Recent theories of lightness perception assume that lightness (perceived reflectance) is computed by a process that contrasts the target's luminance with that of one or more regions in its spatial surround. A challenge for any such theory is the phenomenon of lightness assimilation, which occurs when increasing the luminance of a surround region increases the target lightness: the opposite of contrast. Here contrast and assimilation are studied quantitatively in lightness matching experiments utilizing concentric disk-and-ring displays. Whether contrast or assimilation is seen depends on a number of factors including: the luminance relations of the target, surround, and background; surround size; and matching instructions. When assimilation occurs, it is always part of a larger pattern in which assimilation and contrast both occur over different ranges of surround luminance. These findings are quantitatively modeled by a theory that assumes lightness is computed from a weighted sum of responses of edge detector neurons in visual cortex. The magnitude of the neural response to an edge is regulated by a combination of contrast gain control acting between neighboring edge detectors and a top-down attentional gain control that selectively weights the response to stimulus edges according to their task relevance.

*contrast,*a class of phenomena that challenges all existing lightness models is that of lightness assimilation, which occurs whenever the target lightness covaries in a positive way with the luminance of the local surround (Helson, 1963; Helson & Joy, 1962; Hong & Shevell, 2004a; Jameson & Hurvich, 1961; Shevell, 2003; Shevell & Cao, 2003): the opposite of contrast. Although assimilation has been studied since at least the 19th century (von Bezold, 1876), it has no generally accepted explanation; nor have the properties of assimilation been explored in the same detail as those of contrast.

*double-decrement*display to refer to DAR stimuli having these luminance polarity relations, meaning that each luminance is a decrement relative to the luminance of its immediate surround.

*D*

_{T}, match ring

*R*

_{M}, and background

*B*were fixed at the values

*D*

_{T}= 0,

*R*

_{M}= 0.7, and

*B*= 1.4 log cd/m

^{2}. The target ring luminance was varied from 0.2 to 1.2 log cd/m

^{2}in six steps separated by intervals of equal RGB units. The experiment was run in three blocks of 36 trials. Each block contained six trials each of the six target ring luminances, presented in random order. Altogether, 18 trials were run at each target ring luminance.

*parabola,*a term that I will use here as shorthand for the more mathematically correct term “second-order polynomial.” In other words, a large percentage of the variance in the matches is accounted for by a regression equation having the form

*D*

_{M}is match disk luminance, and

*R*

_{T}is the target ring luminance, both specified in

*log units*. It will be convenient to always take log units to be the fundamental units of luminance throughout this paper, unless otherwise noted. The equations for the least-squares parabolic models for each observer and ring size are given in the figure. Second, increasing the ring width has the effect of decreasing both the parabola's curvature (parameter

*c*in Equation 1) and its slope

*b*.

*logarithm*of the luminance ratio of the border that generates it. This logarithm of the luminance ratio is the same thing as the signed luminance step in log units. The overall perceived disk color (that is, its shade of gray) is determined by the quantitative sum of the two achromatic color filling-in signals. Support for the quantitative assumptions of the model can be found in the following papers: Rudd (2001, 2003a, 2003b, 2007), Rudd and Arrington (2001), Rudd and Popa (2007), Rudd and Zemach (2002a, 2002b, 2004, 2005, 2007), Vladusich, Lucassen, and Cornelissen (2006a, 2006b, 2007).

*only*be explained by assuming the existence of a cortical mechanism that sums luminance steps across space.

_{D}of either the target of match disk in the DAR stimulus is computed from a weighted sum of the luminance steps in log units at the inner and outer borders of the surround ring. This is expressed formally by the following equation:

*D*−

*R*is the luminance step traversed in crossing the border from the ring to the disk;

*R*−

*B*is the luminance step traversed in crossing from the background to the ring; and

*w*

_{1}and

*w*

_{2}are the weights associated with the disk–ring and ring–background borders in the disk lightness computation. Previous results (see quantitative references above) suggest that the weights assigned to the two edges tend to decrease—or at least do not increase—as a function of distance from the disk; that is,

*w*

_{2}≤

*w*

_{1}.

*w*

_{2}assigned to the outer border of the surround ring is attenuated by the contrast of the disk–ring edge in a contrast-dependent manner. The blockage assumption is formalized by

*w*

_{2}* is the weight that would be assigned to the outer border in the absence of an inner border (the “a priori” weight); [ ]+ denotes the half-wave rectification operator, which returns either the value of the algebraic expression in brackets or zero, whichever is smaller; the function

*δ*= +1 or −1, according to the sign of the luminance step

*D*−

*R*; and the “blockage” parameter

*β*quantifies the degree to which the a priori weight associated with the outer ring border is attenuated as a function of the edge step at the inner (disk–ring) border. Note that the attenuation term (the terms in brackets) saturates (goes to zero) when

*δ*(

*D*−

*R*) = ∣

*D*−

*R*∣ >

*β*

^{−1}; thus, the blockage parameter is also the inverse of the absolute value of the luminance step that saturates the attenuation.

