The primary objective of Experiment 1 was to identify whether geometrically thin parts increase the magnitude of perceived translucency difference between two materials. Experiment 2 was inspired by the results of Experiment 1. It was conducted to determine how sensitivity to translucency differences depends on the optical thickness of the materials.

In order to determine suprathreshold translucency differences for each object’s shape, the method of constant stimuli has been used [10]. This is a popular method for determining suprathreshold color differences [3, 38], and has been used for interpreting translucency as well [39, 40]. The anchor pair, which consists of two Buddhas with a suprathreshold translucency difference and remains fixed throughout the whole experiment, is compared with a test pair. The test pair consists of two similarly-shaped objects – one control point (CP) and one respective test sample (see Section 2.3.1). The task of an observer is to identify which pair has larger difference in perceived translucency. Each CP is compared with different test points that are sampled in four different directions, in the absorption-scattering space. The objective is to identify, how far we need to depart from the CP in each direction in the absorption-scattering space, to obtain the translucency difference similar to that of the anchor pair. The following instruction was given “Please, select a pair, either a top, or a bottom one, with higher translucency difference” without explicitly defining translucency. The observers had to use arrow keys of a standard keyboard to select the respective pair. The pairs were displayed atop each other on a neutral gray background. The two images within a pair were separated with a 3-pixel gap. The position of the pairs (whether the anchor is the top or the bottom one) and within each pair (left or right) was randomized. A sample comparison from the experiment is shown in Figure 1. The only difference between Experiment 1 and Experiment 2 was the anchor pair, as they were based on see-through and non-see-through anchor pairs, respectively. All other procedures were identical between the two experiments. 5 CPs × 4 directions × 5 samples per direction × 5 shapes, amounted to 500 comparisons per experiment.

We used a set of simple virtual materials, similar to those used by Urban et al. [39]. All stimuli were presented on a calibrated display using a Virtual Viewing Booth [39]. The CIE D65 diffuse light source is located on the ceiling, which is typical to what we encounter on a daily basis – both indoors and outdoors (refer to [39] for the full specification of the Virtual Viewing Booth). We used Monte-Carlo Bidirectional Path Tracer in the Mitsuba Physically-based Renderer [25] to solve the RTE. The minimum path depth was set to 20 and “Russian Roulette” termination was deployed afterwards.

In order to keep the degrees of freedom within the manageable range, identical surface roughness, scattering phase function and indices of refraction were used for all objects. In particular, we used perfectly smooth microfacet-scale surface roughness and isotropic phase function. The refractive index of the outer medium was set to 1 (vacuum), while that of materials was fixed to 1.3, which is a characteristic of water and typically has small Fresnel reflection [39] (large portion of the light is refracted towards the subsurface).

27 observers including three co-authors of this article have participated in the experiments. 18 were male and 9 were female, representing 20 different nationalities. The median age was 31 years with standard deviation equal to 10.97. All of them passed a Snellen visual acuity test to make sure that they had normal or corrected-to-normal vision. As wavelength-independent absorption and scattering yield grayscale stimuli, color vision of the observers was not tested. 21 observers had a background in color science, imaging, vision, material appearance or related fields.

As the Experiment 2 was conceived after the preliminary analysis of the Experiment 1, there was a considerable temporal gap between the start of the two experiments, which made it impossible to have all observers available for the both experiments. Besides, in order to avoid exhaustion and lack of concentration among observers, lengthy experiments have been avoided and 1000 comparisons (500 comparisons per experiment) were distributed over three different sessions, where not all observers were able to participate. Eventually, each comparison has been assessed by 20 observers. No comparisons were assessed by all observers, and not all observers were shown all comparisons. 23 participants assessed comparisons from both experiments, while 4 observers assessed comparisons from Experiment 1 only.

Both experiments were conducted on the same color-calibrated display and under the same viewing conditions. The stimuli were displayed on a 24.1 inch EIZO ColorEdge CG246 LCD, which was calibrated to CIE D65 white point with gamma equal to 2.2. Konica Minolta CS-2000 spectroradiometer was used to measure the luminance of the monitor displaying a perfect diffuser patch. The maximum measured luminance was 196 cd∕m2 with 6542 K color temperature. The monitor was warmed up for at least 30 min before each experiment.

