How are different contrast effects on binocular luster spatially integrated?

Observers

Eight subjects participated in the experiment, one of them being the first author of the study (GW). Five of the participants were female and three male, with their ages ranging from 20 to 53 years. All subjects had normal or corrected-to-normal visual acuity. Prior to the experiment, we obtained written informed consent from all subjects. The study was conducted in agreement with the ethical standards of the Deutsche Forschungsgemeinschaft (German Research Foundation, DFG) and was approved by the Central Ethics Committee of Kiel University (ZEK-18/24).

Stimuli and Apparatus

In order to test how the visual system spatially integrates different contrast effects on binocular luster, we used dichoptic center-ring-surround configurations with a circular shape as test stimuli that were mixed versions of the two original contrast conditions Consistent A and Reversed B (see Figure 1; note that we use the term contrast effects in a largely descriptive sense here to refer to the different effects on the lustrous response produced by the two “pure” contrast conditions Consistent A and Reversed B). We used stimuli in which the ring element was subdivided into either 2, 4, 8, or 16 segments, where neighboring segments either differed in luminance only (stimulus set 1, left panel in Figure 2), or both in luminance and width (stimulus set 2, right panel in Figure 2). In addition, we varied the relative length of the two segment types. The ring proportions are defined by the mixing factor α_ring, which represents the relative length of the stronger ring part (based on the Reversed B contrast condition, see Figure 1), whereas the weaker part (based on the Consistent A condition) has a proportion of 1.0 - α_ring. Stimuli with α_ring = 0, 0.25, 0.5, 0.75, and 1.0 were tested. Stimuli with α_ring = 0 and 1.0 are the two original contrast conditions with a uniform ring. Finally, we varied the width of the ring segments, either equally for the two ring parts (stimulus set 1), or for one ring part only, while the other ring part was kept at a constant width (stimulus set 2). Ring widths of 0.0, 0.0625, 0.125, 0.25, 0.375, 0.5625, 0.8125, and 1.0625 dva were used for the varying part of the ring in stimulus set 2. In stimulus set 1, however, the no-ring condition (0 dva) was omitted, because this condition would produce identical center-surround stimuli for all combinations of segment number and ring proportion. The luminance of the common surround, which covered the entire upper half of the monitor in which the test stimuli were presented, always had a constant luminance of 5 cd/m². The circular center patches, representing the target area of the stimuli, had a radius of 2.0 dva and a luminance of 10 cd/m² for one eye and 40 cd/m² for the other (these luminances were swapped between eyes in half of the trials to account for potential imbalances in eye dominance).

The stimuli were displayed on a monitor with a diagonal of 24 inches (EIZO CG243 W) and a resolution of 1,920 × 1,200 pixels. To color calibrate the monitor, a JETI specbos 1211 spectroradiometer was used, following a method described in Brainard (1989). The dichoptic stimulus pairs were presented side by side on the screen and fused using a mirror-stereoscope (ScreenScope) mounted on the monitor. The light paths from the screen to the observer's eyes had a length of 50 cm.

Procedure

In each trial, observers evaluated three properties of the test stimulus. First, they had to determine whether the target area of the test stimulus appeared lustrous at all. To this end, the test stimulus (in the top half of the monitor) was presented together with an anchor stimulus (in the bottom half). While the ring structure and the background of the anchor stimulus were always identical to those of the test, the center patches of the anchor always had the same luminance of 22.5 cd/m² and thus appeared matte. This luminance was chosen such that (a) it was close to the mean luminance of the two center patches of the test stimulus (which was always 25 cd/m²) and (b) it still allowed for spatial discrimination between the center patch area and the ring area in cases where the ring element had a luminance of 25 cd/m². This anchor stimulus was meant to facilitate the detection of weaker lustrous sensations in the test. Using the arrow keys on the keyboard, the observer could switch between the two response alternatives “lustrous” and “nonlustrous.” If “nonlustrous” was selected, the trial was terminated and the next trial began after a dark adaptation period of 3 s. If “lustrous” was selected, the trial proceeded to the second part, in which the test stimulus was presented together with a matching stimulus consisting of a dichoptic center-ring-surround configuration with a square outline (Figure 3). The center patches of the match had a side-length of 2.0 dva and were enclosed by a 0.125 dva wide ring with a fixed luminance of 25 cd/m². The luminance of the common surround, which filled the entire bottom half of the screen, was 10 cd/m². Using the arrow keys on the keyboard, the observer had to adjust the interocular luminance difference |C_l – C_r| between the two center patches such that the perceived luster in the match was indistinguishable from that of the target patch of the test stimulus. This luminance difference was used as a measure for the perceived luster strength (see also Wendt & Faul, 2025). This choice is supported by the strong correlations between the interocular luminance difference |C_l – C_r| and corresponding rating values in our previous studies (Wendt & Faul, 2020, 2022a, 2024), in which the observers additionally rated the strength of the lustrous impression on a scale from 0 to 5. The observers adjusted a parameter p, which was used to determine the two luminances, C_l and C_r, of the centers of the match as follows:

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Figure 3.

Central part of the screen area during the matching task. The dichoptic test stimulus was always presented in the upper half of the screen, the matching stimulus always in the lower half. The task of the subject was to adjust the interocular luminance difference between the two center patches of the match such that the perceived luster was indistinguishable from that of the test stimulus.

C_l,r = 25 · (1 ± p^s) cd/m², for 0 ≤ p ≤ 1 and s = 1/0.77.

The constant s in this equation was selected such that the relationship between p and the strength of perceived luster was approximately linear (see Wendt & Faul, 2019).

Since the two different ring parts of the mixed-ring stimuli have different effects on the lustrous impression, it is possible that the perceived strength of the luster varies spatially within the target patch. For instance, areas close to the ring parts with the stronger effect could be perceived as more lustrous than areas adjacent to the weaker ring parts. This could be problematic for the matching task, as the observer has to match the perceived luster in the test with a match stimulus that always produces a uniform lustrous appearance. In these cases, a perfect match would be difficult to achieve and the observer may then use an alternative (conscious or unconscious) strategy to complete this task, for instance, by basing the luster judgment exclusively on those areas with the stronger luster or by using a weighted mixture. To check whether such effects could play a role in our study, we asked the observers in the third part of a trial to assess the perceived spatial homogeneity of the luster within the target area. To this end, the same test stimulus as in the previous two parts of the trial was shown in isolation and the observer had to select an integer rating value between 0 (= “extremely inhomogeneous”) and 10 (= “perfectly homogeneous”) using the arrow keys on the keyboard.

Prior to the experiment, each observer was carefully instructed and a series of test trials had to be performed under the supervision of the experimenter. The stimuli used in these test trials were selected from the entire set of stimulus conditions used in the experiment and provided a wide range of different luster impressions. In total, 210 different stimulus combinations were tested, each of them with four repetitions. The entire set of trials was presented in random order.

Calculating the Predictions for the Mixed-Ring Conditions

The empirical luster curves were evaluated with respect to two alternative integration mechanisms: a winner-takes-all integration and a balanced integration.

Results

Since all mixed-ring stimuli combine the pure contrast conditions Consistent A and Reversed B, we will first take a look at the luster curves for these basic stimulus sets. The luster settings averaged across all eight subjects are shown in the top row of Figure 4 as colored symbols (left diagram, yellow for the Consistent A and blue for the Reversed B condition). Also shown are the corresponding predictions and best-fitting parameter values of the LoG (middle diagram, see also Appendix A) and the oriented ON–OFF model (right diagram, see also Appendix B). The luster curve for the Reversed B condition (blue) is very similar to those observed in our previous studies (see Figure 1). The curve for the Consistent A condition, however, deviates considerably from previous results. The general shape of the curve is similar: it is non-monotonic with a steep increase between the no-ring condition and a ring width of 0.0625 dva, and a smoother decrease at larger ring widths. However, the peak of the curve is shifted to a much larger ring width (at 0.375 dva) and the decrease of the luster settings for subsequent ring widths is much smoother than we previously found.

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Figure 4.