*square*of the ring luminance over the range of ring luminances in which the attenuation term is nonsaturating. Over this range, the model predicts that the target disk lightness will exhibit a parabolic dependence on the target ring luminance.

*direction*of curvature for the parabolic matching plots.

*negative*: the local slope of the plot tends to decrease as the ring luminance is increased. The tangent slope is positive in the assimilation regime of the plot and negative in the contrast regime. The blockage model predicts this negative curvature because the coefficient multiplying the

*R*

^{2}term in the polynomial expansion of Equation 4—

*βδw*

_{2}*—must be negative for a double-decrement stimulus. This is because the weight

*w*

_{2}* is constrained to be positive, the sign

*δ*of the contrast polarity of the disk–ring border is −1 for the double-decrement stimulus, in which

*D*<

*R,*and the blockage parameter

*β*must be positive in order for Equation 4 to represent a process by which the contrast of the inner ring border partially

*blocks*(attenuates) the contribution of the outer ring border to the disk lightness computation.

*β*in Equation 4 to be negative for the double-decrement stimulus and positive for the disk polarity-reversed stimulus, but then the process modeled by Equation 4 would not represent blockage in the latter case. Equation 4 would represent either blockage (attenuation) or “anti-blockage” (amplification) of the remote induction signal originating from the outer border, depending on the contrast polarity of the inner border (Rudd & Zemach, 2007).

*VLC model*.

*relationship*between the contrast polarities of the inner and outer ring borders. If the contrast polarities are the same, blockage occurs; if the border contrast polarities differ, anti-blockage occurs. From Equation 4, it can be seen that the direction of the curvature in the matching plot depends strictly on the contrast polarities of the edges and the sign of the blockage constant. Because the VLC model associates a unique blockage constant sign with each pair of edge polarities, the model predicts that only one direction of curvature should be observed for any given set of edge contrast polarities.

*Wallach stimuli*.

*r*

^{2}-change = 0.003,

*t*(102) = −1.380,

*p*= 0.019; JL:

*r*

^{2}-change = 0.015;

*t*(105) = 4.174,

*p*< 0.0005; AD:

*r*

^{2}-change = 0.001,

*t*(105) = 0.886,

*p*= 0.378; IKZ:

*r*

^{2}-change = 0.005,

*t*(105) = −3.557,

*p*= 0.001). However, the additional variance accounted for by the parabolic model is quite small, ranging from 0.1 to 1.5% for those three observers. For this reason, Rudd and Zemach fit their matching data with a linear model. For each of the four observers, the slope of the least-squares linear model was about −0.7 (Figure 6). We will return to this finding of −0.7 matching plot slopes below because it also shows up in other studies involving DAR stimuli having the same size as these Wallach stimuli but different combinations to edge contrast polarities.

*p*-values associated with the percent improvement in the fit of the parabolic model over the linear model are subject to a Bonferroni correction in order to take into account the number of statistical tests performed in the study (

*p*

_{crit}= 0.013). The 95% confidence intervals for the curvature (the coefficient that multiplies the

*R*

_{T}

^{2}term in the parabolic regression model) do not include any values whose sign is inconsistent with the sign of the coefficient estimate (confidence intervals = LT: −0.599, −0.054; JL: 0.414, 1.162; IKZ: −0.761, −0.216); so we can be reasonably confident that observer JL's true matching plot curvature is positive, while the curvatures in IKZ's and perhaps LT's matching plots are both negative.

*double-increment*DAR stimuli: that is, DAR stimuli in which the disk luminance exceeded the ring luminance and the background had the lowest luminance in the display. The stimuli used by Rudd and Zemach in their 2005 study were the same size as the Wallach stimuli that produced the matching plots shown in Figures 5 and 6. The matching plots from the study using double-increment stimuli are shown in Figure 9, along with least-squares parabolic models of the data. The parabolic models accounted, on average, for 96.3% of the variance in the match disk settings made by the three observers, compared to 94.3% for the linear model. For two of the three observers, the parabolic model accounted for a very large proportion of the match setting variance: 98.6% and 99.1%. Nevertheless, the improvement in the fit of the parabolic model over the linear model was statistically significant for only one observer (AD:

*r*

^{2}-change = 0.041,

*t*(71) = −4.522,

*p*< 0.0005) and borderline significant for another (JL:

*r*

^{2}-change: = 0.005;

*t*(105) = −1.793,

*p*= 0.076). For the third observer, no statistical improvement was found (LT:

*r*

^{2}-change = 0.001,

*t*(105) = 0.313,

*p*= 0.755).

*only*negative values (−0.704, −0.273). For the plot having borderline-significant curvature (JL), the majority of the 95% confidence interval for this coefficient (−0.242, 0.012) comprises negative values. Thus, we find no evidence from this study to suggest that the matching plots for double-increment stimuli curve in different directions for different observers.

