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Article  |   November 2019
The influence of Fresnel effects on gloss perception
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Journal of Vision November 2019, Vol.19, 1. doi:https://doi.org/10.1167/19.13.1
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      Franz Faul; The influence of Fresnel effects on gloss perception. Journal of Vision 2019;19(13):1. doi: https://doi.org/10.1167/19.13.1.

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Abstract

Glossy surfaces reflect a mirror image of the environment. The perceived gloss depends (a) on the blurriness of this mirror image, which is a function of surface roughness, and (b) the strength of the mirror reflection, which, according to Fresnel's equations, is a function of the material's refractive index and the angle of the incident light. Investigations on gloss perception often used simplified reflection models, e.g., the Ward model (Ward, 1992), which do not correctly account for Fresnel effects. Here, possible perceptual consequences of this simplification are investigated in three experiments, in which the gloss impression produced by a physically more plausible reflection model (Fresnel–bidirectional reflectance distribution function [BRDF]) is compared to the gloss produced by two variants of the Ward model under identical conditions. The results show that it is, in general, not possible to match the gloss impression elicited by a Fresnel-BRDF with a Ward-BRDF. Furthermore, compared with the Ward-BRDF, the gloss impression produced with the Fresnel-BRDF under identical conditions is, in general, stronger, more vivid, and more realistic. Gloss constancy is also improved, i.e., the gloss impression depends less on the type of illumination, the presence and properties of a floor, and surface shape. These differences are especially evident with relatively homogeneous illuminations. The results of a fourth experiment, which tested gloss constancy under changes in illumination and shape with a matching task, confirm an improved gloss constancy with a Fresnel-BRDF. Together, these findings suggest that Fresnel effects are used as a cue in gloss perception.

Introduction
Many surfaces in the environment are perceived as glossy. It is, therefore, somewhat surprising that, for a long time, there was relatively little research on gloss perception (Adelson, 2001; Chadwick & Kentridge, 2015; Fleming, 2014). The situation has improved in recent years, not least because of the availability of fast computers that make it possible to generate the complex stimuli necessary for these investigations. 
Most of the existing work on gloss perception either focuses on the stimulus conditions that determine the occurrence, strength, and quality of gloss impressions or on gloss constancy, i.e., the degree to which a gloss impression is stable across changes in context, for instance, across different object shapes or illuminations. A further important topic is the influence of gloss on the perception of other scene properties, for instance, on the recognition of the shape or color of objects (Fleming, Torralba, & Adelson, 2004; Xiao & Brainard, 2008). 
In an early influential study on the stimulus conditions for perceived gloss, Beck and Prazdny (1981) showed that adding a highlight to a diffuse surface in a static image may elicit a spatially extended gloss impression in the neighborhood of the highlight. Berzhanskaya, Swaminathan, Beck, and Mingolla (2005) later determined the extent of the induced glossy-appearing region. A very important property of a highlight is the sharpness of its border, which is positively correlated with perceived glossiness (Di Cicco, Wijntjes, & Pont, 2019; Marlow & Anderson, 2013; Sève, 1993). However, many more highlight properties, conditions that systematically influence them, and correlated changes in perceived gloss have been identified and investigated. Some of these factors are the illumination type (Pont & te Pas, 2006; Wendt & Faul, 2017), highlight color (Hanada, 2012; Nishida et al., 2008; Wendt & Faul, 2018; Wendt, Faul, Ekroll, & Mausfeld, 2010), object motion (Doerschner et al., 2011; Wendt et al., 2010), binocular disparity in stereo vision (Blake & Bülthoff, 1990; Kerrigan & Adams, 2013; Wendt, Faul, & Mausfeld, 2008), and the consistency with form information (Beck & Prazdny, 1981; Kim, Marlow, & Anderson, 2011; Marlow, Kim, & Anderson, 2011; Todd, Norman, & Mingolla, 2004). Most of these works used only a few localized light sources, often point lights, whose mirror images on the surface are isolated highlights. 
Since Debevec (1998) introduced “image-based lighting” into computer graphics, investigations on gloss perception increasingly used more realistic global illuminations, which have the property that the light comes from all directions. From a technical point of view, this approach uses high dynamic 360° panoramic images of a 3-D scene (“illumination maps”) to describe the global light field. Consequently, the mirror image of the illumination is no longer a sparse pattern of isolated highlights, but covers the whole surface of the object and shows a (usually distorted) image of the surround. Only global illumination makes it possible to depict highly glossy metallic objects, such as a polished chrome sphere, and it also considerably increases the realism of the gloss impression in highly glossy nonmetallic opaque objects, such as a billiard ball. Some highlight properties that were known to influence gloss perception could be easily transferred to the global illumination case. For example, the fuzziness of isolated highlights translates into the blurriness of the mirror image, and both increase with surface roughness. However, the complexity and variability of the mirror image made it much more difficult to identify critical properties that influence perceived gloss. One approach referred to specific statistical properties of either the illumination (Dror, Willsky, & Adelson, 2004; Fleming, Dror, & Adelson, 2003) or the resulting image (Motoyoshi, Nishida, Sharan, & Adelson, 2007; Sharan, Li, Motoyoshi, Nishida, & Adelson, 2008). But such approaches seem to have some limitations, at least in their present form (for a critical evaluation of the image statistics approach, see Anderson & Kim, 2009; Kim & Anderson, 2010). In a different approach, Marlow and colleagues (Marlow & Anderson, 2013; Marlow, Kim, & Anderson, 2012) tried to identify image cues that are predictive of a surface's gloss level, such as the contrast, sharpness, and “coverage” of specular reflections. Kim, Marlow, and Anderson (2012) also point to the fact that not only highlights but also dark regions influence the gloss impression. 
A second, closely related line of research focused more on gloss constancy. In a pioneering study, Nishida and Shinya (1998) found that the gloss impression depends strongly on object shape. More specifically, the perceived glossiness of an object of fixed material tends to increase with the curvature of the surface. These authors used static gray-level stimuli and point light sources. Wendt et al. (2010) found that adding color, motion, and disparity information improved gloss constancy in such situations, but the overall degree of constancy remained low. Later investigations that used more natural global illuminations again confirmed these results (Olkkonen & Brainard, 2011; Vangorp, Laurijssen, & Dutré, 2007). In addition, relatively low degrees of gloss constancy across different illuminations were found, especially with relatively localized “artificial” illuminations (Fleming et al., 2003). But, even with natural global illuminations, this type of gloss constancy was found to be limited (Adams, Kucukoglu, Landy, & Mantiuk, 2018; Olkkonen & Brainard, 2011). 
Aim and outline of the present study
The present investigation was motivated by an observation made in attempts to replicate some published results on gloss constancy, namely that the stimuli used in these previous studies often produced a rather weak, unrealistic, and inhomogeneous gloss impression. Informal tests led to the hypothesis that this is due to the fact that a simplified reflection model proposed by Ward (1992) was used to render the stimuli. This model ignores “Fresnel effects,” i.e., shape-dependent intensity variations of the mirror image of the surround that can be seen on glossy surfaces. In line with this hypothesis, the gloss impression was often much improved if a more realistic reflection model was used that correctly takes Fresnel effects into account (see Figure 3 right). 
It was clear that this observation, if confirmed, could be relevant for both lines of research outlined above: Regarding the stimulus conditions for perceived gloss, it suggests that the systematic changes in the strength of the mirror image visible on the surface of specularly reflecting objects are used by the visual system as a cue for glossiness. This cue could complement other cues that have already been described in the literature and would presumably be especially relevant in situations with global illumination because it should be relatively easy to detect variations in specular reflection strength when the mirror image of the surround covers the whole surface of the object. The observation also seemed to be of high relevance with regard to gloss constancy, as the neglect of Fresnel effects—or, from a different perspective, the violation of a fundamental physical regularity in the stimuli—can also have detrimental effects on gloss constancy. In particular, it is well possible that the low degrees of gloss constancy that have been observed with global illumination do not point to limitations of gloss perception, but are instead the result of a physically incorrect stimulation. 
In the following, I first summarize some relevant physical regularities of glossy materials. Then possible problems of the Ward model are dealt with as this specific model has been used in many psychophysical studies on gloss perception. Subsequently, three experiments are reported in which the gloss impressions achievable with a Ward-type model and with a physically more plausible reflection model with correct Fresnel effects (“Fresnel model”) are compared under otherwise identical conditions. The results of these experiments reveal that the gloss impressions elicited with the Ward model and the Fresnel model under comparable conditions can deviate substantially in both qualitative and quantitative ways. They also suggested that the degree of constancy across several types of context changes tested in the experiment is much larger with a Fresnel model. This prediction was tested (and confirmed) in a fourth experiment, in which the degrees of gloss constancy that can be achieved with and without correct Fresnel effects were directly compared. Finally, the present results are briefly discussed with respect to their relations to previous findings, their significance, and possible consequences for further research. 
Physical properties of glossy surfaces
From a physical point of view, the materials involved in gloss perception can be assigned either to the class of (electrically conductive) metals or to the class of dielectric (nonconductive) surfaces. Typical examples of glossy dielectrics are transparent materials, such as glass, water, clear lacquers, or transparent plastics. In nonmetallic opaque glossy surfaces, the gloss impression is often due to the fact that an opaque base material is seen through a thin transparent layer. By combining different types of base materials (including metallic ones) with one or more transparent layers, materials with complex reflection characteristics can be produced (Jakob, d'Eon, Jakob, & Marschner, 2014; Weidlich & Wilkie, 2007). 
The specular reflections of metals and dielectrics are similar in many respects but differ significantly in how the strength of the reflection depends on the angle of incidence of the light. This angular dependence of the reflection strength is described by Fresnel's equations. In the following, the term “Fresnel effects” always refers to this directional dependence of specular reflection strength. 
The present investigation is primarily concerned with gloss perception in dielectric materials and, within this class, with opaque surfaces in which a diffuse reflecting base material is combined with a transparent coating. The central question is whether and in what way the directional dependence of the specular reflection strength—i.e., Fresnel effects—influences perceived gloss. 
As already mentioned, many influential studies on gloss perception with global illumination (e.g., Ferwerda, Pellacini, & Greenberg, 2001; Fleming et al., 2003; Marlow & Anderson, 2013; Olkkonen & Brainard, 2010, 2011; Pellacini, Ferwerda, & Greenberg, 2000; Vangorp et al., 2007) used reflectance models, which do not correctly account for Fresnel effects, in particular variants of the Ward model. It has long been known that simple reflection models, such as the Ward model, do not describe real materials very accurately. In an experimental study in which different reflection models were compared with each other and also with measured data of real objects, Ngan, Durand, and Matusik (2005, p. 6) found clear deficiencies of such models: “The Ward, Ward-Duer and Blinn-Phong images are all noticeably different from the original data. The highlights near the center of the sphere are significantly brighter than the original, and near grazing angle the highlights are much less pronounced. Of the three models, Ward-Duer has the strongest highlight near grazing angle, but it is still noticeably dimmer than the measured data.” 
Reflection properties of glossy surfaces
The reflection properties of surfaces are theoretically well understood and can be simulated relatively easily in good approximation. Such a simulation can be performed using freely available render programs, e.g., the Mitsuba renderer (Jakob, 2010). A detailed description of the computational, geometrical, and physical principles underlying such simulations can be found in Pharr, Jakob, and Humphreys (2016) and Glassner (1995). Here some selected aspects, which are relevant for further considerations, are summarized. 
The reflection behavior of opaque surfaces can be described by the so-called bidirectional reflectance distribution function (BRDF) f(p,ωi, ωo, λ). The BRDF indicates how a light beam from direction ωi incident on surface point p is spatially distributed into the hemisphere above the surface. Specifically, the value of f corresponds to the relative amount of light that is reflected in direction ωo. Considered across all ωo, this is the reflection distribution for the given direction of incidence. The BRDF also depends on the wavelength λ of the light, but often, the same function type is assumed for all wavelengths (and positions p). This justifies the use of the short form: f(ωi, ωo) = f(θi, ϕi, θo, ϕo), which only takes the dependency on the directions into account. In the right term, the directions are alternatively expressed by the polar and azimuth angles θ and ϕ
A physically correct BRDF has two essential properties, namely (a) energy conservation, i.e., the total reflected light energy cannot be larger than the energy of the incident light, and (b) the so-called Helmholtz reciprocity, which means that the directions of incidence and reflection of the light can be swapped in the BRDF. In this reading, i.e., with reversed incidence and reflection directions, the BRDF indicates to what extent light incident from different spatial directions contributes to the light reflected from the surface in a certain direction. 
The BRDF of an “ideal specularly reflecting” surface is particularly simple: For each direction of incidence, the light is reflected in exactly one direction (“mirror direction”), which lies in the plane spanned by the incident light beam and the surface normal, whereby the angle of reflection is equal to the angle of incidence. This ideal reflection only occurs when the surface is completely smooth. If the surface is roughened, then an incident light beam is still primarily reflected in a direction close to the “mirror direction,” but the result is a lobe-shaped distribution with a diameter that increases with roughness. Analytical models, which describe this dependence of specular reflection on roughness (e.g., Cook & Torrance, 1982; He, Torrance, Sillion, & Greenberg, 1991), consider a surface as composed of planar microfacets with ideal specular reflection. Surface roughness is expressed by the variance of the directions of the microfacets' normal: It is zero for a completely smooth surface and increases with increasing roughness. The directional dependence of the specularly reflected light caused by surface roughness is the same for metals and dielectric surfaces. 
On an ideal specularly reflecting surface, a sharp mirror image of the environment can be seen, which, in general, is distorted depending on the shape of the surface. With increasing roughness, the mirror image becomes increasingly blurred, and the surface appears correspondingly increasingly matte. The perceptual correlate of increasing surface roughness ranges from highly glossy to semi-glossy to matte and is an essential dimension of the perceived gloss strength of a surface. 
Metallic surfaces reflect light only specularly and the associated BRDF is, therefore, a “lobe” of varying width. The glossy materials that are of interest here, on the other hand, also have a diffusely reflecting component. Because the BRDF of a perfectly diffuse surface is a hemisphere for each direction of incidence, which means that it scatters incident light equally in all spatial directions, the combined BRDF that describes both diffuse and specular reflection has roughly the shape of a hemisphere with a lobe (see Figure 1). 
Figure 1
 
Typical BRDF of a glossy surface that combines diffuse and diffuse specular reflection. The color of the surface depends on the diffuse (hemispherical) part of the BRDF, the strength and blurriness of the mirror reflection on the specular part (lobe). The narrower the lobe, the sharper the edges in the mirror image and the glossier the surface appears.
Figure 1
 