The experiment took place in a completely dark room, where the display was the only light source. The distance between an observer and the display was approximately 60 cm. 148 × 348 pixel images were used to display the anchor pairs, occupying 3.81∘ of the visual field horizontally and 8∘ vertically. The height of the test pair images was 348 pixels, while the width varied depending on the shape. The largest image with 348 × 348 pixels was used to display spheres with the longest spikes. All test pairs occupied 8∘ vertically and 6.20∘, 6.48∘, 8.10∘, and 8.96∘ horizontally, for spheres with no, short, medium and long spikes, respectively (we ensured that the distance from the objects to the vertical edges of the image was the same for all shapes; cropping the non-occluded part of the background is unlikely to have affected the results, as the point-wise difference in these regions is negligible).

The results for all individual CPs and types of anchor are illustrated in Fig. 7.

For CP1, which is optically thin and see-through, the T50 point is determined in the vast majority of the cases. When non-see-through anchor pair was used, the T50 point was determined for all objects in all directions. The distances from the control point are generally short and they are shorter when these see-through test pairs are compared with a non-see-through anchor pair. The distance is usually largest for a spherical shape, while no clear difference is visible among other shapes.

For CP2, the distances are usually larger for a spherical shape, while no clear trend emerged for other shapes. Interestingly, T50 point is never determined for the more opaque direction (σa = σs) and the difference in both anchor pairs was usually judged larger than the difference between a CP and a test sample with higher absorption and scattering. This implies that when the mean free path is sufficiently short, shortening it further does not yield any perceptual difference.

For CP3, the estimated T50 point coordinates for a spherical object have been negative, thus, physically implausible when the experiment was conducted on a see-through anchor pair. Also, T50 is never reached when scattering decreases and absorption goes up. This can be explained with the fact that observers considered all test samples already opaque and increasing absorption, simply affected their lightness, not their translucency cues. However, we cannot rule out that the reason for failure of reaching T50 could be due to improper selection of the sample points, i.e. the range might be too small. Interestingly, this trend changes and T50 points are reached more often when a non-see-through anchor pair is used instead, which can indicate that sensitivity to see-through and non-see-through anchor pairs differs.

For CP4, the distances to T50 points are rather large and generally consistent among all shapes. For a sphere with thin long spikes, T50 was determined only when non-see-through anchor-pair was used. Further observation is that the decrease in absorption leads to more noticeable appearance difference than the decrease in scattering. This could be rooted in the fact that the magnitude of absorption of a CP is considerably smaller than the magnitude of its scattering.

For CP5, similarly to other CPs, T50 distances have been shorter and easier to determine when samples were judged against a non-see-through anchor pair. On many occasions, a scattering estimate has been negative, which can be attributed to the CP’s proximity to the scattering axis (i.e. the CP is too close to the axis and it is not sufficiently different from the sample point located on the axis; it is worth mentioning that this is the case when decreasing direction is examined; negative estimates are not obtained if all sample points are in the increasing direction, as for CPs 1 and 2). In both experiments, pairs of objects with thin parts have higher magnitude of perceived translucency difference than pairs of spherical objects. For this CP, the object with the medium-sized spikes has been the best one to detect translucency differences on.

On many occasions, T50 distances are neither reached, nor estimated with high confidence. Refer to Table II – if the T50 point was reached for a given CP in a given direction, a respective cell is marked green, otherwise, it is marked red. The table shows that T50 point was not reached 38 times (number of red cells in the respective half of the table) out of possible 100 (5 shapes × 5 CPs × 4 directions) when see-through anchor pair was used and 29 times when the experiment was based on non-see-through anchor pair. As the test pairs have been identical in both experiments, this is an indication that observers are more sensitive to background distortion cues of the see-through anchor pair when comparing it to non-see-through test materials. Additionally, the T50 point was reached for optically thin CP1 in all but one occasion in both experiments.