Left column: empirical luster settings for the pure Consistent A (yellow disks, error bars represent the SEM in both directions) and Reversed B (blue disks) contrast conditions and their corresponding predictions based on the LoG (orange lines) and the oriented ON–OFF model (dashed black lines). The middle and the right diagrams show the relationship between the empirical luster settings and the conflict measure C for the LoG (middle diagram) and the oriented ON–OFF model (right diagram), respectively, based on the parameters of the best fit. Note that conflict measure C is not the final output of the model (which is C’, as it was used for the prediction lines in the left diagrams, s. also Appendix A), but an intermediate result which we use here to demonstrate the shape of the transducer functions (red curves). The top row shows the results for the entire sample of eight subjects, the middle and bottom rows those for the two sub-groups A+ and A−, respectively.

The reason for this unexpected shape of the Consistent A luster curve becomes clearer when comparing the empirical luster settings of the individual subjects (see Appendix C). In the Reversed B condition, the luster curves of all subjects have a similar shape: The lustrous response increases sharply with increasing ring width until it reaches an asymptote, with slight variations in the steepness of the increase and the absolute level of the asymptote. However, the subjects show clear differences in their sensitivity to luster in the Consistent A condition. There appear to be two main trends, which are also reflected in the parameter values of the corresponding model fits. One group of subjects (S01, S02, and S04), who showed high luster sensitivity for stimuli from the Consistent A condition (which we will refer to as the A+ group, see the middle row in Figure 4), produced luster curves very similar to those we have found in our previous studies. For this group, the best fit was achieved with a large radius of the filter kernel (20 and 26 pixels for the LoG and the Gabor kernel, respectively) and a nonlinear decelerating transducer function (see the red curves in the respective diagrams in Figure 4). The other group (referred to as A−, see the bottom row in Figure 4), which includes subjects S03, S06, and S07, turns out to be nearly insensitive to stimuli from the Consistent A condition. For this group, the best fit was achieved with a comparatively small radius of the filter kernel (8 and 11 pixels for the LoG and the Gabor kernel, respectively) and an accelerating transducer function. The two remaining subjects S05 and S08, although showing a high sensitivity for both stimulus sets, cannot be clearly assigned to either of these two groups as they produced rather unusual and, at least partly, somewhat unsystematic luster curves for the Consistent A condition, in combination with comparatively large standard errors. Interestingly, despite large differences in the shapes of the luster curves and corresponding differences in the model parameters, the predictive quality of the model remains very high for both the LoG and the Gabor variant (with R² values for the individual datasets between 0.927 and 0.99, except for subject S08 with an R² of about 0.85). Since groups A+ and A− appear to indicate different types of sensitivity to the luster phenomenon, we will conduct our further analyses separately for these two groups.

Figures 5 and 6 show the empirical results of the matching task for the A+ and the A− group, respectively. For each of the two stimulus sets (see the top and bottom row in each figure), the results are shown separately for the four different numbers of ring segments (columns). Each diagram shows the luster curves for the three mixed-ring conditions with the different colors representing the different proportions of the two ring parts. In addition, the two curves corresponding to the pure ring conditions are also added to each diagram in a row (see the blue and yellow curves).

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Figure 5.

Results of the matching task of group A+. For each of the two stimulus sets (top and bottom row), the empirical luster settings are shown as a function of ring width, separately for the different segment numbers (columns) and the different spatial proportions for two ring parts (see the differently colored lines in each diagram; the transparent areas represent the SEM in both directions).

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Figure 6.

Results of the matching task of group A−. For each of the two stimulus sets (top and bottom row), the empirical luster settings are shown as a function of ring width, separately for the different segment numbers (columns) and the different spatial proportions for two ring parts (see the differently colored lines in each diagram; the transparent areas represent the SEM in both directions).

