Anchoring Theory of Lightness
Lightness refers to the white, black, or gray shade of a surface. The basic problem is that the light reaching the eye from a given surface, as the product of surface reflectance and illumination, does not specify the reflectance of the surface. A black in sunlight can reflect more light than a white in shadow. No computer program exists that can identify the reflectance of an object in a photo or video. Any possible solution must exploit the context surrounding the object. Anchoring theory proposes to solve the problem by grouping patches of the retinal image into regions of equal illumination, called frameworks, then computing lightness values within each framework using rules of anchoring combined with luminance ratios.
Lightness refers to the perceived shade of white, gray, or black of a surface and is sometimes called perceived reflectance. The reflectance of a surface is the percentage of light it reflects. White surfaces reflect about 90 % of the light that illuminates them while black surfaces reflect only about 3 %. Given the light-sensitive receptor cells in the eye, one might think that the perception of lightness could be easily explained. But the amount of light reflected by a surface, called luminance, depends not only on its reflectance but also on the intensity of light illuminating it. And the intensity of illumination varies from place to place and time to time. Thus, a black surface in sunlight can easily reflect more light than a white in shadow. In fact, any shade of gray can reflect any amount of light.
The ability to perceive the correct shade of gray of a surface despite variations (both spatial and temporal) in illumination is called lightness constancy, or more specifically, illumination-independent lightness constancy. The ability to perceive the correct shade of gray despite variations in the luminance of the background surrounding a target surface is called background-independent constancy. Neither type of constancy is complete. Variations in illumination level produce failures of constancy as described in the traditional literature while variations in background produce failures of constancy that have been called contrast effects.
Summary of the Theory
According to anchoring theory , the lightness of a target surface is codetermined by its relationship to the field of illumination in which it is embedded, called the local framework, and its relationship to the entire visual field, or global framework. The values computed within these local and global frameworks are combined in a weighted average. Within each framework the value is based on the ratio between the luminance of the target surface and the highest luminance in the framework, called the anchor. The weight of a framework depends on its size and complexity. Framework boundaries include corners, occlusion contours, and penumbrae.
Helmholtz  suggested that given the luminance of a target surface, the visual system would have to somehow take into account the intensity of light illuminating that surface in order to compute its reflectance. But this idea has never been adequately operationalized. Koffka  suggested that fields of illumination can be regarded as frames of reference. The same luminance value that would be computed to be a light gray if it appears within a shadow would be computed to be a dark gray if it appears in sunlight. Anchoring theory borrows Koffka’s concept of frames of reference.
In 1948, Wallach  proposed the idea that lightness depends simply on the ratio between the luminance of a surface and the luminance of the surrounding region. There is clearly much validity in this proposal. A homogenous disk of constant luminance can be made to appear as any shade of gray from white to black simply by varying the luminance of a homogeneous ring that surrounds it. However, relative luminance per se can only produce relative shades of gray. For example, consider two adjacent surfaces whose luminance values stand in a 5:1 ratio. These could be white and middle gray. But they might also be light gray and dark gray, middle gray and black, or any number of reflectance pairs. To determine the specific shade of a surface requires, in addition to relative luminance, an anchoring rule – that is, a rule that links some value of relative luminance to some value on the gray scale. Anchoring theory invokes a highest luminance rule according to which the highest luminance within a region of illumination is perceived as white, and this value serves as the standard to which lower luminance values are compared.
Simple and Complex Images
Anchoring theory makes an important distinction between simple images and complex images. A simple image is one in which a single level of illumination fills the entire visual field while a complex image, typical of most images we see, contains multiple levels of illumination. The theory also describes how the rules for simple and complex images are related to each other.
In a simple image, the formula used to predict lightness value of a given target surface is Lightness = T/H * 0.9, where T is the luminance of the target, H is the highest luminance in the image, and 0.9 is the reflectance of white.
Geometric relationships, in addition to photometric relationships, also play a role in anchoring. Lightness depends, to a limited extent, on the relative area of surfaces. In general, lightness increases as relative area increases. But according to anchoring theory, this effect of “the larger the lighter” applies only to surfaces darker than the highest luminance and then only when that darker surface fills more than half of the area of the visual field.