Most frequently, the T50 point is not reached for the spherical object and for the one with long thin spikes, while it is mostly reached for Buddha and the sphere with medium-sized spikes. A compact spherical object lacks thin parts and respective translucency cues, while Buddha per contra possesses finer details, i.e. broader range of translucency cues (as shown by [41]). These two observations are consistent with our hypotheses. However, the results for a sphere with the thinnest spikes is largely counter-intuitive. While the luminance gradient characteristic for translucency is visible on wider and thicker medium-sized spikes, it might be harder to detect on thinner spikes due to contrast sensitivity limitations – they simply occupy a smaller portion of the field of view (e.g. ref. to Fig. 4). A second explanation for this result can be found in the remarks made by the observers – some of them noted that the spikes look so different from the spherical core that they thought they were made of different materials and decided to assess just the major body of the object. While these hypotheses deserve further study, the relation between structural thickness and sensitivity to translucency differences is evidently neither straightforward, nor monotonic.

Figure 8 shows the mean distance needed to reach the T50 point for different shapes and materials. We can observe that the distance is considerably shorter for optically thin test materials and also when non-see-through anchor pair was used, being consistent with our hypotheses. Being aligned with our hypothesis, the mean distance to the T50 point is larger for a sphere than it is for Buddha and spiky objects with short- and medium-sized spikes (although not with long spikes). Figure 9 illustrates average Alpha distances to T50 points. We can see that ΔA largely accounts for the sensitivity difference between optically thin and optically thick materials. However, shape-specific adjustments are needed.

The histograms are shown in Figure 10. The summary statistics of the histograms and how they correlate with the mean T50 distances are given in Tables III–IV, respectively. Mean and median distance to the medial axis, as well as the percentiles, are significantly correlated with the T50 distances, except for the case, when the optically thin anchor pair is compared with the optically thick test pairs. However, standard deviation, skewness and kurtosis turned out to be poor correlates of the psychophysical data. The medial axis of a perfect sphere is its center and the distance to it is equal to its radius, for all surface voxels. This makes it challenging to compare a perfect sphere with other shapes in terms of histogram statistics. The data for spheres has been only considered when a given metric was applicable to spheres.

While these metrics assume a single Gaussian distribution, there are in fact two distributions of thicknesses in spiky objects – the core sphere with larger distances and spikes with shorter distances. We fitted a two component Gaussian mixture model to the histogram and report the ratio between component proportions, as well as the difference and ratio between the means of the two Gaussians. Against our expectations, neither of the three correlated with the observer data. We want to highlight that the accidental similarity in the magnitudes of geometrical thickness and T50 distances can produce high Pearson correlation, while the rank order correlation can still be insignificant (e.g. see the difference between the means of the two Gaussians when optically thin test and optically thick anchor pairs are compared).

In addition to the random human error, which will be discussed later in Section 4.7, there are three primary explanations for failure to determine the T50 points: proximity of the CPs to the axes, improperly selected sample points and non-existence of a T50 point.

In the decreasing absorption and/or scattering direction (e.g. D1 and D3 for CP5), if the CP is not noticeably different from the sample point located on the axis, the estimated T50 point will be negative and physically implausible. This has been most commonly observed for CPs 4 and 5, that have been close to absorption and scattering axes, respectively, ending in negative estimates in decreasing directions. This means that CPs, especially in optically thick regions, should be sufficiently far away from the axes, if decreasing direction is to be studied.

If sample points are improperly selected, and either all or none of the sample pair differences are noticeably larger than that of the anchor pair, the experiment is not adequate for estimation of the T50 points. We have observed that when we failed to identify a T50 point in our experiment, this was because all sample pair differences were deemed smaller than that of the anchor pair. There can be two reasons for this: either the span of the sample points we selected have been too narrow and we had to test the sample points further away from the CP in the absorption-scattering space, or because it is impossible to produce noticeable translucency difference in a given direction and the realistic T50 point does not exist. We have observed that in optically thick regions, many T50 estimations have been unrealistically high and failed goodness-of-fit test. A broader range of sample points need to be studied in the future to identify whether T50 points can be ever reached and whether the reason for failure in this case was the narrow span of the sample points.

