**In 1986, Paul Whittle investigated the ability to discriminate between the luminance of two small patches viewed upon a uniform background. In 1992, Paul Whittle asked subjects to manipulate the luminance of a number of patches on a uniform background until their brightness appeared to vary from black to white with even steps. The data from the discrimination experiment almost perfectly predicted the gradient of the function obtained in the brightness experiment, indicating that the two experimental methodologies were probing the same underlying mechanism. Whittle introduced a model that was able to capture the pattern of discrimination thresholds and, in turn, the brightness data; however, there were a number of features in the data set that the model couldn't capture. In this paper, we demonstrate that the models of Kane and Bertalmío (2017) and Kingdom and Moulden (1991) may be adapted to predict all the data but only by incorporating an accurate model of detection thresholds. Additionally, we show that a divisive gain model may also capture the data but only by considering polarity-dependent, nonlinear inputs following the underlying pattern of detection thresholds. In summary, we conclude that these models provide a simple link between detection thresholds, discrimination thresholds, and brightness perception.**

*I*) and the reference (

_{t}*I*) patches. Luminance discrimination thresholds were investigated using a reference luminance that was either above (positive/bright pedestals:

_{r}*I*>

_{r}*I*) or below (negative/dark pedestals:

_{b}*I*<

_{r}*I*) the background luminance level

_{b}*I*. A subset of Whittle's discrimination data is shown in Figure 1, and it can be seen that substantially different results are obtained for the two polarities. For positive/bright pedestals, thresholds increase approximately in proportion to the pedestal luminance, which is defined as the luminance difference between the background and the reference patch (

_{b}*I*= |

_{p}*I*–

_{b}*I*|). However, for negative/dark pedestals, a more complex pattern of thresholds is observed with the form of an inverted “U.”

_{r}*W*is determined by a fixed denominator for positive/bright pedestals and a variable denominator for negative/dark pedestals, a polarity-dependent result is obtained that is broadly consistent with the experimental data. To understand why, it is helpful to reformulate Whittle's model in the form Δ

*I*=

*f*(

*I*), and the calculations for doing so can be found in appendix A of Kingdom and Moulden (1991); for positive pedestals, this gives

_{p}*I*and the descending variable

_{r}*I*produces an inverted “U.” The predictions from these two equations are shown by the solid line in Figure 1B.

_{p}*I*), and the innermost patch was maximally dark (

_{max}*I*). Subjects were asked to manipulate the luminance of the intermediary patches until each luminance step appeared to be of equal magnitude. In Figure 2, we reproduce the data from two conditions. In the leftmost figure, the test patches were yellow, and the background green and the resulting nonlinearity is compressive with no inflexion. This form is common to most of the nonlinearities reported in the literature (Wyszecki & Stiles, 1982). However, when the test patches and the background were both achromatic, the function steepened around the background luminance level: This is known as the “crispening” effect (Takasaki, 1966) and mirrors the high sensitivity around the background luminance noted in the discrimination threshold data set. To make this point, Whittle plotted the discrimination thresholds alongside the luminance intervals, and we replot this data in Figure 3. The luminance intervals (blue dots) are consistently 4.2 times greater than the discrimination thresholds (green triangles). This strong correlation indicates that the two experimental paradigms were able to probe the same underlying mechanism. This is highly encouraging as it suggests that (under some circumstances) the suprathreshold percept can be reliably measured and, conversely, that the task of discriminating between the luminance of two spatially separated patches is a meaningful way to probe luminance perception. Additionally, the work indicated that the effect known as crispening could be studied using threshold measures, which are generally considered to be more reliable than absolute or relative judgments of suprathreshold luminance.

_{min}*brighntess*,

*lightness*, or local

*brightness contrast*(all suprathreshold judgments) cannot be distinguished (Arend & Goldstein, 1987; Arend & Spehar, 1993; McCourt & Blakeslee, 2008). We shall use the term “brightness” to be consistent with Whittle, but we note that our interpretation is that, for simple stimuli such as that used by Whittle, suprathreshold judgments probe the same mechanism as threshold judgments and that the use of the term “brightness” could be interchanged with the term “lightness.”