While it is not hard to exhaust the rules by which lightness is computed in simple images, the more important goal of a theory of lightness is to explain how lightness is computed in complex images typical of the real world. Anchoring theory makes the claim that complex images are related to simple images according to two principles. The first is that the same rules of lightness computation found in simple images apply to frameworks embedded in complex images. The second involves a kind of crosstalk between frameworks called codetermination.
The term codetermination was coined by Kardos , a gestalt psychologist from Budapest. He proposed a friendly amendment to Koffka’s idea of frameworks, observing that the lightness of a target is not computed exclusively within its framework but is influenced by neighboring frameworks as well. Kardos argued that the lightness of a surface is partly determined by its relative luminance in the field of illumination to which it belongs, called the relevant field, and partly by its relationship to a neighboring field of illumination, which he called the foreign field.
Anchoring theory invokes the closely related concepts of local and global frameworks. The global framework is the entire visual field. Unlike relevant and foreign frameworks, which are exclusive, local and global frameworks are hierarchical. The global framework includes the local.
In general, the visual field is segregated into frameworks by identifying illumination boundaries. More specifically, these include cast illumination edges, such as the border of a shadow or spotlight, corners, where the planarity of a surface changes, and occlusion boundaries, where one object partially obscures a more distant object.
To implement the concept of codetermination, a provisional lightness value is computed for the target within each framework and then a weighted average of these values is computed. The weight of a framework in this average depends on the size of the framework and the degree of complexity within it.
The factors of framework size and articulation come from the early work of David Katz . Having invented some of the fundamental methods for studying lightness, Katz conducted experiments testing the factors that lead to strong and weak constancy. He found that the degree of constancy varies in proportion to the size of a framework, which he called field size, and its level of articulation. In his experiments with Burzlaff, level of articulation was operationally defined simply as the number of distinct elements within it.
The concepts of field size and articulation are given a slightly different interpretation in anchoring theory. Rather than factors consistently leading to good constancy, they are treated as factors that determine the relative weight of a framework. This twist resolves an empirical challenge for Katz, namely, that under certain conditions, increasing the articulation level of a framework leads to weaker constancy.
Staircase Gelb Effect
Anchoring theory was inspired by an empirical finding called the staircase Gelb effect. Gelb had shown that a piece of black paper appears white when suspended in midair and illuminated by a spotlight. In the staircase version, a row of five squares arranged in an ascending series from black to white is placed within the spotlight. This produces a dramatic compression of the perceived range of grays. Although the white square is perceived correctly as white, the black square appears somewhat lighter than middle gray. This finding seriously undermined several earlier decomposition theories of lightness but suggested an interaction between the bright illumination on the five squares and the dimmer illumination of the room.
According to anchoring theory, each of the squares is assigned its correct value within the local spotlight framework. But relative to the surrounding room, as long as the spotlight is at least 30 times brighter than the illumination level in the room, each square is assigned the same (global) value of white, given that its luminance is as high as, or higher than, a white surface in the room light. The perceived compression of the lightness range is explained by these equal values assigned to the squares relative to the room illumination.
The Scaling Problem
The mapping of luminance values onto lightness values within a given framework involves both anchoring and scaling. If anchoring concerns which value of relative luminance is linked to which value of lightness, scaling concerns which range of luminance values is linked to which range of lightness values. Much as a normal distribution can be characterized by both a measure of central tendency and a measure of dispersion, so a set of luminance values within a framework can be converted to lightness values using an anchoring rule and a scaling metric (although the anchor is located at the top, rather than in the center, of the range).
Wallach’s ratio principle assumes a one-to-one scaling metric. That is, if the luminance ratio between two adjacent surfaces is 3:1, then the ratio of their two perceived reflectance values must also be 3:1. This scaling metric is implicit in the formula given above: Lightness = T/H * 0.9. It can be considered the default scaling metric and applies to large, well-articulated frameworks typical of everyday scenes.
However, under specific conditions, the range of perceived lightness values can be either expanded or compressed, relative to the range of luminance values. This occurs when the luminance range within a framework is either much less or much greater than the canonical 30:1 range between white and black (from 90 % to 3 %). Anchoring theory thus includes a scale normalization principle, according to which the range of perceived lightness values tends toward the white–black range. When the luminance range within a framework is less than 30:1 the lightness range is expanded, and when the luminance range exceeds 30:1 the lightness range is compressed. The degree of expansion or compression is proportional to the deviation of the range from 30:1, but the normalization is only partial.
