In this section, we perform our analysis and describe our findings. Since the dependent variables (attributes) are 7-point Likert items and the collected observations are not normally distributed (\(p < 0.05\) for the Kolmogorov–Smirnov test), we use CLMM to analyze ordinal-scale observations without treating these variables as quantitative. The Estimated Marginal Means (EMM) with Bonferroni correction are selected for the post hoc analysis. The illumination, geometry, and material category are treated as fixed factors, and the user and material as mixed effects for our CLMM model. We use this model for analyzing the effect of material category, geometry, and illumination in Sects. 4.1 and 4.2, respectively.
Then, we compute another CLMM model for analyzing the complex interactions of material category, geometry, and illumination. We use a subset of the data to allow the model to converge, since using all possible pairwise interactions between these three factors would become intractable. After our analysis in Sect. 3.3, we use a subset of four illuminations, in particular A, E, G, and I, as well as four geometries: dragon, buddha, blob, and ghost. For materials, we focus on the MERL dataset [30] since it has the most diverse material categories. After our initial analysis in Sect. 4.1, we select four material categories: fabric, plastic, ceramic, and metal that cover the largest range of all perceptual attributes. We analyze this interaction model in Sect. 4.3.
Effect of material category
There is a significant effect of the material category on the perceptual material attributes (\(p<0.001\)). We show the post hoc Estimated Marginal Mean (EMM) effects of material categories on our four attributes in the first row of Fig. 3. Note that the EMM effects show relative changes with respect to the average model: they account for the estimated effects of material category on isolation, across all variety of geometries and illuminations. There are no significant differences observed between wood, leather, fabric, paper, and natural materials. Those five material categories can be treated as one group, which together with plastic, ceramic, and metal presenting a monotonic trend of increasing attribute rating. The large confidence interval for ceramics is due to a small number of such materials in our dataset (Table 1).
We also analyze Spearman rank correlations between the users’ ratings for the perceptual attributes on each material category, shown in the bottom row of Fig. 3. Our perceptual data do not show obvious correlations for wood, leather, fabric, paper, and natural materials. As expected, for plastic, ceramic, and metal, the gloss-related attributes (glossiness, contrast of reflection and sharpness of reflection) show high correlations. Based on the outcome of EMM and correlation analysis, fabric has been selected as a material category that represents well wood, leather, paper, and natural materials with respect to the gloss-related attributes.
Additionally, we compute the average BRDFs for fabrics, plastics, ceramics, and metals to get some insights about the basic characteristics of each material category. In Fig. 5-right we show the profiles of averaged BRDFs in polar coordinates. We parameterize the averaged BRDFs to the half-angle \(\theta _h\) representation proposed by Rusinkiewicz [37] to get a better insight about the reflectance transition from the specular peak to grazing angles. A slice with the difference angles \(\theta _d=45\) and \(\phi _d=90\) [4] for each averaged BRDF is shown in Fig. 5-left. Metal has the highest specular peak when \(\theta _h=0\), but the reflectance falls down stronger than for other material categories with increasing \(\theta _h\). Plastic and ceramic show similar trends except the grazing angles (\(\theta _h~\text {near}~90^\circ \)), where Fresnel effects can be observed. Fabric shows the lowest specular peak but highest Fresnel effect near grazing angle. Figure 4 demonstrates the rendered images using our four averaged BRDFs on two extreme geometries (dragon and ghost) under two extreme illuminations (A and I).
We also explore further the relationships between each perceptual attribute on the collected data for different material categories. Thus, we average the subjective ratings on each stimuli and create 2D embeddings of perceptual attributes pairs for different material categories, shown in Fig. 6. An interesting nonlinearity can be observed on some combinations: glossiness–sharpness of reflections and glossiness–contrast of reflections for plastic, which is also the case for ceramic and metal (refer to the supplementary for all embedding combinations). This suggests that the sharpness of reflections and contrast of reflections start to be observed when glossiness is high enough for specular materials such as plastic, ceramic, and metal [39]. The embedding of sharpness of reflections and contrast of reflection on metals (Fig. 6-right) indicates that sharpness becomes more visible only for larger contrast of reflections. Such larger contrast is characteristic for more specular material, where the surround of typically clamped highlights is darker and makes the reflected environment better visible [41]. There is no obvious relationship among the gloss-related attributes on fabric. Our embedding reveals the relationships among perceptual attributes that can be useful for perceptual material editing and design. Please refer to the supplementary for more results.
