**Abstract**:

**Abstract**
In transparency perception the visual system assigns transmission-related attributes to transparent layers. Based on a filter model of perceptual transparency we investigate to what extent these attributes remain constant across changes of background and illumination. On a computational level, we used computer simulations to test how constant the parameters of the filter model remain under realistic changes in background reflectances and illumination and found almost complete constancy. This contrasts with systematic deviations from constancy found in cross-context matches of transparent filters. We show that these deviations are of a very regular nature and can be understood as a compromise between a proximal match of the mean stimulus color and complete constancy as predicted by the filter model.

*m*(

*λ*), 0 ≤

*m*(

*λ*) ≤ 1, the filter thickness

*x*> 0, and the refractive index

*n*. The Bouguer-Beer law

*I*

_{1}/

*I*

_{0}=

*θ*(

*λ*) = exp[–

*m*(

*λ*)

*x*] describes how the inner transmittance

*θ*(

*λ*) (the ratio of the amount

*I*

_{1}of light reaching the bottom of the filter to the amount

*I*

_{0}entering at the top) depends on absorption and thickness. Fresnel's equations describe how the relative amount

*k*of light that is specularly reflected at each air-filter interface depends on the angle of the incoming light and the refractive index. For normal incidence as assumed here,

*k*= (

*n*–1)

^{2}/(

*n*+1)

^{2}.

*r*(

*λ*) and total transmittance

*t*(

*λ*) (i.e., the relative amounts of light leaving the filter after multiple inner reflections at the illuminated side and the opposite side, respectively) can be given in closed form:

*r*(

*λ*) =

*k + k*(1–

*k*)

^{2}

*θ*

^{2}(

*λ*)/[1–

*k*

^{2}

*θ*

^{2}(

*λ*)], and

*t*(

*λ*) = (1–

*k*)

^{2}

*θ*(

*λ*)/[1–

*k*

^{2}

*θ*

^{2}(

*λ*)]. If the filter is placed in front of a background with reflectance

*a*(

*λ*), then the virtual reflectance

*p*(

*λ*) of the filter surface (i.e., the relative amount of incident light that is reflected from the filter area) can be written as: Given

*p*(

*λ*), the cone excitation

*P*,

_{i}*i*=

*L, M, S*can be computed in the usual way, that is,

*P*= ∫

_{i}*(*

_{λ}p*λ*)

*I*(

*λ*)

*R*(

_{i}*λ*)

*dλ*, where

*I*(

*λ*) is the illumination spectrum and

*R*(

_{i}*λ*) the sensitivity spectrum of cone class

*i*.

*A*and

*B*denote the color codes of the bipartite background region and

*P*and

*Q*the color codes of the same regions viewed through the filter (see equations 17 and 18 in Faul & Ekroll, 2002). In the following we will always assume that the color codes are cone excitations where the index indicates one of

*L, M,*or

*S*.

*A, B, P,*and

*Q*describe the input from which the other model parameters

*I*,

*τ*, and

*δ*must be inferred.

*I*is the color of the illumination. A comparison of the model equations with a simplified version of the image generation model suggests (see Faul & Ekroll, 2002, p. 1086) that the vector

*τ*is related to the squared total transmittance

*t*

^{2}(

*λ*) and that

*δ*is related to the direct reflection factor

*k*. This motivates the parameter restrictions 0 ≤

*τ*≤ 1, and

_{i}*δ*≥ 0. The remaining parameter

*μ*controls the relative amount of directly reflected light of first order that is reflected from the top surface of the filter to higher order contributions that traveled through the filter and are thus affected by its transmissive properties. Here it mainly has a technical meaning and is used to distinguish between a “full” model with

*μ*= 1 and a “reduced” model with

*μ*= 0, which we considered in Faul and Ekroll (2002, 2011).