Typical BRDF of a glossy surface that combines diffuse and diffuse specular reflection. The color of the surface depends on the diffuse (hemispherical) part of the BRDF, the strength and blurriness of the mirror reflection on the specular part (lobe). The narrower the lobe, the sharper the edges in the mirror image and the glossier the surface appears.
Fresnel's equations describe the dependence of the strength of the specular reflection on the viewing angle and the refractive index of the material. With metals, the reflection is hardly dependent on the viewing angle. The mirror image of the environment seen in a chrome sphere is therefore of approximately the same strength at all surface locations. However, the reflection strength is clearly wavelength-dependent, which causes the characteristic color of different metals. Conversely, in the case of dielectrics considered here, the reflection intensity depends strongly on the direction of incidence of the light and the refractive index but hardly on wavelength. The latter property is due to the fact that the refractive index of dielectrics is almost constant in the wavelength range of visible light. For this reason, the color of the specularly reflected light is, to a first approximation, the same as the color of the incident light. 
For dielectric materials, the reflection strength R is minimal for orthogonal incidence (incidence angle α = 0), then increases strictly monotonously and nonlinearly with the angle of incidence and reaches 100% at the maximum angle of 90° (see Figure 2). Only the remaining light T = 1 − R that has not already been reflected can then be reflected diffusely. This means that the ratio of diffuse and specular reflection changes with the angle of incidence. As shown in Figure 2, a variation of the refractive index mainly changes the initial level at an angle of incidence of 0°, whereas the rest of the curve is quite similar. In particular, at 90° always 100% of the light is reflected. The refractive index of real materials lies between 1.0 (approximately air, no reflection) and somewhat below 3.0. Common refractive indices in everyday life are 1.33 for water and 1.5 for typical (acrylic) glass. 
Figure 2
 
The strength of the reflection of a ray of light hitting a dielectric surface from a medium with a refractive index of n0 = 1 (air) as a function of its angle of incidence relative to the surface normal and the refractive index n1 of the material.
Figure 2
 
The strength of the reflection of a ray of light hitting a dielectric surface from a medium with a refractive index of n0 = 1 (air) as a function of its angle of incidence relative to the surface normal and the refractive index n1 of the material.
Figure 3
 
Comparison of the specular (left) and the diffuse (center) components of the BRDF of a highly glossy material that reproduces Fresnel effects correctly (top) or incorrectly (Ward model, bottom). The specular reflections on the left side are exaggerated in brightness to make the differences easier to see. If the components are displayed correctly, the sum of the left and middle images results in the right image. The geometry of the reflection layer is identical in both cases (and, if applicable, also its blurriness that depends on surface roughness).
Figure 3
 
Comparison of the specular (left) and the diffuse (center) components of the BRDF of a highly glossy material that reproduces Fresnel effects correctly (top) or incorrectly (Ward model, bottom). The specular reflections on the left side are exaggerated in brightness to make the differences easier to see. If the components are displayed correctly, the sum of the left and middle images results in the right image. The geometry of the reflection layer is identical in both cases (and, if applicable, also its blurriness that depends on surface roughness).
It has repeatedly been suggested to consider the ratio of specular to diffuse reflection as a further dimension of the gloss impression (“contrast gloss,” Hunter, 1937), whereby it was assumed that perceived glossiness increases with an increasing ratio of specular to diffuse reflection. This is problematic, however, because the (relative) strength of the two types of reflection is not a material constant and cannot be assigned to a specific location on the surface, but depends on the angle of incidence of the light. Despite the varying reflection intensity due to Fresnel effects—and presumably even because of this—the gloss impression is usually very similar at all points on the surface of an object. 
For real materials, the ratio of specular to diffuse reflection can change in two ways depending on the material properties: (a) by changing the refractive index, whereby the specular portion increases with the refractive index, and (b) by changing the albedo, i.e., the strength of the diffuse portion. The darker the material, the more dominant the specular component in the mixture. In both cases, however, these are global changes that do not eliminate the spatial variation of the reflection intensity. 
The Ward reflection model
As already mentioned, the reflection model of Ward (1992) was frequently used in studies on gloss perception. However, this model has some limitations that can lead to severe deviations from the physically correct situation. 
The model proposed by Ward (1992) was intended as a simple and efficient approximation to physically more plausible models, which could be used in so-called raytracing methods for the photorealistic rendering of three-dimensional scenes. This simplification was justified in view of the limited computing power of the computers at that time, especially because the model often leads to visually appealing results. 
The Ward-BRDF for isotropic materials (to which the investigation is limited here) has, in addition to the diffuse component ρd, which determines the color of the surface, two further parameters for the gloss component, namely a roughness parameter α and a specular reflection parameter ρs:  
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\begin{equation}f({\theta _i},{\phi _i},{\theta _o},{\phi _o}) = {{{\rho _d}} \over \pi } + {\rho _s}{{\exp ( - {{\tan }^2}h/{\alpha ^2})} \over {4\pi {\alpha ^2}\sqrt {\cos {\theta _i}\cos {\theta _o}} }}{\rm {,}}\end{equation}
where h is the half angle between ωi and ωo. Roughly speaking, the roughness parameter determines the diameter of the lobe and the reflection parameter its length. In the retinal image of the surface, they influence the blurriness and the brightness of the mirror reflection, respectively.  
The original Ward model has the problem that it is not energy conserving. However, this deficiency can be remedied by a modification of the model (Geisler-Moroder & Dür, 2010). 
Another problem is that the strength of the specular component takes Fresnel effects not correctly into account. The energy-conserving variant proposed by Geisler-Moroder and Dür (2010) extends and is very similar to an earlier modification of the Ward model by Dür (2006). The latter was found to approximate some aspects of the correct Fresnel effects more accurately but to still deviate substantially from the measurements (see the reference to the Ward–Duer model in the earlier quote from Ngan et al., 2005). 
A further, closely related deficiency is that the specular component is simply added to the diffuse component instead of convexly mixing the diffuse and specularly reflective components according to a factor determined from Fresnel's equations (see the illustration in Figure 3). 
A “psychophysical” Ward model
A problem with most physical models of material properties is that their parameters are difficult to relate to their perceptual correlates. In the case of the Ward model, this potentially affects both the roughness parameter α and the specular reflection parameter ρs. Pellacini et al. (2000) have, therefore, tried to construct a perceptually equidistant reparameterization of the Ward model. 
They used images of spherical objects placed in a closed cube whose inner walls had a checkerboard pattern. They simulated 27 objects that differed only in the parameters of the Ward model used, i.e., in ρd, ρs, and α. Their subjects assessed the perceived differences in gloss in all 278 possible pairs of these objects. They then applied multidimensional scaling to this data to search for appropriate perceptual dimensions of the gloss impression. In this way, they identified two dimensions, which they called d for “distinctness of image gloss” and c for “contrast gloss,” in accordance with the categories of gloss proposed by Hunter (1937). In a second step, they determined an equidistant scale along these dimensions. For d, the result was d = 1 − α, i.e., this dimension is directly linked to the roughness parameter α and controls how blurry the contours of the mirror image are. For the contrast parameter c, the result was Display Formula\(c = \root 3 \of {{\rho _s} + {\rho _d}/2} - \root 3 \of {{\rho _d}/2} \), i.e., c depends on the relative strength of the specular and diffuse reflections. 
In the abovementioned studies, the Ward model was used not least because the psychophysical gloss model promised a simple construction of the stimuli and a reliable adjustment of the gloss by the subjects. Vangorp et al. (2007, p. 3) explicitly write, “The main reason for choosing the Ward model over the tabulated measurement data or other analytical models is because a perceptually meaningful reparameterization exists [Pellacini et al. 2000]. It can be used to generate perceptually uniform gloss variations on the base materials.” However, because this is only a reparameterization of the original Ward model, it inherits all the weaknesses of the original model. 
Empirical study on the influence of Fresnel effects on gloss perception
The central aim of the first three experiments of the present study was to determine how Fresnel effects influence gloss perception. To this end, the Ward model in its original form (Ward, 1992, “Ward-BRDF”) and a modified version of it with energy conservation (Geisler-Moroder & Dür, 2010, “balanced Ward-BRDF”) were compared with a physically more plausible model with correct consideration of energy conservation and Fresnel effects (Walter, Marschner, Li, & Torrance, 2007, “Fresnel-BRDF”). In these comparisons, all parameters of the scene, the object, and the material were chosen to be as similar as possible for all BRDFs to guarantee that the only difference between the models concerns the spatial distribution of the reflection strength across the surface. With the Fresnel-BRDF, this distribution is physically correct, i.e., conforms to Fresnel's equations, whereas this is not true for the distributions produced with the Ward-BRDFs. 
The balanced Ward model was included in the investigation to distinguish between possible detrimental effects of a violation of energy conservation (which is fixed in the balanced model) and deviations from correct Fresnel effects that result (albeit to different degrees) from both Ward models. All three BRDFs are implemented in the rendering software Mitsuba (Jakob, 2010) so that all can be calculated under comparable conditions. In Mitsuba, they are called “Ward,” “balanced Ward,” and “roughplastic,” respectively. 
Informal observations and previous investigations suggest that the influence of a specific reflection model on the gloss impression depends on numerous context factors. This is also to be expected for the Ward model. The very fact that it has been widely used in applications and research indicates that there are many situations in which it leads to convincing gloss impressions. The following six factors were considered plausible candidates for a moderating influence on the gloss impressions and were, therefore, systematically varied: the shape of the object; the type of illumination; the existence and nature of a floor on which the object rests; and the material properties themselves, i.e., the strength of the diffuse reflection (“albedo”), the value of the refractive index, and the roughness of the surface. 
In Experiment 1, the subjects were asked to match the gloss impression produced by the Fresnel-BRDF with both Ward models by adjusting the specular reflection strength parameter ρs of the Ward-BRDFs. They also rated how well they succeeded in matching the gloss impression and overall brightness of standard and comparison stimulus. This resulted in 1,512 pairs of stimuli with matched gloss impressions (756 Fresnel–Ward pairs and 756 Fresnel–balanced Ward pairs). In Experiment 2, the subjects had to decide for each of these pairs which of the BRDFs produced a stronger or more natural gloss impression. In Experiment 3, a subset of the 756 triples of matched stimuli (Fresnel, balanced Ward, Ward) that resulted from Experiment 1 were further investigated. The subjects rated the gloss impression resulting from each BRDF with regard to gloss strength (on the dimension matte/glossy) and gloss quality (on the dimension natural/unnatural). 
General methods
Stimuli and presentation conditions were similar in all three experiments and are, therefore, described in advance. 
Stimulus material
The same stimuli were used in all three experiments. They were created with the Mitsuba renderer as high dynamic range (HDR) images with 768 × 768 pixels and transformed into low dynamic range (LDR) images using the tone-mapping method of Reinhard and Devlin (2005). For stimuli rendered with the Ward models, the diffuse part D and the specular part S with ρs = 0.1 were calculated separately as HDR images. During the presentation, they were combined to D + βS and transformed into an LDR image with previously determined tone-mapping parameters. In this way, the results of the Ward models for any ρs set by the subjects could be created from only two rendered HDR images. 
Objects
In order to detect a possible influence of object shape, seven different objects were used (see Figure 4). For gloss perception, surface curvatures are of particular interest. In this respect, the sphere is a boundary case because it has a single constant curvature. The two blobs are globally convex, like the sphere, but have several local curvatures, which are more pronounced in “blob3” than in “blob2.” The object “dog” (obtained from https://free3d.com/3d-model/dog-19722.html) has a natural and known form. The “vase” was chosen because it is a rotationally symmetric object and has an opening. Both sinus surfaces have a spatially relatively homogeneous curvature distribution but differ in the amplitudes. Furthermore, due to their orientation, the reflection from the sinusoidal surfaces is only slightly influenced by the floor pattern. 
Figure 4
 
The seven object shapes used in the experiments.
Figure 4
 
The seven object shapes used in the experiments.
Illuminations
Only global illumination was used (“image-based lighting,” Debevec, 1998). In addition to a completely uniform illumination, two “natural” illumination maps from the “Extreme Highres” series of Dosch Design (https://doschdesign.com/) were used, the first showing an urban outdoor scene (DH209) and the second an interior view of an office (DH206; see first column in Figure 5). 
Figure 5
 
Three global illumination maps (first column), three floor conditions (second column), and three different values for the roughness parameter (third column) were used. The objects in the first three columns are calculated with a refractive index of 1.7 and an albedo of 0.5. The last column shows the object labeled α = 0.001 in the third column with a different level of the factors refractive index (ior = 1.5) and albedo (albedo = 0.3).
Figure 5
 
Three global illumination maps (first column), three floor conditions (second column), and three different values for the roughness parameter (third column) were used. The objects in the first three columns are calculated with a refractive index of 1.7 and an albedo of 0.5. The last column shows the object labeled α = 0.001 in the third column with a different level of the factors refractive index (ior = 1.5) and albedo (albedo = 0.3).
Floor
In natural scenes, objects often rest on a floor or ground plane that is mirror reflected in the object. This reflection can potentially provide cues about the three-dimensional shape of the object and the material properties of its surface. This information seems especially useful if the illumination map is rather homogeneous. Three floor conditions were realized, namely “no floor,” “checker,” and “uniform” (see second column in Figure 5). 
Material parameters
With respect to the material itself, three parameters were varied, namely the roughness in three steps (0.001, 0.06, 0.12; see third column in Figure 5), the albedo of the diffuse part in two steps (0.3, 0.5), and the refractive index also in two steps (1.5, 1.7; see last column in Figure 5). The roughness parameter α of the Fresnel-BRDF is related to the Beckmann distribution (Beckmann & Spizzichino, 1987) and is with regard to the perceptual effect comparable to the α parameter of the Ward model. 
Stimulus presentation
Combining the levels of the six context factors: 7 (object) × 3 (illumination) × 3 (floor) × 3 (roughness) × 2 (refractive index) × 2 (albedo) results in a total of 756 different conditions. 
Before the experiment, the tone-mapping parameters for transforming an HDR image into an LDR image were manually tuned for each of the nine combinations of illumination and floor conditions to create a similar overall impression of the scenes. 
The stimuli had a size of 768 × 768 pixels and were presented on a 24-in. Eizo monitor with a resolution of 1,920 × 1,200 pixels in a completely darkened room. The subjects' responses were entered via a standard computer keyboard. 
Experiment 1: Matching the gloss impression
In the first experiment, the subjects had to match the material impression, in particular, the perceived gloss, that results with a Fresnel-BRDF in a certain situation with a Ward-BRDF. To this end, the subjects adjusted the strength of the specular reflection, i.e., the parameter Display Formula\({\rho _s}\), of the Ward model. This was done for various context conditions that were always identical for the two compared reflection models. 
From a theoretical perspective, Experiment 1 had two purposes: A first goal was to test to what extent the two types of BRDF could be matched at all. If this were not always possible, then this would point to qualitative differences in the gloss impressions evoked by both types of models. A second goal was to exclude differences in the absolute specular reflection strength as a possible explanation for differences in perceived gloss that may result in Experiments 2 and 3
Figure 6 illustrates the logic of the task used in Experiment 1: For a stimulus with Fresnel effects shown in the upper row, say the one with a refractive index of n = 1.5, the subjects had to determine the specular reflection strength Display Formula\({\rho _s}\) of the Ward-BRDF (lower row) such that the material impressions, in particular, the gloss impressions, were as similar as possible in both objects. Afterward, they had to evaluate the quality of the best achievable match result. In keeping with the goal to silence the influence of all variables not related to Fresnel effects, the roughness parameter α was always set to an identical value in both types of BRDF. 
Figure 6
 