Effect of geometry and illumination
Geometry In order to derive insights about the observed effects of geometry, we compute a set of geometry features for describing our selected geometries. Based on the curvature distributions computed in Sect. 3.3, we compute the standard deviation (\(G_\mathrm{std}\)), kurtosis (\(G_\mathrm{kurt}\)), and median (\(G_\mathrm{median}\)) to characterize our four geometries. We choose median over mean to improve robustness toward outliers. In Fig. 7-right, we show the impact of these features in the estimated marginal effects for each of our four attributes. All the four perceptual attributes analyzed monotonically increase as the median curvature (\(G_\mathrm{median}\)) decreases. The decreasing median curvature indicates an overall smooth surface that will enlarge the coverage of specular highlight and further enhance the perceived glossiness and other gloss-related attributes [26]. Although the \(G_\mathrm{std}\) and \(G_\mathrm{kurt}\) do not show so determinate correlation to the perceived appearance as \(G_\mathrm{median}\), they follow a similar trend.
Illumination Similarly, we compute a set of illumination features and relate them to the observed effects in perceptual attributes. As we use HDR environment maps for illumination during rendering, we directly calculate the following statistics on four environment maps (A, E, G, and I): standard deviation (\(I_\mathrm{std}\)), skewness (\(I_\mathrm{skew}\)), kurtosis (\(I_\mathrm{kurt}\)) light source coverage (\(I_\mathrm{area}\), where small values correspond to point-like light sources), contrast between light sources and the rest of image (\(I_\mathrm{range}\)), and high-frequency content (\(I_\mathrm{hfc}\)). Results are shown in Fig. 7-left. Our data confirm literature findings that gloss perception becomes stronger for more directional and higher intensity light sources (low \(I_\mathrm{area}\) and high \(I_\mathrm{hfc}\)) [6, 36, 45]. The perceived glossiness is also consistent with increasing environment map contrast (high \(I_\mathrm{std}\) and \(I_\mathrm{range}\)) as also reported in [1, 31]. A positive correlation between the skew of environment map histogram and perceived gloss has been reported by Adams et al. [1]. The skew and other statistical moments in Fig. 7-left also correlate well with perceived gloss. However, the environment map (G) is an outlier here as its statistical moments are consistently higher than for (E), while the perceived gloss is similar. This different behavior can be better explained by variations in \(I_\mathrm{area}\) and \(I_\mathrm{hfc}\).
Interaction between illumination, geometry, and material category
In this section, we compute and analyze a second CLMM model with the interaction between material category, illumination, and geometry.
Glossiness
Perceived gloss increases with the highlight coverage, sharpness, contrast, and brightness that in complex ways depend on illumination spatial structure, material properties, and surface geometry [26]. For example, the highlight coverage can be extended by blurring environment maps, expanding lobes of glossy reflection (material roughness), reducing surface curvature, making the viewpoint position perpendicular to the surface, to name just a few possible manipulations to achieve this goal. Here, we consider gloss variations due to all those factors for metals, ceramics, plastics, and fabrics.
As seen in Fig. 8-a, the gloss perception is strongest for metals, but some inconsistencies can be clearly observed. For example, the blob illuminated by A and E environment maps might be perceived even comparably glossy to ceramics and plastics. In contrast to more complex geometries such as the dragon and buddha, the blob features larger smooth surface regions (smaller \(G_\mathrm{median}\) in Fig. 7-right) that enable a more intuitive judgment of reflection sharpness (Fig. 12) and contrast (Fig. 10). This can be observed in Fig. 11. Note that for slowly changing illumination A and E (smaller \(I_\mathrm{std}\) and \(I_\mathrm{range}\) in Fig. 7-left) such reflections appear blurry, of low contrast and similar coverage, regardless whether this is metal, ceramic, or plastic material. Note that for ghost, the highlight coverage for metal is enlarged with respect to the ceramic and plastic, in particular, due to the wavy regions surrounding the central hemi-sphere that somehow enhances perceived glossiness. Overall slowly changing illumination (as A and E) in interaction with slowly changing, smooth surfaces (as the blob) might reduce the differences in glossiness perception between metals and plastics/ceramics. On the other hand, the absolute magnitude of perceived glossiness is the highest for blob and ghost, mostly, regardless of illumination (refer also to Fig. 8-b).
