Although the ambiguous relationship between the luminance of a surface and its perceived lightness is widely understood, there has been little appreciation of the fact the relative luminance values are scarcely less ambiguous than absolute luminance values. For instance, consider a pair of adjacent regions in the retinal image whose luminance values stand in a five to one ratio (see Figure 1). This five to one ratio informs the visual system only about the relative lightness values of the two surfaces, not their specific or absolute lightness values. It informs only about the distance between the two gray shades on the phenomenal gray scale, not the specific location of either on that scale. There is an infinite family of pairs of gray shades that are consistent with the five to one ratio. For example, if the five represents white then the one represents middle gray. But the five might represent middle gray, in which case the one will represent black. Indeed it is even possible that the one represents white and the five represents an adjacent self-luminous region. So the solution is not even restricted to the scale of surface grays.
The anchoring problem is the problem of how the visual system ties relative luminance values extracted from the retinal image to specific values of perceived black, white, and gray.
2.2. Mapping luminance onto lightness
To derive specific shades of gray from relative luminance values in the image, one needs an anchoring rule. An anchoring rule defines at least one point of contact between luminance values in the image and gray scale values along our phenomenal black to white scale. Lightness values cannot be tied to absolute luminance values because there is no systematic relationship between absolute luminance and surface reflectance, as noted earlier. Rather, lightness values must be tied to some measure of relative luminance.
The anchoring problem, closely related to the issue of normalization (Horn, 1977, 1986), is illustrated in Figure 2. Consider the following remark by Koffka (1935, p. 255) in relation to Figure 2: "...the stimulus gradient...alone does not determine the absolute position of this apparent gradient.... This whole manifold of colours may be considered as a fixed scale, on which the two colours produced by the two stimulations.... keeping the same distance from each other, may slide, according to the general conditions. I have called this the principle of the shift of level." Anchoring concerns where this sliding relationship comes to rest. Koffka suggests here that the relative lightness of two regions in an image can remain fully consistent with the luminance ratio between them, even though their absolute lightness levels depend on how the luminance values are anchored.
2.3. Wallach on anchoring: Highest Luminance Rule
Apparently Wallach made no systematic study of the anchoring problem. He did mention in passing that the value of white is assigned to the highest luminance in the display and serves as the standard for darker surfaces (Wallach, 1976, p. 8). Land, McCann, and Horn (Horn, 1977; Land & McCann, 1971; McCann, 1987, 1994) have also adopted this rule, which we will call the Highest Luminance Rule.
2.4. Helson on anchoring: Average Luminance Rule
Although Helson neither discussed the anchoring problem per se, nor made any systematic investigation of it, his entire adaptation-level theory is built, in effect, on an anchoring rule. His rule is that the average luminance in the visual field is perceived as middle gray, and this value serves as the standard for both lighter and darker values. This average luminance rule, which is closely related to the gray world hypothesis (Hurlbert, 1986) has found its way into more recent models (Buchsbaum, 1980), especially in the chromatic domain. It is implicit in the concept of equivalent background (Bruno, 1992; Bruno, Bernardis, & Schirillo, in press; Brown, 1994; Schirillo & Shevell, 1996). Land (1983) reverted to the average luminance rule in a later version of Retinex theory.
No one has in fact proposed a rule whereby the lowest luminance in the field is seen as black, but in principle such a rule is possible.
2.5. Anchoring in intrinsic image models
None of the intrinsic image models contains any specific anchoring rule. However the task is simplified because the various factors that can modulate luminance within the image, that is reflectance, illuminance, and three dimensional form, have already been segregated into separate layers. The obvious next step would be to apply something like the Highest Luminance Rule solely to the reflectance intrinsic image.
2.6. Absolute lightness versus absolute luminance.
The issue of relative versus absolute lightness, central to the anchoring problem, is independent of the issue of whether relative or absolute luminance values are encoded at input. The former applies to the output, the latter applies to the input. Normalization typically refers to the fact that absolute luminance values are lost in the encoding of retinal luminance contrasts, whereas the anchoring problem concerns lightness values, not luminance values. It should be noted that even if absolute luminance values were encoded at input (a dubious assumption; see Shapley and Enroth-Cugell, 1984; Whittle & Challands, 1969) or somehow recovered later (perhaps by combining relative luminance information with adaptive state information), the anchoring problem would remain completely unsolved, due to the lack of correlation between luminance and lightness.
Anchoring concerns the mapping of relative luminance values from the image onto perceived shades of surface gray. A parallel aspect of this mapping, that we call the scaling problem, is illustrated in Figure 3. It is distinct from the anchoring problem, though possibly less important. It concerns how the range of luminances in the image is mapped onto a range of perceived grays. Re-scaling can take one of two forms: compression, in which the range of perceived surface grays is less (on a log scale) than the range of luminances in the image; and expansion (Brown & MacLeod, 1992, use the term gamut expansion), which is just the reverse.
While anchoring can be said to concern the constancy of absolute shades of gray, scaling can be said to concern the constancy of relative shades of gray. Wallach was more explicit about scaling than about anchoring. His ratio principle is a scaling rule; it asserts a one-to-one mapping from relative luminance to relative lightness.
Contrast theories, by comparison, reject this kind of one-to-one mapping in favor of expansion. In contrast theories, the relative luminance values encoded at input are magnified (Hurvich and Jameson, 1966, p. 85), exaggerated (Leibowitz, 1965, p. 57), or amplified (Cornsweet, 1970, p. 300) by the lateral inhibitory component of the encoding process itself.
Helson, unlike Wallach, was more explicit about anchoring than about scaling. His adaptation level represents an anchor; he did not specify how regions lighter and darker than the average are scaled, but we presume that the ratio principle would apply.