*I*is determined from the input. But whatever method is used, it follows from Equations 2 and 3 that which shows that the

*transmittance factors τ*do not depend on the illumination. The computation of the factor

_{i}*δ*, in contrast, requires knowledge about the illumination: In the following, we will assume that

*μ*= 0 and that the illumination color is determined from the mean of the background colors within a framework, that is, a region of common illumination in the proximal stimulus (for the framework concept, see Gilchrist et al., 1999). As will be shown below, the latter would be a reliable strategy if the gray-world assumption (Buchsbaum, 1980; Land, 1977) is approximately true.

*A*and filter colors

*P*. They use the following equations: where the index

*i*refers to color channel

*i*. In this previous work we also showed how problems with singularities [sd(

*A*) = 0 in some channels

_{i}*i*] can be avoided and discussed pros and cons of this approach.

*τ*, which in the original model was interpreted as channel-wise transmittance, is transformed to the alternative filter parameters “hue” (

*H*), “saturation” (

*S*), and “overall transmittance” (

*V*), whereas the parameter

*δ*, which in the original model represents the amount of direct reflection, is transformed to a bounded “clarity” parameter (

*C*).

*m*(

*λ*), the refractive index

*r*, and the thickness

*t*, whereas the context is given by the spectrum of the illuminant and the reflectance spectra of the background colors. To get a realistic picture of the degree of constancy to expect under natural viewing conditions, we drew representative samples from all these physical values.

*τ*and

*δ*of the filter model were estimated by applying one of the robust estimation procedures (Case 3,

*f*) proposed in Faul and Ekroll (2011, p. 6f). From these values, the alternative parameters

_{L}*H*,

*S*,

*V*, and

*C*were also computed. The variance of the estimated parameter values across contexts was then used as a measure of constancy, with zero variance indicating complete constancy. An important variable that influences estimation accuracy is the number

*N*of background and filter color pairs that enter into the estimation procedure. We therefore also considered the degree of constancy as a function of

*N*. In all simulations reported below, the absorption spectra of the filter and the reflectance spectra of the backgrounds were randomly generated frequency limited spectra (see Faul & Ekroll, 2011; Wyszecki & Stiles, 1982) with limiting frequencies of 1/150 cycles/nm and 1/75 cycles/nm, respectively. The absorption spectra were rescaled to the range [0.1, 0.8] to introduce a bias for highly saturated filter colors, and the reflectance spectra were multiplied with a random factor between 0.1 and 1 to increase the variance in albedo. All spectra were sampled from 400 to 700 nm in steps of 5 nm.

*δ*(or, in the alternative parameterization, the clarity of the filter). A simple assumption often made in our earlier papers is that the arithmetic mean of the background colors is used to estimate the illumination color. To investigate the accuracy of this estimate, we computed 2 ≤

*N*≤ 20 background colors resulting from randomly generated reflectance spectra under a certain illuminant. It was then determined how far the chromaticity of the mean of

*N*background colors

*◯*deviated from the expected chromaticity

*x*that was directly computed from the spectrum of the illuminant. All computations were done in the CIE UCS 1976 (

*u*′,

*v*′)-chromaticity space (Wyszecki & Stiles, 1982, p.165).

*x*=

*◯*−

*x*in samples of 500 estimates. The illuminations were daylight spectra of different temperature, ranging from reddish (3000 K) over neutral (6500 K) to blueish (30,000 K). The distributions appeared unbiased, that is, roughly centered at zero. The two left panels illustrate the distribution for two special cases in detail and the right panel summarizes the complete results. The latter shows that the error measure (length of the major semi-axis of the 2

*σ*error ellipse) shrinks rapidly with

*N*and flattens out at approximately

*N*= 10 at a value of less than 0.01. Expressed in Δ$Eab*$ the mean deviation approaches a value of approximately 6. This is clearly above detection threshold, which was found by Stokes, Fairchild, and Berns (1992) to be approximately 2.3 in successive image comparisons, but nevertheless a rather small and hardly noticeable value.

