The AR simulator, from our previous experimental setup [23], was housed in a dark room and comprised three main components: a physical background, a digital virtual foreground, and an observation assembly. Figure 1 provides a schematic overview of the apparatus. The physical background pattern was inkjet-printed and mounted at the rear of a custom wooden booth. Two spectrally tunable light sources (Thouslite LEDCube) were positioned at the top and bottom of the booth to provide uniform illumination across the background surface. The virtual foreground was a professional display (EIZO ColorEdge CG277) positioned to match the optical depth plane of the background pattern.

The observation assembly comprised a plate beamsplitter, a magnetic black occlusion panel, an aperture, and a chin rest. Specifically, a plate beamsplitter (Edmund Optics Plate Beamsplitter #68-268, 75:25 reflection/transmission ratio) optically superimposed the self-luminous foreground emission onto the reflective background pattern. Directly behind the beamsplitter, a magnetic black occlusion panel could be deployed to block the light from the background booth for display-only conditions. Progressing toward the observer, a square aperture and a chin rest jointly restricted the observer’s viewpoint and effective field of view. The entire apparatus was enclosed within a black felt curtain to eliminate extraneous visual distractions. The viewing distance was 1 m for both the foreground and the background, with care taken to ensure precise alignment in depth. Observers performed color adjustments via a keyboard placed outside the curtain.

All colorimetric measurements were taken from the observer’s eye position through the beamsplitter using a spectrophotometer (Konica Minolta CS-2000), thereby ensuring accurate measurements of the actually perceived stimuli.

Within each experiment, trials were organized into distinct phases primarily dictated by the physical constraints of toggling certain environmental hardware (e.g., swapping presentation media or modifying ambient luminance). Within each phase, the presentation order of target colors was randomly shuffled to ensure that no identical color target appeared consecutively. Each phase evaluated a predefined set of target colors, with each target repeated twice.

The multifactorial design isolated and evaluated seven environmental, spatial, and procedural factors across the four experiments. Specifically, Experiment 1 primarily investigated the effects of stimulus size and viewing mode (binocular or monocular viewing); Experiment 2 examined the presence of aperture, foreground spatial offset (the presence of a minor positional change of the foreground patch), and the starting point (the initial color configuration of the test patch); Experiment 3 focused on ambient luminance levels (the interior luminance of the booth in AR or its digitally simulated counterpart on screen) and the presentation medium (AR simulator or display-only); and Experiment 4 investigated the starting point and presentation medium. Table II summarizes the manipulated factors, their levels, and the variable names used in subsequent statistical modeling along with operational descriptions clarifying the technical and perceptual definitions of complex procedural factors.

Specifically, for the display-only medium condition, the physical background pattern was reproduced on the screen by sampling the measured tristimulus values at the four corners of each physical patch and applying a linear RGB gradient within each patch boundary to preserve a natural paper-like appearance. The surrounding region outside the patches was rendered a uniform neutral gray.

An aggregate of eight foreground–background target combinations (target_id) was evaluated. Six combinations, labeled 1, 3, 4, 6, 7, and 9 according to their spatial layout on the background pattern (ordered top to bottom, left to right), mapped six background patches to six distinct intended target colors and were shared across multiple experiments. Two additional combinations, labeled 13 and 19, were evaluated exclusively in Experiment 1. Specifically, Target 13 features a red foreground on a green background while Target 19 features a green foreground on a red background. These two targets were designed to have the same intended target chromaticities, thereby allowing a direct probe into background chromatic effects. The complete colorimetric specifications of all background patches and their corresponding target mixtures are summarized in Table I.

A total of 12 observers (aged 22–32 years, three females and nine males) participated in the study. Each experiment involved eight participants, meaning that individual observers completed between one and four experiments. All observers passed the Ishihara test to confirm normal color vision, and all had prior experience with general color-matching experiments.

