It has been hypothesized that lightness is computed in a series of stages involving: (1) extraction of local contrast or luminance ratios at borders; (2) edge integration, to combine contrast or luminance ratios across space; and (3) anchoring, to relate the relative lightness scale computed in Stage 2 to the scale of real-world reflectances. The results of several past experiments have been interpreted as supporting the highest luminance anchoring rule, which states that the highest luminance in a scene always appears white. We have previously proposed a quantitative model of achromatic color computation based on a distance-dependent edge integration mechanism. In the case of two disks surrounded by lower luminance rings, these two theories—highest luminance anchoring and distance-dependent edge integration—make different predictions regarding the luminance of a matching disk required to for an achromatic color match to a test disk of fixed luminance. The highest luminance rule predicts that luminance of the ring surrounding the test should make no difference, whereas the edge integration model predicts that increasing the surround luminance should reduce the luminance required for a match. The two theories were tested against one another in two experiments. The results of both experiments support the edge integration model over the highest luminance rule.

*scaling*and

*anchoring*. In Retinex theory, the edge integration serves to establish the ratio scale of relative lightness. Anchoring theory does not commit to a specific mechanism for computing the ratio scale. The main focus of Anchoring theory is on the anchoring rules that map relative lightness onto absolute lightness values.

*relative area*, which states that larger areas appear lighter and that the lightness of a target region within an image will be increased by increases in the areas, but not the luminances, of regions that are lower in luminance than the target region (Li & Gilchrist, 1999). Thus, according to Anchoring theory, anchoring is achieved by a combination of local and global applications of the highest luminance rule and the relative area rule.

*lightness*(perceived reflectance) and

*brightness*(perceived physical intensity) of the target when viewing similar displays consisting of square-shaped targets surrounded by frames, if specifically instructed to do so. In the present paper, we will investigate the properties of both

*naïve*matches (i.e., matches in which the observer is given to no special instructions to judge either brightness or lightness) and lightness matches, specifically. (Thanks to Prof. Davida Teller for suggesting the term “naïve match.”) Brightness matches will be studied, and their properties compared to those of lightness and naïve matches, in an upcoming paper (Rudd, Zemach, & Heredia, 2005). Ultimately, the properties of all of these types of matches—as well as the relationships between them—need to be understood if the extant literature is to be fully addressed by a theoretical model.

*achromatic color*to refer to the attribute that is matched in naïve appearance matching experiments. This term avoids the ambiguity of the common alternative term “brightness,” which is sometimes used to refer to perceived luminance (as in the present study) and at other times used to refer to the attribute that is matched in naïve matching experiments. We will also use the term achromatic color when we wish to refer to the general category of neutral color percepts that includes lightness and brightness as specific dimensions or attributes. The meaning should be clear from the context. When we wish to refer to lightness or brightness, in particular, we will be careful to use those labels.

*B*.

*w*decay with distance (Reid & Shapley, 1988; Rudd, 2001, 2003a, 2003b; Rudd & Arrington, 2001; Rudd & Zemach, 2002a, 2002b, 2003, 2004; Shapley & Reid, 1985; Zemach and Rudd, 2002, 2003). We will refer to an edge integration model that makes this ancillary assumption as the distance-dependent edge integration model. For additional ideas about the rules governing edge integration, the reader is referred to papers by Gilchrist (1988) and Ross and Pessoa (2000).

_{i}^{2}(

^{2}(

*B*was 0.10 cd/m

^{2}(

^{2}(−0.3 to 0.25 log units) in six equal RGB unit steps. In each block of trials, each of the six

^{2},

^{2},

^{2},

*p*< .0001, two tailed; JL:

*p*< .0001, two tailed; LT:

*p*= .0005). The hypothesis that the true intercept equals 0.5 log units is also rejected for observers JL and LT (AD:

*p*= .435, two tailed; JL:

*p*< .0001, two tailed; LT:

*p*< .0001, two tailed). The predictions of the global highest luminance anchoring rule are thus strongly disconfirmed by the data from Experiment 1.

*p*< .0001, one tailed; JL:

*p*< .0001, one tailed; LT:

*p*= .0003).

