Transparent layer constancy improves with increased naturalness of the scene

The extent to which hue, saturation, and transmittance of thin light-transmitting layers are perceived as constant when the illumination changes ( transparent layer constancy , TLC) has previously been investigated with simple stimuli in asymmetric matching tasks. In this task, a target filter is presented under one illumination and a second filter is matched under a second illumination. Although two different illuminations are applied in the stimulus generation, there is no guarantee that the stimulus will be interpreted appropriately by the visual system. In previous work, we found a higher degree of TLC when both illuminations were presented alternately than when they were presented simultaneously, which could be explained, for example, by an increased plausibility of an illumination change. In this work, we test whether TLC can also be increased in simultaneous presentation when the filter ’ s belonging to a particular illumination context is made more likely by additional cues. To this end, we presented filters in differently lit areas of complex, naturalistically rendered 3D scenes containing different types of cues to the prevailing illumination, such as scene geometry, object shading, and cast shadows. We found higher degrees of TLC in such complex scenes than in colorimetrically similar simple 2D color mosaics, which is consistent with the results of similar studies in the area of color constancy. To test which of the illumination cues available in the scenes are actually used, the different types of cues were successively removed from the natu-ralistically rendered complex scene. A total of eight levels of scene complexity were examined. As expected, TLC decreased the more cues were removed. Object shading and illumination gradients due to shadow cast were both found to have a positive effect on TLC. A second filter had a small positive effect on TLC when added in strongly reduced scenes, but not in the complex scenes that already provide many cues about the illumination context of the filter.


Introduction
In previous work on the perception of thin colored transparent layers mostly limited constancy has been observed under context variations (Faul & Ekroll, 2012;Falkenberg & Faul, 2019;Falkenberg & Faul, 2021;Faul & Falkenberg, 2015).In contrast, Khang and Zaidi (2002a) found almost complete constancy in performance-based identification of optical filters.However, the findings of Faul and Falkenberg (2015) suggest that this apparently high degree of constancy is due to the choice of distractors that were too clearly different from a filter with constant properties.So far several factors have already been identified that may increase transparent layer constancy, e. g. moving filters, stereo disparity of filters, and regular structures of the background pattern, i. e., cues that promote a decomposition of the stimulus into a transparent layer and a background.However, the effects are rather moderate and transparent layer constancy (TLC) remained below 50 % even when these cues were used simultaneously (Falkenberg & Faul, 2019).A possible explanation for this limited TLC lies in the nature of the stimuli used, which mostly consist of a small number of homogeneous background colors that are partially overlapped by the transparent layer.Since Faul andEkroll (2002, 2011) presented a psychophysical model of chromatic perceptual transparency according to which, in principle, the perceptual properties of a filter can be computed very reliably from relatively few uncovered and covered colors given in the image, it did not seem necessary to use more complex scenes.However, there are numerous studies from the related field of color constancy that observed improved constancy performance with more complex and natural stimuli (e. g., Brainard, Brunt, & Speigle, 1997;De Almeida, Fiadeiro, & Nascimento, 2004;Foster, Amano, & Nascimento, 2006;Kraft & Brainard, 1999;Nascimento, De Almeida, Fiadeiro, & Foster, 2005;Radonjić, Cottaris, & Brainard, 2015;Xiao, Hurst, MacIntyre, & Brainard, 2012; for a review, see e. g.Falkenberg & Faul, 2021).However, the underlying mechanisms and the effects of presenting specific stimulus features at the image level that are potentially exploitable have not yet been fully elucidated.
In Falkenberg and Faul (2021), we made a first attempt to develop a taxonomy of relevant features in natural scenes based on the literature on color constancy, which could also contribute to a constant perception of transparent layers.At one end of the taxonomy, we located scene features that have been examined primarily in studies that consider scene complexity from a "low-level" stimulus-centered perspective, such as the number of background colors or scene statistics such as the chromatic mean or variance of colors.At the other end of the taxonomy we located scene features such as object shading, shadow cast, texture gradients, or perspective convergence, which seem relevant from a more scene-centered perspective that emphasizes the "high-level" interpretation of the scene, for example, in terms of prevailing illumination or three-dimensional surface orientation.In Falkenberg and Faul (2021) we started by investigating TLC with highly reduced stimuli and scene features that belong to the "low-level" end of this taxonomy.We observed that an increased number of randomly drawn background colors led to higher degrees of TLC.However, the positive effect of numerosity could be nullified by keeping the chromatic mean of the stimuli constant.In essence, the degree of TLC was determined by the chromatic mean of the background colors: The more the chromatic mean of the test stimulus differed from the neutral mean of the standard stimulus, the lower the TLC, regardless of the number of background colors.This means that the effect of increasing the sample of randomly drawn reflectances is only that it leads to an improved estimate of the actual chromatic mean.As a possible explanation for these findings, we discussed that in such simple scenes the chromatic mean of the colors may be taken as the illumination color, although it may deviate significantly from the actual illumination color used in the stimulus generation.
To investigate illumination-invariant filter constancy, a target filter is shown in a first illumination context and matched to a filter in a simultaneously presented second illumination context ("asymmetric filter matching").However, it is unclear whether an observer experiences the two scenes as being under two different illuminations, as desired, or as one larger scene under a single, global, illuminant (Falkenberg & Faul, 2019).The alternating presentation of the two displays makes a single global illuminant less likely and does increase TLC compared to simultaneous presentation (Faul & Ekroll, 2012;Falkenberg & Faul, 2019).In fact, the perception of a non-veridical single global illumination in the case of simultaneous presentation could lead to a non-veridical estimation of the filter properties according to the filter by the model of Faul andEkroll (2002, 2011) that might explain the lower degrees of TLC (Falkenberg & Faul, 2019).
To investigate the impact of the perceived illumination on TLC, the present study starts at the scene-centered end of the suggested taxonomy and examines naturalistically rendered complex scenes.Complex threedimensional (3D) scenes naturally contain cues about the scene geometry, such as linear perspective, texture gradients, and occlusion, as well as cues about the prevailing illumination.It has been suggested that the visual system creates an internal representation of the position, intensity, and spectral composition of the prevailing illumination based on various cues, such as cast shadows, illumination gradients on curved surfaces, or specular highlights (e. g.Koenderink, Pont, Van Doorn, Kappers, & Todd, 2007;Xia, Pont, & Heynderickx, 2014).In this context, the illumination interpretation of a scene and thus the perceived color and brightness of objects interact strongly with the depth and orientation interpretation of surfaces (e. g.Gilchrist, 1977;Radonjić & Gilchrist, 2013;Lotto & Purves, 2000, 2002).
In the domain of color constancy, Radonjić et al. (2015) found significantly higher constancy in naturalistically rendered 3D scenes than in colorimetrically similar 2D color mosaics.An obvious assumption related to this finding is that in natural scenes the existing illumination change is more likely to be perceived as such, which then leads to a more constant perception of the object color.If the previously observed limited constancy of transparent layers is due to an ambiguous or incorrect illumination perception, then the achievable degree of TLC should also be higher in such naturalistically rendered scenes than in colorimetrically comparable reduced stimuli.If only the local pairs of covered and uncovered colors at the filter border were used for the estimation of the filter parameters, TLC should be independent of the perceived illumination context of the filter.But the higher TLC for alternating presentation indicated that this is not the case (e.g.Falkenberg & Faul, 2019).However, if the local color shifts at the filter border were kept constant, an additional improvement in the degree of TLC with an increasing number of illumination cues could then be attributed to the relevance of a correctly recognized illumination context.
The primary goal of the study was therefore to examine whether TLC improves in naturalistically rendered scenes in a similar way as Radonjić et al. (2015) found for the constancy of surface colors.For this purpose, we basically used the approach of Radonjić et al. (2015), i. e., complex naturalistically rendered scenes were compared with simple abstract stimuli.However, we have slightly extended and modified their method.First, the generation of the reduced stimuli (color mosaics) was modified to ensure better comparability with the naturalistically rendered scenes.Radonjić et al. (2015) generated the reduced stimulus with similar colors as the complex scene, but the color mosaic used had a very high texture density, so that the individual surface colors adjacent to the test field were no longer recognizable (cf.Fig. 5 in Radonjić et al., 2015).In our stimulus material, the texture density of the background colors was kept comparable across the two complexity levels, so that the relations between background colors and filtered background colors were similarly recognizable.Furthermore, the naturalistically rendered scenes were additionally enriched, e. g. by textured surfaces and additional objects, so that the recognition of the illumination change based on numerous different cues was even more likely.
Because it is still unclear exactly which stimulus features increase the perceptual plausibility of an illumination change, the second goal of this study was to identify those cues to the prevailing illumination in naturalistically rendered scene that actually contribute to the enhancement of TLC.