The T50 distances that passed the goodness-of-fit test have oftentimes been larger for spherical and thicker objects than for those with thin parts. Moreover, on many occasions, the T50 distances have not been determined at all for a perfect sphere, while they had been found for other shapes.

We observed that mean and median surface-to-medial-axis distances have been better predictors of the experimental data than the measures quantifying asymmetry and distribution of thick and thin parts. However, the average thickness of the object does not adequately reflect presence or absence of the thin parts and can depend more on the scale of the object. Our observation is consistent with other research that postulates that the areas where a photon needs to travel shorter distance, such as edges [12, 22] and other fine details on the surface [31, 34, 41], contain the vital portion of the information about material translucence (refer to Figure 11). Although the two optically thick materials are far away in the absorption-scattering space, the pixel-wise difference shows that they differ in thin parts, while remain relatively similar in thick areas.

While this trend is easier to characterize qualitatively, proper quantitative modeling of the impact of shape and geometry seems to range from very difficult to infeasible at this stage. First of all, missing T50 points, possibly due to poor selection of the sample points, might have affected the existing correlation between shape descriptors and T50 distances. More importantly, as long as we do not know exactly which image regions the HVS relies on and how it weights the luminance information present in the image, we are not able to construct a shape descriptor that will correlate well with the perceived translucency differences. For instance, the thinnest spikes, against our expectations, turned out counter-productive in assessment of translucency differences. Moreover, a histogram of surface-to-medial-axis distances is viewpoint-blind and accounts for all spikes of the shape. However, we do not know, whether they are equally important [31], as for instance, the spikes located on the backside of the object might have a negligible role. However, this limitation is less important in real-life scenarios, as humans can interact with the objects and inspect them from all different geometries [18].

Thin parts that are represented as bumps considerably affect the surface topography. They generate shadows and provide additional cues about the surface geometry of the object (refer to Figure 12). Although the respective objects are made of the identical materials in both pairs, the difference becomes more apparent in the right pair, as the bumps of an optically thick material produce sharper shadows than its optically thinner counterpart. Similar image contrast resulting from a surface relief has already been shown by Xiao et al. [42] to be diagnostic for optical thickness of the material. Marlow et al. [28] have demonstrated that when surface and shading co-vary (as in the right image of the right pair in Fig. 12), the material is perceived as opaque, and when they do not, perception of translucency is evoked (left image in the right pair, Fig. 12). However, a simple spherical shape with no concavities or bumps leaves less room for observing this kind of surface-shading co-variance and leaves more room for interpretation. This may have implications for material appearance research. If a sphere lacks translucency cues, it may be a bad idea to have it as a shape of choice in psychophysical experiments, being consistent with [23].

As the HVS supposedly relies on low level image cues for assessing translucency [12], we believe that careful analysis of the image structure in future works will provide answers as to why geometrical and optical thickness affect perceived translucency differences. It has been proposed that the most informative parts about translucency are edges and thin parts (as shown in Fig. 11), as well as the concavities and convexities that are shadowed in opaque materials and lighter in translucent ones (as shown in Fig. 12) [20]. Although the in-depth analysis of the image statistics is beyond the scope of this work, in Figure 13 we demonstrate how the difference in luminance distribution varies across shapes and optical thickness levels. The figure shows that the pixel-wise difference is larger in optically thin materials, even though the Euclidean distance in the absorption-scattering space is shorter than for illustrated optically thick pairs. For optically thick materials, the difference is smaller and exhibited near the concavities (thin parts) only.