*n*randomly chosen background patterns, (b) estimate the parameters of the filter model for each of the

*n*input patterns, and (c) compute the standard deviation of the individual estimates. Complete scene constancy would hold if the estimates were identical for different backgrounds, which implies a standard deviation of zero. The standard deviation is here used as a measure of constancy, because the true value of these perceptual variables is not known.

*t*(0.5, 1, 1.5) and three refractive indexes

*r*(1 = no refraction, 1.3 ≈ refraction of water, 1.6 ≈ refraction of glass). The backgrounds consisted of

*N*= 2 to

*N*= 10 regions, and the color of each region was determined from a frequency limited reflectance spectrum that was randomly selected from a set of 200 precomputed spectra. For each

*N*,

*n*= 50 random backgrounds were generated. The spectra of the 50 illuminants were randomly generated frequency limited spectra (

*ω*= 1/75 cycles/nm) that were normalized to have the same luminance. Thus, in total, 180×9×50×50 = 4,050,000 stimuli were computed. Figure 4 illustrates the range of illuminations, reflectances, and transmittances used in the simulations.

*τ*and its alternative representation in terms of hue

*H*, saturation

*S*, and overall transmittance

*V*, we not only determined the standard deviation of the estimates of

*H*,

*S*, and

*V*but also applied

*τ*estimates to a nominally white background color

*W*; that is, we computed

*Ŵ*=

*τW*and determined the standard deviation of the resulting color codes in a uniform color space (CIE UCS, 1976). The nominal range of the

*H*,

*S*,

*V*, and

*C*parameters is the interval [0, 1] and a standard deviation of 0.1 of parameter estimates may thus be interpreted as an error of about 10 percent. It should be kept in mind, however, that the hue scale is perceptually nonuniform, that the range of saturation values for a given hue is often restricted to values smaller than one, and that the clarity scale starts at values greater than zero (see Faul & Ekroll, 2011).

*t*= 1 and a refractive index of

*r*= 1.3 in detail. The first and second columns show the distribution of the deviation of

*Ŵ*around the mean estimate in the chromaticity space and the corresponding 2

*σ*-error ellipse. The scatter plots shown in the second column enable one to see the total range of values found in the simulation, whereas the contour plot of the 2D histograms shown in the first column is better suited for judging the form of the distribution. The contour lines in the latter plots mark 1/11 to 10/11 of the histogram height and the 1/11 height contour is very close to the error ellipse. Figure 6 shows, in a more condensed form, how the variability of the parameter estimates depends on the thickness and the refractive index of the simulated optical filter. The results indicate a slight increase in variability with increasing values in both the thickness and the refractive index. The left panel in Figure 7 shows that the variability of the parameter estimates decreases rapidly with increasing numbers

*N*of background regions used in the estimation procedure.

*n*different illuminations, (b) estimate the parameters of the filter model for each input, and (c) compute the standard deviation of the parameter estimates. Complete constancy under illumination changes holds if this deviation is zero. All conditions realized in the simulation were identical to those used in the previous section.

*N*of background regions used in the estimation procedure.

*t*and a refractive index

*r*of 1. The variability of the estimates increases (but is still low) for larger values of

*r*and

*t*. Increasing

*r*leads to an increase in the amount of direct reflection at the top surface of the filter, which solely depends on the illumination. Thus, inaccuracies in the illumination estimation increasingly add to the variability of the estimated filter properties, especially that of the clarity parameter. The observed effects of changes in filter thickness

*t*can be understood from the fact that the hue and the saturation parameter depend only on the ratio of the mean color in the background and the mean color in the filter region of the stimulus (see Faul & Ekroll, 2011, figure 3). For very thin filters (i.e., high transmittance values) these two means are (whatever the background colors) very similar and thus lead to almost identical estimates. With thicker filters, the two means get increasingly different and thus inaccuracies of the model in describing the color changes caused by optical filters have a more pronounced influence on estimates of the hue and saturation parameter. This analysis suggests that the variability of the parameter estimates under changes in background reflectance depends mainly on the variability of the mean color in the background and the filter region of the stimulus. This also explains why the constancy rapidly increases with increasing numbers of background colors.