^{2}(AD); .527 ± .002 cd/m

^{2}(AD); .563 ± .003 cd/m

^{2}(LT). The hypothesis that

^{2}was statistically rejected for observers JL and LT. Both of these observers set the matching disk luminance to be higher than the value predicted by the edge integration model. The percent error in

*B*were consistent with the values given above. The revised luminance estimates imply that the right side of the display was, in fact, about 14% more luminant than the left side. However, the estimates should be considered approximate. In light of this fact, an adjustment of the luminance values of about this magnitude is probably appropriate.

^{2}. As a consequence, the horizontal dotted line in Figure 2, representing the prediction of the highest luminance anchoring rule, should be redrawn at a height of 0.557 log cd/m

^{2}on the

*y*-axis.

^{2}. Thus, the distance-dependent edge integration model predicts that the matching function intercepts should be positively correlated with the matching function slopes. This pattern is, in fact, observed in the data (

*p*= .016,

*cf*on the left side of the equation to the right side yields

*w*coefficients signify the distances between the borders with which they are associated and the corresponding test or matching disk perimeter:

_{i}*w*(0°);

*w*(0.35°); and

*w*(0.70°).

^{2}(

^{2}(

^{2}(

*B*was 0.10 cd/m

^{2}(

^{2}(−0.8 to −0.4 log cd/m

^{2}) in six equal RGB unit steps, that is, from 0.8 to 0.4 log units below the test disk luminance and from 0.5 to 0.1 log units below the inner ring luminance.

^{2},

^{2},

*p*< .0001 two tailed; KS:

*p*< .0001, two tailed). The predictions of the global highest luminance rule are thus strongly disconfirmed by the data from Experiment 2.

*p*< .0001 one tailed; KS:

*p*< .0001, one tailed).

^{2}(−0.112 log units). Matches were performed at six

^{2}(−0.326, −0.233, −0.134, 0.013, 0.125, and 0.201 log cd/m

^{2}). As in Experiment 1, the test disk luminance

^{2}(

*B*was fixed at 0.10 cd/m

^{2}(

*p*< .0001, two tailed; MER:

*p*< .0001, two tailed). The actual slopes were about −.75 for both observers (AH: −.725 ± .044; MER: −.780 ± .026), which indicates a large magnitude contrast induction from the surround. A ratio match would be indicated by a slope of −1, as illustrated by the diagonal black line shown in the figure.

*p*< .0001, one tailed; MER:

*p*< .0001, one tailed).

*p*< .0001 one tailed; KS:

*p*< .0001, one tailed). We thus conclude that the results of this lightness matching experiment, like those of the naïve matching Experiments 1 and 2, are consistent with the distance-dependent edge integration and inconsistent with the highest luminance rule.

*decrease*with increases in the test surround luminance (contrast), and more specifically, (2) the log luminance of the matching disk would decrease linearly with increases in the log luminance of the test surround. Thus, the distance-dependent edge integration model not only successfully predicted the existence of violations of the highest luminance rule that depend on the intensities of lower luminance regions in the surround, it also successfully predicted the direction of these effects and some of their mathematical properties.

*gray world hypothesis*(Helson, 1943, 1964)—both rely on the implicit assumption that there exists a one-to-one mapping between at least one particular luminance and the physiological state associated with a particular lightness. According to the highest luminance rule, a one-to-one mapping exists between the highest luminance and the physiological state corresponding to white. According to the gray-world hypothesis, a one-to-one mapping exists between the average luminance and the physiological state corresponding to middle gray.

*increased*as a function of the surround luminance over the lower part of the surround luminance range—an assimilation effect—then decreased over the higher part—a contrast effect. A recent study by Bressan and Actis-Grosso (2001) found the opposite trend: a U-shaped matching function with a contrast effect at the lower end of the surround luminance range studied and an assimilation effect at the higher end.

^{2}, as in our study, while Bressan and Actis-Grosso reported significant contrast effects only for increments above 20.59 cd/m

^{2}. The latter authors suggested that the range of luminances investigated may be the key factor accounting for the difference in the patterns of results obtained in the two studies. But this conclusion cannot be correct because the stimulus luminances employed in the present study were within the range of those studied by Heinemann, yet the pattern of results observed in our study was consistent with the pattern obtained by Bressan and Actis-Grosso.

*D*

_{M}versus log

*R*

_{T}plots and induction signal “blockage”

*p*< .001) and a borderline significant improvement was obtained for observer JL (

*p*= .071). For both observers, the least-squares estimate of the coefficient multiplying the

*p*= .750).

*p*= .0003), but not in the case of observer MER (

*p*= .857). For observer AH, the coefficient multiplying the