*I*

^{–1}can provide a meaningful estimate of the perceived luminance for intervals greater than a single threshold (Fechner, 1966). Δ

*I*

^{–1}may refer to the detection threshold or the discrimination threshold. It is well established that integrating over detection thresholds produces a compressive nonlinearity (with no inflexion) and provides a reasonable estimate of the global perception of luminance under many circumstances (Fechner, 1966; Bartleson & Breneman, 1967). The theory has had considerable influence upon the design of response nonlinearities used in industry for the storing and transmission of visual content as it forms the basis of the CIE color space (Fairchild, 2013) and the recently developed PQ curve (Miller, Nezamabadi, & Daly, 2013) for high dynamic range content. Clearly then, detection thresholds have proven useful to predict luminance perception under some circumstances, but as described above, luminance discrimination thresholds are more appropriate for the stimulus configurations investigated by Whittle (1986, 1992). As such, a model that can account for both types of phenomena would be able to unify the two sets of results, and this was the idea behind the work in Kane and Bertalmío (2017), which proposes a relationship between detection thresholds and discrimination thresholds as follows:

*I*is the discrimination threshold,

*e*and

*c*are constants to be fit to the data, and

*C*some function of the contrast between

*I*and

_{b}*I*.

_{r}*I*=

_{r}*I*, discrimination thresholds reduce to detection thresholds, but when

_{b}*I*≠

_{r}*I*, contrast gain decreases sensitivity and gives rise to the crispening effect. In the original model

_{b}*C*=

*I*; however, we can see in Figure 3 that, although it captures the basic effect, it does not produce a particularly accurate fit. The alternative is to consider nonlinear processing of luminance and, more specifically, that there is good reason to expect contrast to be computed differently depending on the polarity of the reference patch. To estimate the initial contrast function, we begin with a detection threshold function Δ

_{p}*I*derived from the psychophysical data by Blackwell (1946; see details in Appendix 2), which is used to estimate the initial model contrast as the integral of one over thresholds, as follows:

_{det}*s*is a normalization constant (the number of samples per troland required to ensure a result that is independent of the sampling rate). The contrast

*C*in the modified model is then

*c*

_{2}= 0 and

*e*

_{2}= 1, and we use this model from this point forth.

*G*:

*W*),

*G*could also predict the pattern of discrimination thresholds.

*I*=

*f*(

*I*), the model can be thought of as a Weber's term for detection thresholds multiplied by a gain term that models the contrast between

_{r}*I*and

_{b}*I*.

_{r}*h*allows the prediction of the non-Weber behavior of detection thresholds. We use this latter formulation throughout the paper, and we later note that it is critical to the modeling of data we present in the section “The background luminance.”

*R*to a single grating, where

*x*is the grating contrast (Albrecht & Hamilton, 1982) and

*s*the semisaturation constant. Because the numerator and the denominator are both driven by the same variable

*x*, this is a model of self-divisive gain, and the response

*R*will saturate with increasing contrast even without the presence of a masking stimulus. To predict Whittle's data, we substitute

*x*for

*C*, which represents some measure of contrast between the background and the reference stimulus; we also use different exponents in the denominator and the numerator, and add the constant

*β*to the denominator as follows:

*I*are proportional to the inverse of the derivative of the response:

*C*in Figure 5. If

*C*=

*I*, the model produces the black curves and cannot accurately model thresholds. However, if we use the nonlinear formulation of contrast described previously in Equation 6, the model produces accurate predictions. Thus, as with the model of Kane and Bertalmío, an accurate modeling of discrimination thresholds requires a polarity-dependent input; for negative/dark pedestals, the function must be expansive, and for positive/bright pedestals, it must be compressive.

_{p}*c*

_{1}controls the degree of contrast gain. For the achromatic condition, the best-fitting parameter is

*c*

_{1}= 0.08, and for the equivalent luminance yellow–green condition, the parameter is much lower at

*c*

_{1}= 0.02. Recall from Equation 7 that at

*c*

_{1}=

*c*

_{2}= 0 the discrimination thresholds reduce to the detection thresholds. An analogous pattern of behavior is noted in the model of Kingdom and Moulden (1991). If we return to Equation 9, we can see that, as the exponent

*n*decreases, the impact of the contrast term is reduced, leaving only a Weber function of luminance. In keeping with this, the best-fitting values are

*n*= 0.48 for the achromatic condition and

*n*= 0.17 for the yellow–green condition.

*m = n*= 1. Doing so considerably reduces the complexity of the resulting threshold model, as follows:

*β*is found in the denominator; when this is set to zero, then the denominator reduces to a divisive constant. The best-fitting values are

*β*= 0.00075 for the achromatic condition and

*β*= 0.00040 for the yellow–green condition.

*I*= [10

_{b}^{1.35}, 10

^{2.35}, 10

^{3.35}, 10

^{4.35}]). For each condition, Whittle probed a very wide range of

*I*. We note that the lowest level of

_{r}*I*tested was constrained by optical scatter, which, in effect, limits the lowest luminance level that can be seen to some fraction of the background luminance level. We take this into account in all modeling, and the rationale for doing so is described in Appendix 2.