*t*and especially in

*r*are here less pronounced. If the model describes the changes in background colors caused by optical filters perfectly, then according to Equation 4 changes in illumination would have no effect on the transmittance parameter

*τ*and the alternative parameters

*H*,

*S*, and

*V*. The increase in parameter variability with increasing thickness thus indicates that the accuracy of the model prediction is worse for thicker filters. This is plausible, because due to the exponential decrease in transmittance with filter thickness the transmittance spectrum does not only have a smaller maximum but gets also increasingly more wavelength selective (

*spectral sharpening*) and is therefore more susceptible to changes in the illumination spectrum. This effect is especially pronounced for the highly saturated (i.e., highly wavelength-selective) filters used in the simulation. These errors are more of a systematic nature and do not decrease with increases in the number of background regions.

*scene constancy*) and illumination (

*illumination constancy*) in six experiments.

*ω*= 1/100 cycles/nm were randomly generated and afterwards multiplied with a random factor between 0.275 and 0.725 to increase the range in albedo. After applying one of the illuminants, the cone excitations were computed and slightly adjusted (by shifting all color coordinates by a color vector) to guarantee that the mean chromaticity was exactly identical to that of the illuminant. Finally, all colors were scaled such that the mean of (L + M) was 12 (which corresponds roughly to a luminance of 12 cd/m

^{2}).

*H*= 0.2, 0.64, 0.98) and clarity (

*C*= 0.5, 0.75, 1) parameters. The transmittance parameter was always set to

*V*= 0.75 and the saturation was set to 35% of the maximally possible saturation at the chosen hue on a background with gray mean (see Figure 11). The filtered colors in the standard stimulus were computed by applying the filter model with the chosen parameters to the background colors. We repeated this procedure if necessary until all colors in the standard stimulus and the colors in the comparison stimulus predicted by the model under the assumption of complete constancy were realizable on the CRT monitor used in the experiment.

*H*,

*S*,

*V*, and

*C*parameters of the filter model. Since the subjects' view was restricted by the mirror stereoscope they had to press the response keys without visual control. To facilitate this, we restricted the number of keys to the four arrow keys (to change the values of either

*H*and

*S*or

*V*and

*C*up and down), the space key to switch between both input modes, and the return key. Feedback on the currently active input mode was given on screen. The starting value for the

*V*parameter was always chosen to be 0.75 (the expected value under complete constancy), whereas all other parameters were given random values within their admissible range. The subjects were instructed to first try to match the filter by only adjusting

*H*,

*S*, and

*C*and to use the adjustment of

*V*only as a last resort. This instruction was given because, in the simple stimuli used in the experiment, there are cases where the perceptual effect of a change in

*C*can partly be compensated by a change in

*V*, and vice versa. A decrease in

*C*, for instance, which increases the contribution of the additive term (direct reflection) and thus the overall brightness in the filter region, can partly be compensated by a decrease in

*V*, which leads to a lower overall transmittance. Focusing on

*C*first helps to avoid possible confusions during the setting procedure, but may lead to distortions in the balance of

*V*and

*C*in the aforementioned cases.

*H*are only noticeable for saturations

*S*> 0 and a second one that lowering the clarity parameter

*C*decreases the maximally possible value for

*S*. Due to the latter restriction it may be necessary to first decrease

*S*to a lower value in order to be able to reduce the value of

*C*.

*V*parameter is very small and of comparable size in all experiments. Actually, adjustment of the

*V*coordinate was almost never needed to achieve a match and in the rare cases where the starting value of

*V*= 0.75 was changed at all, it was adjusted only very slightly. We may therefore conclude that the context changes used in the experiment have almost no effect on perceived transmittance (but see the remarks at the end of the next section). This result is in line with the findings of Gerbino et al. (1990) in the achromatic domain.