Method
In order to test the influence of scene complexity on transparent layer constancy (TLC), we simulated naturalistic 3D scenes (condition Complex) containing many cues for the scene geometry and the prevailing light sources, such as texture gradients, illumination gradients on object surfaces (object shading), and shadow cast.Second, for each complex scene, a colorimetrically comparable but maximally reduced scene was created (condition Minimal), which corresponds to the 2D mosaics of homogeneous color surfaces used so far (Fig. 1).In order to identify the illumination cues that are actually used by the visual system in the complex scenes, the available cues were successively removed.Between the two extremes of scene complexity (Complex and Minimal), four intermediate levels were created, which contain successively fewer cues to scene geometry and prevailing illumination.Furthermore, for two of the complexity levels the scenes were enriched with another filter that possessed the same physical properties as the target filter but was presented in a differently illuminated part of the scene.To determine the degree of TLC as a function of scene complexity, a matching task was conducted to measure the perceived properties of different target filters in all scene types.

Experimental design and stimulus generation
Initially, 3D scenes were modeled and rendered using the 3D modeling software Blender (Blender Foundation, 2019) (Fig. 1a).The floor and back wall of the scene were given a checkerboard pattern to support spatial interpretation.
In the foreground of the scene, a cube with rounded edges was positioned so that the two sides viewed frontally had the same angle to the viewer and occupied significantly larger areas in the image than the top of the cube.The dimensions of the cube object in the scene were 0.8 × 0.8 × 0.8 scene units.Three different chromatic textures and one grayscale texture were used for the surface of the cube, each consisted of eight colors or gray values in the latter case (Fig. 2a).The texture maps of the eight colors or gray values were arranged in such a way that two identical surface reflectances were continued at each of the three visible edges of the cube (Fig. 2b), with the result that the color shifts at the edges contain information about the illumination context.The filters were positioned in front of these textures so that all eight surface colors of the cube were partially occluded by the filter.Both the grayscale texture and the three chromatic surface textures of the cube had an achromatic mean.The chromatic cube textures (C1 -C3) were conceptually identical and contained eight hues from all directions of the chromaticity diagram at random saturation levels within the displayable range.For this purpose, the hues for each chromatic texture were drawn at equidistant directions around the white point of the u′v′ chromaticity diagram of the CIE 1976 UCS (L*u*v*) color space (Wyszecki & Stiles, 1982, p. 165f) with a different random starting angle between 0 • and 360 • .
The scene was provided with a basic illumination by an infinitely distant spherical emitter, which was set to homogeneous white (surface shader "Background" with parameters "Color" = RGB(1, 1, 1) and "Strength" = 0.4).In addition to the homogeneous achromatic background illumination, the scene was illuminated by two chromatic lights ("Area"-Lights with areas of 20 × 20 scene units), each of which vertically illuminated only one of the two sides of the cube.The shortest distance between the corresponding surface of the cube and the light surfaces was seven scene units.In this way, the left side of the cube was lit by a bluish illumination ("Emission" shader with a "Blackbody" parameter of 12,000 Kelvin, and a "Strength" of 15) and the right side of the cube with a reddish illumination (corresponding to a color temperature of 4,000 Kelvin and a "Strength" of 15).The filter was presented in one of these two illuminations.Furthermore, the background of the scene was enriched with two other objects, a torus and a sphere, which both had a neutral and homogeneous reflectance and thus provided further cues to the two illuminations through shading and shadowing.
All scenes were rendered as high dynamic range images using the Cycles renderer (Blender Foundation, 2019) with an image size of 720 × 600 pixels with 2048 samples/pixel at a color depth of 16 bits/channel.They were then converted to low dynamic range images (color depth 8 bits/channel) following the procedure described by Reinhard and Devlin (2005), to ensure that they are displayable on the monitor used (EIZO ColourEdge CG243W 24.1 inch, 1920 × 1200 pixels, 60 Hz refresh rate, driven by an NVIDIA Quadro 600 graphics card with a bit depth of 8 bits per channel.The emission spectra of the monitor were measured with the JETI specbos 1211 spectroradiometer and used in the monitor calibration).All images were adjusted with the same initial parameters of the tone mapping procedure (intensity = 2, light adjustment = 1, color adjustment = 0.1, contrast = 0.1) and with the same image-dependent parameters initially obtained for a high dynamic range stimulus.For this purpose, a scene similar to the one shown in Fig. 1a was rendered with a filter that had higher luminance values due to specular highlights (surface shader "Glass-BSDF").
The complex scene was simulated with each of the four different cube textures shown in Fig. 2 and the three filters were each presented in either the reddish or bluish illumination on the corresponding side of the cube.This resulted in a total of 24 filter presentations in the Complex condition, from which stimuli of lower complexity were generated by systematically removing cues from the complex scene (Fig. 3 and Fig. 4).
In the first step of the reduction from Complex to condition Reduced1 (Fig. 3b), the cue object shading of the cube was eliminated by using sharp instead of rounded cube edges.In the condition Reduced2 (Fig. 3c), the cube was presented with rounded edges as in Complex, i. e., the cue object shading of the cube was retained, but the cue background illumination gradients was removed, i. e., all illumination cues provided by the wall and floor textures, as well as by the additional objects and shadow cast in the background of the scene.To this end, the background was replaced by 60 × 60 pixel homogeneous color patches, each patch reflecting the mean value of the underlying pixels.Thus, the condition Reduced2 corresponds in its level of complexity to that of the stimuli of Radonjić et al. (2015).In the condition Reduced3 (Fig. 3d), the background was identical to Reduced2, but the cube object had sharp edges as in Reduced1, thus, compared to Reduced2 the cue of object shading on the cube was missing.In the condition Reduced4 (Fig. 4a), all that remains of the sharp-edged cube object is one of the frontal planes, but which still has slight color gradients due to the corporality of the filter.To create the background pattern, a corresponding scene with the cube texture without the filter was rendered.Then the technique already described was used to generate a grid texture in which each homogeneous patch of 60 × 60 pixels preserves the average scene color in the corresponding image region.