Objects with an increasing specular reflection strength rendered with Fresnel-BRDF (top) and Ward-BRDF (bottom). In the case of the Fresnel-BRDF, this is achieved by increasing the refractive index from 1.4 to 2 and in the case of the Ward-BRDF by increasing the specular reflection strength parameter ρs from 0.056 to 0.14. With the Fresnel-BRDF, both the gloss impression and the object color are spatially more homogeneous, and the shape of the object stands out more clearly than with the Ward model. With the Ward-BRDF, a darkening occurs toward the rim, and with increasing ρs, the color and brightness of the material changes: It appears both brighter and more desaturated. An exact visual match of the material impression across the different BRDFs is, therefore, generally not possible.
Figure 6
 
Objects with an increasing specular reflection strength rendered with Fresnel-BRDF (top) and Ward-BRDF (bottom). In the case of the Fresnel-BRDF, this is achieved by increasing the refractive index from 1.4 to 2 and in the case of the Ward-BRDF by increasing the specular reflection strength parameter ρs from 0.056 to 0.14. With the Fresnel-BRDF, both the gloss impression and the object color are spatially more homogeneous, and the shape of the object stands out more clearly than with the Ward model. With the Ward-BRDF, a darkening occurs toward the rim, and with increasing ρs, the color and brightness of the material changes: It appears both brighter and more desaturated. An exact visual match of the material impression across the different BRDFs is, therefore, generally not possible.
Methods
Two stimuli rendered with different BRDFs were displayed side by side with a small gap between them on an otherwise black screen. On the left, a fixed image rendered with the Fresnel-BRDF was shown. The image displayed on the right was rendered with a Ward-BRDF and could be adjusted by the subject. 
In a first step, the subject's task was to choose a specular reflection strength ρs such that the material impression in both objects was as similar as possible. Actually, the subject varied a factor β in steps of 0.04, which was multiplied with the prerendered reflection strength of ρs = 0.1. Thus, the real increment was ρs = 0.004. Subsequently, the subjects had to evaluate the match. First, the similarity of the gloss impression was judged on a scale from zero to five. In particular, a value of zero should be given if a gloss impression was present with the Fresnel-BRDF but not with the Ward-BRDF. Finally, the similarity of the brightness distribution across each surface was also judged on a scale from zero to five. 
Because the Fresnel-BRDF was compared with both the original and the balanced Ward-BRDF, a total of 1,512 trials were performed by each subject in randomized order. 
Subjects
Three subjects participated in the experiment. Two of them were student assistants who were familiar with similar experiments; another subject who was paid for the experiment was naïve about the research question. The experiment was divided into several sessions, and the total duration was between 12 and 16 hr per subject. 
Results
The interesting two aspects of the results are (a) the strength of the reflective component that the subjects had chosen in the match, and (b) the evaluation of the best achievable match with regard to gloss and brightness. 
Settings for the specular reflection strength
Figure 7 shows the main effects of the six context factors on the specular reflection strength settings (see also Appendix Table A1 for the results of an ANOVA). The general pattern of the results is very similar for both Ward-BRDFs, but the values are systematically lower for the balanced model. 
Figure 7
 
Main effects of the six context factors on the specular reflection strength settings (ρs) for both Ward-BRDFs. The error bars correspond to ±2 SEM. The number of observations belonging to each data point is given by n = 2,268/#levels and ranges from n = 326 in panel A to n = 1,134 in panels C and F. Note: Here and in subsequent plots, lines between levels of categorical variables are added to emphasize the pattern of the results although, in such cases, the intermediate values are meaningless.
Figure 7
 
Main effects of the six context factors on the specular reflection strength settings (ρs) for both Ward-BRDFs. The error bars correspond to ±2 SEM. The number of observations belonging to each data point is given by n = 2,268/#levels and ranges from n = 326 in panel A to n = 1,134 in panels C and F. Note: Here and in subsequent plots, lines between levels of categorical variables are added to emphasize the pattern of the results although, in such cases, the intermediate values are meaningless.
For both Ward-BRDFs, the reflections strength settings increase with the refractive index (top right panel in Figure 7). This is to be expected, because in the Fresnel-BRDF the refractive index controls the overall strength of specular reflections (see Figure 6). All other factors, however, should not systematically influence the settings if the approximation by the Ward model is always possible and the parameter ρs of the Ward model is a valid measure of perceived gloss strength. This expectation is clearly not fulfilled: All factors except “albedo” have significant main effects. The effect of object shape is particularly pronounced, with clearly larger settings for “icosphere” and “dog,” especially with the original Ward model. The settings also increase systematically with increasing roughness. The conditions without a floor and with a uniform illumination also have a strong influence on the settings. 
Figure 8 shows the distributions of the individual settings of each subject in relation to the corresponding mean settings across the subjects for both BRDFs and the two levels of the refractive index. The vertical variation about the diagonal is, thus, mainly due to random fluctuations in the settings, whereas the variation along the x-axis contains also systematic variation. There are also systematic deviations from the diagonal (the distribution of subject 1 lies mainly below the diagonal, that of subject 3 mainly above), which may be attributed to individual differences in the match criterion (see discussion). 
Figure 8
 
The settings for the specular reflection strength parameter ρs of each subject plotted against the mean setting across all subjects in the same condition. The distributions for the two levels of the refractive index and the two Ward-BRDFs are plotted separately.
Figure 8
 
The settings for the specular reflection strength parameter ρs of each subject plotted against the mean setting across all subjects in the same condition. The distributions for the two levels of the refractive index and the two Ward-BRDFs are plotted separately.
Although the settings for the higher refractive index are systematically shifted to larger values, there is also a considerable overlap between the two distributions, which is mainly due to the systematic effects of the context factors. A comparison of the distributions obtained with the two Ward-BRDFs shows that the settings with the original Ward-BRDF are more susceptible to context effects and also less reliable than those with the balanced BRDF. 
Gloss similarity of the best match
Overall, rather low mean gloss similarity ratings were given, namely 2.45 (1.2) for the Ward-BRDF and 2.95 (1.19) for the balanced Ward-BRDF (on a scale from zero to five). In 244 (about 5.4%) of the 4,536 gloss similarity judgments, a value of zero was given. According to the instruction, this means that the subject has seen gloss with the Fresnel-BRDF but was not able to find a parameter setting for the Ward-BRDF, which also led to a gloss impression. As Figure 9 shows, such cases were predominantly found with the two “blob” objects in conditions with uniform illumination, low surface roughness, and without a floor. Together, these findings indicate that the matches were often far from perfect and sometimes even impossible. 
Figure 9
 
Distribution of “impossible gloss matches” (gloss similarity = 0, i.e., gloss with Fresnel-BRDF but not with Ward-BRDF) across different experimental factors and subjects.
Figure 9
 
Distribution of “impossible gloss matches” (gloss similarity = 0, i.e., gloss with Fresnel-BRDF but not with Ward-BRDF) across different experimental factors and subjects.
Figure 10 shows the main effects of the six context factors on the gloss similarity ratings for both Ward-BRDFs. With respect to the main effects, only two of the six factors, namely the shape of the object and the illumination type have a clear influence on the settings. There are, however, some strong interaction effects between object shape, illumination, surface roughness, the type of floor, and the type of the BRDF (see Figure 11 and the results of a seven-way ANOVA in Appendix Table A2). 
Figure 10
 
Main effects of the six context factors on the gloss similarity ratings for both Ward-BRDFs. The error bars correspond to ±2 SEM. The number of observations belonging to each data point is given by n = 2,268/#levels and ranges from n = 326 in panel A to n = 1,134 in panels C and F.
Figure 10
 
Main effects of the six context factors on the gloss similarity ratings for both Ward-BRDFs. The error bars correspond to ±2 SEM. The number of observations belonging to each data point is given by n = 2,268/#levels and ranges from n = 326 in panel A to n = 1,134 in panels C and F.
Figure 11
 
Interaction effects with respect to ratings of the gloss similarity of the match between the Fresnel-BRDF and the Ward-BRDFs. There is an interaction between object shape and the type of illumination that can be seen in all subplots A–C: Under the indoor illumination, the gloss similarity is relatively high throughout. With the outdoor illumination, the general level of gloss similarity decreases, and the dependence on object shape increases. This tendency is even more pronounced with a uniform illumination. This interaction is further influenced by the type of BRDF (A), the type of floor in the scene (B), and surface roughness (C). The error bars correspond to ±2 SEM.
Figure 11
 
Interaction effects with respect to ratings of the gloss similarity of the match between the Fresnel-BRDF and the Ward-BRDFs. There is an interaction between object shape and the type of illumination that can be seen in all subplots A–C: Under the indoor illumination, the gloss similarity is relatively high throughout. With the outdoor illumination, the general level of gloss similarity decreases, and the dependence on object shape increases. This tendency is even more pronounced with a uniform illumination. This interaction is further influenced by the type of BRDF (A), the type of floor in the scene (B), and surface roughness (C). The error bars correspond to ±2 SEM.
Again, a systematic effect of the BRDF type can be seen: The ratings of gloss similarity are, in general, significantly higher with the balanced Ward-BRDF than with the original Ward-BRDF, whereas the general pattern of the results is very similar. However, there are significant interactions between the BRDF and the type of illumination (see Figure 11A) and between BRDF and the type of floor. These interactions are mainly due to a reduced difference between the BRDFs in low information conditions, i.e., when the illumination is uniform or the scene does not have a floor. 
Brightness similarity of the best match
In the present context, the brightness similarity of the best match is of lower interest. The main reason why corresponding ratings were also gathered was the fact that in the Ward-BRDF an increase of the specular reflection strength parameter ρs is, in general, accompanied by an increase in overall brightness (see Figure 6). This may lead to a conflict when trying to match the perceived material properties because an attempt to increase the strength of visible highlights to make them more similar to those observed with the Fresnel-BRDF usually also increases the perceived difference in overall brightness (and saturation). That this antagonistic relationship between gloss and brightness similarity has indeed influenced the settings is suggested by the fact that the correlation rb = −0.13 between the ρs settings and the brightness similarity ratings has a different sign than the correlation rg = 0.12 between ρs and the gloss similarity ratings (these correlations include only cases with a gloss similarity rating greater than zero). The systematic differences between the ρs settings of the three subjects (see Figure 8) may at least partly be due to a different compromise criterion used in this conflict. This hypothesis is supported by clear individual differences in these two correlations (rb, rs), which are (−0.32, 0.11), (−0.07, 0.15), and (−0.12, 0.17) for the three subjects. For instance, the highly negative correlation rb = −0.32 of subject 1 indicates that this subject gave a greater weight to brightness deviations than the other subjects and, thus, tended to use lower reflection strength settings. 
Mean brightness similarity ratings of 2.96 (0.71) for the Ward-BRDF and 3.3 (0.8) for the balanced Ward-BRDF were observed. These mean values are slightly higher than the corresponding mean gloss similarity ratings, and the standard deviations are much lower. There seems to be only a minor influence of the context factors on the brightness similarity settings. A single exception is “object shape,” but even this influence is only moderate (see Appendix Figure A1 and Appendix Table A3). 
Discussion
The results of Experiment 1 show that there are cases in which the gloss impressions produced by a Fresnel-BRDF can be well approximated with the Ward model if the reflectance strength ρs is chosen appropriately. For example, this was true for high-gloss materials with the “indoor” illumination map. Under this condition, the highest mean similarity ratings were given for the sphere. However, even in this particularly favorable case, the match is by no means perfect as can be seen in Figure 6, which depicts this factor combination. There remain qualitative differences in the gloss impression, which are particularly obvious in direct comparison. 
On the other hand, there were also numerous problematic cases in which no setting of the reflectance strength parameter ρs could be found that led to a satisfactory match. According to the present data, the degree of similarity that can be achieved is most strongly influenced by the three factors: “illumination,” “shape,” and “roughness.” Already the difference in perceived gloss between the two “natural” illuminations is surprisingly large, indicating that illuminations within this class should not be considered equivalent. Particularly problematic, however, is a uniform illumination, with which it is often no longer possible to achieve a convincing gloss impression at all with the Ward-BRDFs, whereas the Fresnel-BRDF is much more robust in this respect. With the Fresnel-BRDF, the gloss impression is often also less clear in this case but usually remains clearly visible. The only exception to this rule was found in the condition in which a sphere was presented free-floating, i.e., without a floor, under uniform illumination. In this specific case, the object no longer looks glossy even if a Fresnel-BRDF is used (for an illustration, see Figure 20 in the general discussion). The fact that the gloss impression under illuminations with weak or no structure is more robust with a Fresnel-BRDF than with a Ward-BRDF explains the decline in the match quality with less structured illuminations. 
Figure 12 demonstrates the just-described differences using the example of the “blob3” object, which is placed on a uniform floor. The gloss impression with the Fresnel-BRDF is already under the “indoor” illumination (left) clearly more convincing than with the Ward-BRDF. This difference becomes even greater under a uniform illumination (right): With the Fresnel-BRDF, the gloss impression is somewhat reduced but still clearly visible, whereas it almost disappears with the Ward-BRDF and especially does not elicit the impression of a highly glossy surface. 
Figure 12
 
Comparison of the Fresnel-BRDF (positions 1 and 3) with the original Ward-BRDF (positions 2 and 4) under the “indoor” (left) and the uniform (right) illumination. With Fresnel effects, the gloss impression is significantly stronger and more similar across the illuminations.
Figure 12
 