*reflectance*(lightness); perceived

*luminance*(brightness); or perceived

*contrast*with respect to the surround (referred to by those authors as “brightness contrast”). Furthermore, an observer who is asked to produce a reflectance match can do so by making either of two assumptions regarding the nature of the illumination falling on the match and target disks. The observer could assume that the illumination across the entire display is constant and remains unchanged when the target ring luminance is varied, in which case experimental manipulations of the target ring luminance must signal changes in the target ring

*reflectance*. Otherwise, the observer could assume that a change in the target ring luminance signals a change in the

*illumination*on the target side of the display only—including the target disk and ring—while the target ring reflectance remains constant.

*brightness*matching, the observer is instructed to match the two disks on perceived luminance, so an ideal observer would perform a

*luminance*match. For

*brightness contrast*matching, the observer is instructed to match the disk–ring contrast on the two sides of the display, so an ideal observer would perform a

*ratio*match. For

*one-spotlight lightness*matching, variations in the target ring luminance are to be interpreted as changes in the target ring reflectance, which the subjects were told explicitly to think of as a paper surface. Such changes should not influence the perceived reflectance of the target

*disk,*so an ideal observer would perform a

*luminance*match. For

*two-spotlight lightness*matching, variations in the target ring luminance are to be interpreted as changes in the illumination on both the target disk and its surround. Here, ideal observer would match the disk–ring

*ratios*on the two sides of the display. The ideal observer predictions for the four sets of matching instructions are illustrated in Figure 11.

*D*

_{T}= 0.5,

*R*

_{M}= 0.0, and

*B*= −1.0 log cd/m

^{2}. The target ring luminance was varied from −0.326 to 0.201 log cd/m

^{2}in six steps separated by intervals of equal RGB units. The experiment was run in three blocks of 36 trials. Each block contained six trials each of the six target ring luminances, presented in random order. Eighteen trials were run at each target ring luminance, for a grand total of 108 trials.

*D*

_{T},

*R*

_{T},

*R*

_{M}, and

*B*and the ones reported by Rudd and Zemach (2005) for the same stimuli. Before performing Experiment 2, it was discovered that the lookup table used in the earlier study contained an error that shifted the values in the table by one line, as discussed in the previous section. This problem was corrected before running Experiment 2. At the same time, the monitor was recalibrated and the luminances

*D*

_{T},

*R*

_{T},

*R*

_{M}, and

*B*were measured directly with a photometer. The directly measured values are reported here. Rudd and Zemach instead reported the nominal luminance values that were targeted by the MATLAB program used to generate the stimuli. Experience in modeling the data from experiments such as these has underscored the importance of precise luminance measurement.

*square*of the target ring luminance (both in log units). A linear model was adopted in place of the parabolic model if the second variable could be removed without producing a statistically significant decrease in the amount of variance in the match disk settings accounted for by the regression model. For one analysis only (AH, brightness matching), neither predictor variable was significant. The results from that condition have been fit with a parabolic model in Figure 12. This procedure resulted, for both observers, in fitting the brightness and one-spotlight lightness matches with a parabolic model and the brightness contrast and two-spotlight lightness matches with a linear model.

*b*

_{0}), slope (

*b*

_{1}), and curvature (

*b*

_{2}) of the second-order polynomial model. The results of the parabolic regression analyses are as follows:

*Brightness*(AH:

*b*

_{0}= 0.505 ± 0.003,

*p*< 0.001;

*b*

_{1}= 0.002 ± 0.015,

*p*= 0.916;

*b*

_{2}= 0.102 ± 0.080;

*r*

^{2}= 0.457; MER:

*b*

_{0}= 0.500 ± 0.011,

*p*< 0.001;

*b*

_{1}= 0.265 ± 0.051,

*p*= 0.015;

*b*

_{2}= 0.886 ± 0.270,

*p*= 0.046;

*r*

^{2}= 0.901);

*brightness contrast*(AH:

*b*

_{0}= 0.422 ± 0.034,

*p*= 0.001;

*b*

_{1}= −1.065 ± 0.152,

*p*= 0.006;

*b*

_{2}= −0.730 ± 0.805;

*r*

^{2}= 0.960; MER:

*b*

_{0}= 0.444 ± 0.010,

*p*< 0.001;

*b*

_{1}= −0.790 ± 0.045,

*p*< 0.001;

*b*

_{2}= −0.069 ± 0.241,

*p*= 0.794;

*r*

^{2}= 0.994);

*one-spotlight lightness*(AH:

*b*

_{0}= 0.497 ± 0.004,

*p*< 0.001;

*b*

_{1}= 0.084 ± 0.020,

*p*= 0.025;

*b*

_{2}= 0.331 ± 0.107,

*p*= 0.053;

*r*

^{2}= 0.856; MER:

*b*

_{0}= 0.491 ± 0.007,

*p*< 0.001;

*b*

_{1}= 0.254 ± 0.030,

*p*= 0.003;

*b*

_{2}= 0.865 ± 0.158,

*p*= 0.012;

*r*

^{2}= 0.961);

*two-spotlight lightness*(AH:

*b*

_{0}= 0.466 ± 0.039,

*p*= 0.001;

*b*

_{1}= −0.854 ± 0.175,

*p*= 0.016;

*b*

_{2}= −1.023 ± 0.929,

*p*= 0.351;

*r*

^{2}= 0.913; MER:

*b*

_{0}= 0.449 ± 0.012,

*p*< 0.001;

*b*

_{1}= −0.776 ± 0.052,

*p*= 0.001;

*b*

_{2}= 0.033 ± 0.278,

*p*= 0.912;

*r*

^{2}= 0.992).