Veridicality Versus Error
In general, theories of lightness have sought to account for lightness constancy, and the degree to which perceived lightness corresponds to physical reflectance. However, an adequate theory of lightness must also explain failures of constancy. The Kardos theory of codetermination was the first lightness theory to confront this problem. The failure of constancy within a relevant field is due to the unwanted influence of the foreign field. Anchoring theory endorses this construction but also broadens it to include a larger class of what can be called lightness errors. This class includes lightness illusions in addition to failures of constancy. Illusions and constancy failures have traditionally been treated separately but have nonveridicality in common.
The most basic and familiar illusion of lightness is called simultaneous contrast. Two identical gray patches are placed, respectively, on adjacent black and white backgrounds. The patch on the black background appears lighter than the patch on the white background. The application of anchoring theory to this illusion is rather simple. The illusion is obviously composed of two frameworks, corresponding to the two backgrounds. These two frameworks taken together can be considered the global framework. For each gray patch, both a local and a global value are computed. In the global framework, both gray patches are computed to be middle gray. But they get different values when computed locally. The patch on the black background is given a value of white in its local framework, because it is the highest luminance. However, the patch on the white background is given a value close to middle gray (but somewhat darker due to scale normalization). When the local and global values for each patch are combined in a weighted average, the patch on the black background ends up with a slightly higher value. According to anchoring theory, this “contrast effect” is relatively weak because the two local frameworks are weak, due to their lack of articulation.
According to anchoring theory, the simultaneous contrast illusion is not caused by lateral inhibition or any exaggeration of edge differences but rather by the fact that one patch is perceptually grouped with the black background while the other patch is grouped with the white background. Consistent with this grouping approach, several authors have produced reverse contrast illusions. These illusions are variations of simultaneous contrast in which a gray patch on the black background appears darker, not lighter, than an identical gray patch on the white background. This reversed effect is created by making the gray patch a member of a group of patches that are opposite in lightness from the black background. Thus, the gray patch on the black background is part of a matrix of white patches of the same size and shape. The gray patch on the white background is likewise part of a group of black patches. These reverse contrast illusions support the concept of grouping that is fundamental to gestalt theories in general and anchoring theory in particular.
Grouping by Illumination
Although Helmholtz suggested that the visual system must take into account the level of illumination on a given target surface, in fact this is overkill. To compute lightness, the visual system needs to know only which surfaces are under the same level of illumination. Thus the segregation of the retinal image into frameworks is equivalent to grouping together those retinal patches that lie under homogeneous illumination. This kind of perceptual grouping must be distinguished from the more traditional gestalt idea of grouping by which objects are segregated. In fact, grouping by common illumination is orthogonal to the traditional kind, which can be thought of as grouping by common reflectance.
Anchoring in Other Domains
The anchoring problem arises in other perceptual domains including perceived size and perceived motion. Motion is a good example. There is a good deal of evidence that perceived motion depends on relative motion, but to achieve specific values of motion, relative motion must be anchored. Here, the metaphor of an anchor is even more relevant than in lightness. In a matrix of relative motions, which elements will be perceived as stationary? In his brilliant writing on perceived motion, Duncker  stressed the importance of frames of reference. And he talked specifically about anchoring rules in a way that appears highly consistent with the anchoring theory of lightness.
Strengths and Weaknesses of the Theory
Anchoring theory has wide applicability within lightness perception. It is consistent with an extensive range of empirical data. In general, it accounts for veridical perception to the degree that perceived lightness is veridical. But unlike most other theories, it also accounts for a wide range of known lightness errors, that is, lightness illusions and failures of constancy. This is important because the pattern of errors shown by humans in lightness perception must be the signature of our visual software.
But anchoring theory is a work in progress. It is not clear whether the current rubric of local and global or the Kardos rubric of relevant and foreign is best. Local/global works effectively for lightness illusions like simultaneous contrast and reverse contrast but creates some difficulties for failures of constancy. Relevant/foreign works better for failures of constancy but creates difficulties for contrast illusions. This and other problems with the theory are discussed in Gilchrist (, pp. 354–357).
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