We have observed that when the magnitude of absorption and scattering is low, a smaller change is needed in absorption and scattering properties to notice a translucency difference. This observation is consistent with the findings by Urban et al. [39] and is accounted for by A (see constant A curves in Fig. 3 of [39]). This fact is also consistent with Stevens’ law [37] (similarly to [40]) and involves important implications for translucency perception research. Seemingly, the HVS is more sensitive to background contrast and background blur cues present in see-through objects and materials [35] than it is to luminance contrast variation, which proposedly is a translucency cue for objects and materials that do not permit seeing through [12, 22, 26, 28, 31, 41]. See-through cues overtake and outweigh luminance contrast variation cues when both are present. This has the following implications:

The observers noted in the post-experiment interviews that they relied on background distortion in see-through images, while there was hardly any cue apart from lightness in non-see-through objects and materials. Several observers noted that they always considered the difference in the see-through anchor pair to be larger, because “the test pair was composed of two opaque materials, meaning that both objects in a pair had zero translucency and thus, were not different by translucency”. To some the test pairs look like “two fully opaque billiard balls with two different colors”. This supports the proposal by Fleming and Bülthoff [12] that the HVS has poor ability to invert optics. It also seems that if the mean free path is very short, subsurface light transport is not noticeable at all. However, it remains an interesting open question why observers relied on lightness or brightness as a translucency cue. Lightness is natural to highly scattering media and it is characteristic to many translucent materials, such as snow and milk. Therefore, we cannot rule out that lightness itself is a cue for perceived translucency. Geometric information [28] can be important if the surface geometry is complex. For instance, observers might have compared lightness of the spikes, when they were present. However, for simple shapes, such as spheres, regional variation of the luminance distribution is minimal. This can explain why some observers use global mean luminance for assessing translucency, while others simply consider the material to be opaque.

We hypothesize that there might be an interaction between geometrical thickness and optical thickness effects. This hypothesis is derived from the notion that background distortion is a strong cue for assessing translucency. However, visibility of the background does not only depend on the optical thickness of a material, but also on its surface geometry. For some object shapes, the background is not visible even if the material is optically very thin (see Figure 14).

The special case of the lower visual sensitivity to absorption-scattering changes in optically thick materials is when both absorption and scattering are increased. In this direction the T50 point was never reached. As discussed above, luminance contrast is a cue used for distinguishing levels of translucency between the objects. While scattering and absorption coefficients are positively correlated with brightness and blackness, respectively [9], if both of them are increased equidistantly, there is a diminishing return effect in the resulting luminance contrast and the overall look remains roughly unchanged (refer to the left pair in Fig. 12). This is consistent with the lightness reflectance measurements conducted by Urban et al. [39] (see Fig. 3 in [39]). When the mean free path is short, a photon cannot traverse through a thick body of the material and the penetration depth into the volume is negligibly small. Therefore, shortening the mean free path further does not yield any perceptual difference. However, the mean free path becomes increasingly important as thinner and finer details are introduced in the shape.

The visualization of the results in absorption-scattering and Alpha spaces shows that ΔA accounts for sensitivity differences between optically thin and optically thick regions relatively well. On the other hand, ΔA is not an adequate metric to reflect shape-specific effects on perceived translucency differences. The value of the ideal translucency difference metric should be identical for all shapes and all pairs of control-T50 points. A is a shape-independent material property. However, the arrangement of A in space depends on the shape. The psychometric function used in the definition of A was measured on Buddha shapes [39]. Translucency cues that are present in the Buddhas may be absent in a sphere or other shapes – therefore, their psychometric function may be different. ΔA should be adjusted so that it could accommodate shape-dependent effects on perceived translucency differences.

If we draw a parallel with colors, CIELAB is suitable to quantify color for defined and fixed viewing conditions. However, for color differences, various “parametric factors” need to be taken into consideration [24, 27] (such as distance between the patches, luminance level, presence of texture etc.). For instance, the CIEDE2000 color difference formula [27] can be fitted to parametric factors using k-values. We believe that a phenomenologically similar parametric factor for perceived translucency differences is the object’s shape, which ΔA should be parametrized for. We hypothesize that the necessary features for parametrization can be extracted from shape descriptors in future work.

Interestingly, T50 distances in the Alpha space have not been identical even for Buddha shapes that A was defined on. This can be an indication that besides parametric factors, inter-observer differences also exist.