*C*parameter are also rather similar across experiments. That the errors in the control condition are somewhat smaller than in the other conditions is to be expected because, in that case, the subject just had to reproduce an identical stimulus, whereas a more difficult abstract match was required in all other conditions. The largest deviations are found in conditions with illumination changes and simultaneous presentation mode. But even in these cases the mean error is relatively moderate and never exceeds 0.08, which is three times the error made in the control condition and about 1/3 of the difference between two consecutive clarity steps realized in the experiment (see Figure 11). A closer inspection of the deviations for

*C*= 0.5 and

*C*= 0.75 revealed an almost symmetric distribution of the settings around the predicted value. This suggests that the observed deviations are best interpreted as random errors. We may thus conclude that the subjects were, in general, well able to isolate and match the clarity aspect of the filters across different contexts. To test for possible erroneous compensations of mismatches in

*V*by adjusting

*C*that may result from the instruction to first adjust

*C*(see methods section), we visually checked the transmittance part of the mean matches in isolation, that is we set always

*C*= 1 in standard and match instead of the actual values. In these partial matches a potential crosstalk between the

*C*and

*V*parameters would manifest itself in a noticeable mismatch between the overall transmittance in the standard and the filter match. There were indeed such cases, especially in the

*C*= 0.5 condition, and the mean relative adjustment of

*V*needed to reestablish a match was approximately 8%. Thus, parts of the errors found in the settings of the

*C*parameter can be attributed to this problem.

*H*and

*S*parameters. Here, the error is small in both the CON and SC conditions, but huge in comparison in all conditions involving illumination changes. The latter result is surprising, given that the transmittance parameter of the filter model is illumination invariant (see Equation 4) and that a high degree of constancy under illumination changes was found in the simulation study (see Figure 8).

*A*(component-wise) with the transmittance vector

_{i}*τ*, that is, the transformed color

*Â*is given by

_{i}*τA*. In the following, we will call these values

_{i}*raw filtered colors*. The chromaticity of the mean of

*Â*is shifted away from the mean of

_{i}*A*and this shift depends in a unique way only on the hue

_{i}*H*and saturation

*S*parameter of the filter. The position of the mean raw filtered color of the standard stimulus under the reddish, greenish, and blueish filters are shown in the diagram as small red, green, and blue crosses, whereas the positions of the mean raw filtered colors in the comparison stimulus predicted for the same filters are marked by large crosses. Thus, under complete constancy the subjects' settings should coincide with the large crosses in each diagram. In most cases, the actual settings shown as colored symbols in the diagrams deviate systematically from this prediction and these deviations increase slightly with a decreasing

*C*-parameter of the filter. These deviations are obviously of a very regular nature: In each case the mean of the filtered colors derived from the settings can be described as

*a compromise between the predicted mean color under complete constancy and the mean color presented in the standard stimulus*. Figure 15 demonstrates the meaning of this compromise on the stimulus level.

*D*–

*S*‖/‖

*P*–

*S*‖ can be used (Leibowitz, 1956), where

*D*denotes the actual setting,

*P*the predicted setting under complete constancy,

*S*the predicted setting under a proximal match, and ‖

*x*–

*y*‖ the distance between

*x*and

*y*in the uv-chromaticity space. The index is one if the subject's setting coincides with the prediction under complete constancy and zero for an exact proximal match.

*proximal*and a

*constancy*match actually predicts the subjects' settings. To this end we first specify concrete mixture models that are then fitted to our data and compared with respect to the size of the residuals.

*A*,

_{j}*P*) and (

_{j}*B*,

_{j}*Q*) denote the background and raw filtered color codes in the standard and the comparison stimuli, respectively, then this core part of the model simply states that for the

_{j}*j*th color pair, it holds that

*P*=

_{ji}*τ*

_{0i}

*A*and

_{ji}*Q*=

_{ji}*τ*

_{1i}

*B*, with

_{ji}*i*=

*L,M,S*. The proximal match criterion is achieved if the mean color of the filter region is identical in both stimuli (

*Q¯*=

*P¯*, where

*Y¯*stands for mean(Y)), whereas the constancy match criterion requires that the transmittance vectors are identical (

*τˆ*

_{1}=

*τ*

_{0}).