We will now compare the findings of both groups, starting with the results of the A+ group. In Figure 5, one can see that the luster curves of the mixed-ring conditions lie between the two curves representing the pure contrast conditions. The order of these curves is in the expected direction, that is, the higher the spatial proportion of the ring part with the stronger effect, the more lustrous the stimulus appears. One can also immediately see that the data cannot be explained by a strict winner-takes-all mechanism. Otherwise, all luster curves for these conditions would have to be at the same level as the blue curve, which always represents the ring part with the stronger contrast effect. The luster curves of the A− group (Figure 6) have the same, expected order. However, compared to those of the A+ group, these curves much stronger tend to the blue curve, particularly at lower segment numbers. Generally, the luster curves of the mixed-ring conditions show a strong dependency on the number of segments, an effect that is only faintly discernible in the data of the A+ group: As the number of ring segments increases, the luster curves shift increasingly downward, that is, the more fragmented the ring is, the weaker the lustrous impression becomes. In the discussion section, we will give two alternative explanations for this effect.

So far, it is obvious that the data cannot be explained by the assumption that the stronger of the two contrast effects fully determines the lustrous appearance in the mixed-ring stimuli (winner-takes-all). But how well are the empirical luster curves predicted by the alternative integration process, which assumes equal weights for the two effects? In Figure 7, we give a first impression of how the data of the two different subject groups A+ and A− match with the corresponding predictions, as they were calculated on the basis of the LoG ON–OFF model. In Appendix D, we show the full set of prediction curves for the three different methods (i.e., the combined-data as well as the two different model-based methods) together with corresponding empirical luster settings, separately for group A+ (Figure D1) and A− (Figure D2). In the present example in Figure 7, one can see that, in general, the empirical settings for the mixed-ring conditions are above their corresponding predictions—a trend that is even more pronounced for group A− than for group A+. This means that also the alternative integration method, that is, the balanced integration, fails to explain the lustrous appearances in the mixed-ring stimuli.

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Figure 7.

Example luster settings (colored disks) obtained with stimulus set 1 for groups A+ (top row) and A− (bottom row) and their corresponding predictions (colored lines) based on the LoG ON–OFF model, separately for the different segment numbers (columns) and the different ring proportions. Generally, the empirical data points for the mixed-ring conditions (with ring proportions of 0.25, 0.5, and 0.75) lie clearly above their predictions.

To provide a more condensed overview of how the empirical luster data relate to the corresponding predictions for the two alternative integration methods, we calculated an index k that indicates how strongly each luster curve deviates from its prediction for balanced integration, relative to the prediction for the winner-takes-all integration. This means that index k lies in an interval between 0.0 and 1.0, where a value of 0.0 means that the given luster curve is fully consistent with the assumption of balanced integration, whereas a value of 1.0 means that the data perfectly align with a winner-takes-all integration process. Values between these poles mean that the two different contrast effects are integrated by an “unbalanced” mechanism, in which the stronger contrast effect has a greater weight than the weaker effect, that is, the higher the index k, the greater the weight for the stronger contrast effect.

For the combined-data method, in which each prediction curve is a convex mixture of the two pure luster curves with a spatially uniform ring (the blue and the yellow curves in the respective diagrams, see Section 2.4), we determined the mixing factor α_emp that best fits the empirical luster curve. We then set α_emp in relation to the two values representing the two different integration methods, where α_bal represents balanced integration (which is identical to the spatial proportion α_ring of the mixed-ring stimuli) and α_win = 1.0 represents winner-takes-all integration (see the left diagram in Figure 8): k_data  = (α_emp – α_bal) / (1.0  – α_bal).

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Figure 8.

Illustration of how the indices k_data (left) and k_model (right) are calculated. As an example, we use the luster curve from subject S01 of the first stimulus set where the mixed-ring condition had 16 segments and a spatial proportion of α_ring = 0.5 (stimulus with black frame). For the combined-data method, the empirical luster curve (green) was fitted with a convex combination of the two curves that represent the pure effects (yellow for the weaker effect, blue for the stronger effect). For the model-based method, the parameter α_emp represents the spatial proportion of that specific stimulus (with varying ring widths), for which the model calculates a curve (red) that best fits the empirical luster curve (green). In both cases, index k represents how strongly the empirical luster curve deviates from the curve representing the prediction for the balanced integration method (α_bal, black), relative to the prediction for the winner-takes-all integration method (α_win, blue, which always has a value of 1.0): k = (α_{emp −} α_bal) / (1.0 – α_bal).