The generation of the most reduced scenes (condition Minimal, Fig. 4b) was carried out in three steps.(1) For the background, the simulated scene was rendered without the cube and again fields of 60 × 60 pixels were replaced by the average value of the respective pixel colors.(2) For the eight colors of the cube, an orthographic image of the left and right sides of the cube was rendered for all four textures under the corresponding illumination without any filter.For each of the eight surface colors of each cube texture, the RGB values of all pixels were read out via masks, converted to LMS coordinates (LMS color space according to Stockman, MacLeod, & Johnson, 1993), averaged in LMS color space, and converted back into RGB values.These eight colors were then used in the stimulus generation for homogeneously coloring the eight patches of the central square.(3) In a final step, homogeneous colors for the filter region were calculated.For this purpose, we used the psychophysical filter model of Faul andEkroll (2002, 2011) which is closely related to a physical filter situation.In the case of clear filters (i.e., filters without direct reflection) as used in this study, the relationship between the covered and uncovered colors in the image can be described by the perceptual parameter τ i , with i = L,M,S, which is directly related to the transmittance of the physical filter.Thus, the perceptual filter parameters τ i of the simulated filters in the complex scenes were estimated from the rendered images and then applied to the eight background colors to draw the filters.This was achieved by reading out the colors in the filter area pixel by pixel from the images of the complex scenes and also converting them to LMS values.The perceptual filter parameter τ i was then calculated from the colors of uncovered background patches calculated in (2) and the filtered colors read out in (3) using the method described in Faul and Ekroll (2011) via the ratio of the mean color in the filter area to the mean color of the uncovered background (see also Fig. 6 in the results section).The estimated τ values for each of the three filter hues red, green, and blue-the physical filter properties from the image generation-are as expected quite similar across the four background patterns and the two illuminations.However, the estimated filter parameters are not exactly the same for each hue due to context variations, which is especially true for the estimates of the red and blue filters under the different illuminations.Therefore the filter parameters were calculated separately for all 24 situations of filter presentation.In order to approximately match the perceived size of the frontally displayed filters in the Minimal condition to the perceptual filter size in the perspective presentation in the other conditions, the size of the filter diameter in the minimal scenes has been ad hoc set to be 3 % larger at the image level compared to the more complex scenes.
In the conditions Complex and Reduced4, the potential cues provided by an additional filter were also examined.The corresponding scenes ComplexFilt and Reduced4Filt are shown in Figs.4d and 4c.These two conditions were chosen as exemplary because they represent the most extreme conditions of the rendered scenes.In each case, this additional reference filter had the same material parameters as the target filter, because this should maximize a potential constancy-increasing effect of an additional filter.

Experimental procedure
Fig. 5 shows a screenshot of one of the stimuli used in the experiment.In the upper half of the screen, the target filter was presented in one of the eight types of scenes.In the lower half, the match filter was presented in front of an achromatic background consisting of overlapping ellipses of different sizes, orientations, and grayscales.Target scene and adjustable scenes each had a visual angle of 22 • in width and 18.4 • in height at an eye-monitor distance of approximately 50 cm.
For the assessment of the perceptual filter properties, we use the perceptual filter model of Faul andEkroll (2002, 2011), according to which the properties of transparent filters can be described by the perceptual dimensions of hue, saturation, total transmittance (i.e., value), and clarity.The task of the participants was to adjust the three filter parameters Hue (H), Saturation (S), and Total Transmittance (V) of the match filter using the arrow keys on the keyboard so that they corresponded to the perceived properties of the target filter (for the conversion routines between the τ i and HSV parameters, see Faul & Ekroll, 2011, Appendix B).To facilitate the matching, we used the "Uniform Parameter Space for Filter Transparency" (UniformHSVC) proposed in Faul (2017), because in this representation the parameters are approximately perceptually equidistant and largely independent from each other.In each trial, the parameters had random starting values.In four practice trials, the participants familiarized themselves with the navigation in the transparency space.The space key could be used to switch between the three parameters at any time and there was no time limit for the settings.Only clear filters were used and thus the clarity parameter (C) was set to 1 and kept constant.In the conditions ComplexFilt and Reduced4Filt where two filters are visible the target filter to be matched was indicated.
During the instruction, we illustrated the task by showing the participants real, flat, colored filtering objects made of glass and plastic.It was said that the scenes on the screen contain such filters and that they have uniform transmittance in the entire filter area, although they may locally appear different due to different background colors.The participants were instructed to judge and match the overall impression of the filters.The instruction aims for an abstract matching criterion, i. e., the properties of the filter (corresponding to a "paper-match" instruction in color constancy).
In each of the eight realized levels of scene complexity (Complex, Reduced1, Reduced2, Reduced3, Reduced4, Minimal as well as ComplexFilt and Reduced4Filt), 24 context combinations of illumination, background texture and filter color were assessed.These combinations consisted of the two illuminations (reddish and bluish), in which the filters were presented, the three filter colors (red, green, and blue), and the four cube textures (three chromatic and one achromatic texture).Each of the 24 context combinations was tested twice per participant in order to be able to check the validity of the settings, resulting in a total of 48 trials at each level of scene complexity.