_{r}*I*against

*I*for the four models. In B, D, F, and H, we plot Δ

_{r}*I*/

*I*against

_{b}*I*/

_{r}*I*. The latter plots normalize the functions such that they superimpose. The plots emphasize that the functions are more or less identical for positive pedestals but substantially heterogeneous for negative pedestals. This heterogeneity for negative pedestals cannot be captured by the (unmodified) model of Whittle. The other three models can capture the heterogeneity. In the case of our model and the divisive gain model, this is captured because the function

_{b}*C*is dependent on the underlying detection functions, which are heterogeneous as a function of

*I*. In the case of the Kingdom and Moulden (1991) model, the parameter

_{b}*h*can play the same role if it is allowed to vary with the background condition. The best-fitting parameters for each model are shown in Table A1. Both the models of Whittle (1986) and Kingdom and Moulden report zero thresholds at zero pedestals; this feature can easily be addressed by the addition of a constant as Whittle (1992) proposes. In our model, this feature is addressed by the incorporation of the detection function.

*I*, he excluded a number of data points near the background luminance level; however, these data points were included when he plotted his data as a function of

_{r}*I*. This reveals the dipper effect, which is an increase in thresholds as the pedestal approaches zero, and this data is replotted in Figure 8. We plot the data for both positive/bright and negative/dark pedestals together.

_{p}*m*>

*n*. The requirement for unequal exponents means the more complex derivative must be used (Equation 13 rather than Equation 15). The finding that

*m*must be greater than

*n*is consistent with previous research that probed the dipper effect using contrast discrimination thresholds (Watson & Solomon, 1997).

*e*

_{2}+

*c*

_{2}

*C*), which can be thought of as a gain term that decreases thresholds near the background.

*I*for positive/bright pedestals, we illustrate this first term with a plateau at one when

_{p}*I*>

_{r}*I*.) The second component of each model can be considered a contrast term that increases monotonically with

_{b}*I*. Those models that can capture the dipper effect include a third term that monotonically decreases with

_{p}*I*. It is worth recalling at this stage that this third term is only required for the divisive gain model if

_{p}*m*≠

*n*.

*h*is novel as is the finding that this model can predict brightness functions without obvious crispening by manipulating the parameter

*n*.

*I*) for a nonlinear, polarity-dependent input based on the underlying pattern of detection thresholds. The requirement for this polarity-dependent input has been obscured from many previous studies of the dipper effect because the majority of studies use a contrast pedestal rather than a luminance pedestal. When a contrast pedestal is used, the resulting contrast must be the average of the two polarities (McIlhagga & Peterson, 2006), thus the impact of polarity is hidden.

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*Color science*(vol. 8)*I*were from 0 to 100,000 trolands without considering optical scatter. Whittle modeled optical scatter as follows:

_{r}*s*= 0.035. This equation means that the lowest illuminance levels presented to the retina depends on the background luminance level and are 0.88, 7.93, 78.53, and 783.65. Without considering the impact of optical scatter, none of the models can capture how thresholds plateau at very low reference luminance levels. This is shown in Figure A1 for Whittle's model. As such, in all the models tested in this article, we use the scatter-corrected values

*I*=

_{r}*I*

_{r}_{(}

_{retina}_{)}.

^{–2}. To convert to retinal illuminance (trolands), one needs a model of the pupil size. We use the model of Moon and Spencer (1944):

^{–2}. The central field was designed to be maximally bright, and the brightness in the surround fell off slowly. A number of features of the data set make it appropriate for our use. First, the stimuli used were broadband as in Whittle's study. Second, various stimulus sizes were used (0.59, 3.60, 9.68, 18.20, 55.20, 121, and 360 arcmin). We use the patch size of 59 arcmin to model the data in Whittle's discrimination study, which used a stimulus size of 55.8 arcmin, and we use the 121 arcmin data to model the data from Whittle's “brightness” study, which used a stimulus size of either 126 or 150 arcmin. Third, the study used an effectively infinite response time, and Whittle used a stimulus duration of 200 ms for the discrimination study because he found that the pattern of thresholds did not change greatly for durations greater than this (see figure 6 of the original Whittle, 1986 publication), and the perceived luminance magnitude function used an infinite stimulus duration. Each observer was trained and was fully adapted to each background luminance level prior to testing. On each run, the luminance values were checked with a photometer.

*I*/

*I*), but at low luminance levels, the Weber fraction rises as thresholds enter the square root region, and this transition occurs earlier when small test patches are used.

_{b}