For the model-based methods, we had to a use a different approach to determine the best fit for the empirical luster curve, as this method requires actual stimuli to calculate a model output (see Appendix A). We therefore needed additional stimuli in addition to those used in the experiment: For each segment number and each ring width, we generated mixed-ring stimuli with spatial proportions α_i for the two ring parts between 0.0 and 1.0 in steps of 0.01 leading in each case to 101 candidate curves. We then determined for each empirical luster curve, which of the 101 curves best fits the empirical curve and set α_emp to the corresponding α_i and calculated the index k_model in the same way as k_data (see the right diagram in Figure 8).

Figure 9 shows the different types of k indices (columns) separately for the two groups A+ (top row) and A− (bottom row). For each combination of the four different segment numbers (differently colored lines) and the three different ring proportions (abscissa), the data points represent the average k index across the three subjects of a group and the two stimulus sets, as these two sets produced very similar trends in the luster data (see Appendix D). A first look at the data confirms our previous observations: The stronger contrast effect in the mixed-ring stimuli contributes more to the overall luster impression than the weaker effect, as all k values are above the zero line, which represents a balanced integration process. However, the weight of the stronger effect differs significantly between the two groups of subjects. In group A−, the weights are generally much higher than in group A+. This is particularly evident in the k indices calculated using the model-based method. For both the LoG kernel (middle column) and the Gabor kernel (right column), the k_model values in group A+ are always less than 0.3 and in group A− always greater than 0.8. Generally, this trend can also be observed in the k indices calculated with the combined-data method (first column). However, especially for group A−, the range of k_data values is considerably larger than that of the k_model values. At least in group A−, the k_data values also show a clear dependence on the number of ring segments—an effect that is much less pronounced in the corresponding k_model indices. Specifically, increasing the number of ring segments leads to lower k_data indices, indicating that the weights of the stronger and the weaker effect become more balanced as the number of segments in the mixed-ring stimuli increases. We will provide two alternative explanations for this effect in the discussion section (Section 2.6).

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Figure 9.

The different types of k indices calculated using either the combined-data method (left column) or the model-based method (middle column for the LoG, right column for the oriented ON–OFF model), separately for the two subject groups A+ (top row) and A− (bottom row). In the diagrams, the data points represent the k values for each combination of the three different ring proportions (abscissa) and the four different segment numbers (colored lines), averaged across the three subjects in a group and the two stimulus sets. Transparent areas represent the SEM in both directions.

Figure 10 shows the results of the homogeneity rating task, separately for the two groups A+ and A− (note that homogeneity ratings were only obtained for those stimuli that were classified as lustrous during the first part of each trial, see Section 2.3). For the statistical testing of the rating data, we performed a three-way ANOVA on the combined datasets that include both groups with the factors group (A+ and A−), number of segments (1, 2, 4, 8, and 16) and ring proportion (0.0, 0.25, 0.5, 0.75, and 1.0), which revealed significant effects for all three main effects and all three first-order interaction effects (see Table E1 in Appendix E). In addition, we performed a two-way ANOVA separately for groups A+ and A− with the factors number of segments and ring proportion. As we were particularly interested in how these factors interact with regard to the mixed-ring conditions, we tested the ring proportion variable only with the levels 0.25, 0.5, and 0.75 and the segment number variable with the levels 2, 4, 8, and 16. Again, all main effects and interaction effects were significant (see Table E1).

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Figure 10.

Results of the homogeneity rating task. Diagrams A and B show the mean homogeneity ratings depending on the spatial proportion between the two ring parts (A) and the number of ring segments (B), separately for the two groups A+ (green lines) and A− (red lines). Diagrams C and D show the interactions between these two stimulus parameters exclusively for the mixed-ring conditions for group A+ (diagram C) and group A− (diagram D). The transparent areas represent the SEM in both directions.