Participants
A total of 14 individuals participated in the experiment, all of them were unaware of the experimental question.Six participants completed all eight complexity levels (384 trials) in six to seven sessions on different days, each session lasting between one and a half to two hours.Eight participants completed one of two sets each containing only four of the eight complexity levels (192 trials) in three to four sessions: Four of them completed the conditions Complex, ComplexFilt, Reduced3, and Minimal, whereas the other four completed the conditions Reduced1, Reduced2, Reduced4, and Reduced4Filt.In this way every complexity condition was completed by 10-partly different-participants, and because each participant completed the filter matching twice in every experimental condition, there are in total 20 measurements in each experimental condition.
All participants reported normal or corrected-to-normal visual acuity and had normal color vision according to a test with the Ishihara color plates (Ishihara, 1967).Participation was voluntary and written informed consent was obtained from the participants.This research is a minimal-risk psychological study that did not require institutional approval at the time it was conducted (09-2019/02-2020).However, all ethical principles formulated in the Declaration of Helsinki were strictly adhered to.

Results
The total of 3840 recorded filter settings were first checked for outliers.For this purpose, filter settings were identified, which fell in the u′v′ chromaticity diagram outside a two-dimensional 99% confidence ellipse around the true value 1 .This affected a total of 58 settings (1.5 %), which were excluded from further data analysis.After this exclusion, the mean retest reliability was calculated for each participant.Because the mean deviations were all below a threshold of Δu′v′ = 0.03, which is twice the mean deviation observed for a trained observer (cf.Falkenberg, 2022) and also for conscientious observers like participant 12 and 13, the retest reliability of all participants was considered acceptable (Fig.

A.1 in Appendix A).
To avoid different sample sizes per experimental condition after the elimination of outliers, the two measurements per person and condition were averaged, if available.Given the high retest reliability this seems unproblematic.Since for each person there was at most 1 outlier per condition, the final data set contains 10 measurements for each experimental condition, i. e., 1920 filter settings in total.
To evaluate the filter settings of H, S, and V made by the participants in the UniformHSVC space, they were converted into τ i parameters, with i = L, M, S, and compared to the τ parameters of the target filters that were estimated from the complex scenes (Fig. 6a).Transferred to the adjustment scene, a filter that has the identical filter parameters as the target filter is called the constancy filter according to our filter model (Fig. 6b).A filter with different parameters, but which leads to the same average color in the filter area of the adjustment scene as in the area of the target filter is referred to as the proximal filter (Fig. 6c).Typically, the match filters adjusted by the participants are located between those two criteria of equality on an abstract and a proximal level (Faul & Ekroll, 2012;Falkenberg & Faul, 2019, 2021).This pattern is also observed in the current filter settings (Fig. As a measure of the degree of TLC, the projected Brunswik ratio (BR ϕ ) was calculated by projecting each observed filter setting in u′v′ chromaticity space onto a straight line through the chromaticities of the constancy filter and the proximal filter, and then forming a ratio between the two equality criteria (i.e., equal filter parameters vs. equal mean color in filter area) for each filter setting (see Fig. 6d).
The projected Brunswik ratio typically takes values between 0 and 1:  The confidence interval was constructed using scatter ellipses whose semiaxes correspond in their orientation and length to the eigenvectors and eigenvalues of the covariance matrix of the u′ and v′ values of the observed filter settings, respectively.
BR ϕ = 1 corresponds to maximum filter constancy, although the projected Brunswik ratio can also take values greater than 1 if the filter setting lies beyond the constancy filter in a direction pointing from the proximal to the constancy filter.This is sometimes referred to as "overconstancy" (cf.Brainard, Cottaris, & Radonjić, 2018, p. 3), which can be interpreted as compensating for an illumination difference larger than the one actually present.BR ϕ = 0 occurs when the filter setting corresponds to the proximal filter.Values with BR ϕ < 0 can also occur, such settings then lie beyond the proximal filter in a direction pointing from the constancy filter to the proximal filter.The projected Brunswik ratio was determined for each of the 1920 observed filter settings (see Table 1 for the mean and standard deviation of BR ϕ in all eight complexity conditions).
As expected, we observed a lower Brunswik ratio of BR ϕ = 0.69 in the Minimal condition, while the filter settings in the Complex condition were much closer to the constancy prediction with a mean Brunswik ratio of BR ϕ = 0.83 (Fig. 7a).The difference in scene complexity in the two extreme conditions Minimal and Complex leads to a positive effect of increased scene complexity on filter constancy (t(478) = 4.37,p < .001,d = 0.40, one-sided t-test for independent samples).
There was a clear difference in the settings of the filters in front of the achromatic (BR ϕ = 0.55) versus the chromatic (BR ϕ = 0.85) cube textures: The filter constancy was significantly lower for the achromatic texture than for the chromatic textures (t(1918) = 15.07,p < .001,d = 0.79, two-sided t-test for independent samples).In addition, the variance in the filter settings was clearly lower in front of the achromatic texture than in front of the chromatic textures (Fig. 7b).However, the positive effect of increased scene complexity was found similarly for both the achromatic and the chromatic texture (Fig. 7c).
For all filter-illumination-combinations, the Complex condition numerically shows a higher degree of transparency constancy than the Minimal condition, but this difference is significant only for the red and green filters but not for the blue filter in both illuminations (Fig. 8).
A total of eight participants completed both extreme complexity levels Minimal and Complex and all but one show a tendency toward a positive effect of the increased scene complexity.However, the size of the effect of scene complexity varies and for some participants it is relatively weak and not statistically significant (Fig. 9).Participant 01 shows a particularly pronounced effect of scene complexity with, on average, complete filter constancy in the Complex condition.But this seems not essential, because the positive effect of scene complexity is slightly smaller but still significant (t(430) = 3.19, p < .001,d = 0.31, one-sided t test for independent samples) if this participant is excluded.To analyze the effects of each intermediate level of scene complexity on TLC, the eight complexity conditions were sorted by the observed degree of filter constancy across all filter-illumination-background combinations.In the complexity condition Minimal, the lowest degree of filter constancy was expected.If there is a monotonic relationship between the availability of illumination cues and the degree of filter constancy, then the resulting order of the complexity levels should reflect the number of cues used by the visual system, with the Minimal condition at the lowest position and the Complex or ComplexFilt condition at the highest.The relationship between scene complexity and filter constancy is broadly consistent with these expectations (Fig. 10a).This observed order remains more or less the same if the data is separately analyzed for the levels of the context factors illumination color, color of the cube texture, and filter color (Fig. 10b-10d).A 4-way ANOVA was performed to evaluate the main effects of the scene complexity and the three context factors.The results indicate a significant main effect for the complexity level of the scene, F(7,1908) = 4.65,p < .001,partialη 2 = 0.017; a significant main effect for the color of the cube texture, F(1, 1908) = 246.42,p< .001,partialη 2 = 0.114; a significant main effect for the color of the illumination, F(1, 1908) = 125.63,p < .001,partialη 2 = 0.062; a significant main effect for the color of the filter, F(2, 1908) = 8.06, p < .001,partialη 2 = 0.008.However, within the observed ranking of all eight complexity conditions, the degrees of constancy do not differ significantly for each adjacent pair of complexity conditions.In order to get specific information about the relevant cues that are used by the visual system in the filter matching task, we therefore analyzed the effects of the three cues (1) object shading of the cube, (2) illumination gradients in the background, and (3) an additional filter across the individual complexity level.
For this purpose, we combined the respective two complexity conditions in which the corresponding cue is present or absent (Fig. 11).
The relevance of the object shading of the cube was tested via a comparison of the edge types of the cube, i. e., round versus sharp edges.In the conditions Complex and Reduced2, which contained the cue object shading of the cube, slightly larger degrees of transparency constancy were shown (BR ϕ = 0.81) than in the Reduced1 and Reduced3 conditions, which did not contain this cue due to sharp edges (BR ϕ = 0.75).The positive effect of object shading on TLC was small but significant (t(958) = 2.34, p = .01,d = 0.15, one-sided t test for independent samples).
To test the effect of background illumination gradients on TLC, the conditions Complex and Reduced1, which contained corresponding illumination gradients due to cast shadows as well as additional objects, were compared with the conditions Reduced2 and Reduced3, in which the background was replaced by a simple mosaic of homogeneous colored patches.Filter constancy was slightly higher in the pooled conditions with illumination gradients in the background (BR ϕ = 0.80) than in the pooled conditions without illumination gradients in the background (BR ϕ = 0.76).This effect of the additional cue of illumination gradients in the background is small but statistically significant (t(958) = 1.80, p = .04,d = 0.12, one-sided t test for independent samples).The effect of an additional filter was examined in the naturalistically rendered condition Complex and in the Reduced4 condition within subjects.When the two complexity conditions with additional filters are combined, TLC is slightly higher in the conditions with filter (Reduc-ed4Filt and ComplexFilt: BR ϕ = 0.81) than in the identical conditions projected Brunswik ratio BR  without filter (Reduced4 and Complex: BR ϕ = 0.79), but the difference is not significant (t(479) = 1.47, p = .07,one-sided t test for dependent samples).In fact, in the naturalistically rendered scenes, the degree of filter constancy remained equally high regardless of whether an additional filter was presented (ComplexFilt) or not (Complex).In contrast, in the very reduced scenes of condition Reduced4 an additional filter led to a small but significant improvement of filter constancy (t(239) = 1.92, p = .03,d = 0.12, one-sided t test for dependent samples) (Fig. 11c).