Comparison of the Fresnel-BRDF (positions 1 and 3) with the original Ward-BRDF (positions 2 and 4) under the “indoor” (left) and the uniform (right) illumination. With Fresnel effects, the gloss impression is significantly stronger and more similar across the illuminations.
Due to the small selection of illuminations used in the experiment, it is not possible to draw clear conclusions as to which properties of the illumination are particularly critical. It seems, however, that the gloss impression with a Ward-BRDF benefits from a global illumination with clear intensity peaks at many different locations, whereas a relatively homogeneous illumination, such as the “outdoor” illumination or the extreme case of a uniform illumination, has a highly detrimental effect. The Fresnel-BRDF, on the other hand, produces (due to the directional dependence of the specular reflection strength) clearly visible highlights even under a uniform illumination, and these help to retain the gloss impression. 
The similarity in the overall brightness of the stimuli is also not very high. According to informal observations and the subjects' reports, a major reason for this is that the reflection in nearly perpendicularly observed surface regions (e.g., in the middle of a convex object) appears too high compared to that in regions observed at a shallower angle (e.g., near the rim of a convex object). Although this effect is attenuated in the balanced Ward-BRDF, it is still present. Because the spatial distribution of the specular reflection strength is almost complementary in the Ward- and Fresnel-BRDF (see Figure 3), a compromise between sufficiently bright highlights and a not too high overall brightness is almost always necessary. The relatively small variability of the brightness similarity ratings indicates that the subjects kept the weights of the two conflicting criteria quite similar across all conditions. 
The data provide additional evidence that the Ward-BRDF can only imperfectly approximate the gloss impressions produced by a Fresnel-BRDF: Although the specular reflection strength in the standard objects with Fresnel-BRDF was varied in only two steps (ior = 1.5 and 1.7), a wide range of different Display Formula\({\rho _s}\) settings for the Ward-BRDF were observed. This means that a certain reflection strength of the Fresnel-BRDF was matched by very different settings of the reflection strength parameter of the Ward-BRDF. The agreement between the settings of different subjects is also only moderate. One reason for this could be that the subjects used slightly different criteria for the compromise required for the match (see previous section). In addition, it might also play a role that, in many cases, no clear gloss impression could be achieved with the Ward-BRDF by varying the reflection strength, which means that, in such cases, the settings are somewhat arbitrary with regard to the gloss impression. 
The comparison of the original Ward-BRDF with the energy-conserving balanced Ward-BRDF indicates that even seemingly subtle differences might play an important role in perception. The settings in the physically more plausible balanced Ward-BRDF can be regarded as “better” in the sense that the similarity to the gloss impression with a Fresnel-BRDF was consistently rated higher and that the variability of the settings was lower than with the original Ward-BRDF. 
Experiment 2: Pair comparisons of Fresnel- and Ward-BRDF
In Experiment 1, the task was to match the gloss impressions produced with a Fresnel-BRDF with two variants of the Ward-BRDF. It turned out that, in many cases, only an unsatisfactory match or no match at all was possible. The choice of the Fresnel-BRDF as the standard in the matching task was not arbitrary, but motivated by the fact that it describes the physical situation more accurately than the Ward-BRDF. Due to this asymmetry, it is not implausible to assume that an unsatisfactory match means that the concomitant gloss impression with a Ward-BRDF is also weaker. However, from a purely empirical point of view, the only conclusion that can be drawn from the results of Experiment 1 is that the gloss impressions produced by the Fresnel-BRDF and the Ward-BRDFs are different, but not which of them is “better” (in the sense of stronger and/or more natural). 
Therefore, the best possible matches of Fresnel- and Ward-BRDF determined in Experiment 1 were evaluated in Experiment 2 in a pair comparison task with regard to the relative quality of the gloss impression. If the gloss impressions with a Ward-BRDF were subjectively equally convincing as those with the Fresnel-BRDF, one would expect a preference probability of 0.5 in these comparisons. 
Methods
In each of the 1,512 pairs that were matched in Experiment 1, the parameter Display Formula\({\rho _s}\) of the Ward-BRDF was set to the mean of the subjects' settings. The presentation of the two stimuli was essentially the same as in Experiment 1, but the horizontal position at which the Fresnel-BRDF was shown was half to the left and half to the right, whereby randomization prevented that the position was predictable for the subject. In each trial, the subject's task was to indicate on which side of the screen the gloss impression is more convincing. 
The subjects were instructed to use as the main criterion for “more convincing” that the gloss impression appears more natural, and if this doesn't suffice to make a decision, as further criteria in descending order, a more homogeneous gloss impression (all areas of the surface appear equally glossy) and a more homogeneous color impression (the surface has the same color everywhere). 
Subjects
Six subjects participated in the experiment. One of them had also performed Experiment 1; the other five were undergraduate psychology students of Kiel University, who were naive with regard to the purpose of the experiment and were compensated with course credits. All participants reported to have normal or corrected-to-normal vision. 
Results
The results are rather unequivocal. Only in 146 of the 9,072 comparisons, i.e., in 1.61% of the cases, was the gloss impression with a Ward-BRDF judged as more convincing than that with the Fresnel-BRDF. Figure 13 breaks down these 146 cases according to the experimental conditions and the subjects. 
Figure 13
 
The distribution of the 146 out of 9,072 cases in which the Ward-BRDFs were judged as glossier, depending on the experimental factors and the subjects. The blue parts are the 56 cases that occurred under the uniform illumination, which includes cases in which a gloss impression is lacking even with the Fresnel-BRDF.
Figure 13
 
The distribution of the 146 out of 9,072 cases in which the Ward-BRDFs were judged as glossier, depending on the experimental factors and the subjects. The blue parts are the 56 cases that occurred under the uniform illumination, which includes cases in which a gloss impression is lacking even with the Fresnel-BRDF.
The cases belonging to the uniform illumination are given separately because this condition comprises some cases in which a gloss impression is lacking even with the Fresnel-BRDF. This concerns 36 cases under the combination of the conditions “uniform illumination,” “sphere,” and “no floor.” Approximately one third (53) of the preferences of the Ward-BRDF are attributable to one of the six subjects. 
Discussion
The results show that, in almost all cases (>97%), the objects rendered with Fresnel-BRDF elicited a more convincing gloss impression than those rendered with a Ward-BRDF, whose specular reflection strength parameter was determined in Experiment 1 in such a way that the gloss impression was as similar as possible to that obtained with the Fresnel-BRDF. This holds even if only the “indoor” illumination condition is considered, in which Fresnel- and Ward-BRDF were rated as most similar. 
When evaluating the results under uniform illumination, it must be taken into account that this condition includes some cases in which the object rendered with the Fresnel-BRDF also does not appear glossy (see general discussion). In such cases, the subjects can only guess or use other arbitrary criteria. Although this only affects very few cases of the total stimulus set, it seems to account for most of the preferences of the Ward-BRDF observed in this condition. 
Under the two “natural” illuminations, most cases of a preference of the Ward-BRDF involved the balanced Ward model. This is to be expected as this model approximates the Fresnel-BRDF more closely than the original Ward model. The gloss impressions produced by Fresnel-BRDF and balanced Ward-BRDF are rather similar under especially favorable conditions, and a clear choice between them is then difficult. However, as the observed preferences show, these are rare exceptions. Overall, the data indicate that physically more plausible simulations of glossy surfaces that include Fresnel effects are clearly preferred by the subjects. 
Experiment 3: Judging strength and quality of the gloss obtained with different BRDFs
The results of Experiment 2 show that, under otherwise comparable conditions, the Fresnel-BRDF leads to more convincing gloss impressions than both Ward models. However, because even relatively small systematic differences can be sufficient for a clear preference in a pair comparison, it remains unclear how large the perceptual differences between the BRDFs actually are and how the strength of these differences depends on the context conditions. 
To investigate this question, a further experiment was conducted in which the stimuli from Experiment 1, i.e., both the given standard objects with Fresnel-BRDF and the best matches with original and balanced Ward-BRDF, were rated on a scale from zero to nine with regard to perceived gloss strength (matte vs. highly glossy) and perceived gloss quality (realistic vs. unrealistic). 
Methods
In order to somewhat reduce the effort for the subjects, only 569 of the 756 conditions of Experiment 1 were chosen as a representative subset for Experiment 3. Because these were realized with all three BRDFs, a total of 569 × 3 = 1,707 different settings had to be made by each subject. For the reflection strength parameter ρs of the Ward-BRDFs, the mean setting in the corresponding condition of Experiment 1 was chosen. 
Figure 14 shows the central part of a display from Experiment 3. The test stimuli were presented on the left side of the screen. The subject's task was to judge the gloss impression in the test object with respect to “gloss strength” along a dimension with the poles “matte” and “highly glossy” and “gloss quality” on a dimension with the poles “unrealistic” and “highly realistic.” 
Figure 14
 
Screenshot from Experiment 3. The test object whose gloss was to be evaluated was shown on the left side, and as a comparison the “Utah teapot” (rendered with Fresnel-BRDF) on the right side. The subjects judged the perceived gloss strength (matte vs. highly glossy) by choosing with the left/right arrow keys a scale value between zero and nine, which led to a corresponding change in the surface roughness of the Utah teapot. They also rated the perceived quality of the gloss impression (natural vs. unnatural) in the test object on a scale from zero to nine. This selection was made with the up/down arrow keys.
Figure 14
 
Screenshot from Experiment 3. The test object whose gloss was to be evaluated was shown on the left side, and as a comparison the “Utah teapot” (rendered with Fresnel-BRDF) on the right side. The subjects judged the perceived gloss strength (matte vs. highly glossy) by choosing with the left/right arrow keys a scale value between zero and nine, which led to a corresponding change in the surface roughness of the Utah teapot. They also rated the perceived quality of the gloss impression (natural vs. unnatural) in the test object on a scale from zero to nine. This selection was made with the up/down arrow keys.
As a reference object for the gloss strength rating, a Utah teapot rendered with Fresnel-BRDF and refractive index 1.7 was used, whose roughness parameter α was variable from 0.001 to 0.13 in small steps of 0.0025. The scale values zero and nine were mapped to the highest and lowest roughness values 0.13 and 0.001, respectively (for an illustration of the corresponding gloss impressions see Appendix Figure A2). The subject's current setting was approximately reflected by the position of a mark on the brightness scale shown on the screen. This gloss strength rating may appear similar to a normal match procedure. Note, however, that the varied material parameter that changes the gloss impression in the Utah teapot is surface roughness (affecting the dimension “distinctness of images gloss”) not the refractive index (which influences specular reflection strength and, thus, the dimension “contrast gloss”). This comparison object was chosen to decouple the gloss strength rating from the dimension on which the two BRDFs actually differed. If it would have been possible in Experiment 1 to produce a true match then no differences in the gloss strength ratings between the BRDFs would be expected in the present experiment because the objective blurriness of the mirror image was always identical in all BRDFs. The subjects were informed that a match in a strict sense is usually not possible but that they should select the gloss level in the teapot that appears most similar to the gloss or matte impression in the test object. 
With regard to gloss quality, the subjects were asked to assess the realism of the gloss impression, with the values zero, one, and two each being assigned a specific meaning: The value zero should be selected if the object was clearly seen as not glossy, the value one if a gloss impression was uncertain, and the value two if the object appeared clearly glossy but the gloss impression was highly deficient or unrealistic. The eight values from two to nine should then be used to evaluate the degree of realism of a given gloss impression. 
Subjects
Five subjects participated in the experiment. Two of them (subjects 1 and 3) had already participated in the first experiment. Another subject (subject 2) had extensive experience with gloss perception and corresponding stimuli, whereas the remaining two subjects (subjects 4 and 5) were relatively inexperienced with psychophysical experiments. The experiment was divided into several sessions. All subjects reported normal or corrected-to-normal vision. 
Results
For all conditions, there are two assessments of the gloss impression. The first judgment on gloss strength indicates the subject's assessment on the dimension matte to highly glossy. This assessment should, of course, depend on the roughness of the simulated material but—if gloss constancy is assumed—on nothing else. The second judgment of gloss quality, on the other hand, reflects how realistic the subject found the gloss impression. 
Main effects of context factors and BRDF on the gloss impression
Figure 15 summarizes the main effects of the six context variables on the gloss impression depending on the BRDF (see also Appendix Tables A4 and A5 for ANOVAs for gloss strength and gloss quality results). As a general trend, it can be seen that the Fresnel-BRDF was rated significantly higher than both Ward-BRDFs in terms of gloss strength and gloss quality. There is also a systematic advantage of the balanced over the original Ward-BRDF, but the differences are comparatively small. 
Figure 15
 
Main effects of the six context factors on the mean gloss strength (matte to highly glossy; upper panel in subplots A–F) and the mean gloss quality (unrealistic to highly realistic; lower panel in subplots A–F) depending on the BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 15
 
Main effects of the six context factors on the mean gloss strength (matte to highly glossy; upper panel in subplots A–F) and the mean gloss quality (unrealistic to highly realistic; lower panel in subplots A–F) depending on the BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
As expected, the factor roughness has the strongest influence on the gloss strength settings (see Figure 15E). For all BRDFs, the values fall monotonously with increasing roughness. With a Fresnel-BRDF, the gloss strength is consistently higher and the curve approximates a linear relationship with the roughness parameter. If constancy with respect to surface roughness is assumed, this linear relationship is to be expected because the teapot is rendered with the same Fresnel-BRDF. The dependence of gloss quality on roughness is much weaker. With increasing roughness, it decreases for the Fresnel-BRDF and increases very slightly for the two Ward-BRDFs. Accordingly, the difference in gloss quality between Fresnel- and Ward-BRDFs is greatest for perfectly smooth surfaces and decreases with increasing roughness. 
The type of illumination also greatly influences gloss strength and gloss quality (see Figure 15D). Both measures decrease with increasing homogeneity of the illumination. However, this effect is much stronger with the Ward-BRDFs than with the Fresnel-BRDF. For “indoor” illumination, the gloss strength ratings for the Ward-BRDF are only slightly lower than those for the Fresnel-BRDF. But these differences increase considerably with the “outdoor” illumination and especially with a uniform illumination. A very similar picture emerges with regard to gloss quality. The ratings for the Ward-BRDFs break down with the “outdoor” illumination and are, for the original Ward model, scarcely better than with a uniform illumination. With uniform illumination, the quality ratings for both Ward-BRDFs are close to two, which was defined as the lowest value for a reliable gloss impression. The Fresnel-BRDF proves to be much more robust in this respect. Although the mean ratings are clearly lower with a uniform illumination, they remain well above a value of two and are even higher than those given to the balanced Ward-BRDF for the “outdoor” illumination. 
There are also effects of object shape on gloss strength and gloss quality, which are more pronounced for the Fresnel-BRDF (see Figure 15A). The pattern of these effects is similar for the two Ward models but differs from that of the Fresnel-BRDF, which suggests that the underlying causes are different. As is shown below, there are also strong interaction effects between object shape, surface roughness, illumination, and BRDF. 
The main effects of the type of floor are small but consistent (see Figure 15B). Both types of gloss ratings decrease from “checkered” to “uniform” to a missing floor. There are, however, also strong interactions with the illumination type. 
The remaining factors, refractive index and albedo, have only relatively small effects on gloss strength (see Figure 15C and F). With increasing refractive index, the gloss strength also increases slightly. Although gloss strength is here mainly understood as a correlate of surface roughness, a clearly visible specular reflection is a necessary condition for an assessment of surface roughness. Because a higher refractive index leads to stronger specular reflections, this may also indirectly have an influence on perceived gloss strength. The factors roughness and refractive index had no systematic influence on the gloss quality. 
Interactions between object shape, illumination, and roughness
Figure 16 shows, for each BRDF, how gloss strength and gloss quality depend on object shape and illumination type. It is obvious that with the Fresnel-BRDF the strength and quality of the perceived gloss depends much less on the type of illumination than in both Ward-BRDFs. Fresnel- and Ward-BRDFs differ most for the uniform illumination. Whereas a gloss impression is retained with the Fresnel-BRDF under this condition, it is almost completely lost in both Ward-BRDFs. The largest difference between the two Ward-BRDFs is found in the “outdoor” illumination condition, in which the strength and quality ratings are consistently higher for the balanced Ward model. This is in line with the pattern found in Experiment 1 for the gloss similarity of the match (see Figure 11). 
Figure 16
 