*X*of the colors

*Q*in the filter region of the comparison stimulus—denoted

_{j}*X*and

_{P}*X*for the proximal and the constancy criterion, respectively—and because

_{C}*τ*=

_{i}*Q¯*/

_{i}*B¯*, also a specific transmittance vector. The simple mixture model we want to propose states that the compromise is found by computing a weighted average

_{i}*Y*between

*X*and

_{P}*X*and that this value is used to compute a corresponding transmittance vector

_{C}*τˆ*

_{1i}=

*Y*/

_{i}*B¯*. To get an idea how the mixture is actually done, it is helpful to have a look at our data: The left panel in Figure 17 reproduces nine matches already shown in Figure 14. The small and large crosses show the chromaticity of the mean raw filtered color in standard and comparison stimuli which correspond to the above defined colors

_{i}*X*and

_{P}*X*, respectively. The filled symbols are the actual settings of the subjects (for red, green, and blue filters with different values in the clarity parameter). The interesting aspect in the present context is that the subjects' settings seem to deviate systematically from the straight dashed lines in the plot, which correspond to the chromaticities of a convex mixtures

_{C}*Y*= (1 –

*α*)

*X*+

_{P}*αX*, 0 ≤

_{C}*α*≤ 1, between

*X*and

_{P}*X*, and to follow instead more closely the curved dashed lines. These curved lines have the same meaning as in Figure 14, that is, they show how the means of the raw filtered colors change under variations in the color temperature of the illumination. The blue curve corresponds approximately to the daylight locus. If the data are replotted in a log rb-MacLeod-Boynton chromaticity space (see right panel of Figure 17), these curved lines are approximately straight. This observation indicates that the mixture is done in a log space, that is, that

_{C}*Y*= exp[(1 –

*α*)ln(

*X*) +

_{P}*α*ln(

*X*)]. In the following we will use the names Mix and LogMix to refer to these two models.

_{C}*α*that determines the relative weight of

*X*and

_{P}*X*, where a value of

_{C}*α*= 0 indicates a proximal match (

*Y = X*) and a value of

_{P}*α*= 1 a constancy match (

*Y*=

*X*). If the match is actually a compromise between these two extreme cases and if one of the mixture models describes this compromise correctly then it should be possible to reproduce the subjects' settings (up to random errors) by choosing an appropriate mixture weight

_{C}*α*. Furthermore, if one considers the complete set of 27 settings made in each experiment, there are again two extreme cases, namely, on the one hand, that a good approximation requires a different

*α*for each setting or, on the other hand, that a single

*α*suffices to described all settings. Intermediate cases, where certain subsets of data share a common

*α*are also possible.

*α*(one for each filter hue) are allowed instead of 27. Although the increase in the variance of the distribution is clearly noticeable, the prediction is still rather good.

*α*resulting from the fits of the LogMix model with 27 and three different values (the residuals of these fits are shown in the two panels on the right side of Figure 18). These

*α*values can be interpreted as constancy indices and as such pose an alternative to the Brunswick ratios used in Figure 16.

*α*in each setting. This suggests that these residuals are best considered to reflect random errors. A comparison between the residuals for different groupings indicates that only filter hue has a strong systematic influence on the size of the mixture weight

*α*, whereas filter clarity has only a small and illumination color barely any effect. Figure 19 suggests that in our experiments mainly the blue filter differed from the green and red one with respect to the degree of constancy.

*X*in the section called Predicting asymmetric matching results) is taken as the real object and the mean color in the filter region (denoted

_{C}*X*) as the proximal information. The finding that the subjects' settings always lay on a simple connecting curve between

_{P}*X*and

_{P}*X*supports this interpretation.