Further challenge is that the success or failure of the fitting procedure does not depend only on the observers’ sensitivity. We cannot assume that the experiment participants are ideal observers, i.e. identical observers always giving an identical response to the identical stimuli. Usually there are sources of variation in the observer responses that we have not accounted for. The most significant source of such kind of noise can be the dissimilarities in internal representations of translucency among different observers. This could potentially explain why the authors’ visual judgements in the stimuli selection process did not generalize to a larger group of naïve observers. For instance, the authors being biased by the familiarity with the stimuli rendering process, considered lightness difference as a cue to translucency difference, because they knew surface reflectance had been identical. Naïve observers, unaware of this fact, attributed lightness differences to surface reflectance instead of subsurface scattering (“two fully opaque billiard balls with two different colors”). And the fact that different observers assessed different stimuli does not permit us to assess the statistical power of the study. We do not exactly know to what extent the failure to determine T50 points can be attributed to random error in observer responses. Inter-observer variability is accounted for in the Probit model, which assumes that observer responses are normally distributed. The analysis presented above is limited with this assumption. Intra-observer variability can be measured in future studies if the same experiment is repeated with the same observers.

However, examining the raw data might provide deeper insight into the reasons of T50 estimation failure. Table II shows the number of observers that considered the difference between a CP and the test samples in a given direction to be larger than that of the anchor pair. In many cases, observer responses exhibit apparently non-monotonic behavior, which is counter-intuitive, because it is usually expected that a larger difference in subsurface scattering properties yields a larger, or at least similar, but not smaller translucency difference. In this case, random error due to observer confusion and differences in internal representations of translucency are very likely to have played an important role in the failure to determine a T50 point. For instance, a spherical shape for CP2 compared with an opaque anchor in the D4 direction exhibited high non-monotonicity, which was the reason why the estimated T50 point failed a goodness-of-fit test. Although non-monotonicity can be to some extent filtered away with a method used by Berns et al. [3], collecting more data in the future will enable us conduct bootstrapping, cross-validation and other resampling methods to shed more light to the extent of the noise.

Despite these non-monotonous results, there are particular cases, where the overwhelming majority agreed that the anchor pair difference was larger even for the furthermost sample. For instance, for spherical shape of CP2, all but 1 observer (95% of all observers) considered a transparent anchor pair difference to be larger in D3, while the number decreased for other shapes. In this case, a random error is unlikely to have played a central role. It is possible that the T50 point does not at all exist for a spherical shape, but may exist for other shapes, which was not reached due to ill-selection of the sample points. This is intuitive, because if the mean free path is already short enough for a thick compact object to appear fully opaque, shortening it further will never impact luminance contrast in a perceptually meaningful way, while shortening mean free path might have considerable impact if the object has thin parts.

This study comes with multiple limitations that need to be addressed in the future. All experiments were conducted with the same scattering phase function, index of refraction, microfacet-level surface roughness and illumination geometry. These factors are known to modulate perceived magnitude of translucency [12–14, 23, 41] and we cannot rule out that our observations do not hold for other materials. We also assumed wavelength-independent absorption and scattering. Although it has been demonstrated earlier that translucency can be assessed without chromatic information [6, 12], in our daily lives we hardly ever encounter materials with wavelength-independent absorption and scattering; the HVS might not be properly trained to assess this kind of stimuli [6, 7].