Mean ratings for gloss strength (A) and gloss quality (B) in dependence on object shape, illumination type, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 16
 
Mean ratings for gloss strength (A) and gloss quality (B) in dependence on object shape, illumination type, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
A rather similar pattern emerges with respect to the interaction of illumination and surface roughness shown in Figure 17. With a Fresnel-BRDF, the illumination has a relatively small additive influence on gloss strength, i.e., the approximately linear relationship between gloss strength and surface roughness is retained. The gloss quality observed with a highly structured illumination falls slightly with increasing roughness, whereas it remains approximately constant for the uniform illumination. With a Ward-BRDF, both gloss strength and gloss quality depend more on surface roughness. The detrimental effect of a uniform illumination is especially pronounced for smooth surfaces, which appear relatively matte (low gloss strength) and produce, at best, an unrealistic gloss impression (gloss quality ratings of about two). 
Figure 17
 
Mean ratings for gloss strength (A) and gloss quality (B) in dependence on roughness, illumination type, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 17
 
Mean ratings for gloss strength (A) and gloss quality (B) in dependence on roughness, illumination type, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Interactions between object shape and floor under uniform illumination
The differences between the BRDFs are particularly apparent with a relatively uniform illumination, in which the specular reflection lacks the structure, which normally serves as a strong cue for gloss. In such cases, the effects of other cues on perceived gloss become more obvious. Figure 18 shows the interaction of object shape and floor condition in the case of a completely uniform illumination. 
Figure 18
 
Mean ratings for gloss strength (A) and gloss quality (B) under uniform illumination in dependence on object shape, type of floor, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 18
 
Mean ratings for gloss strength (A) and gloss quality (B) under uniform illumination in dependence on object shape, type of floor, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
In the case of a Fresnel-BRDF, there are, in general, only relatively small differences between the different floor types. But there is one notable exception: Without a floor, the sphere, which is characterized by a constant curvature, no longer appears glossy at all. In the case of a Ward-BRDF, on the other hand, the gloss impression is consistently very low. The mean gloss quality ratings are close to the value of two, which marks the limit below which a gloss impression is uncertain or missing. The values are especially low in scenes without a floor. 
On the stimulus level, replacing the uniform floor with a patterned one leads in all objects other then the “sin” surfaces to recognizable contours in the mirror image seen on their surface. With this additional information, ratings of gloss strength and gloss quality are slightly higher, especially in the case of the sphere. 
Interindividual differences
In most cases, the settings of gloss strength agree very well across different subjects, whereas there is more variability with respect to gloss quality. Figure 19 illustrates this with an example in which these differences in the gloss quality settings are especially strong, namely the interaction between BRDF and surface roughness under illumination “outdoor.” In a sense, the subjects' settings may be considered to be similar because, in each case, the ratings with a Fresnel-BRDF are higher or equal to those made with a Ward-BRDF but the absolute values differ considerably. 
Figure 19
 
Mean ratings for gloss strength (A) and gloss quality (B) of different subjects under the “outdoor” illumination in dependence of surface roughness and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 19
 
Mean ratings for gloss strength (A) and gloss quality (B) of different subjects under the “outdoor” illumination in dependence of surface roughness and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 20
 
Some demonstrations of how gloss perception under a uniform illumination differs when the stimuli are rendered with Fresnel-BRDF (top row) or the original Ward-BRDF (bottom row). All objects are nominally of highly glossy material. A free-floating sphere (second column) does not appear glossy with either of the two BRDFs, whereas an object with different curvatures under the same condition (left column) appears somewhat glossy with Fresnel-BRDF but completely matte with the Ward-BRDF. The only difference between the scenes shown in columns 2 and 3 is that, in the latter, a uniform floor was added. This improves the gloss impression considerably with the Fresnel-BRDF, whereas the corresponding gloss impression with Ward-BRDF is much less vivid and appears rather unrealistic. Adding a texture to the floor (fourth column) improves the gloss impression with Fresnel-BRDF further, whereas this has only a minor effect on the gloss impression with the Ward-BRDF.
Figure 20
 
Some demonstrations of how gloss perception under a uniform illumination differs when the stimuli are rendered with Fresnel-BRDF (top row) or the original Ward-BRDF (bottom row). All objects are nominally of highly glossy material. A free-floating sphere (second column) does not appear glossy with either of the two BRDFs, whereas an object with different curvatures under the same condition (left column) appears somewhat glossy with Fresnel-BRDF but completely matte with the Ward-BRDF. The only difference between the scenes shown in columns 2 and 3 is that, in the latter, a uniform floor was added. This improves the gloss impression considerably with the Fresnel-BRDF, whereas the corresponding gloss impression with Ward-BRDF is much less vivid and appears rather unrealistic. Adding a texture to the floor (fourth column) improves the gloss impression with Fresnel-BRDF further, whereas this has only a minor effect on the gloss impression with the Ward-BRDF.
Subjects 1 and 2 were very experienced psychophysical observers, but this cannot be the only explanation because the settings of subject 5, who was not more experienced than subjects 3 and 4, produced results that are highly similar to those of subject 1. Furthermore, the settings of subjects 3 and 4 also have much in common in that their gloss quality ratings for the different BRDFs are rather similar. 
Discussion
The results of the third experiment show that, under otherwise identical conditions, surfaces rendered with Fresnel-BRDF consistently produce a significantly stronger gloss impression than surfaces rendered with the Ward-BRDFs. These differences cannot be attributed to an inappropriate parameter setting for the Ward models because the best-fitting parameters, which make the gloss impression with a Ward model most similar to that experienced with a Fresnel-BRDF in the same condition, were empirically determined in Experiment 1
The results of Experiment 3, therefore, suggest that there are qualitative differences between the gloss impressions elicited by Fresnel- and Ward-BRDFs that cannot be compensated for by an adjustment of the reflection strength of the Ward models. This would mean that the Ward-BRDFs fail to capture an important regularity of specular reflection that is used in gloss perception. The comparison with the Fresnel-BRDF makes it clear that this lacking aspect concerns Fresnel effects. 
That the procedures used in Experiments 1 and 3 are valid and the conclusions justified is also supported by the observed differences between the two Ward-BRDFs: Even the relatively small discrepancies between these two variants of the same model are clearly reflected in the gloss strength and gloss quality settings with consistently better results for the physically more plausible balanced Ward model. The results for both Ward-BRDFs are largely “parallel” to each other, i.e., the results for the balanced Ward model are, in most cases, almost identical to those observed with the original Ward model apart from an additive constant. There is also a plausible relationship between the gloss similarity settings made in Experiment 1 and the gloss strength and gloss quality settings made in Experiment 3 in the sense that a higher match similarity in Experiment 1 corresponds to a higher similarity in gloss strength and gloss quality ratings between Fresnel and Ward-BRDFs in Experiment 3. Together, these findings indicate a high reliability of the settings made in Experiments 1 and 3
Although the surface roughness and, thus, the blurriness of the edges in the reflected mirror image are virtually identical in all BRDFs, the surfaces appeared usually clearly more glossy with the Fresnel-BRDF than with both Ward-BRDFs. Under nearly optimal lighting conditions, which, in the present experiment, were approximated by the “indoor” illumination, the gloss strength is only slightly reduced with the Ward models; for more homogeneous lighting as with the “outdoor” illumination or with the extreme case of a completely uniform illumination, considerable differences result. Under uniform illumination, surfaces (in particular convex ones) rendered with a Ward-BRDF appear either completely matte or the gloss impression is weak and unrealistic (see Figure 20 for examples). In this respect, the Fresnel-BRDF is much more robust. In the experiment, there was only one condition in which a gloss impression was lacking with the Fresnel-BRDF, namely when the sphere was presented free-floating without a floor under uniform illumination. In all other cases, the gloss impression was sometimes significantly reduced but still clearly present. This result challenges the generality of the conclusion derived from a small number of examples that gloss cannot be seen with diffuse light sources (Dror et al., 2004; Pont & te Pas, 2006; Sève, 1993). 
As an aside, the rather vivid gloss impression observed in the combination of a uniform illumination with a uniform floor, i.e., in a condition in which the environment is almost completely unstructured, demonstrates that gloss perception is also possible without complex illumination maps. This result suggests that a proper image statistic of the illumination map, for example, a conformance with statistics occurring in natural environments (see Fleming et al., 2003), is not necessary for a clear gloss impression. 
A very similar picture emerges with respect to gloss quality, which is almost always “better” with a Fresnel-BRDF, i.e., both the perceived gloss and the diffuse base color of the object appear more homogeneous and overall more realistic. This is true even in conditions in which Fresnel- and Ward-BRDFs produce comparably strong gloss impressions (see, for example, Figure 6). This basic pattern is evident in the data of all subjects although their specific settings may differ considerably in some cases. These interindividual differences observed in Experiment 3 could be explained in different ways. First, they could reflect real differences in perception. That the subjects with more experience in the investigation of gloss phenomena tended to report larger differences between the BRDFs could point in this direction. Another possibility is that the subjects used different internal references for an “optimal gloss quality” and, therefore, assigned different values. It seems also possible that the concepts of “gloss quality” and “gloss strength,” which are here understood as orthogonal to each other, are not always clearly distinguished by some subjects. 
Degrees of constancy with Fresnel- and Ward-BRDFs
The results of Experiment 1 imply that the degree of gloss constancy differs between Fresnel- and Ward-BRDFs. This follows from the fact that, depending on context, very different values were chosen for the reflection strength parameter ρs of the Ward-BRDF to match the gloss impression elicited by a Fresnel-BRDF with a fixed refractive index. The results of Experiment 3, on the other hand, can be interpreted as evidence that the degree of constancy is higher with a Fresnel-BRDF. This is suggested by the weak influence of the varied context factors on the gloss ratings (see Figure 15), which can be attributed to the Fresnel-BRDF that was used as the standard in Experiment 1
Experiment 4: Comparing the degrees of constancy between the BRDFs
To more explicitly test whether a Fresnel-BRDF leads to improved gloss constancy relative to the Ward-BRDF, a fourth experiment was conducted, in which the subjects were asked to match the gloss impression of two objects under different global illuminations and with more or less different shapes. Both stimuli involved in a given matching trial were either rendered with a Ward-BRDF with varying values of the specular reflection strength ρs or with a Fresnel-BRDF with varying values for the refractive index. A certain value of the reflection strength parameter (ρs or ior) was given in the fixed standard stimulus, and perfect constancy across changes in illumination and shape would be achieved if the subjects were able to exactly reproduce this value in the adjustable comparison stimulus. 
Regarding stimuli, design, and task, Experiment 4 bears some similarity to an experiment reported in Olkkonen and Brainard (2011). These authors used a Ward-BRDF to render their stimuli and found severe deviations from constancy in matches of gloss strength across illuminations and shapes. 
Methods
Stimuli
Figure 21 shows the type of stimuli used in the experiment. Three objects, blob1, blob2, and blob3, with increasing surface complexity were rendered under the “indoor” and “outdoor” illuminations (used already in Experiments 13) for 51 different levels of the specular reflection strength parameter. The refractive index, which controls reflection strength in the Fresnel-BRDF, ranged from 1.0 to 2.0, the corresponding parameter ρs of the Ward-BRDF from 0 to 0.28. 
Figure 21
 
Some of the stimuli used in Experiment 4. Upper and lower rows show the same stimuli rendered with Fresnel- and Ward-BRDFs, respectively. In total, there were three different blobs (blob1, blob2, and blob3 shown in this order in the last three columns) rendered in two orientations under two different illuminations and with varying reflection strength. A stimulus in one illumination (first column) was compared with objects shown in the other illumination. The shape was either identical (second column), identical but viewed from another angle (rotation about the vertical axis), or different with two degrees of dissimilarity; here the fourth column shows the more similar blob, the fifth the more dissimilar one. All stimuli within each row have the same reflection strength, i.e., the four stimuli on the right represent “perfect constancy” matches to the standard depicted in the leftmost column.
Figure 21
 