*b*

_{0}) and slope (

*b*

_{1}). The results of the linear regression analyses, where appropriate, are:

*brightness contrast*(AH:

*b*

_{0}= 0.399 ± 0.022,

*p*< 0.001;

*b*

_{1}= 0.975 ± 0.112,

*p*= 0.001;

*r*

^{2}= 0.950; MER:

*b*

_{0}= 0.442 ± 0.006,

*p*< 0.001;

*b*

_{1}= −0.782 ± 0.030,

*p*< 0.001;

*r*

^{2}= 0.994);

*two-spotlight lightness*(AH:

*b*

_{0}= 0.434 ± 0.027,

*p*= 0.001;

*b*

_{1}= −0.728 ± 0.136,

*p*= 0.006;

*r*

^{2}= 0.877; MER:

*b*

_{0}= 0.450 ± 0.007,

*p*< 0.001;

*b*

_{1}= −0.780 ± 0.034,

*p*< 0.001;

*r*

^{2}= 0.992).

*lightness*matches are not always independent of the surround luminance in the case of double-increment stimuli. Contrary to the predictions of Anchoring Theory, the effect of manipulating the luminance of the lower luminance surround in a double-increment stimulus depends on the assumptions that the observer makes about the nature of the illumination. Overall, ideal observer models provide a better account of the results of Experiment 2 than does anchoring to the highest luminance.

*no*effect of surround luminance on the matches made with double-increment stimuli (Jacobsen & Gilchrist, 1988), observers were explicitly told to match the stimuli on

*brightness*when the target was an increment relative to its surround luminance. Despite this fact, the results of Jacobsen and Gilchrist's study have subsequently been cited (erroneously) as evidence in favor of highest luminance anchoring in

*lightness*perception (Gilchrist, 2006; Gilchrist et al., 1999).

*lightness*matching conditions of Experiment 2 exhibit regularities that mirror findings obtained in our 2004 study with Wallach stimuli and with double-decrement stimuli in Experiment 1 of the present study. First, roughly linear matching plots having slopes of about −0.7 were found in both the two-spotlight matching condition of Experiment 2 and in our earlier study employing Wallach stimuli having surround rings of the same size. The fact that strong contrast effects were obtained both with Wallach stimuli in our 2004 study and with two-spotlight instructions here is consistent with Wallach's interpretation of his own results as an attempt on the part of the observers to achieve lightness constancy under the assumption that the two sides of the display are lit by separate illuminants. Nevertheless, in both our 2004 study and in the two-spotlight condition of Experiment 2, our observers failed to make the exact ratio matches predicted for an ideal observer under two-spotlight lightness matching instructions, as indicated by the fact that the matching plot slopes were about −0.7, rather than −1. It seems likely that that this deviation from exact ratio matching is due to the small size of the 0.35 deg surround used in both experiments. The plots obtained with 0.35 deg Wallach stimuli in our 2004 study were about −0.7, but the slopes tended to decrease toward −1 as the ring size was increased in that study (Rudd & Zemach, 2004). Further work is required to determine whether a perfect ratio match would hold for double-increment and two-spotlight instructions if the surround size were increased.

*naive*matches made with double-decrement stimuli in Experiment 1 of the present study and for

*one-spotlight lightness*matches made with double-increment stimuli in Experiment 2. In the next section, a neural model that accounts for this regularity is proposed. The model also accounts for the rest of the results discussed to this point, including both the parabolic and approximately linear matching plots, and their dependence on luminance polarity, surround size, and matching instructions. The model is able to account both for the aspects of the data that approximate ideal observer behavior (i.e., constancy) and for the pattern of errors. By incorporating attentional feedback to adjust the model parameters in a task-specific way, the model is able to fit the results of both lightness matching conditions of Experiment 2 in the context of a single unified neural theory.

*relative*strengths of the two gain controls determine the direction of curvature in the matching plot.

*achromatic color (AC) neurons*. The population response of the AC neurons determines the achromatic color assigned to each point in the image array. AC neurons come in two types:

*lightness neurons*and

*darkness neurons*. Each AC neuron performs a weighted spatial summation of the responses of the ED2 neurons located within its receptive field. The receptive field weighting function falls off linearly from the receptive field center to its outer perimeter. AC neuronal receptive fields must be much broader than the receptive fields of the edge detector neurons in Layers 1 and 2 to account for the fact that achromatic color can be influenced by remote edge contrast (Bressan & Kramer, 2008; Shevell et al., 1992).