Panels A and B in Figure 10 reveal two very clear trends. First, both groups perceived the luster in the target area of the mixed-ring stimuli (with spatial proportions between 0.25 and 0.75 or segment numbers greater than 1, respectively) as significantly less homogeneous than in the two pure conditions with a full ring. Second, spatial homogeneity of the lustrous impression was generally rated significantly lower by group A− than by group A+. This difference is evident even for the two pure conditions, for which spatial luster homogeneity was rated at an average of only 7.77, whereas group A+ rated these stimuli as almost perfectly homogeneous, with an average rating of 9.41. When considering only the mixed-ring stimuli (Panels C and D in Figure 10), a general dependence on the number of ring segments emerges, similar to that observed for the k indices (see Figure 9): The greater the number of ring segments, the more homogeneous the luster in the target area appears. In addition, at least for group A−, perceived homogeneity appears to increase systematically with increasing ring proportion (Figure 10D).

Discussion

The results of the matching task suggest that the two subject groups A+ and A− differ significantly in how they spatially integrate different contrast effects on binocular luster. Neither a strict winner-takes-all integration nor a pure balanced integration, which assumes equal weights for the two different contrast effects, can explain these results.

To test the assumption of a balanced integration, we used different methods to calculate corresponding luster predictions for the mixed-ring stimuli (see Appendix D), namely a combined-data method and two model-based methods, which are based on the LoG ON–OFF model and the oriented ON–OFF model variant, respectively (see Section 2.4.). To determine how well the empirical luster data obtained with the mixed-ring stimuli could be predicted by the two model-based methods, we calculated the corresponding prediction curves using the same model parameters that had previously been determined by fitting the data of the two pure contrast conditions Consistent A and Reversed B (see Figure 4), separately for the two subject groups A+ and A−. As we have seen, the empirical data of these two contrast conditions could generally be predicted very accurately for both groups (Figure 4) as well as for the individual subjects (see Figure C1), although they showed large differences, particularly in the shapes of the luster curves obtained with the pure Consistent A stimuli (see the yellow data symbols in Figure C1). This could be an indication of overfitting, meaning that the models are too flexible due to an excessive number of free parameters (which are five for both model variants, see Appendices A and B, respectively). However, when comparing the different shapes of the Consistent A luster curves, it becomes apparent that they all share some common features: They all start at a low luster level (with no-ring stimuli), then show a more or less steep increase at small ring widths, after which the curves more or less decline, finally reaching a nearly constant level at large ring widths. This means that most of these curves can be regarded as scaled versions of the same general shape (except for the curve of subject S08, which rather has an inverted U-shape). In our previous work (Wendt & Faul, 2025), we have shown that this general curve shape is very similar to the so-called Westheimer functions, which are thought to reflect specific properties of the receptive fields of the underlying contrast detector cells (see Ransom-Hogg & Spillmann, 1980; Spillmann, 2014). We therefore do not believe that our models are overfitted, but rather that all model parameters represent specific characteristics of different (monocular and binocular) physiological mechanisms involved in the phenomenon of binocular luster (see Appendices A and B).

However, if these models (including the combined-data method) are applied to the mixed-ring data, none of them is able to accurately predict the empirical luster settings. The best results, in terms of the coefficient of determination R², were obtained for group A+, which shows a much stronger tendency toward balanced integration than group A− (see Figure 9): The luster settings for the mixed-ring conditions of group A+ could best be predicted by the oriented ON–OFF model (with R² = 0.31), while the predictions using the LoG ON–OFF model (R² < 0) and the combined-data method (R² = 0.01) were significantly worse. For the mixed-ring data of group A−, all of these methods yielded an R² < 0. Note that R² values can be negative if the mean of the dataset (in our case, the luster settings) provides a better prediction than the model, that is, if the sum of squared errors (SSE) is greater than the total sum of squares (SST).

The failure to predict the mixed-ring data with these models (which assume balanced integration) therefore suggests an unbalanced integration of the two contrast effects, where both effects contribute to the lustrous impression, but with different weights. In both groups, the stronger of the two contrast effects has a higher weight than the weaker one; however, this weighting asymmetry is considerably more pronounced for group A− than for A+. This trend is consistent across the different analysis methods employed (see Figure 9).