This study was conducted in accordance with the Declaration of Helsinki. Given the noninvasive nature of these psychophysical procedures, which involved no greater risk than everyday visual activities, formal ethics approval was not required under institutional guidelines. All observers provided written informed consent prior to participation and were debriefed after completing the sessions.

Prior to each experimental session, the experimenter preheated the background light sources for at least 1 h and the display for at least 30 min. Upon entering the dark room, observers adapted for at least 5 min while the experimenter explained the general workflow, trial counts, and control mechanisms. Observers were then allowed to practice with the keyboard freely. Following adaptation, observers completed the white-point adjustment (for sessions where this feature was active) and initiated the matching trials at their own discretion. Observers were permitted to take breaks or leave the room at any time, provided that they underwent readaptation upon returning.

Experiments 3 and 4 involved switching between AR and display-only media. Observers experiencing this transition for the first time were requested to look away during the phase switch to keep them unaware of the medium transition. In the follow-up experiment, they were directly notified of the change. After completion of the psychophysical tasks, a validation program read the recorded RGB values and reproduced the matched patches at their identical positions and sizes. The corresponding XY Z tristimulus values were then systematically measured using the spectrophotometer while the background occlusion board was closed to ensure radiometric accuracy. The display was verified to be sufficiently stable to allow these measurements to be conducted following the completion of each observer’s experimental session.

The four experiments were not executed in a strict chronological sequence. Individual white-point adjustments were performed by three participants in Experiment 1, six participants in Experiment 2, and all participants in Experiments 3 and 4. Regarding medium transition, all observers were completely naïve to the purpose and existence of the change during their first exposure (either in Experiment 3 or 4). In the subsequent follow-up experiment, observers were explicitly notified of the transition as part of the experimental protocol, resulting in one observer in Experiment 3 and seven observers in Experiment 4 being aware of the display-only medium during their respective second test sessions.

The recorded measurements were transformed into a uniform color space combining the CIECAM16 color appearance model [18] and the CAM16-UCS conversion algorithm [22], designated CIECAM16-UCS throughout this paper. Reference white points for high- (31.8 cd/m²) and low-luminance (7.06 cd/m²) conditions were derived from mean observer-adjusted chromaticity coordinates (Appendix A.2, Supplementary Material). Because only the nine color patches within the central matrix were visible during most of the trials, the relative background luminance Y _b was set at their mean luminance value of 26.28, and the viewing condition parameter was specified as “dim” for the CIECAM16 model configuration.

To eliminate obvious procedural matching blunders, multivariate outlier removal [14] was performed using Mahalanobis distance within subgroups (8 observers × 2 repetitions per target combination). This procedure identified and excluded 12 extreme outliers out of 1088 total observations (Appendix A.3, Supplementary Material). To resolve the unbalanced design in Experiment 3 caused by procedural constraints, corresponding observations were supplemented from Experiments 1 and 2 during preliminary factor screening (Appendix A.4, Supplementary Material).

The statistical analysis began with descriptive metrics to evaluate empirical matching precision and target deviation, followed by a structured GLMM framework to isolate independent factor effects and assess their relative contributions to perceptual color shifts.

The CIECAM16-UCS coordinates were computed using observer-adjusted, condition-specific white points (established in Section 2.5; detailed in Appendix A.5, Supplementary Material). Based on these, observer precision (MEDM) and target deviation (MEDT) were quantified across Experiments 1–4 to establish baseline response quality.

In Experiment 1, raw matches showed the highest precision (MEDM = 2.61) and smallest target shifts (MEDT = 4.36), driven by the large stimulus size condition (MEDM = 2.05, MEDT = 3.48). In Experiments 2–4 (conducted under small-stimulus conditions), observer variability (MEDM = 4.39–4.60) and target shifts (MEDT = 6.19–7.85) were larger but more consistent. Across targets, Target 4 (BG: gray, FG: purple) exhibited the largest shift (MEDT = 11.53) whereas Targets 9, 13, and 19 maintained higher fidelity (MEDT = 3.13–3.46).