*edge detectors,*the AC neurons function as

*edge integrators*. Darkness neurons sum the outputs of ED2 units that fire in response to dark-inside edges. Lightness neurons sum the outputs of ED2 units that fire in response to light-inside edges (Figure 13). The achromatic color assigned to a given image location depends on the relative strengths of the lightness and darkness neuronal responses at that location (i.e., the difference

*L*(

*i*) −

*D*(

*i*), where

*i*is the array index). To keep things simple, it is assumed in what follows that the lightness of either disk in any given DAR display is determined solely by the difference in the responses of the lightness and darkness neurons whose receptive field centers are located at the disk

*center*. A more sophisticated version of the model would assign an achromatic color to each point within the disk interior, according to the computational rules described above. In that case, lightness could vary within the disk, along its radius as a function of distance from the disk edge.

*absolute*lightness values to the disks—that is, values like “white,” “mid-gray,” etc.—the model would need to be supplemented with an “anchoring” rule. It might be assumed, for example, that the maximum firing rate across the Layer 3 array always corresponds to the value “white” (that is, a reflectance of 90%). The perceived reflectance corresponding to any other arbitrary point in the image would then be computed from the ratio of the neuronal firing rate associated with the arbitrary point to the firing rate associated with the white point.

*lightness*in the image will always be white (Kingdom, in press; Rudd & Zemach, 2005). The highest lightness rule is closely related to, but not identical with, the well-known hypothesis that highest

*luminance*is always seen as white (Gilchrist, 2006; Gilchrist & Radonjić, 2009; Gilchrist et al., 1999; Li & Gilchrist, 1999). In the few cases where these two alternative anchoring rules have been tested against one another by experiment, the psychophysical data have supported the highest lightness rule over the highest luminance rule (Bressan & Actis-Grosso, 2001; Rudd & Zemach, 2005).

*fails*to bridge the distance between the inner and outer edges of the surround ring, the matching plot will be a straight line. If the contrast gain control successfully spans the ring width, the matching plot will have the form of a parabola. The contrast gain control will fail to span the distance if the ring is too wide; or the luminance ratio of the edge that generates the contrast gain control is too weak; or if the top-down attentional gain applied to the edge that generates the gain control is damped down sufficiently. If the contrast gain control does span the ring width, the degree of curvature in the matching plot will depend on the strengths of the contrast and attentional gain controls, as well as on the ring width. Larger gain factors and smaller widths produce greater curvature.

*δ*

_{21}and

*δ*

_{12}represent the signs of the inwardly and outwardly directed gain controls;

*g*1

_{1}and

*g*1

_{2}are the neural gains applied to the inner and outer ring edges at Layer 1 of the model;

*ω*

_{1T}and

*ω*

_{2T}are the AC neuronal receptive field weights applied to the ED2 neuronal responses to the inner and outer ring edges; and

*α*

_{T}and

*β*

_{T}are the strengths of the inwardly and outwardly directed gain controls associated with the target side of the display.

*D*

_{T}+

*B*). For the double-increment DAR stimulus used in the experiments, −(

*D*

_{T}+

*B*) = −(0.5 − 1.0) = 0.5 log cd/m

^{2}. Thus, the model makes a strong parameter-free prediction that depends only on the physical properties of the stimulus.

*not*conform the model prediction, one might wonder if the results that

*do*conform to the model prediction are somehow a fluke, or perhaps that the model appears to work only because I selected a opportunistic subset of the points in Figure 19 to fit with the linear regression model. To address this potential concern, I reanalyzed the data from the other three contrast polarity experiments analyzed by Rudd and Zemach (2007) to see whether the predicted relationship between the slope and curvature of the matching functions holds for those three additional data sets. The additional data sets include the matches obtained with Wallach stimuli by Rudd and Zemach (2004), and with double-decrement and incremental disks and white background DARs by Rudd and Zemach (2007). As can be seen from the plots shown in Figure 20, the model prediction is verified for each of these three additional data sets. Thus, the data from the two-spotlight matching condition of Experiment 2 is the only data that I have analyzed to date that fails to conform to the model prediction relating the matching plot slope to its curvature.

*along*this line do not also vary the line's intercept. Any parameter change that shifts the line's intercept for a particular observer or set of matching instructions would result in an

*apparent*violation of the model. The reader can verify from inspection of the systems of Equations B4–B4c in 2 that two and only two model parameters can be varied to shift the data along the regression line in Figure 19 without simultaneously changing the line's intercept. Those are the parameters

*α*

_{T}and

*β*

_{T}, which model the strengths of the inwardly and outwardly directed gain controls associated with the target side of the display. Thus, if the neural lightness computation model is correct, the differences between the observers and matching instructions represented by the purple circles in Figure 19 might result from variations in contrast gain control strengths across observers and instructions. However, the possibility cannot be ruled out that variations between observers and conditions instead reflect differences in combinations of neural parameters that

*together*vary in such a way that the intercept of the slope versus curvature plot is held constant.