It is worth noting that the two parametric models (i.e., the LoG ON–OFF model and the oriented ON–OFF model) provide significantly better predictions if we include the data obtained under the two pure contrast conditions and then redetermine the best-fitting model parameters based on these complete datasets for the two subject groups. The goodness of fit is very similar for the two model variants; however, as before, the predictions for group A+ (with R² = 0.848 for the LoG ON–OFF model and R² = 0.858 for the oriented ON–OFF model, respectively, see Figure F1 in Appendix F) are significantly better than those for group A− (R² = 0.565 for the LoG and R² = 0.557 for the oriented ON–OFF model, respectively, see Figure F2). Again, these differences in the overall quality of the model predictions between the two groups can be explained by the trend that the luster settings of group A+ are much closer to balanced integration than those of group A−, which show a stronger tendency toward winner-takes-all integration (Figure 9). In both cases, however, this alternative procedure yields unusually high values for the w_OFF−OFF parameter, which translates into a significantly poorer prediction for the pure Consistent A condition in particular (see the yellow curves in Figures F1 and F2).

We next investigate why the k_data and the k_model indices differ. By construction, the combined-data method produces luster curves that are equally spaced and do not depend on the number of ring segments (see the lines in the top panels of Figures D1 and D2 in Appendix D). In contrast, the model-based predictions show shifts that are not only influenced by the number of ring segments but also clearly differ between groups A+ and A−. They also differ to some extent between the two variants of the model, especially at higher ring segment numbers (compare the corresponding lines between the middle and the bottom panels of Figures. D1 and D2). These shifts are caused by two different components of the model. To illustrate the respective effects in more detail, Figure 11 shows the predictions calculated with the LoG ON–OFF model for stimuli of set 2 using eleven different ring proportions between 0 and 1 in steps of 0.1, separately for the two groups A+ (top row) and A− (bottom row).

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Figure 11.

Curve shifts in the model-based predictions. The diagrams show the predicted luster curves for stimuli of set 2 based on the LoG ON–OFF model for eleven different ring proportions (between 0.0 and 1.0 in steps of 0.1) and the four different segment numbers (columns), separately for group A+ (top row) and A− (bottom row). The higher the ring proportion, the higher the magnitude of the model prediction. However, the increase in this magnitude is not proportional to the increase in ring proportion but depends on the different transducer functions of the groups and is also affected by the number of ring segments.

The first effect is most clearly observed in the diagrams showing the prediction curves for stimuli with only two ring segments, because these curves are only weakly affected by the second effect, which depends on the number of ring segments (left column in Figure 11). For group A+, the curves become denser with increasing ring proportion (that is, at higher luster strengths), whereas for group A−, the curves become denser with decreasing ring proportion (at lower luster strengths). These effects are caused by the different transducer functions of the two groups (see Figure 4). Due to the decelerating transducer of group A+, weaker interocular conflicts are weighted higher compared to stronger conflicts, whereas for group A−, the accelerating transducer causes the opposite effect. This implies that the transducers reflect group-specific sensitivities to interocular discrepancies. Group A+ is much more sensitive to weaker than to stronger interocular conflicts, whereas the reverse holds for group A−.