Discussion
The aim of the present study was to investigate the influence of factors facilitating a correct recognition of an illumination change on transparent layer constancy (TLC).To this end, we started from a naturalistically rendered scene, which presumably contains a large number of illumination cues and then systematically reduced the number of cues.This resulted in eight scene types of different complexity levels.Despite the differences in the available cues to the illumination context of the filters, the local color shifts at the filter border were kept constant in all complexity conditions.If the perceived illumination context was relevant for the veridical perception of the filter properties, TLC should increase with the availability of illumination cues.
The first key finding of this study is that TLC is clearly better in complex naturalistically rendered scenes than in simple abstract color mosaics of homogeneous color patches that we used in previous work (Falkenberg & Faul, 2019, 2021;Faul & Falkenberg, 2015).This observed superiority of the naturalistically rendered scenes over the highly reduced simple color mosaics is in line with the results of Radonjić et al. (2015) in the area of color constancy, although we observed significantly higher overall degrees of constancy.Radonjić et al. (2015) report a mean color constancy index (CCI) of 0.47 in their complex scene, which can be assumed to be equivalent in complexity level to our Reduced2 condition.In contrast, our data already showed a TLC of 0.79 in the Reduced2 level of scene complexity.In our most complex scene (Complex), which in our design contained even more cues for the illumination by a more complex scene geometry, TLC even increased to 0.83.The values of TLC we observed in naturalistically rendered scenes are more in line with other work on color constancy in natural scenes, such as e. g. by Kraft and Brainard (1999) with a mean CCI of 0.83 or by De Almeida, Fiadeiro, and Nascimento (2010) with a mean CCI of 0.82.
However, the degree of color constancy was not only significantly lower than the degrees of filter constancy for the higher levels of scene complexity, but also in the minimal scenes.Radonjić et al. (2015) report a CCI of 0.10 for the simple color mosaics, while we found a TLC of 0.69 in the minimal condition.This significant difference may be partly due to the changes we made to the properties of the minimal scene, in which, unlike in Radonjić et al. (2015), individual color patches were clearly discernible.This alteration of the color mosaics allows for an interpretation of surface colors (and filtered surface colors), whereas in Radonjić  2015) the pointillistic background texture may have suggested a proximal match.On the other hand this difference is also in line with our previous result that under comparable conditions the degree of constancy is much higher for transparent objects than for opaque objects (Falkenberg & Faul, 2019;Faul & Falkenberg, 2015).We argued that the systematic color shifts at the border of a transparent layer directly available in the proximal stimulus facilitates the decomposition of the color signal into background and transparent layer, whereas in the opaque case the decomposition of illumination and object color is clearly less determined.An obvious explanation for this positive effect of complex scenes on TLC is the presence of cues in such scenes, which allow inferences about the illumination of the filters and the environment.This seems surprising at first, because the perceptual model of filter properties proposed by Faul andEkroll (2002, 2011) suggests that the illumination is irrelevant for the estimation of filter properties in the case of filters without direct reflection as they were used in this study.However, this only applies for scenes with homogeneous illumination.If the illumination changes spatially within a scene, it is crucial to correctly identify the regions of different illumination.Since, according to the model, the filter parameters are calculated using the ratio of the uncovered background colors and the ones covered by the filter, the estimation of the filter properties depends on the background colors.In order to correctly split the confounded signal in the filter region into a filter component and a background component, an estimate of the background component must be made (in the case of unknown filter properties) using the uncovered background colors.To ensure that the background component actually corresponds to the properties of the reflectances behind the filter, only those background colors should be used that occur in the same illumination context as the filter itself.Optimal for the estimation of the filter parameters is a situation in which all background colors occur both covered and uncovered.In this case, local estimation from related color pairs can take place and be integrated.However, in natural scenes, reflectances occur in the surrounding of the filter that do not occur in the filter area and vice versa.For this reason, among others, Faul and Ekroll (2011) propose a global estimation using the mean values in the filter area and in the area of the uncovered background (for a discussion of the advantages and disadvantages of local and global estimation, see Faul & Ekroll, 2011).Such a global approach is relatively robust; it only becomes problematic if, for example, colors from other illumination contexts are included, as these can then lead to systematic biases in the estimation of the filter parameters (Falkenberg & Faul, 2021).In the scenes with minimal complexity, the illumination context of the filter is unclear, while the complex scenes contain potentially useful cues about the illumination properties, making it easier to recognize the background regions relevant for estimating the filter parameters.A key difference between these two scene types is that the naturalistically rendered scenes contain illumination gradients, while the chromatically adjusted color mosaics consist only of homogeneous color patches.The inhomogeneity in natural scenes arises, e. g., from the three-dimensional shape of objects and correspondingly the different orientations of the object surface to the light source (and to the viewer).The resulting object shading can serve as a cue to the prevailing illumination context (see e. g.Koenderink, Van Doorn, Kappers, Te Pas, & Pont, 2003).This is supported by the finding that brightness and color constancy are increased when this cues is available (e. g.Boyaci, Doerschner, & Maloney, 2006;Xiao et al., 2012).In addition, shadow cast in the scene leads to illumination gradients that contain information about illumination direction, intensity, and color.These illumination cues are inextricably linked to the three-dimensional interpretation of the scene, which is also supported by object shading in addition to perspective cues, texture gradients, occlusion, etc.
There are also interindividual differences in the filter settings in the extreme conditions Complex and Minimal: While some participants reach a rather lower level of constancy in both complexity levels, others already show relatively high filter constancy in the Minimal condition.Such interindividual differences in overall constancy levels have been observed previously and may be explained by a different weighting of the constancy criterion over the proximal equality criterion (Faul & Falkenberg, 2015;Radonjić & Brainard, 2016;Radonjić et al., 2015).The missing effect of scene complexity in participants with an overall high level of filter constancy could be due to a ceiling effect.
Which cues are being used?The second goal of this study was to analyze in more detail specific stimulus features of complex scenes that lead to an increase in TLC in complex scenes.Since it is not clear a priori which of the cues in the naturalistically rendered scenes actually contribute to a more constant perception of filter properties, the scene complexity, and thus the number of potential cues, was successively reduced.The observed degrees of TLC in the realized complexity levels were largely in line with expectations based on the number of cues available in each complexity condition: The Minimal condition led to the lowest degrees of TLC whereas the Complex and the ComplexFilt conditions led to the highest degrees of TLC.Although we did not observe a clear fixed order of the intermediate complexity conditions, we found clear effects of cue level when multiple complexity conditions were combined.
There were TLC-enhancing effects of the object shading of the cube and of illumination gradients of the background.This indicates that object shading is a useful cue for the prevailing illumination, at least when the reflectance properties continue across the illumination change as it was implemented in our stimuli.The gradual color changes at rounded cube edges seem to indicate an illumination change much more clearly than the abrupt color changes at sharp cube edges.The results also suggest that illumination gradients of the background are used in scenes with higher levels of complexity.As the illumination gradients in   The participants are arranged in ascending order according to the mean deviation between the two filter settings in all measurements under identical conditions (black line).The red dashed line shows the applied limit of the mean measurement repeatability of Δu′v′ = 0.03.Apparently, the filter parameter settings were easier in front of the achromatic cube texture, since better retest reliability was consistently obtained here.