*other*model parameters. In the previous section, I proposed that changes in the lightness matching instructions cause the observer to change the attentional gain applied to the outer edge of the surround ring at Layer 1 of the model (see Figure 16d). Since the prescription for ideal matching behavior in the two-spotlight condition is to match the disk–ring ratios on the two sides of the display, all of the weight in the lightness computation should ideally be given to the inner edge. Thus, the observer in the two-spotlight condition ideally ought to completely damp down the weight associated with the outer ring edge. In principle, this might be accomplished either by changing the gain applied to that edge at Layer 1 or by altering the shape of the AC neuronal receptive fields in Layer 3.

*sufficient*to account for the changes in the shape of the matching plots shown in Figure 12 when the matching instructions are changed, I fit the data from each of the two observers in the one-spotlight and two-spotlight lightness matching conditions by assuming that the only model parameter that changes across the two instructional conditions is the attentional gain applied to the outer ring edge in Layer 1. In the one-spotlight lightness matching condition, I assumed that both observers set the neural gains applied to the inner and outer edges of the surround ring to be

*equal*. In the absence of both contrast gain control and the spatial falloff in the AC receptive field weighting function in Layer 3, this choice of Layer 1 gain settings would produce a luminance match, which is ideal behavior in the one-spotlight condition. To model the results of the two-spotlight condition, I assumed that the gain applied to the outer ring edge in Layer 1 was reduced relative to the gain applied to that edge in the one-spotlight condition. This reduction in the attentional gain applied to the outer ring edge reduces the strength of the gain control directed from the outer edge to the inner edge in the pathway from Layer 1 to Layer 2. The resulting gain change in Layer 2 in turn reduces the curvature of the theoretical matching plots associated with the two-spotlight condition relative to those associated with the one-spotlight condition, while simultaneously making the slope of the matching function steeper and more negative. The effect of changing the Layer 1 gain on the slope–curvature pairs for “simulated” two-spotlight matches is illustrated in Figure 21. Details of the procedure used to compute the simulated matches are given in 2. It is not possible to simulate two-spotlight matching plots having negative curvature. However, the 95% confidence intervals of the curvature estimates for the least-squares parabolic models of the actual matching plots from this matching condition include positive values for both observers. Thus, a mechanism based on attentional modulation of the neural gain in Layer 1 is shown to be sufficient to account for all aspects of the instructional effects seen in Experiment 2, down to quantitative details, within measurement error.

*ω*

_{2}= 0.98

*ω*

_{1}), it appears that the observers in the two studies set the weights on the two ring edges to be approximately equal in both of the one-spotlight and naive matching experiments. Setting the inner and outer edge weights to be equal is ideal behavior for both one-spotlight lightness matching and brightness matching, so the modeling results suggest that the observers in both of these conditions interpreted disk and rings as papers lit by a single global illuminant or, alternatively, that they matched the stimuli on brightness, rather than lightness.

*luminance*anchoring, it should be clear from the forgoing discussion that—in the context of Retinex theory at least—it is technically not the highest luminance that is mapped to white but instead the highest

*reflectance*. Under some—but not all—conditions—the highest reflectance and the highest luminance are equivalent (see Rudd & Zemach, 2005 for further details).

*equal*weights to all edges in the scene; whereas, in subsequently proposed algorithms the weights are assumed to fall off with distance from the patch whose reflectance is being computed (Gilchrist, 1988; Reid & Shapley, 1988; Rudd, 2001, 2003a, 2003b, 2007; Rudd & Arrington, 2001; Rudd & Popa, 2007; Rudd & Zemach, 2002a, 2002b, 2004, 2005, 2007; Shapley & Reid, 1985). Because of its “equal weights for all edges” assumption, the Retinex model exhibits a property that Gilchrist et al. (1999) referred to as Type II lightness constancy: that is, constancy with respect to changes in the reflectances of other surfaces in the target's surround. Altering this assumption leads to an edge integration model in which the surround influences the target lightness, which of course is required in order to model contrast and assimilation.

*partially*integrated in the relative reflectance computation, which would result in a partial failure of Type I constancy. In either case, the illusion would be explained at least qualitatively. von Helmholtz (1866/1924) suggested that discounting the illuminant results from a process of “unconscious inference.” In the case of the checker-shadow illusion, the “unconscious inference” might be achieved mechanistically by a biological strategy that excludes gradient edges from the edge integration computation.

*all*of the edges in the display should be treated as reflectance edges. In that case, the outer edges or the rings should be included in the edge integration computation and all edges should be given equal weights, as in Retinex theory. Consistent with this idea, the results from the one-spotlight condition indicate that observers

*do*tend to give the outer edge more weight when they are instructed to interpret this edge as a reflectance edge.