In addition, the raw matching distribution reveals a subtle procedural anchoring effect. As demonstrated in Figure 2, adjustments starting from “low” starting points systematically fall between the mean match from the “mid” condition and the starting point itself. The displacement between matches from the “low” and “mid” starting points ranges from ΔE  ′ = 0.26 to 1.82 across targets.

These preliminary observations from the raw data revealed two empirical tendencies that motivated the subsequent GLMM analysis. First, observer variability and target deviation were substantially larger under the small-stimulus conditions than under the large-stimulus conditions, indicating that stimulus size strongly influenced matching performance. Second, matches initiated from the low starting point consistently remained closer to their initial values than those from the mid starting point, suggesting a procedural anchoring tendency (Fig. 2).

Although these descriptive observations provide an intuitive overview of the data, they do not isolate the individual contributions of the multiple experimental factors because of the unbalanced factorial design and the overlap among experiments. Therefore, a multivariate GLMM framework was implemented using signed coordinate differences as responses to quantify the independent effects of each predictor.

Following the two-stage model selection strategy outlined in Section 2.6.3, top-down refinement was first applied within each individual experiment dataset to identify statistically significant predictors and interaction terms. Model comparison based on AICc yielded the following optimal fixed-effects specifications for the four experiments (with detailed selection tables provided in Appendix Tables B3.1–B3.4, Supplementary Material):

Experiment 1: Diff  ∼ axis * (view_mode + stim_size * target_id)

Experiment 2: Diff  ∼ axis * (start_point + target_id)

Experiment 3: Diff  ∼ axis * (ambient_lum * target_id + medium)

Experiment 4: Diff  ∼ axis * (start_point + target_id) + medium

Across the pooled dataset (comprising 1076 trials, corresponding to 3228 long-format observations across the three coordinate axes after outlier removal), the significant predictors identified from the individual experiments were consolidated into a single pooled framework. The final pooled model retained viewing mode, stimulus size, ambient luminance, starting point, and presentation medium as the primary predictors, while aperture presence, foreground offset, and the aforementioned interactions were excluded due to lack of statistical contribution. The fixed-effects formula for the consolidated pooled model is specified as follows (Appendix Table B3.5, Supplementary Material):

Diff   ∼  axis  *  (target_id  *  (stim_size  +  ambient_lum) +  start_point  +  view_mode  +  medium)

In this specification, color-related factors interacting with target combinations are placed first within the formula. All individual and pooled models maintained the baseline random-effects structure (0 + axis | observer) and conditional dispersion model ∼axis * target_id defined in Section 2.6.2. Fitting this consolidated framework captured the systematic underlying perceptual patterns while effectively filtering out random interobserver adjustment noise. Comprehensive residual diagnostics confirmed the structural adequacy, distributional robustness, and statistical validity of the consolidated GLMM (Appendix A.6, Supplementary Material).

Model performance was quantified by MAEs, yielding values of 2.76, 1.05, and 1.35 along the J  ′, a ′, and b ′ axes, respectively. Structured random variability across observers was summarized by standard deviations of 1.69 (J  ′), 0.64 (a ′), and 0.68 (b ′). The low ICC (0.019) indicated that modeled random effects accounted for only a minor fraction of the total residual variance—a pattern attributable to the limited number of repeated measures per observer across a highly multifactorial experimental design. Overall, these diagnostics confirm that the consolidated GLMM provides a stable and valid statistical foundation for subsequent inference. Full model parameters and diagnostics are reported in Appendix Table B4 (Supplementary Material).