We show results of glossiness pairwise comparison on interaction between material category and geometry in Fig. 8-c. Four material categories show similar trend as in Fig. 3 under the same geometry. When observing across different geometries, the value differences between fabric and other materials show negative correlation with the complexity of geometry. It seems the tesselated-like surfaces on dragon reduce the perceived glossiness on all materials, especially on dielectric specular materials such as plastic and ceramic. This is consistent with the finding of Vangorp et al. [43]. It seems the geometry complexity has less influence on metal compared to other materials.
Metallicness
As expected, the metallicness attribute is consistently strongest for metal over other material categories regardless of geometry and illumination (Fig. 9). The large gap between metal and other material categories lies in the reflectance differences between metal and dielectrics as seen in Fig. 5. Despite display brightness limitations and contrast-compressive tone mapping, the intensity of specular reflection is consistently the highest for metals (refer also to Figs. 4 and 11) due to largest magnitude of BRDF for specular angles (small \(\theta _h\)), which is one of the key factors that enable to distinguish metals from plastic and other materials [41].
It seems the illumination has limited influence on metallicness when observing across different environment maps on all materials (Fig. 9-b). The geometry shows clear influence on metallicness (Fig. 9-c). It is believed that the perception of metallicness is related to the coverage of highlight (specular sheen), which can be enlarged by reducing the local surface curvature [26, 41]. Our results match with this theory closely: an decreasing local surface curvature (\(G_\mathrm{median}\)) results in increasing metallicness in Fig. 9-c.
Contrast of reflections
As shown in Fig. 10-a, the rating of metal under illumination I is significantly higher when compared to other material categories in terms of reflection contrast. This phenomenon can easily be observed in Fig. 4. Obviously, under illumination I, the highlight coverage on metal is much larger than for other material categories. Comparing with other environment maps, I has the largest dynamic range (the highest \(I_\mathrm{range}\) and \(I_\mathrm{std}\) in Fig. 7) and is dominated by only one strongest point light (the smallest \(I_\mathrm{area}\)), which results in wide and high contrast highlights on metal but narrow and lower-contrast highlight on other materials (refer to ghost under illumination I in Fig. 4). This effect is amplified due to the different profiles of respective BRDFs shown in Fig. 5.
Another interesting observation holds for the blob. Except for illumination I, the blob gets similar rates on plastic, ceramic, and metal. Actually, this phenomenon is pronounced on all gloss-related attributes (Figs. 8 and 12). We show the rendered result for four geometries using the averaged BRDFs for these three material categories under illumination E in Fig. 11 (similar for illumination A and G). As discussed in Sect. 4.3.1, the gently changing smooth surface of blob enlarges the coverage of highlight and consequently boosts the perceived glossiness and all the other gloss-related attributes.
For the interaction between material category and illumination shown in Fig. 10-b, there are no significant differences observed between the combinations: fabric-A, plastic-A, and ceramic-A. This means the highly diffuse illumination like A minimizes the differences between material categories [23].
Sharpness of reflections
Similar observations as for contrast of reflection under illumination I also hold for sharpness of reflection in Fig. 12-a. The strong directional light I consistently results in significantly higher rating for metal than for other material categories, where as can be seen in Fig. 5 the BRDF magnitude is larger for small \(\theta _h\) and at the same time it is smaller for larger \(\theta _h\). This contributes to stronger contrast between highlight and its surround for metals. As mentioned before, the highlight areas for metal tend to be overexposed due to the high reflectance property of metal, and the limit of current display system and tone mapping algorithm make this problem even more pronounced. Thus, one should avoid the highlight areas on metal when make the judgment on sharpness of reflection, which was explained by Serrano et al. [39] to experiment participants during the rating collection.
Except for I, the sharpness of reflection of dragon under other illuminations shows unintelligible patterns on different material categories (Fig. 12-a, c). It seems the tessellated-like bumpy surfaces on dragon greatly affect people’s perception judgment on this attribute. We believe this is consistent with Marlow et al. [26] in their Experiment 2.