*all*of the phenomenal properties of lightness filling-in. Consider the Kanizsa triangle figure shown in Figure 25 (Kanizsa, 1955, 1979). Most observers, when shown this figure, report seeing a bright illusory triangle having sharply defined edges that appears to stand out in front of three partially occluded disks and a partially occluded black triangular outline. Many observers also report that the illusory square appears whiter or brighter than background, despite the fact that most of the illusory edge is not supported by a real edge having signed contrast in the image.

*same*contrast polarity as the inducer edges. This edge detector activation would not only create the seen border but also produce light and dark filling-in percepts on either side of the border, which would explain why the square looks lighter than the background. Although further work would be required to flesh out this model in detail, this description should suffice to make the basic point that filling-in does not necessarily have to be produced by a combination of separate BCS and FCS activations nor by a diffusive filling-in process that stops at figural borders.

*not*comprising homogeneous patches demarcated by sharp luminance borders and lying in a single depth plane. In previous papers, I have discussed ways in which edge integration models could be generalized to deal with images having blurred edges at arbitrary spatial scales: in other words, to account for lightness percepts in arbitrary two-dimensional grayscale images (Rudd & Popa, 2007; Rudd & Zemach, 2004, 2005, 2007; Zemach & Rudd, 2007). Spatially oriented steps in log luminance could be computed at a range of spatial scales by a collection of two-dimensional oriented band-pass spatial filters, such as the Gabor functions that have been used extensively to model simple cell receptive fields in cortical area V1 (e.g., Daugman, 1985; Jones & Palmer, 1987a, 1987b; Marcelja, 1980; Ringach, 2002) operating on a logarithmically compressed neural image of the stimulus luminance profile (Cornsweet, 1970). To compute lightness in three-dimensional scenes, depth cues such as stereopsis could be used as control signals for determining whether or not edges will be spatially integrated.

*intends*to make. In other words, the visual system can “morph” into a range of different visual measuring devices. An implication of this idea is that there is no such thing as “low-level” vision.

_{ k }

^{+}(0) and AC

_{ k }

^{−}(0) denote the activities of the lightness and darkness neurons in Layer 3, respectively, whose receptive fields are located at origin, which is taken to be the center of the disk on the

*k*th side of the display,

*k*=

*M, T*.

*E*2

_{1k }

^{+}and

*E*2

_{2k }

^{+}represent the activities of the Layer 2 edge detectors located at the positions of the inner and outer edges—edges 1 and 2—of the surround ring; and

*ω*

_{1k }

^{+}and

*ω*

_{2k }

^{+}represent the weights given to those activities by the lightness neuron centered at the origin. Similarly, darkness neurons sum only the responses of Layer 2 neurons that fire in response to edges whose dark sides point inward:

*E*2 neural responses to edges of contrast polarity

*ρ*= +, − are assumed to fall off linearly with distance from the AC neuron's receptive field center. Thus

*d*

_{ k }and

*r*

_{ k }are the disk and ring radii on the

*k*

^{ th }side of the display.

*ρ*and located at the position

*n*= 1, 2 is given by the products of the Layer 1 neuron having the same polarity sensitivity and located at the same position, and the Layer 2 neuronal gain factor:

*n*of contrast polarity

*ρ*is initially set to the value

*n*. The magnitude of the adjustment is assumed to fall off as a linear function of distance such that

*ρ*

_{2}to the Layer 2 neuron responding on an edge 1 having contrast polarity

*ρ*

_{1}; and similarly,

*ρ*

_{1}to the Layer 2 neuron responding on an edge 2 of contrast polarity

*ρ*

_{2}.

*k*th side of the display:

*g*2

_{1k }* and

*g*2

_{2k }* of the Layer 2 edge detectors have been absorbed into the expressions for the weights

*ω*

_{1k }and

*ω*

_{2k }.

*D*

_{M}in terms of the target ring luminance

*R*

_{T}that satisfies Equations A8 and B1. For present purposes, we can write Equation A8 in the following simplified form:

*d*

_{ k }and

*r*

_{ k }denoting the disk and ring radii have been dropped because the disks and rings on the two sides of the displays were the same sizes in Experiments 1 and 2; and it has been assumed that identical Layer 1 gains are applied to target and match sides of the display.

*θ*is the size of the Layer 3 AC neuronal receptive field.

*Mathematica*5.2 (Wolfram Research, Cambridge, MA).