_{C}*D*–

*S*‖/‖

*P*–

*S*‖. A first problematic point is that the empirical value is determined on a nominal scale and not on the (unknown) perceptual scale, which is the actually relevant one. If the relation between the nominal and the perceptual scale is nonlinear, then equal constancy indices computed on the nominal scale are in general different on the perceptual scale and vice versa. The larger the variation in ||

*P*–

*S*|| the more pronounced the effect of a nonlinearity would be and the larger the observed variance in the constancy index. A second, more fundamental problem may be that the distortion of the phenomenal impression in the direction of the true object is essentially constant in absolute terms and not—as is implicitly assumed in the construction of the constancy index—in relative terms. This was, for instance, roughly the case in the experiment on form constancy conducted by Thouless (1931). As a consequence, the constancy index as defined above would decrease with an increasing distance between

*P*and

*S*. Thus, if one assumes that the phenomenal distance between

*P*and

*S*(i.e., between

*X*and

_{P}*X*as defined in the section called Predicting asymmetric matching results) was different across the 27 conditions, then the two mentioned factors could at least partly explain the observed variation in the constancy index. This assumption does not seem unreasonable, given that differences in filter hue explain a large portion of the constancy index variance in our experiment. However, due to the unknown metric of color space, it is difficult to explicitly test this hypothesis based on our data.

_{C}*X*in our notation) and the no scission predictions (corresponding to our

_{C}*X*) are compared with the actual settings reveals a pattern that is roughly compatible with our findings: The actual settings are also in most cases intermediate between

_{P}*X*and

_{C}*X*. Their results seem slightly less regular, though. This may be due to the fact that the filter model predicts perceived transparency better than the physical model (see Faul & Ekroll, 2011, Experiment 1). Thus, the predictions based on the filter model that we used may be more reliable than the ones based on the physical model that were used by Khang and Zaidi (2002a).

_{P}*g*(

*X*,

*Y*,…). Arend and Reeves (1986), for instance, found much lower levels of constancy in a task that requires to match the local colors (both targets “have the same hue and saturation”) than in a more abstract task that referred to scene interpretation (both targets “are cut from the same piece of paper”). Craven and Foster (1992) report that subjects can quickly and with high reliability decide whether color changes between two scenes are due to a change in illumination or due to a change in reflectance. In this task, which mainly refers to scene interpretation, also high levels of constancy have been found (Reeves et al., 2008).

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*α*value was as follows: (a) Compute

*X*and

_{P}*X*from the known background colors and

_{C}*τ*

_{0}, the transmittance factor, computed from the standard stimulus. (b) Compute an intermediate color

*Y*=

*f*(

*α*,

*X*,

_{P}*X*), where

_{C}*f*is one of the mixture models. (c) Compute an estimate

*τˆ*

_{1}of the transmittance factor with

*τˆ*

_{1i}=

*Y*/

_{i}*B¯*. If max(

_{i}*τˆ*) > 1, which would violate a constraint on

_{i}*τ*, divide

*τˆ*by max(

*τˆ*). (d) Compute the predicted filter color using

_{i}*Fˆ*=

_{ji}*τˆ*

_{1i}(

*B¯*+

_{ji}*δ*

_{0}

*B¯*), with

_{i}*δ*

_{0}= 1/

*C*

_{0}– 1 where

*C*

_{0}is the clarity parameter used in the standard stimulus. Note that this is the reduced filter model including the direct reflection component. (e) Compute the loss function

*Err*(

*α*) = Σ

*(*

_{ji}*Fˆ*−

_{ji}*F*)

_{ji}^{2}, where

*Fˆ*is the predicted color in the filter region of the comparison stimulus and

_{j}*F*the corresponding mean color set by the subjects. To compensate for different scales in the three color channels, both

_{j}*Fˆ*and

*F*were first component-wise divided by the relative cone excitations values of an equal energy white.