest luminance goes up. But all of the constantly visible surfaces remain constant in luminance. Apparently each such surface votes against changing the anchor, and the more such votes, the slower the new anchor is applied.

## Multi-Lit Objects

According to the anchoring theory, lightness is computed within illumination frames of reference. But what about a surface that lies partly in one field of illumination and partly in another? Anchoring theory effectively predicts the lightness of each patch, or separately illuminated part of the object, but has not been able to predict the lightness of the multi-lit object as a whole. This is a serious problem (p.332) because lightness is the property of an object, not the property of a patch. A patch is merely the intersection between a region of uniform illumination and a region of uniform reflectance. Whether appropriate or not, observers can readily assign lightness values to these patches, and they always assign a different value to the two patches (even as they insist that the object has only a single lightness). Experts do the same thing. Of course observers can also assign a single value to the object as a whole, and indeed this is more natural, as Kardos (1934) found many years ago.

Suncica Zdravković and I (Zdravković & Gilchrist, in press) reasoned that there must be some systematic relationship between the lightness of the separate patches and the lightness of the object as a whole, and we set out to determine that relationship. In a series of experiments we discovered that the lightness of the object as a whole is in agreement with the lightness assigned to the patch in the highest of the two regions of illumination, but also with the lightness of the patch in the larger of the two regions of illumination. Thus, if a shadow falls on half of an object, its lightness will be the same as the lightness assigned to the illuminated half, because that region of illumination is both the largest and the highest. On the other hand, if a spotlight falls on half of an object, its lightness will represent a compromise between the lightness values assigned to the two halves, because one half lies in the highest illumination while the other half lies within the largest region of illumination.

Why these two rules? Perhaps the visual system has implicit knowledge of the optimal conditions for perceiving lightness. Katz (1935) established his laws of field size, showing that lightness constancy is stronger in larger fields of illumination. It is also known that visual acuity increases with increasing levels of illumination.

These two rules—highest and largest—are strikingly reminiscent of the two basic rules of anchoring lightness within a framework: surface lightness is anchored by the highest luminance and the luminance with the largest area. Finding a similar pair of rules for multi-lit objects may not be a coincidence. Perhaps the multi-lit object also presents a kind of anchoring problem: anchoring with the domain of illumination. The visual system may be trying to determine which region of illumination should be considered normal.

# SUMMARY

The decomposition story, it turns out, was too good to be true. Lightness constancy is not that good, and the representation of the external world is not that complete. Most lightness errors are not explained by the decomposition models, nor have efforts succeeded to explain those errors through failures of the various components of these models. Lightness now appears to be strongly influenced by factors that seem (p.333) inconsistent with the logic of inverse optics. Take the strong influence of perceived area on lightness, for example. Such a function would mirror the process of image formation only if the physical reflectance of objects changed with a change in object size.

So it appears we must reject both the structure-blind approach of the contrast theories and the complete representation of the distal stimulus implicit in the decomposition theories. Lightness and perceived illumination represent their physical counterparts more crudely. The lightness system takes short cuts that presumably have turned out to be good enough for survival. This new look in lightness theory is sometimes called mid-level. In fact, it takes us back to Gestalt theory. Concepts like belongingness, grouping principles, and frames of reference are essential. The structure of the image is acknowledged, if not fully represented.

Fields of illumination exist in the external world. Without some representation of them, lightness constancy would not be possible. But to be represented internally, fields of illumination must be operationally defined in proximal terms. The two main factors by which the image is segregated into frameworks appear to be penumbra and depth boundaries, as Kardos said, with weaker frameworks spawned by virtually all the Gestalt grouping factors. For a given object, lightness is not computed exclusively within the framework to which the object primarily belongs. Rather, its lightness is computed partially in relation to that framework and partially in relation to a foreign, or global framework. This is the very important Kardos doctrine of co-determination.

So the framework that takes part in lightness computation is a surrogate for the field of illumination in the external world. These are not totally isomorphic. Segregation factors often appear in the image, without a corresponding external field. The obvious example is the black and white backgrounds of the simultaneous contrast display. These frameworks function as weak fields of illumination. Within each framework, the level of illumination is operationally defined in terms of the highest luminance (with largest area factored in). Again the correspondence between actual level of illumination and the functional level, by which lightness is computed, is rough. Mismatches occur.

## Notes:

(1.) The name “staircase Gelb effect” was suggested by Bill Ross and Luiz Pessoa.

(2.) This mistake was later corrected by using steps equal in log reflectance. The Munsell scale is not a log scale.

(3.) A look at Figure 5 from Cataliotti and Gilchrist (1995) suggests that there might be a real but tiny effect of viewing distance. But even this effect might reflect a small failure of size constancy, as other work shows that perceived size does influence the amount of compression.

(4.) Again, the data hint at a slightly greater darkening for adjacent surfaces that might reach significance with greater power.

(5.) Diamond (1953) did not take Munsell matches, but his brightness matches show the same shallow drop that we obtained in our data.

(6.) Some would appear self-luminous as well. But, as other work we conducted made clear, that is a separate matter.

(7.) This was achieved by turning off the room lights and placing the observer in a closed booth so that the five squares were seen through an aperture.

(8.) An exception to this rule occurs when the local framework does not contain a white surface

(9.) Bringing in global anchoring doesn't help either, because the values so produced are likely to be even further in error.

(10.) The relevant/foreign construction cannot explain the simultaneous lightness contrast illusion without invoking some additional assumptions.

(11.) Possibly also self-luminous, but that is a different matter.

(12.) Virtually all occlusion edges are compound; they represent a change in both reflectance and illuminance.

(13.) This analysis is essentially the same as that given in Chapter 10 for why illumination-independent constancy is poor with dark backgrounds.

(14.) This is similar to proposals by Anderson (1997) and Todorović (1997), but Anderson's proposal involves transparency and Todorović's proposal is couched in terms of contrast.

(15.) If the strength is due to articulation it shows, as we have found elsewhere, that articulation is defined only by the number of distinct surfaces within a group, not by the number of different gray levels.

(16.) Gillam (1987) has shown that a stimulus very much like our scrambled group produces stronger subjective contour than a regular group (as long as the endpoint alignment is preserved). This suggests that the net effect of our manipulation did not weaken the group.

(17.) Scale normalization affects only the darker region in each local frame because locally the lighter region anchors on white.

(18.) The Munsell value equivalent for a given reflectance can be obtained from a table (Judd, 1966, p. 849) or by a formula (Wyszecki & Stiles, 1967, p. 478).

(19.) There are some good reasons to believe that the figure belongs much more strongly to the surrounding background than vice versa (see Fig. 11.11).