*ρ*

_{1}=

*ρ*

_{2}= −1;

*D*

_{T}= 0;

*R*

_{M}= 0.7;

*B*= 1.4;

*d*= 0.35;

*r*= 0.12 (small ring), 0.583 (medium ring), 1.05 (large ring). Model parameters (both subjects):

*g*1

_{1}= 1.0;

*g*1

_{2}= 1.2;

*δ*

_{21}=

*δ*

_{12}= −1. Model constraints (to ensure contrast gain control spans the ring width):

*α*

_{M}=

*α*

_{T}= 2.698 × 10

^{−2}

*ν*/

*g*1

_{2};

*β*

_{M}=

*β*

_{T}= 2.698 × 10

^{−2}

*λ*. Subject-specific parameters: HK,

*θ*

_{M}=

*θ*

_{T}= 5.667,

*α*

_{M}=

*α*

_{T}= 0.452116;

*β*

_{M}=

*β*

_{T}= 0.157049;

*ν*= 1.017 × 10

^{−1};

*λ*= 2.945 × 10

^{−2}; MER,

*θ*

_{M}=

*θ*

_{T}= 4.1333,

*α*

_{M}=

*α*

_{T}= 0.263923;

*β*

_{M}=

*β*

_{T}= 0.332915;

*ν*= 5.278 × 10

^{−2};

*λ*= 5.549 × 10

^{−2}. Regression coefficients for model fits (actual coefficients, where different, in parentheses): HK (small ring)

*c*

_{0}= −0.6832 (−0.684 ± 0.092),

*c*

_{1}= 1.6967 (1.814 ± 0.295),

*c*

_{2}= −1.0296 (−1.283 ± 0.207); (medium ring)

*c*

_{0}= −0.3161 (−0.307 ± 0.077),

*c*

_{1}= 1.0008,

*c*

_{2}= −0.7847; (large ring)

*c*

_{0}= −0.0095 (−0.120 ± 0.034),

*c*

_{1}= 0.4257 (0.520 ± 0.108),

*c*

_{2}= −0.5888 (−0.563 ± 0.076); MER (small ring)

*c*

_{0}= −0.6832 (−0.865 ± 0.060),

*c*

_{1}= 2.0575 (2.181 ± 0.192),

*c*

_{2}= −1.3114 (−1.529 ± 0.135); (medium ring)

*c*

_{0}= −0.1922 (−0.033 ± 0.032),

*c*

_{1}= 0.9152,

*c*

_{2}= −0.7847; (large ring)

*c*

_{0}= 0.2584 (0.308 ± 0.124),

*c*

_{1}= 0.0726 (0.155 ± 0.398),

*c*

_{2}= −0.6311 (−0.820 ± 0.280). Notes: Model gets

*c*

_{1}and

*c*

_{2}exactly for medium ring (by design);

*g*1

_{2}had to be >1 and values in the range of 1.1–1.2 gave acceptable fits;

*θ*

_{M}=

*θ*

_{T}had to be >2.5 (deg) for HK and >2.0 (deg) for MER.

*ρ*

_{1}=

*ρ*

_{2}= 1;

*D*

_{T}= 0.5;

*R*

_{M}= −0.112;

*B*= −1;

*d*=

*r*= 0.35. Model parameters (both subjects):

*δ*

_{21}= −1;

*δ*

_{12}= 1;

*g*1

_{1}= 1.0. Subject-specific parameters: AH,

*θ*

_{M}=

*θ*

_{T}= 35.7;

*g*1

_{2}= 1.0 (one-spotlight), 0.166181 (two-spotlight);

*α*

_{M}= 0.266348;

*β*

_{M}= 0.0122513;

*α*

_{T}= 0.278583;

*β*

_{T}= 0.0122513;

*ν*= 0.0892039;

*λ*= 0.0420291; MER,

*θ*

_{M}=

*θ*

_{T}= 35.7;

*g*1

_{2}= 1.0 (one-spotlight), 0.272765 (two-spotlight);

*α*

_{M}= 0.522321;

*β*

_{M}= 0.0141519;

*α*

_{T}= 0.552814;

*β*

_{T}= 0.0141519;

*ν*= 0.213039;

*λ*= 0.048549. Regression coefficients produced by model fits (actual regression coefficients, where different, in parentheses): AH (one-spotlight)

*c*

_{0}= 0.497,

*c*

_{1}= 0.084,

*c*

_{2}= 0.331; (two-spotlight)

*c*

_{0}= 0.402681 (0.466 ± 0.039),

*c*

_{1}= −0.854,

*c*

_{2}= 0.0445812 (−1.023 ± 0.929); MER (one-spotlight)

*c*

_{0}= 0.491,

*c*

_{1}= 0.254,

*c*

_{2}= 0.865; (two-spotlight)

*c*

_{0}= 0.406399 (0.449 ± 0.012),

*c*

_{1}= −0.776,

*c*

_{2}= 0.154514 (0.033 ± 0.278).

*Proceedings of SPIE-IS&T Electronic Imaging, SPIE, 4299,*400–413.

*Proceedings of SPIE-IS&T Electronic Imaging, SPIE, 5007,*170–181.

*Journal of the Optical Society of America A: Optics, Image Science, and Vision, 24,*3335. [CrossRef]

*Zur Farbenlehre*by C. L. Eastlake). Cambridge, MA: MIT Press.