The statistical contribution of the experimental factors within the consolidated pooled GLMM was evaluated using Type III Wald χ² tests. Table III summarizes the statistically significant main effects and interaction terms (p < 0.05) while the full test statistics—including nonsignificant terms and random-effects estimates—are detailed in Appendix Table B4 (Supplementary Material). While coordinate axis and target combination served as necessary control variables, the statistical analysis focused on five primary predictors: viewing mode, stimulus size, starting point, ambient luminance, and presentation medium. Each of these five factors achieved statistical significance through main effects, higher-order interactions, or both (p < 0.05).

To interpret these significant terms and quantify both the magnitude and direction of the resulting color shift, EMMs were calculated for the 11 unique experimental conditions present in our design. Pairwise contrasts were evaluated under a standardized reference condition (composed of the default baseline levels for all factors): binocular viewing, small stimulus size, high ambient luminance, mid starting point, and AR presentation medium. The full parameter configurations defining the 11 unique conditions are detailed in Table IV. Key pairwise contrasts for significant factors are summarized in Table V; the full set of EMMs and contrast estimates are provided in Appendix Table B5 (Supplementary Material).

Figure 3 illustrates GLMM estimates alongside raw data points for Condition 3 (large stimulus) and Condition 4 (small stimulus) in the a ′–b ′ plane, where all other factors are at their default baseline levels. Raw matches (small circles) cluster tightly around the observed means (small diamonds) and GLMM estimates (large rhombuses). While model estimates aligned closely with observed means, subtle model regularization occurred in specific cases (e.g., Target 19 in Condition 4; see inset) by filtering random interobserver noise.

In the a ′–b ′ plane, the straight line between the background chromaticity and the target also provides a reference for the additive mixture trajectory. The target color itself is close to that line by construction while the matched colors for several targets, particularly Targets 3 and 7, deviate from it in a way that is visually apparent and consistent across conditions. Similar deviations are observable for other targets (e.g., Target 4) across the complete set of trajectory plots (Appendix Figure C1, Supplementary Material).

In what follows, we describe the specific perceptual color shifts driven by these primary factors.

For the background-swapped pair, Targets 13 (BG: green, FG: red) and 19 (BG: red, FG: green), a highly significant discrepancy was observed along the a ′ axis (0.93 ± 0.21, p < 0.001) under the small-stimulus condition (Condition 4), where the background occupancy was higher. Specifically, Target 13 appeared redder—a direction complementary to its green background—than Target 19 on the red background as visually highlighted in the inset of Fig. 3b. No significant differences were found along the J  ′ (p = 0.822) and b ′ (p = 1.000) axes. Conversely, under the large-stimulus condition (Condition 3, Fig. 3a inset), the a ′ axis discrepancy diminished and became statistically nonsignificant (p = 0.056). Instead, a significant difference emerged along the b ′ axis (0.41 ± 0.16, p = 0.0498) whereas the J  ′ axis remained stable (p = 0.939).

The results from our model indicate that stimulus size is the most powerful predictor affecting the color appearance of the foreground–background mixture. As shown in Table V, the effect of changing the stimulus size is much larger than any other experimental factor. For example, in Target 7 (BG: white, FG: blue), the Euclidean distance between the matching results for the small and large stimulus sizes reached 6.80. This result follows the findings by Blackwell and Buchsbaum [2], which showed that the surround effect in SC increases with the surround size. In our experiment, a small stimulus size meant the physical background occupied a larger relative area of the field of view, which strengthened the chromatic induction from the background to the virtual foreground.

Beyond the influence of size, our data shows that these color shifts are highly systematic and follow the classical complementarity law of SC. By analyzing the “small minus large” contrast values in the a ′b ′ plane, we observed that the direction of the perceived shift was consistently opposite to the coordinates of the background where the shift attained statistical significance. A clear example is Target 3 (BG: green, FG: green; BG: a ′ < 0, b ′ > 0): reducing the stimulus size drove a significant positive shift along the a ′ axis and a negative shift along the b ′ axis. This sign inversion demonstrates that as the induction effect was intensified by the smaller stimulus size, the perceived color moved toward the complement of the background. Similar complementary evidence was observed for Targets 7 and 19.