the background occurred both in form of object shading on the two background objects with neutral reflectance, and also in form of shadow cast, the relative contribution of these two components of the background cannot be conclusively determined.
Presenting an additional filter on the differently illuminated side of the cube had an positive effect on TLC only in condition Reduced4 with the most reduced scene, whereas the degree of constancy in the naturalistically rendered scenes of the Complex condition remained unaffected.Given the high TLC of 0.83 in Complex and ComplexFilt this could be due to a ceiling effect.Following Foster and Nascimento (1994) a positive effect of presenting an additional filter with identical properties could be explained by assuming that it represents a relational cue, because the ratio of the filtered colors in the filter area to the unfiltered colors of the cube is approximately the same under both illuminations.
A possible explanation for the small or absent effect of an additional filter could be that we omitted specular reflections from the filter surface in our scenes.This was done in order to keep the filters in the complex and the minimal scene comparable.Specular reflections, which always occur in transparent objects like glass or water, contain direct information about the illumination that seems to be used by the visual system, for example to improve color constancy (Snyder, Doerschner, & Maloney, 2005;Wedge-Roberts et al., 2020;Yang & Maloney, 2001).Not only does the omission of specular reflections exclude important information about illumination, but this unnatural condition could also complicate the interpretation of other filter properties.
The reason why we nevertheless decided to omit specular reflections lies in the properties of the minimal scenes, which in this study, as in the previous ones, are composed of homogeneous color regions both in the background and in the filter area.Under this condition, the direct reflection at the filter surface could only be implemented as a homogeneous additive constant under the assumption of diffuse homogeneous illumination.This leads to a contrast reduction in the filter area, which according to Gigilashvili, Thomas, Hardeberg, and Pedersen (2021) is a cue for translucency.The perceptual dimension of such an additive constant is also described in the model of Faul and Ekroll (2011) as the degree of clarity or "milkiness" of the filter and is not directly comparable to the percept of specular highlights, which are typically evoked by direct reflection.Accordingly, an investigation of the influence of Each plot contains the results for the three filter hues (green, red, blue, respectively) and both Illuminations (triangle: red illumination; circle: blue illumination).The big filled symbols refer to the constancy prediction, i. e., the position of the constancy filter in the match scene with identical filter parameter as the target filter.The small filled symbols mark the position of the proximal filter.This is the same position as the average color in the filter area as in the target scene.The open symbols are the filter settings of each participant in the respective condition.
specular reflections and specular highlights as cue stimuli on transparency constancy is still pending.
Advantages and disadvantages of neutral spectra We observed significantly higher degrees of TLC compared to previous studies (Falkenberg & Faul, 2019, 2021;Faul & Ekroll, 2012;Faul & Falkenberg, 2015) not only in the condition with the naturalistically rendered complex scenes, but also in the minimal scenes.The generally higher filter constancy may be due to an easier task, because of the achromatic background of the adjustable match filter.
In previous studies that used a chromatic background, chromatic properties of the background and the illumination could be incorrectly attributed to the filtering object, or vice versa.This could happen at two stages during the task: First, when estimating the filter properties in the target situation, and, second, when applying the estimated parameters to the match filter in the adjustment situation.In the present study, however, this potential problem existed only in the target situation.Due to the neutral background reflectances and achromatic illumination in the scene in which the adjustment filter was matched, the filter properties were straightforward to recognize.The higher degrees of constancy in a filter matching task against an achromatic background are consistent with the observation of Khang and Zaidi (2002b).As comparable constancy levels were not observed in previous studies even when the colored background had a neutral mean indicating a neutral illumination (Faul & Falkenberg, 2015), the gain in TLC is most likely due to the reflectance spectra also being neutral.
Conversely, our data also suggest that neutral reflectance spectra alone are also insufficient to resolve the confounding of the spectra at the image level: We unexpectedly found significantly worse filter constancy when filters were presented in front of the achromatic cube texture than when shown in front of the chromatic background textures.In fact, physically identical filters that are presented in two different illumination contexts in the Complex condition, look more different in front of the achromatic cube texture than in front of the chromatic textures (Fig. 12).In the case of an achromatic cube texture, the reflectance in the distal scene is achromatic-which is also typically interpreted as such by the visual system-, but the scene is not achromatic at the proximal level as it is the case in the matching context.However, it is unclear why the perceived filter properties are more affected by the properties of the illumination when presented in front of achromatic reflectance spectra than when shown against a background comprised of chromatic reflectance spectra, because the illumination color should be easier to recognize in the achromatic case.Nevertheless, also in the case of achromatic reflectances, the different illuminations are apparently taken into account in the scene interpretation to a considerable degree, since higher degrees of TLC are found in the Complex condition than in the Minimal condition.The higher degrees of TLC in the conditions with chromatic cube texture might also be due to the specific selection of colors in our study.These were drawn uniformly from the different directions around the neutral point in the u′v′ chromaticity diagram to maximize color variability while maintaining a neutral mean.Experience has shown that in this case the filter material itself appears somewhat less homogeneous, but the eight available pairs of filtered and unfiltered colors contain potentially more information as an achromatic pattern, so that the estimate of the filter properties can possibly be improved.Furthermore, in the conditions with chromatic reflectances, we observed larger variance in the settings than in the conditions with achromatic reflectances.The lower variance of the settings in front of the achromatic texture can probably be attributed to the fact that the filter properties are putatively easier to recognize due to the more homogeneous colors in the filter area due to the lack of chromatic variance in the background.
Conclusions Overall, it became apparent that naturalistically rendered scenes clearly improve TLC compared to the simple abstract scenes used in earlier studies.An obvious explanation for the positive effect of scene complexity on TLC is the increased number of available cues to the illumination context of the filter in complex scenes.We expected that the size of the effect observed in a specific complexity condition depends on how many cues about the prevailing illumination it contains that are used by the visual system to estimate the illumination context.Although we observed a ranking of the complexity levels that is largely consistent with this expectation, i. e., the Minimal condition at the lowest position and the Complex and the ComplexFilt condition sharing the highest position, no distinct order of the intermediate complexity levels could be determined.
However, when the data were grouped according to the type of cues included in multiple complexity conditions, higher degrees of TLC were observed for object shading of the central object in the scene as well as for illumination gradients in the background.This suggests the high relevance of object shading and illumination gradients due to shadows cast for a valid estimation of the illumination context and thus for TLC.Additional filters, on the other hand, only led to a slight increase in filter constancy in otherwise strongly reduced scenes but not in the naturalistically rendered scenes, which contained already various cues to the prevailing illumination.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.Extrema of investivated scene complexity.(a) Naturalistically rendered scene for the Complex condition, exemplified with a red filter under reddish illumination (4,000 Kelvin).The left side of the cube is illuminated by a bluish illumination (12,000 Kelvin).(b) Stimulus of condition Minimal, which colorimetrically corresponds to the complex scene in (a) and also shows the red filter in front of a chromatic background texture under the reddish illumination.