Some of the stimuli used in Experiment 4. Upper and lower rows show the same stimuli rendered with Fresnel- and Ward-BRDFs, respectively. In total, there were three different blobs (blob1, blob2, and blob3 shown in this order in the last three columns) rendered in two orientations under two different illuminations and with varying reflection strength. A stimulus in one illumination (first column) was compared with objects shown in the other illumination. The shape was either identical (second column), identical but viewed from another angle (rotation about the vertical axis), or different with two degrees of dissimilarity; here the fourth column shows the more similar blob, the fifth the more dissimilar one. All stimuli within each row have the same reflection strength, i.e., the four stimuli on the right represent “perfect constancy” matches to the standard depicted in the leftmost column.
Each row in Figure 21 shows in the leftmost column a standard stimulus and in the remaining columns the four comparison stimuli, which had to be matched to this standard. The comparison stimulus was always shown in another illumination than the test stimulus. Its shape was either identical, identical but rotated, or more or less different from the shape of the standard. Each of the six possible shape × illumination combinations were used as standard, resulting in 6 × 4 = 24 different match pairs. Each standard stimulus was presented with one of four different values of the reflection strength parameter, which were selected at the same position (10, 20, 30, 40) along the 51 steps scale for the Ward-BRDF (ρs = 0.0504, 0.1064, 0.1624, 0.2184) and for the Fresnel-BRDF (ior = 1.18, 1.38, 1.58, 1.78). This resulted in 24 × 4 = 96 comparisons for each BRDF and 192 comparisons in total. 
The stimuli had a size of 512 × 512 pixels and were presented side by side using the same equipment as in Experiments 13. The standard stimulus was always presented in the middle of the left half of the screen, the adjustable stimulus in the middle of the right half. Again, all stimuli were rendered with the Mitsuba renderer in HDR format and afterward tone-mapped using the procedure proposed by Reinhard and Devlin (2005). The image-based parameters for this procedure were determined once for one case (blob2, Ward-BRDF, ρs = 0.28, “indoor” illumination) and then used in all other cases. 
All 192 stimuli were shown in random order, that is, the randomization did not only affect illumination and shape, but also the type of BRDF. No information about the type of BRDF used in a certain trial was given to the subjects. The subjects used the arrow keys to choose from the 51 prerendered images of the comparison stimulus the one that, in their view, best matched the perceived material of the fixed standard stimulus. 
Subjects
Seven subjects participated in the experiment, including the author (subject 1). One subject (subject 2) had participated in all the other experiments. The remaining subjects were psychology students of Kiel University and were naive with respect to the purpose of the experiment. Four subjects repeated the experiment once, two further subjects twice. All reported normal or corrected-to-normal visual acuity and had normal color vision according to the Ishihara test. 
Results
To be able to directly compare deviations from constancy across the BRDFs, all reflection strength values were transformed to a standard scale r, with r = ior − 1 in the Fresnel case and r = ρs/0.17 in the Ward case. These standardized reflection strength values have a similar relationship to perceived gloss for both BRDFs (see section A.3 in the Appendix). Note that, in a first approximation, differences in r can be interpreted as differences in the refractive index. 
Reflection strength errors
Figure 22 shows the mean reflection strength errors [r(setting) − r(standard)] obtained with Ward- and Fresnel-BRDFs depending on the reflection strength in the standard stimulus (“standard value”) and the levels of “shape dissimilarity” between standard and comparison objects. The means are across all subjects and are given separately for both standard illuminations. Shape dissimilarity is defined as zero for identical shapes, one for identical but rotated shapes, and as |ij| + 1 for different shapes, where i, j are the blob numbers of standard and comparison (i.e., it is three for blob1 vs. blob3 and two for blob2 vs. blob1 and blob2 vs. blob3). 
Figure 22
 
Mean reflection strength errors for all standard values depending on the BRDF (blue vs. red), the standard illuminations (top vs. bottom), and the degree of shape dissimilarity (left to right in the order of increasing dissimilarity; see text for the exact definition). A reflection strength error of zero means perfect gloss constancy. The shaded error regions correspond to ±1 SEM. The dashed vertical lines enclose the standard values from 0.3 to 0.63 for which the error made with different BRDFs can best be compared. Please note that the sample sizes differ in the four dissimilarity conditions. The proportions from zero to three are 1, 1, 4/3, 1/2.
Figure 22
 
Mean reflection strength errors for all standard values depending on the BRDF (blue vs. red), the standard illuminations (top vs. bottom), and the degree of shape dissimilarity (left to right in the order of increasing dissimilarity; see text for the exact definition). A reflection strength error of zero means perfect gloss constancy. The shaded error regions correspond to ±1 SEM. The dashed vertical lines enclose the standard values from 0.3 to 0.63 for which the error made with different BRDFs can best be compared. Please note that the sample sizes differ in the four dissimilarity conditions. The proportions from zero to three are 1, 1, 4/3, 1/2.
In those regions in which the standard values used with each BRDF overlap, the general pattern of errors observed with both types of BRDF is quite similar, but the absolute error level is, in most cases, higher with a Ward-BRDF. This is particularly true for the interval 0.3 ≤ r ≤ 0.63, which comprises two standard values of similar size for each BRDF and in which the BRDFs can, therefore, best be compared (Figure 23). At least for standard values less than r < 1, the mean reflection strength errors do not exceed a value of 0.2. With respect to the Fresnel-BRDF, this corresponds to a deviation of the mean refractive index settings from the true value by less than 0.2. 
Figure 23
 
The standard deviation of the reflection strength errors in Experiment 4 for each subject. The values in the upper row are computed for all standard values, those in the lower row only for the standard values between 0.3 and 0.63 that were similar in both BRDFs.
Figure 23
 
The standard deviation of the reflection strength errors in Experiment 4 for each subject. The values in the upper row are computed for all standard values, those in the lower row only for the standard values between 0.3 and 0.63 that were similar in both BRDFs.
Illumination effect
In the experiment, the objects that had to be matched differed in both their illumination and their shape, and each factor potentially contributes to the overall error shown in Figure 22
The illumination effect can be isolated by considering the differences between the reflection strength settings made with different standard illuminations under otherwise identical conditions. The expected value of this difference is zero if there is no systematic effect of illumination. Figure 24 shows the absolute values of the mean of these differences across all subjects. In line with the hypothesis of an improved constancy with a Fresnel-BRDF, the illumination effects observed with the Fresnel-BRDF are systematically lower than with the Ward-BRDF. The illumination effect decreases with increasing shape dissimilarity for both BRDFs. 
Figure 24
 
The effect of the illumination on gloss constancy depending on BRDF and standard value. The data points are absolute values of the mean differences between the reflection strength errors made with different standard illuminations under otherwise identical conditions. For additional information, see the caption of Figure 22.
Figure 24
 
The effect of the illumination on gloss constancy depending on BRDF and standard value. The data points are absolute values of the mean differences between the reflection strength errors made with different standard illuminations under otherwise identical conditions. For additional information, see the caption of Figure 22.
Shape effect
To isolate shape-dependent effects from errors related to an illumination change, one may consider the means across the two standard illuminations shown in Figure 22. These mean values are depicted in Figure 25
Figure 25
 
The effect of shape on gloss constancy. The data points show the mean reflection strength errors depending on the standard values. They correspond to the means of the values given in Figure 22 across the two standard illuminations. For additional information, see the caption of Figure 22.
Figure 25
 
The effect of shape on gloss constancy. The data points show the mean reflection strength errors depending on the standard values. They correspond to the means of the values given in Figure 22 across the two standard illuminations. For additional information, see the caption of Figure 22.
Again, the general pattern of the results across the different levels of shape dissimilarity is rather similar for both BRDFs. When standard and test objects have identical shapes (shape dissimilarity zero), the systematic errors made with the Ward- and Fresnel-BRDFs are almost identical. This is plausible because, in this case, the effects of both illumination and shape are absent. With increasing shape dissimilarity, the reflection strength errors increase slightly more with a Ward-BRDF than with a Fresnel-BRDF. Again, this is consistent with the assumption of improved gloss constancy with a Fresnel-BRDF. 
Another prominent feature of these results is an effect of the standard value with low standard values leading to a systematic overestimation in the match and high standard values to a systematic underestimation. 
Interindividual differences
There are considerable interindividual differences in the accuracy of the settings. This can be seen in Figure 23, which shows the standard deviation of the reflection strength errors for each subject. The values in the top row are calculated over all standard values; those in the bottom row for standard values between 0.3 and 0.63, where the standard values of both BRDFs are similar and one would, therefore, expect similar error levels if the BRDF did not matter. The consistently larger values with a Ward-BRDF in the lower row are also in line with the assumption of an improved gloss constancy with a Fresnel-BRDF. 
Discussion
The results of Experiments 1 and 3 indicate that the constancy of perceived gloss across different illuminations and shapes is much higher when the objects are rendered with correct Fresnel effects. This led to the expectation that the degree of constancy found in a gloss-matching task, in which the subjects adjust the reflection strength, is also higher with a Fresnel-BRDF. The results of Experiment 4 confirm this hypothesis. 
This finding is in line with the informal observation made in the experiment that the matching task was subjectively easier with a Fresnel-BRDF. In stimuli with correct Fresnel effects, one has the impression to modify the properties of a homogeneous material and of possessing a well-defined criterion as to when the two materials are maximally similar. For stimuli rendered with the Ward-BRDF, by contrast, a clear impression of a homogeneously glossy material is often lacking. Subjectively, this makes it more difficult to decide on which part of the surface the match should be based. As a consequence, in trying to find the best match, one often tends to base the judgment on some presumably related low-level image attributes, such as the mean brightness, local contrasts, or properties of conspicuous highlights. 
The settings made in Experiment 4 with the Ward-BRDF deviated much less from the standard values than in the similar experiment reported in Olkkonen and Brainard (2011). For standard ρs of 0.04 and 0.16, they found mean matches of more than 0.15 and 0.4, respectively (see figure 3 of their paper). Translated to the standardized values used here, these values would correspond to reflection strength errors of (0.15 − 0.04)/0.17 = 0.650 and (0.4 − 0.15)/0.17 = 1.47, respectively, which are up to seven times larger than those observed in Experiment 4. The causes for these large differences are, at present, unclear. Possible candidates are their use of an HDR display or of more different illumination maps. It seems also possible that these are artifacts of the procedure because settings of ρs near 0.5 as they are reported by these authors seem unrealistically high. For comparison, Pellacini et al. (2000) regard values of ρs = 0.099 as “high specular energy,” and the maximum value of ρs = 0.28 used in the present study already leads to clearly stronger reflection strengths than an ior of 2.0 in the Fresnel model (see Appendix Figure A3). 
The present investigation revealed also some potential methodological problems. A first problem is that the accuracy of the settings varied considerably between subjects. Although it is clearly possible that the subjects were not all equally motivated to really find the best possible match, an alternative hypothesis is that some subjects were confused by the fact that trials with Ward- and Fresnel-BRDFs appeared randomly intermixed in the experiment. Especially for naive subjects, this may have prevented them from developing an appropriate matching strategy for each type of BRDF. 
Furthermore, comparing the degree of gloss constancy across the BRDFs is complicated by the fact that the psychophysical functions that relate the reflection strength parameters ior and ρs to perceived gloss strength are not yet known. If the corresponding functions differ markedly in form, threshold, and saturation level, this could also influence the results. For future experiments in which settings are compared between BRDFs, it seems, therefore, advisable to explicitly determine the corresponding psychophysical functions for Fresnel- and Ward-BRDFs beforehand. 
A general methodological problem associated with matching experiments using complex stimuli, such as those presented here, is that it remains unclear whether the subjects actually used the intended comparison criterion. In the present case, the subjects were asked to match perceived glossiness. However, when a clear uniform gloss impression is lacking, the subject may be tempted or even forced to use a simpler criterion that presumably is related to the target property. The problem is that such strategies may lead to settings that do not deviate much from the standard values even though a material match in the actual sense was not possible. Given that the gloss impression is generally less vivid with the Ward-BRDF, it is, in this case, more probable that the subjects relied on such strategies with the possible consequence that the actual degree of gloss constancy is overestimated in this condition. 
Aside from such methodological issues, there is a more fundamental theoretical problem concerning the matching task conducted in Experiment 4 because it is implicitly based on the assumption that the global reflection strength is used to discern different materials. However, this assumption seems not very plausible because, in typical environments, the refractive index of real glossy materials lies in the narrow range between 1.33 (water) and 1.7 (glass). In general, differences in the apparent overall specular reflection strength depend more on the albedo of the diffuse reflection and properties of the illumination than on variations in the refractive index. Thus, although the strength of the specular reflection is clearly a material property, it seems more appropriate to regard it as a material constant than as a material parameter. This consideration could partly explain the relatively large and systematic dependence of the reflection strength settings on the standard values. The result pattern for the Fresnel-BRDF shown in Figure 25 is to be expected if one assumes a general tendency to choose refractive indices close to the middle of the abovementioned range between 1.33 and 1.7 that gets even more pronounced if a comparison of the actual strength of the specular reflection is difficult, for example, due to a large difference in shape. 
Summary and general discussion
Simple reflection models for glossy surfaces, such as the Ward model, often neglect Fresnel effects, i.e., the change in the strength of the specular reflection with the direction of the incident light. Although the retinal image computed with such models can deviate quite strongly from a physically correct rendering, it often leads to a quite convincing gloss impression. Of course, this is the very reason for their existence, but this doesn't imply that such models are also well suited for the investigation of gloss perception. One reason for being suspicious in this respect are the many cases in which the gloss impression produced by such models appears strikingly weak and unrealistic. 
The aim of the present study was to investigate how Fresnel effects influence perceived gloss and gloss constancy. The general approach was to compare the gloss impressions produced by BRDFs with and without Fresnel effects. Two variants of the Ward model, the original and an improved version (Geisler-Moroder & Dür, 2010) that guarantees energy conservation, were used as representatives of models without Fresnel effects. This choice was also motivated by the fact that the original Ward model was often used in investigations on gloss perception. To consider possible moderating influences, several context factors were varied, namely the roughness, albedo, and refractive index of the material; the shape of the glossy object; the type of illumination; and the kind of floor on which the object rests. 
Important findings
In Experiment 1, a stimulus calculated with a Fresnel-BRDF (Walter et al., 2007) was given as standard. A test stimulus rendered with a Ward-BRDF was shown simultaneously under identical conditions, and the subjects were asked to set the strength of the specular reflection of this test stimulus such that the resulting gloss impression was as similar as possible to that of the standard. Afterward, the subjects rated the similarity of this “best match” to the standard with respect to perceived gloss and overall brightness. 
It turned out that a complete match was virtually never possible. The reported deviations concerned both the gloss impression and the overall brightness. The perceived difference in overall brightness was relatively constant, whereas the perceived gloss difference depended strongly on the context. The illumination type, object shape, and surface roughness had the largest effects. The values of the reflection strength parameter chosen for the best match were not constant for a fixed Fresnel-BRDF, but varied also considerably with context. This dependency on the context was more pronounced for the original Ward model than for its improved energy-conserving variant, and the similarity ratings were also systematically lower for the original Ward model. 
These results show that the gloss impressions produced with Fresnel- and Ward-BRDFs are qualitatively different. However, they do not allow direct conclusions as to which gloss impression is “better” in the sense of stronger, more vivid, or more natural. To overcome this limitation, a second experiment was conducted in which the stimulus pairs matched in Experiment 1 were directly compared with respect to the relative quality of the gloss impression (two-alternative forced choice). In more than 97% of the pair comparisons, the subjects judged the perceived gloss produced by the Fresnel-BRDF as more realistic, more vivid, and spatially more homogeneous. 
To gain a more thorough picture about the relative strength of the gloss impression with and without Fresnel effects, a third experiment was conducted in which the subjects rated the objects rendered with Fresnel-BRDF and with both Ward-BRDFs with respect to gloss strength (matte vs. highly glossy) and gloss quality (unrealistic vs. highly realistic). All objects were taken from Experiment 1, the ones with the Fresnel effects were used in Experiment 1 as standard objects; the ones with the Ward-BRDFs were the corresponding best matches. In almost all cases, the gloss with a Fresnel-BRDF was judged as stronger and more realistic than the gloss produced by the matched Ward-BRDFs. Also, in this case, strong context effects were found. Of particular importance is the type of illumination. The differences between the BRDFs are particularly strong if the illumination is relatively homogeneous. In the extreme case of a uniform illumination, a gloss impression is usually retained with a Fresnel-BRDF, whereas it mostly vanishes with a Ward-BRDF. 
The results of Experiments 1 and 3 indicate that the degree of constancy of the gloss impression across a variation in context is larger with a Fresnel-BRDF than with a Ward-BRDF. This hypothesis was tested in a fourth experiment in which constancy was measured in a gloss-matching task across different illuminations and shapes. The degree of gloss constancy in this task was indeed found to be higher with a Fresnel-BRDF. In particular, with a Fresnel-BRDF, the settings seem more robust against changes in illumination than with a Ward-BRDF. 
According to the subjects' reports made in interviews after the four experiments, the main problem with the Ward model was that the brightness of the diffuse color was typically markedly reduced near the rim of the object (or, more generally, at surface areas, at which the surface normal is nearly perpendicular to the viewing direction) and that the gloss impression varies considerably across the surface. Both of these factors seem to contribute to an unrealistic gloss impression. 
Fresnel effects as a cue for gloss perception
An important goal of the design of Experiments 13 was to isolate a possible effect of the specular reflection strength distribution on perceived gloss from other factors that may also have an influence. To this end, all scene factors and the remaining material parameters (diffuse color, roughness) were held constant across all BRDFs. To also exclude a difference in the absolute level of specular reflection strength as a possible explanatory factor, the subjects chose in Experiment 1 for each of the tested conditions the specular reflection strength of the Ward-BRDF that maximized the similarity of the gloss impression resulting with Ward- and Fresnel-BRDFs. The reflection strength leading to the best match was then also used in Experiments 2 and 3
This strict control of other potential influencing factors makes it highly probable that the improved gloss impression and higher degree of gloss constancy observed with the Fresnel-BRDF in Experiment 3 can actually be attributed to the presence of correct Fresnel effects. This suggests that the shape-related variation of the strength of specular reflections across a surface can be used by the visual system as an information source for gloss perception. The relevant aspect of Fresnel effects is the correlated information in the retinal image that is directly available to the visual system. The nature of these “proximal” Fresnel effects can be directly derived from the well-understood image-generation process and consists in a shape-dependent intensity variation of the mirror image of the global illumination that is formed inside the object's retinal projection. With a uniform illumination, this intensity increases monotonously with surface inclination relative to the viewing direction and reaches a maximum near grazing angle. Figure 26 illustrates the corresponding intensity pattern for some objects. In real scenes, however, there are several factors that complicate the estimate of this pattern from the stimulus: (a) Actual illuminations are seldom completely homogeneous, (b) there is often some form of self-shadowing, and (c) the contribution of the mirror image to the total local brightness must be estimated, which requires that diffuse and specular reflections differ in some respect, e.g., in color or texture (for the similar problem of separating texture and shading, see Kim, Marlow, & Anderson, 2014). For such reasons, this regularity can only be used heuristically, and it is at present an open empirical question how exactly the visual system evaluates this cue. In principle, many different strategies are possible. A simple strategy could be to check for a rapid increase in brightness near the rim of (approximately convex) objects and relative low brightness levels in the central area, whereas more sophisticated strategies could relate to an actual estimate of the brightness of the mirror image depending on surface inclination. The latter strategy requires at least a rough estimate of object shape. This information could be gained from the object's outline (Hayward, 1998), shading (Zhang, Tsai, Cryer, & Shah, 1999), or the distortion pattern in the mirror image (Fleming et al., 2004). Because the brightness distribution itself provides shape information, as can be seen in Figure 26, a cooperative computation of form and gloss seems possible, in which the final estimation needs to be compatible with information provided by different cues (Kersten, 1991). 
Figure 26
 