To further confirm that these shifts are driven by SC, we compared Target 13 (BG: green, FG: red) and Target 19 (BG: red, FG: green), which share the same target color. Their background lightness (J  ′) and yellow-blueness (b ′) were similar, but they differed significantly in red-greenness (a ′ < 0 for Target 13 vs a ′ > 0 for Target 19). In the small-stimulus condition, where SC is strongest, a highly significant contrast emerged between Targets 13 and 19 exclusively along the a ′ axis, with Target 13 perceived as redder—a direction complementary to its green background. We also performed an ancillary check using the Hunt model [10]—currently the only model that accounts for SC against chromatic backgrounds. These systematic complementary shifts are also consistent with the predictions from the Hunt model when the parameter p is varied from 0 to −1 toward increasing SC.

However, under the large-stimulus condition, this a ′ axis discrepancy diminished and became nonsignificant while a small difference appeared along the b ′ axis. This shift in significance suggests that as the visual angle increases, the interaction between the axes and the spatial context becomes more complex even as the overall induction effect tends to weaken.

The systematic shifts we observed provide a clear link between the physiological mechanism of SC and the theory of layer scissioning often used in AR research. Notably, the results from Hassani and Murdoch’s [16] data plots also show clear trends of complementarity in SC, where the matching points consistently shift away from the physical mixture in a direction complementary to the chromaticities of the backgrounds. This observation is mathematically consistent with the concept of background discounting. In the αβ model, using a coefficient for the background (β) that is smaller than 1 is identical to subtracting a portion of its color or moving the perceived color along the complementary direction.

Our findings suggest that the background discounting mentioned in scissioning models can be explained as a manifestation of SC at the perceptual level. Hassani and Murdoch [16] attempted to include SC in their modeling to improve color appearance predictions, which did provide some improvement over simple additivity. Although the modeling of SC in their work was relatively simple, our results indicate that a more comprehensive understanding of these inductions can provide a useful alternative perspective for explaining color shifts in AR.

A further link can be seen in the effect of stimulus complexity. Hassani [15] reported that the texture and complexity of the stimulus heavily influenced the weights in her transparency-based model, where increased complexity led to more significant background discounting. This observation aligns with recent findings by Grasso et al. [13], who demonstrated that high-complexity stimuli trigger stronger chromatic induction compared to simple rectangles. These consistencies suggest that SC provides a functional bridge between low-level physiological induction and the cognitive separation of layers. Supporting this view, informal post-experiment debriefings revealed that a subset of observers reported no perceived transparency in certain trials, yet their matching data exhibited the systematic complementary bias. This observation suggests that the background discounting mechanism can operate in the absence of explicit layer decomposition, consistent with a primarily low-level chromatic induction account.

We also examined the difference between matching results obtained in the AR simulator and the display-only medium. The contrast analysis between these two presentation media showed no statistically significant differences along either the J  ′ or a ′ axis. A significant but small difference was found exclusively along the b ′ axis, with the overall Euclidean distance remaining minor (ΔE  ′ = 0.564). This trivial effect of the presentation medium remained consistent across three alternative reference white-point definitions evaluated during analysis (observer-adjusted, literature-based nominal [15], and their intermediate).

Although the current setup adopted the configuration established in the prior literature [15]—characterized by uniform background patches with smaller foreground stimuli rather than the explicit geometric alignment employed in our previous work [23]—the optical depth was carefully matched between the foreground and the background. The negligible effect of the presentation medium observed here therefore demonstrates that under sufficiently aligned perceptual conditions, including matched optical depth, uniform illumination, and high-fidelity background reproduction, the human visual system exhibits highly consistent perceptual behavior when processing additive color mixtures in AR and simulated mixtures on a display. This robustness diverges from the larger effects of the presentation medium reported by Hassani [15] and suggests that subtle differences in visual context or observer adaptation state could have contributed to the inconsistencies with prior results.