Fig. 2 .
Fig. 2. Three chromatic (C1-C3) and one achromatic (BW) cube textures were investigated, each consisting of eight different colors or gray values.(a) shows the UV maps of the textures that contain the identical colors on each side of the cube.(b) shows the chromatic texture C2 projected onto the cube object.Identical surface reflectances were continued at each of the three visible cube edges (marked in (a) and (b) by green, orange, and purple lines), so that the smooth color gradients at the cube edges provide typical cues to the illumination changes in the scene.

Fig. 3 .
Fig. 3. Levels of scene complexity with gradually eliminated cues for scene geometry and illumination.(a) The condition Complex is the naturalistically rendered scene with all cues implemented.(b) The condition Reduced1 is identical to Complex, but rendered with sharp cube edges, so object shading is omitted.In (a) and (b), the green filter is exemplified under the reddish illumination.(c) In the condition Reduced2 the cube object is identical to Complex, but the background is preserved only colorimetrically.(d) In the condition Reduced3 the cube object is sharp-edged as in Reduced1, the background is also preserved only colorimetrically.In (c) and (d), the blue filter can be seen under the bluish illumination as an example.Continuation of the examined complexity levels in Fig. 4.

Fig. 4 .
Fig. 4. Continuation of Fig. 3: levels of scene complexity in the experiment.(a) In the condition Reduced4, there is only one plane of the sharp-edged cube left, in front of which the filter is presented.(b) In the condition Minimal the stimulus consists of a simple 2D color pattern with only homogeneous color areas.In the complexity conditions Reduced4Filt (c) and ComplexFilt (d), another potential cue was added compared to the Reduced4 and Complex conditions: An additional filter with the material properties of the target filter was presented under the second illumination on the other side of the cube.In (c) and (d) the red filter is exemplarily shown under both illuminations.