Relative intensity of the mirror image of a homogeneous diffuse global illumination according to Fresnel's equations, shown for a sphere and the four blobs depicted in the four rightmost columns of Figure 21. The Fresnel effects are computed for a refractive index of 1.45 and displayed with a gamma of 1.8.
Figure 26
 
Relative intensity of the mirror image of a homogeneous diffuse global illumination according to Fresnel's equations, shown for a sphere and the four blobs depicted in the four rightmost columns of Figure 21. The Fresnel effects are computed for a refractive index of 1.45 and displayed with a gamma of 1.8.
The interpretation that the visual system is actually sensitive to a physically correct reflection and that the observed effects on gloss perception are not simply due to an increased availability of some general gloss cues is further corroborated by the demonstration in Figure 27. It shows that a corresponding darkening of the diffuse component is also required and that even slight deviations from the physically correct relationship between the strength of specular and diffuse reflections may deteriorate the impression of an opaque, homogeneously glossy material. The hypothesis that this physical regularity is used as a cue seems also tenable from a computational perspective because it is well known from other domains, for instance, from “shape from shading” (Durou, Falcone, & Sagona, 2008; Horn, 1970), that the visual system is, in principle, able to exploit cues of a similar complexity. In “shape from shading” the shape-related darkening of diffuse surfaces, i.e., the shading pattern, is a rather reliable cue for local surface orientation. 
Figure 27
 
Combining a given “Fresnel reflection pattern” with correct and incorrect diffuse reflections leads to qualitatively different gloss impressions. Panels B and C show the same specular reflections computed according to Fresnel's equations combined with either the correct diffuse component shown in panel A or with an isolated diffuse component shown in panel D, which is incompatible (see also Figure 3). Although the image of the incorrect combination in panel C is everywhere brighter, the quality of the gloss impression seems actually slightly reduced, i.e., the material appears both more inhomogeneous and less realistic (with respect to the interpretation as an opaque glossy surface).
Figure 27
 
Combining a given “Fresnel reflection pattern” with correct and incorrect diffuse reflections leads to qualitatively different gloss impressions. Panels B and C show the same specular reflections computed according to Fresnel's equations combined with either the correct diffuse component shown in panel A or with an isolated diffuse component shown in panel D, which is incompatible (see also Figure 3). Although the image of the incorrect combination in panel C is everywhere brighter, the quality of the gloss impression seems actually slightly reduced, i.e., the material appears both more inhomogeneous and less realistic (with respect to the interpretation as an opaque glossy surface).
The fact that the gloss impressions produced with Ward- and Fresnel-BRDFs are often qualitatively different (Figure 21 provides some examples) is another important observation that speaks in favor of regarding Fresnel effects as a primary gloss cue. Otherwise, it would be hard to explain why it was often impossible in Experiment 1 to match the gloss impressions by adjusting the reflection strength of the Ward-BRDF although all other stimulus factors were identical. On the other hand, this is to be expected when the relevant cue consists of a certain intensity pattern that cannot be produced with a Ward-BRDF. 
This does not imply, however, that this “Fresnel effects cue” is uncorrelated to other gloss cues that have been discussed in the literature. Marlow and Anderson (2013) distinguish three such cues, namely “sharpness,” “contrast,” and “coverage” of specular reflections and discuss several generative constraints for these cues. In particular, they see a relation between Fresnel effects and the contrast cue. This assumption seems plausible, but the present results suggest that it is not only the local contrast (which may vary considerably with position) that matters, but also the shape-dependent contrast pattern across the surface. It is also likely that Fresnel effects affect perceived coverage because a salient feature of surfaces rendered with correct Fresnel effects is that a larger portion appears to be “covered by a glossy layer” than with a Ward-BRDF (see, for instance, Figure 12). It is also highly probable that the contribution of Fresnel effects to gloss perception depends on the situation. The influence should be strong if the intensity variations of the mirror image are easily accessible, for instance, with global illumination and a homogeneous material. A lesser influence is to be expected if only localized illuminations are used because in this case the mirror image of the environment contains only a sparse pattern of isolated highlights. 
A relation of Fresnel effects to the specific image statistics that were proposed by Motoyoshi et al. (2007) as gloss indicators is less likely because these completely ignore shape-related information that plays a central role in Fresnel effects. Furthermore, Olkkonen and Brainard (2011) have shown that these image cues could not predict the settings in their experiment made under conditions similar to those used in the present Experiment 4. It may well be possible, however, that more specific image statistics may be used to detect deviations from correct Fresnel effects. 
These considerations as to how exactly Fresnel effects are exploited in gloss perception and how they relate to other gloss cues are inevitably somewhat speculative and need to be more thoroughly investigated in future research. 
Conclusions and implications
Together, the present findings suggest that the Ward model is often not a good approximation to the physically more realistic Fresnel-BRDF that was used as a reference. This does not only hold for the original Ward model, but also for its energy-conserving variant, which performed only slightly better. 
This suggests that Fresnel effects, which are not correctly handled in the Ward models, strongly contribute to gloss perception. This is especially true in situations in which the illumination is relatively homogeneous. Such cases occur frequently under natural viewing conditions, for instance, outdoors under a cloudy sky, and the perceived glossiness of a surface would be underestimated if the Ward-BRDF were used in simulations of such environments. 
A comparison of the results obtained with the original and the balanced Ward models indicates that the negative effects of an incorrect handling of Fresnel effects on the gloss impression and gloss constancy increase with the degree of deviation from the correct pattern of mirror reflection strength. As the specular reflection strength computed with the Ward model is often complementary to the physically correct one, i.e., it is large were it should be low and vice versa (see Figure 3), its deviation from the true values must be considered rather large and qualitative. Lesser, mere quantitative violations as they are demonstrated in Figure 27 appear to result in correspondingly weaker adverse effects. 
The results of Experiments 1 and 3 indicated that relatively high degrees of gloss constancy can be achieved, even in static scenes, if the surfaces are correctly rendered and that much lower degrees of gloss constancy would have been found with the Ward models. In the fourth experiment, the gloss constancy performance was compared using a gloss-matching task across different illuminations and shapes. The degree of gloss constancy with correct Fresnel effects was found to be systematically higher than without Fresnel effects. 
Given these results, it cannot be excluded that experiments on gloss perception using the Ward model and global illumination have led to partially misleading results and conclusions. This applies not only to statements regarding the necessary and sufficient conditions for the occurrence of gloss, but also to conclusions about gloss constancy performance. It seems, therefore, advisable to verify such results using a BRDF that correctly accounts for Fresnel effects. More generally speaking, it seems reasonable to prefer such physically plausible BRDFs in further research on gloss perception, at least if a global illumination is used. 
The present results suggest also some conclusions on related topics. A first point concerns the very concept of gloss. It is often assumed that gloss has two dimensions, namely an aspect related to surface roughness and a second aspect, specular reflection strength, which is related to the relative amount of diffuse and specular reflection (Chadwick & Kentridge, 2015). This is, for instance, reflected in the corresponding parameters of the Ward model. 
However, in real glossy surfaces, the two dimensions are not independent because the strength of specular reflections depends on the three factors: refractive index, surface roughness, and surface orientation. The first factor, the refractive index, influences the overall strength of specular reflections and is not very informative because, for most glossy materials, the refractive index lies in a narrow range around 1.5. The influence of surface orientation on the strength of specular reflection is due to Fresnel effects and leads to strong variations of the gloss strength across the surface. But surface roughness also contributes to the strength of specular reflections: Given an inhomogeneous illumination, smoother surfaces reflect the incoming light into narrower spatial angles, which leads to brighter and more focused reflections of localized light sources. 
Thus, if the BRDF is energy conserving and the illumination inhomogeneous, the strength of specular reflections in the stimulus could in principle also be used as a cue for surface roughness provided that it is possible to discount the additional effects of surface orientation. However, this cannot be expected with the original Ward-BRDF because it is neither energy conserving nor does it correctly account for Fresnel effects. This may also contribute to failures to correctly interpret specular reflections computed with the Ward-BRDF in terms of material properties. A prediction derived from these considerations is that gloss constancy with respect to the roughness parameter may also be improved with a Fresnel-BRDF. This hypothesis is compatible with the results of Experiment 3 but should be directly investigated. 
A second point concerns investigations of gloss in metallic and dielectric materials, which differ fundamentally with respect to Fresnel effects. The present finding that, in general, Fresnel effects strongly influence the perceived gloss of a surface suggests that gloss perception uses different cues and mechanisms in both material classes. Therefore, the approach used by Vangorp et al. (2007) to investigate gloss perception in both types of material using the original Ward-BRDF appears problematic. 
Informal observations indicate that also shape perception is impaired if Fresnel effects are not correctly accounted for (see, e.g., Figure 6). Apparently, the visual system is able to exploit the relationship between surface orientation and the strength of the specular reflection in shape perception. It is, at present, unclear how strong such effects are, but it seems advisable to use Fresnel-BRDFs also in investigations of shape perception. 
Acknowledgments
This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant number FA425/3-2. I want to thank Gunnar Wendt and Nick Schlüter for their comments and constructive criticism that helped to improve the manuscript and Lili Rathke for her help with conducting the experiments. 
Commercial relationships: none. 
Corresponding author: Franz Faul. 
Address: Institut für Psychologie, Universität Kiel, Kiel, Germany. 
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Appendix: Additional data and results
Ratings of the brightness similarity of the match
Figure A1 shows the main effects of the six context factors on the brightness similarity rating of the matches in Experiment 1
Figure A1
 
Main effects of the six context factors on the brightness similarity ratings for both Ward BRDFs. The error bars correspond to ±2 SEM.
Figure A1
 
Main effects of the six context factors on the brightness similarity ratings for both Ward BRDFs. The error bars correspond to ±2 SEM.
Comparison stimuli for Experiment 3
A subset of comparison stimuli used in Experiment 3 is shown in Figure A2
Figure A2
 
Comparison stimuli for the rating of gloss strength in Experiment 3. Only the images for integer rating values are shown, where s = 0 was the smallest and s = 9 the largest scale value. Intermediate values were also possible.
Figure A2
 
Comparison stimuli for the rating of gloss strength in Experiment 3. Only the images for integer rating values are shown, where s = 0 was the smallest and s = 9 the largest scale value. Intermediate values were also possible.
Comparable gloss scales for Experiment 4
To allow a fair comparison between the errors made with a Ward- and a Fresnel-BRDF, it is essential to take the perceptual gloss ranges related to the nominal reflection strength values into account. Figure A3 shows that the gloss range realized with the Fresnel-BRDF (middle row) is more closely matched with ρs values ranging from 0 to 0.17 (bottom row) than with the range realized in the experiment (top row). Thus, approximately equal perceptual scales result with ρs = 0.17 (ior − 1). 
Figure A3
 
Ranges of specular reflection strength for Ward- and Fresnel-BRDFs and corresponding ranges in gloss strength. Upper row: An equidistant subset of the 51 ρs values ranging from 0 to 0.28 used for the Ward-BRDF in Experiment 4. Middle row: An equidistant subset of the 51 ior values ranging from one to two used for the Fresnel-BRDF in Experiment 4. Lower row: Values for ρs ranging from 0 to 0.17 match the gloss range shown in the middle row more closely than the range shown in the upper row.
Figure A3
 