The most significant limitation of the study is the fact that T50 points have not been determined in 33% of the cases (38 and 29 times for optically thin and thick anchor pairs, respectively, out of 200 total). Therefore, all observations need to be considered with caution. Ideally, if T50 points were determined for all shapes, anchor pair types, center points and directions, the comparison among them would have been a more straightforward task. The work is based on the assumption that at least one sample point was smaller than that of the anchor pair, and larger for at least one other sample point. If that assumption does not hold, e.g. if all test pair differences are noticeably smaller than that of the anchor pair, the conducted experiment is not a suitable for determination of T50 points. This problem has been observed by Urban et al. [39] who note that T50 estimations, which failed the goodness-of-fit test, “were characterized by a test sample choice spanning a too small visual translucency difference interval so that many observers found all test pairs in this direction to have a smaller translucency difference than the anchor pair”. We also observed the similar phenomenon – if T50 was not determined, it has been because all test pair differences where considered smaller than the anchor pair by the vast majority of the observers. We admit that our methodology of stimuli selection is limited and neither Alpha differences, which are defined on Buddha shapes, nor visual judgements of the authors, can be generalized to all shapes and all naïve observers, respectively. However, this study is motivated by the lack of intuitive perceived translucency space and limited understanding of how absorption, scattering and shape affect perceived translucency differences. This leaves visual inspection by authors in a trial-and-error manner still the most reliable technique, to the best of our knowledge. Moreover, as we hypothesize that perceived translucency differences vary among shapes and the levels of optical thickness, we cannot assume that there exists a unique set of test sample materials that allow estimation of T50 points for all shapes and all anchor pairs at the same time. It might well happen that for one shape, a change in absorption and scattering properties in a given direction never produces larger-than-anchor-pair difference thus, the realistic and physically plausible T50 does not exist for this shape, while this is not necessarily true for another shape. In this case, if T50 point is determined for one shape but not for another one, while both shapes are made of the identical sets of materials, this can be an indication that first shape facilitates detection of translucency differences. Similarly, if the difference between the same test pairs is never considered larger than that of see-through anchor pair, but the very same test pair differences are considered larger than non-see-through anchor pair difference, this might imply that the latter anchor pair difference is smaller than the former. In this case, determining T50 point for one anchor pair and the lack of thereof for another can be also an indication of different sensitivities between them. Furthermore, missing T50 points can still potentially provide insight into differences between optically thin and thick regions. If in optically thin region, T50 distance T50thin = X, but in the optically thick region, we have not reached T50 point within the range of Y , where Y > X, because all sample pair differences have been deemed still smaller than the anchor pair difference, this indicates that T50thin≠T50thick, and if T50thick exists, then T50thick > T50thin.

Whether missing T50 points have not been determined because the span of the test sample materials was too narrow, or because the realistic T50 points for a given shape and direction simply do not exist, is an open question that needs to be addressed in future studies. It is also worth noting that T50 points have been determined for the optically thin CP1 with a considerably smaller range of sample points, while for other CPs, T50 points were not reached even with a broader range of sample points, because the vast majority of the observers considered all sample pair differences to be smaller than that of the anchor pair. If T50 points exist for those CPs, they should be larger than the maximum span of the current sample points. In this case, even without reaching T50 points with other CPs, it is evident that T50 distances are shorter in optically thin regions.

Besides, our ability to model the correlation between sensitivity to perceived translucency differences and geometric shape are inherently limited by two factors: our knowledge on how the HVS selects the image regions to deduce translucency; and the availability of proper shape descriptors. For instance, we do not know whether the HVS relies on the information in all visible spikes, or if even a single spike is enough. Nagai et al. [31] have observed that different individuals rely on different image regions to assess translucency that might complicate definition of a robust shape descriptor for translucency differences.

Apart from that, we have used a Buddha shape as an anchor pair. Presence of thin parts in the anchor pair Buddhas might itself have increased sensitivity to anchor pair differences. Additionally, Gigilashvili et al. [18] observed that cross-shape translucency comparison is generally found challenging by observers. Hence, we believe that future studies should consider differently-shaped anchor pairs.

There might be some unintended noise in the data due to the interpretation of the object composition. For some materials the appearance of the sphere proper and its spikes is so different that a couple of observers thought that the spikes were made of a different material and thus, they decided to assess a core sphere only.

Finally, the remarks made by some observers about opacity of optically thick materials opens up a discussion where the conceptual boundary lies between translucency and opacity (see [19] for further discussion).

We believe future work should evolve towards three objectives: firstly, shape descriptors, which could correlate with detectability of perceived translucency differences, should be developed. We hypothesize that identification of image cues to perceived translucency could facilitate construction of such a descriptor. Subsequently, shape-related effects should be incorporated in translucency difference formulae. Finally, a perceptual translucency space should be constructed that could permit navigation in nearly perceptually uniform units instead of highly perceptually non-uniform absorption and scattering coefficients [16].