Fig. 5 .
Fig. 5. Basic stimulus setup in the experiment.The top half of the stimulus is the target scene, where the target filters are presented in the scenes of different complexity levels.In this case, a red filter is presented in the Complex condition under a reddish illumination.The lower half of the stimulus shows the achromatic scene in which the perceived parameters of the target filter are adjusted in the match filter by the participants.

Fig. 6 .
Fig. 6.(a) The parameter τ i , with i = L, M, S, of the target filter are estimated by the ratio of the mean color in the filter area mean(P n ) to the mean color of the relevant background colors in the surround mean(A n ) belonging to the same illumination context as the filter.(b)The constancy filter has the identical filter parameter τ i as the target and if applied to the achromatic match scene the ratio of the mean color in the filter area (mean(Q n )) to the mean color of the surround (mean(B n )) is the same as in the target scene.(c) The proximal filter is a filter that leads to the same mean color in the filter area in the match scene as in the target scene.The parameter τ i of the proximal filter are given via the ratio mean(P n )/mean(B n ).(d) The two postulated theoretical criteria for the filter matching in u′v′ chromaticity space: The constancy filter according to our perceptual filter model (green circle) and the proximal filter (red circle).The exemplary match filter (blue star) can be interpreted as a compromise between the constancy filter and the proximal filter.In this case, the filter setting would reflect more identical filter parameters than proximal equality.The projected Brunswik ratio (BR ϕ ) as a relative measure of constancy (length of the dashed purple line) is calculated by BR ϕ = c cos(ϕ)/a.A match filter corresponding to the constancy filter would have a BR ϕ of 1.A match filter corresponding to the proximal filter would correspond to a BR ϕ of 0.
Fig. A.3 in the Appendix shows the mean Brunswik ratios in both complexity levels with and without participant 01.

Fig. 7 .
Fig. 7. (a) The cumulative relative frequencies of the projected Brunswik ratios of all mean filter settings in the two most extreme complexity conditions Minimal (blue) and Complex (red).(b) The observed distribution of projected Brunswik ratios separately for the achromatic (blue, n = 480) and chromatic (red, n = 1440) background textures.The blue and red lines show, respectively, the approximation of the data by a normal probability distribution.(c) The mean projected Brunswik ratios observed in the complexity conditions Minimal and Complex for achromatic (blue solid line, n = 60) and chromatic (red dashed line, n = 180) cube textures.Individual results for the three chromatic textures C1 -C3 (gray dashed lines, n = 60 each) are shown separately.Error bars show ± SEM.

Fig. 8 .
Fig. 8.The mean projected Brunswik ratios (BR ϕ ) in the two conditions Complex and Minimal separately for the three filter colors for the (a) bluish and the (b) reddish illumination.Each data point includes 40 measurements (10 participants × 4 background textures).Error bars show ± SEM.All p values refer to one-sided t tests for independent samples.

Fig. 9 .
Fig.9.The mean projected Brunswik ratios (BR ϕ ) of the filter settings in the two conditions Complex and Minimal for all participants who completed the filter matches in both extreme conditions of scene complexity.Each bar includes all 24 data points of a participant for the respective condition (3 filters × 2 illuminations × 4 backgrounds).The error bars correspond to ±SEM.

Fig. 10 .
Fig. 10.(a) All scene complexity conditions arranged in ascending order of the observed mean projected Brunswik ratios (BR ϕ ).Each data point includes 240 measurements (10 participants × 2 illumination colors × 4 background textures × 3 filter colors).Data separated by (b) illumination color (each data point n = 120), (c) background texture of the cube (data points in achromatic case n = 60, chromatic case n = 180), and (d) filter color (each data point n = 80).The error bars correspond to ±SEM.

Fig. 11 .
Fig. 11.(a) Assignment of the single conditions of scene complexity to the combined conditions according the three cues object shading of the cube, illumination gradients in the background, and additional filter.(b) Mean projected Brunswik ratios for the scene complexity conditions in which the cue object shading of the cube (black solid line), background illumination gradients (red dashed line), and additional filter (blue dotted line) were absent or present, respectively.Each data point contains 480 measurements from two scene complexity conditions (10 participants × 2 scene complexity levels × 3 filters × 2 illuminations × 4 backgrounds).(c) The influence of an additional filter separated by the complexity levels Complex (dashed line) and Reduced4 (solid line).Note: the ordinates show only a restricted range of BR ϕ values.The error bars correspond to ±SEM.

Fig. 12 .
Fig. 12.Comparison of a chromatic cube texture with the achromatic one.(a) Two identical filters in front of a chromatic cube texture.The identical filters appear very similar under the two differently illuminated sides of the cube (left: bluish illumination, right: reddish illumination).(b) The same filters in front of the achromatic cube texture.The two identical filters appear less similar than in (a).

Fig. A. 1 .
Fig. A.1.Retest reliability of the participants for the filter settings in front of chromatic texture (red crosses) and in front of achromatic texture (blue circles).The participants are arranged in ascending order according to the mean deviation between the two filter settings in all measurements under identical conditions (black line).The red dashed line shows the applied limit of the mean measurement repeatability of Δu′v′ = 0.03.Apparently, the filter parameter settings were easier in front of the achromatic cube texture, since better retest reliability was consistently obtained here.
Fig. A.2.Filter settings observed in front of the achromatic (a) and chromatic (b) cube textures exemplary for the Minimal (left column) and the Complex (right column) condition in the u′v′ chromaticity diagram.Each plot contains the results for the three filter hues (green, red, blue, respectively) and both Illuminations (triangle: red illumination; circle: blue illumination).The big filled symbols refer to the constancy prediction, i. e., the position of the constancy filter in the match scene with identical filter parameter as the target filter.The small filled symbols mark the position of the proximal filter.This is the same position as the average color in the filter area as in the target scene.The open symbols are the filter settings of each participant in the respective condition.

Table 1
Degrees of TLC in all eight conditions of scene complexity and the cues contained in each condition.The group size (n) as well as the mean (M) and the standard deviation (SD) of the projected Brunswik ratios BR ϕ are indicated.The existence of the three cues object shading (OS) of the cube, background (BG) illumination gradients, and an additional filter in each condition are indicated with +, the absence with -.