The matching results demonstrate that while the linear αβ model provides a reasonable baseline for predicting color appearance in AR, our data also reveal nonlinear deviations that are themselves systematic and interpretable through SC. The results for Targets 3 (BG: green, FG: green), 4 (BG: gray, FG: purple), and 7 (BG: white, FG: blue) identify the specific conditions where these deviations become most pronounced. Across our dataset, the Euclidean distance between matched and target colors remained small under most conditions (median = 2.82) but increased to 6.35 for these three specific targets. This supports the qualitative prediction of Kirschmann’s Third Law [19] or the inverse size hypothesis [8]: chromatic induction strengthens as brightness or chromaticity contrast decreases.

As evidenced by the trajectory plots (Fig. 3 and Figure C1, Supplementary Material), the matching results for these targets tend to branch off from the expected linear line connecting the foreground and background chromaticities. Regarding the direction of these shifts, our data show a dominant trend that aligns more closely with the background’s complement than the direction law [9]. Since both the αβ model and the complementarity law assume that perceived color shifts should fall somewhere along the foreground–background axis, the observed deviations suggest the involvement of complex nonlinear mechanisms of SC. These findings highlight a fundamental limitation of simple linear frameworks (including Hassani and Murdoch’s weighted-sum approach [15, 16]) and underscore the necessity of incorporating advanced, nonlinear SC mechanisms in future AR color appearance models.

Regarding ambient luminance, synchronously scaling the display’s peak luminance with ambient illuminance was intended to maintain perceptual invariance. This expectation held reasonably well along the lightness (J  ′) axis. However, chromatic coordinates (a ′, b ′) were less consistent, with significant shifts observed in two of the six target combinations tested. Simple physical metrics, such as background chroma and contrast, failed to provide a straightforward account for why these specific shifts occurred. This suggests that changing ambient luminance introduces complex, luminance-dependent color appearance phenomena that proportional scaling alone does not fully resolve.

Beyond primary perceptual shifts, our analysis revealed two distinct axial patterns corresponding to lower-level physiological constraints and higher-level cognitive decision strategies.

First, specific volatility was found concentrated along the blue–yellow (b ′) axis for the viewing mode and presentation medium. This axial asymmetry likely stems from the higher discrimination threshold intrinsic to the blue–yellow dimension [12]. Within this broader tolerance zone, subtle physical and physiological perturbations are easily amplified. For the presentation medium, this shift may be primarily driven by observer metamerism between broadband backgrounds and narrowband display primaries. For the viewing mode, we tentatively hypothesize that a nonlinear binocular combination [6] interacting with the high baseline noise of the S-cone pathway during monocular viewing [1] underlies the observed systematic shift.

Second, the procedural bias driven by the starting point reflects a cognitive directional anchoring effect. This phenomenon suggests that color matching occurs within a perceptual tolerance range rather than at a single point, and observers tend to terminate the adjustment process at the proximal boundary of the “satisfactory” range, which is a cognitive process rather than a perceptual color shift. The GLMM estimated a generalized offset across targets, which is likely due to small differences across targets and the concentrated chromatic distribution of our stimuli.

Despite the high consistency of our results, certain limitations remain. First, we observed occasional J  ′ axis inversions, where lightness matches exceeded the physical additive sum. This result is unexpected given that the matching patch was presented against a black background, which theoretically should have induced higher perceived brightness. Second, a directional conflict exists between our interpretation and the “foreground discounting” reported by Murdoch [26]. While our data aligns background discounting with SC, the conditions under which the visual system prioritizes one layer over the other remain unclear. Finally, our stimulus set was limited and chromatically concentrated—including the selective detection of significant ambient-luminance-driven shifts in only two target combinations evaluated. A broader and more uniform sampling across both the color gamut and contrast ratios is required to validate the generalizability of the observed effects and axial sensitivities.