The second effect underlying the shifts of the luster curves strongly depends on the number of ring segments and has its origin at the level of the monocular contrast detectors, which in our model are simulated either with an LoG or a Garbor filter kernel. As shown in Figure 11, with increasing number of ring segments the luster curves tend to move downward. That is, the more ring segments, the weaker the lustrous impression becomes—an effect that has the same direction for both subject groups. As mentioned in the model description (Appendix A), the magnitude of the lustrous response mainly depends on the relative proportion of local conflicts produced by ON–OFF pairings, whereas ON–ON and OFF–OFF pairings contribute much less to the overall response (Figure A2 in Appendix A). Figure 12 illustrates how the proportion of these ON–OFF pairings depends on the number of ring segments, using example stimuli from stimulus set 1. Figure 12A illustrates which types of monocular contrast mechanisms are activated at different locations within the target area of the stimulus. OFF-center mechanisms are stimulated at corresponding positions when the filter kernels exclusively cover areas that include the lighter ring part (associated with the weaker Consistent A condition, see the red areas in Figure 12). Stronger ON–OFF signals are generated at those locations, at which the filter kernels cover areas that only contain the darker ring part (associated with the Reversed B condition, see the yellow areas in Figure 12). The critical stimulus locations responsible for the reduction of ON–OFF signals are intersections between the two different ring parts where two different ring luminances are spatially integrated by the filter kernels. In these cases, slightly less ON–OFF pairings than pairings with equal polarities are produced. For the stimulus example with two ring segments shown in Figure 12A, the proportion of ON–OFF pairings is about 48%. With increasing number of ring segments, and therefore with increasing number of such intersections, the relative number of ON–OFF pairings continuously decreases until it reaches a proportion of less than 36% with rings having 16 segments (Figure 12B). Note that this effect also depends on the size of the filter kernel. At some locations within the target area, larger kernels can even cover more than two ring segments. This is especially the case for group A+, whose estimated model parameters include comparatively large kernel radii. The fact that, for both groups, the sizes of the kernels are always much larger for the Gabor than for the LoG filter (see Figure 4), also explains the differences between corresponding prediction curves, particularly for stimuli with more than four ring segments (see Figures D1 and D2 in Appendix D).

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Figure 12.

Why do the model-based predictions depend on the number of ring segments? (A) Example stimulus pair from the first stimulus set with a ring subdivided into two segments. Filter kernels that exclusively cover one ring part either produce weak OFF–OFF pairings (red areas) or strong ON–OFF pairings (yellow areas). At the intersections between two ring parts, at which two different ring luminances are spatially integrated by the filter kernel, more OFF–OFF than ON–OFF pairings are produced. (B) The proportion of ON–OFF signals continuously decreases with increasing number of ring segments, or number of intersections, respectively. Since the magnitude of the model output mainly depends on the number of strong ON–OFF signals, this explains why the model predicts weaker lustrous responses for stimuli with increasing number of ring segments.

The differences between the k_data and the k_model indices are therefore due to the fact the model-based method predicts curve shifts that depend on both the number of ring segments and the different sensitivities to the luster phenomenon, whereas the combined-data method does not take either of these influences into account. At least with regard to the number of ring segments, the empirical data indicate that the magnitude of the lustrous response does indeed depend on this variable. In particular, the results of group A− clearly show that the lustrous impression becomes increasingly weaker with increasing number of ring segments (see Figure 6), a trend that was correctly predicted by the model (Figure D2, see also Figure 11).

However, an alternative interpretation for this effect exists that is not based on the properties of our model. The results of the homogeneity task suggest that the number of ring segments also has an influence on the spatial homogeneity of the luster (Figure 10). As the number of segments increases, the luster within the target area is increasingly perceived as spatially uniform. This could mean that in mixed-ring stimuli with only a few segments, the observers can clearly distinguish between areas with different degrees of luster. As we have already speculated in the Methods section (see 2.3. Procedure), the observers may then (consciously or unconsciously) base their judgments more strongly on those areas with stronger luster. As the number of ring segments increases, the two effects may gradually merge into a spatially more homogeneous luster impression, in which the weighting of the two effects becomes more balanced. The stronger impact of the number of ring segments observed in group A− could be due to the fact that the perceived difference between the two effects is much greater for group A− than for group A+. In group A−, the stronger effect (caused by those ring parts associated with the Reversed B condition) produces a very strong lustrous response, while the weaker effect (associated with the Consistent A condition) hardly evokes any impression of luster. In contrast, in group A+, both effects generally show some degree of luster.

Interestingly, though, the subjects reported that the homogeneity task was the most difficult and the most time-consuming part of a trial. While they had no problems matching the strength of the lustrous impressions, they described the evaluation of the spatial homogeneity of the luster as a very artificial and abstract task that required direct comparisons between different areas of the stimulus. This could indicate that a greater weighting of the stronger effect during the matching task was, at least, not a conscious strategy.