Ranges of specular reflection strength for Ward- and Fresnel-BRDFs and corresponding ranges in gloss strength. Upper row: An equidistant subset of the 51 ρs values ranging from 0 to 0.28 used for the Ward-BRDF in Experiment 4. Middle row: An equidistant subset of the 51 ior values ranging from one to two used for the Fresnel-BRDF in Experiment 4. Lower row: Values for ρs ranging from 0 to 0.17 match the gloss range shown in the middle row more closely than the range shown in the upper row.
Statistical results for Experiment 1
Tables A1A3 summarize the results of a seven-way ANOVA for the gloss strength settings, the gloss similarity ratings, and the gloss quality ratings, respectively. 
Table A1
 
Results of a seven-way ANOVA for the reflection strength settings in Experiment 1. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A1
 
Results of a seven-way ANOVA for the reflection strength settings in Experiment 1. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A2
 
Results of a seven-way ANOVA for the gloss similarity ratings. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A2
 
Results of a seven-way ANOVA for the gloss similarity ratings. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A3
 
Results of a seven-way ANOVA for the brightness similarity rating for the match in Experiment 1. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A3
 
Results of a seven-way ANOVA for the brightness similarity rating for the match in Experiment 1. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Statistical results for Experiment 3
Tables A4 and A5 summarize the results of a seven-way ANOVA for the gloss strength and gloss quality ratings. 
Table A4
 
Results of a seven-way Anova for the gloss strength ratings from Experiment 3. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A4
 
Results of a seven-way Anova for the gloss strength ratings from Experiment 3. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A5
 
Results of a seven-way ANOVA for the gloss quality ratings from Experiment 3. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A5
 
Results of a seven-way ANOVA for the gloss quality ratings from Experiment 3. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Figure 1
 
Typical BRDF of a glossy surface that combines diffuse and diffuse specular reflection. The color of the surface depends on the diffuse (hemispherical) part of the BRDF, the strength and blurriness of the mirror reflection on the specular part (lobe). The narrower the lobe, the sharper the edges in the mirror image and the glossier the surface appears.
Figure 1
 
Typical BRDF of a glossy surface that combines diffuse and diffuse specular reflection. The color of the surface depends on the diffuse (hemispherical) part of the BRDF, the strength and blurriness of the mirror reflection on the specular part (lobe). The narrower the lobe, the sharper the edges in the mirror image and the glossier the surface appears.
Figure 2
 
The strength of the reflection of a ray of light hitting a dielectric surface from a medium with a refractive index of n0 = 1 (air) as a function of its angle of incidence relative to the surface normal and the refractive index n1 of the material.
Figure 2
 
The strength of the reflection of a ray of light hitting a dielectric surface from a medium with a refractive index of n0 = 1 (air) as a function of its angle of incidence relative to the surface normal and the refractive index n1 of the material.
Figure 3
 
Comparison of the specular (left) and the diffuse (center) components of the BRDF of a highly glossy material that reproduces Fresnel effects correctly (top) or incorrectly (Ward model, bottom). The specular reflections on the left side are exaggerated in brightness to make the differences easier to see. If the components are displayed correctly, the sum of the left and middle images results in the right image. The geometry of the reflection layer is identical in both cases (and, if applicable, also its blurriness that depends on surface roughness).
Figure 3
 
Comparison of the specular (left) and the diffuse (center) components of the BRDF of a highly glossy material that reproduces Fresnel effects correctly (top) or incorrectly (Ward model, bottom). The specular reflections on the left side are exaggerated in brightness to make the differences easier to see. If the components are displayed correctly, the sum of the left and middle images results in the right image. The geometry of the reflection layer is identical in both cases (and, if applicable, also its blurriness that depends on surface roughness).
Figure 4
 
The seven object shapes used in the experiments.
Figure 4
 
The seven object shapes used in the experiments.
Figure 5
 
Three global illumination maps (first column), three floor conditions (second column), and three different values for the roughness parameter (third column) were used. The objects in the first three columns are calculated with a refractive index of 1.7 and an albedo of 0.5. The last column shows the object labeled α = 0.001 in the third column with a different level of the factors refractive index (ior = 1.5) and albedo (albedo = 0.3).
Figure 5
 
Three global illumination maps (first column), three floor conditions (second column), and three different values for the roughness parameter (third column) were used. The objects in the first three columns are calculated with a refractive index of 1.7 and an albedo of 0.5. The last column shows the object labeled α = 0.001 in the third column with a different level of the factors refractive index (ior = 1.5) and albedo (albedo = 0.3).
Figure 6
 
Objects with an increasing specular reflection strength rendered with Fresnel-BRDF (top) and Ward-BRDF (bottom). In the case of the Fresnel-BRDF, this is achieved by increasing the refractive index from 1.4 to 2 and in the case of the Ward-BRDF by increasing the specular reflection strength parameter ρs from 0.056 to 0.14. With the Fresnel-BRDF, both the gloss impression and the object color are spatially more homogeneous, and the shape of the object stands out more clearly than with the Ward model. With the Ward-BRDF, a darkening occurs toward the rim, and with increasing ρs, the color and brightness of the material changes: It appears both brighter and more desaturated. An exact visual match of the material impression across the different BRDFs is, therefore, generally not possible.
Figure 6
 
Objects with an increasing specular reflection strength rendered with Fresnel-BRDF (top) and Ward-BRDF (bottom). In the case of the Fresnel-BRDF, this is achieved by increasing the refractive index from 1.4 to 2 and in the case of the Ward-BRDF by increasing the specular reflection strength parameter ρs from 0.056 to 0.14. With the Fresnel-BRDF, both the gloss impression and the object color are spatially more homogeneous, and the shape of the object stands out more clearly than with the Ward model. With the Ward-BRDF, a darkening occurs toward the rim, and with increasing ρs, the color and brightness of the material changes: It appears both brighter and more desaturated. An exact visual match of the material impression across the different BRDFs is, therefore, generally not possible.
Figure 7
 
Main effects of the six context factors on the specular reflection strength settings (ρs) for both Ward-BRDFs. The error bars correspond to ±2 SEM. The number of observations belonging to each data point is given by n = 2,268/#levels and ranges from n = 326 in panel A to n = 1,134 in panels C and F. Note: Here and in subsequent plots, lines between levels of categorical variables are added to emphasize the pattern of the results although, in such cases, the intermediate values are meaningless.
Figure 7
 
Main effects of the six context factors on the specular reflection strength settings (ρs) for both Ward-BRDFs. The error bars correspond to ±2 SEM. The number of observations belonging to each data point is given by n = 2,268/#levels and ranges from n = 326 in panel A to n = 1,134 in panels C and F. Note: Here and in subsequent plots, lines between levels of categorical variables are added to emphasize the pattern of the results although, in such cases, the intermediate values are meaningless.
Figure 8
 
The settings for the specular reflection strength parameter ρs of each subject plotted against the mean setting across all subjects in the same condition. The distributions for the two levels of the refractive index and the two Ward-BRDFs are plotted separately.
Figure 8
 
The settings for the specular reflection strength parameter ρs of each subject plotted against the mean setting across all subjects in the same condition. The distributions for the two levels of the refractive index and the two Ward-BRDFs are plotted separately.
Figure 9
 
Distribution of “impossible gloss matches” (gloss similarity = 0, i.e., gloss with Fresnel-BRDF but not with Ward-BRDF) across different experimental factors and subjects.
Figure 9
 
Distribution of “impossible gloss matches” (gloss similarity = 0, i.e., gloss with Fresnel-BRDF but not with Ward-BRDF) across different experimental factors and subjects.
Figure 10
 
Main effects of the six context factors on the gloss similarity ratings for both Ward-BRDFs. The error bars correspond to ±2 SEM. The number of observations belonging to each data point is given by n = 2,268/#levels and ranges from n = 326 in panel A to n = 1,134 in panels C and F.
Figure 10
 
Main effects of the six context factors on the gloss similarity ratings for both Ward-BRDFs. The error bars correspond to ±2 SEM. The number of observations belonging to each data point is given by n = 2,268/#levels and ranges from n = 326 in panel A to n = 1,134 in panels C and F.
Figure 11
 
Interaction effects with respect to ratings of the gloss similarity of the match between the Fresnel-BRDF and the Ward-BRDFs. There is an interaction between object shape and the type of illumination that can be seen in all subplots A–C: Under the indoor illumination, the gloss similarity is relatively high throughout. With the outdoor illumination, the general level of gloss similarity decreases, and the dependence on object shape increases. This tendency is even more pronounced with a uniform illumination. This interaction is further influenced by the type of BRDF (A), the type of floor in the scene (B), and surface roughness (C). The error bars correspond to ±2 SEM.
Figure 11
 
Interaction effects with respect to ratings of the gloss similarity of the match between the Fresnel-BRDF and the Ward-BRDFs. There is an interaction between object shape and the type of illumination that can be seen in all subplots A–C: Under the indoor illumination, the gloss similarity is relatively high throughout. With the outdoor illumination, the general level of gloss similarity decreases, and the dependence on object shape increases. This tendency is even more pronounced with a uniform illumination. This interaction is further influenced by the type of BRDF (A), the type of floor in the scene (B), and surface roughness (C). The error bars correspond to ±2 SEM.
Figure 12
 
Comparison of the Fresnel-BRDF (positions 1 and 3) with the original Ward-BRDF (positions 2 and 4) under the “indoor” (left) and the uniform (right) illumination. With Fresnel effects, the gloss impression is significantly stronger and more similar across the illuminations.
Figure 12
 
Comparison of the Fresnel-BRDF (positions 1 and 3) with the original Ward-BRDF (positions 2 and 4) under the “indoor” (left) and the uniform (right) illumination. With Fresnel effects, the gloss impression is significantly stronger and more similar across the illuminations.
Figure 13
 
The distribution of the 146 out of 9,072 cases in which the Ward-BRDFs were judged as glossier, depending on the experimental factors and the subjects. The blue parts are the 56 cases that occurred under the uniform illumination, which includes cases in which a gloss impression is lacking even with the Fresnel-BRDF.
Figure 13
 
The distribution of the 146 out of 9,072 cases in which the Ward-BRDFs were judged as glossier, depending on the experimental factors and the subjects. The blue parts are the 56 cases that occurred under the uniform illumination, which includes cases in which a gloss impression is lacking even with the Fresnel-BRDF.
Figure 14
 
Screenshot from Experiment 3. The test object whose gloss was to be evaluated was shown on the left side, and as a comparison the “Utah teapot” (rendered with Fresnel-BRDF) on the right side. The subjects judged the perceived gloss strength (matte vs. highly glossy) by choosing with the left/right arrow keys a scale value between zero and nine, which led to a corresponding change in the surface roughness of the Utah teapot. They also rated the perceived quality of the gloss impression (natural vs. unnatural) in the test object on a scale from zero to nine. This selection was made with the up/down arrow keys.
Figure 14
 
Screenshot from Experiment 3. The test object whose gloss was to be evaluated was shown on the left side, and as a comparison the “Utah teapot” (rendered with Fresnel-BRDF) on the right side. The subjects judged the perceived gloss strength (matte vs. highly glossy) by choosing with the left/right arrow keys a scale value between zero and nine, which led to a corresponding change in the surface roughness of the Utah teapot. They also rated the perceived quality of the gloss impression (natural vs. unnatural) in the test object on a scale from zero to nine. This selection was made with the up/down arrow keys.
Figure 15
 
Main effects of the six context factors on the mean gloss strength (matte to highly glossy; upper panel in subplots A–F) and the mean gloss quality (unrealistic to highly realistic; lower panel in subplots A–F) depending on the BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 15
 
Main effects of the six context factors on the mean gloss strength (matte to highly glossy; upper panel in subplots A–F) and the mean gloss quality (unrealistic to highly realistic; lower panel in subplots A–F) depending on the BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 16
 
Mean ratings for gloss strength (A) and gloss quality (B) in dependence on object shape, illumination type, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 16
 
Mean ratings for gloss strength (A) and gloss quality (B) in dependence on object shape, illumination type, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 17
 
Mean ratings for gloss strength (A) and gloss quality (B) in dependence on roughness, illumination type, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 17
 
Mean ratings for gloss strength (A) and gloss quality (B) in dependence on roughness, illumination type, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 18
 
Mean ratings for gloss strength (A) and gloss quality (B) under uniform illumination in dependence on object shape, type of floor, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 18
 
Mean ratings for gloss strength (A) and gloss quality (B) under uniform illumination in dependence on object shape, type of floor, and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 19
 
Mean ratings for gloss strength (A) and gloss quality (B) of different subjects under the “outdoor” illumination in dependence of surface roughness and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 19
 
Mean ratings for gloss strength (A) and gloss quality (B) of different subjects under the “outdoor” illumination in dependence of surface roughness and BRDF. The error bars correspond to ±2 SEM. Data points below the horizontal gray line in the quality plots indicate cases in which there was no reliable gloss impression.
Figure 20
 
Some demonstrations of how gloss perception under a uniform illumination differs when the stimuli are rendered with Fresnel-BRDF (top row) or the original Ward-BRDF (bottom row). All objects are nominally of highly glossy material. A free-floating sphere (second column) does not appear glossy with either of the two BRDFs, whereas an object with different curvatures under the same condition (left column) appears somewhat glossy with Fresnel-BRDF but completely matte with the Ward-BRDF. The only difference between the scenes shown in columns 2 and 3 is that, in the latter, a uniform floor was added. This improves the gloss impression considerably with the Fresnel-BRDF, whereas the corresponding gloss impression with Ward-BRDF is much less vivid and appears rather unrealistic. Adding a texture to the floor (fourth column) improves the gloss impression with Fresnel-BRDF further, whereas this has only a minor effect on the gloss impression with the Ward-BRDF.
Figure 20
 
Some demonstrations of how gloss perception under a uniform illumination differs when the stimuli are rendered with Fresnel-BRDF (top row) or the original Ward-BRDF (bottom row). All objects are nominally of highly glossy material. A free-floating sphere (second column) does not appear glossy with either of the two BRDFs, whereas an object with different curvatures under the same condition (left column) appears somewhat glossy with Fresnel-BRDF but completely matte with the Ward-BRDF. The only difference between the scenes shown in columns 2 and 3 is that, in the latter, a uniform floor was added. This improves the gloss impression considerably with the Fresnel-BRDF, whereas the corresponding gloss impression with Ward-BRDF is much less vivid and appears rather unrealistic. Adding a texture to the floor (fourth column) improves the gloss impression with Fresnel-BRDF further, whereas this has only a minor effect on the gloss impression with the Ward-BRDF.
Figure 21
 
Some of the stimuli used in Experiment 4. Upper and lower rows show the same stimuli rendered with Fresnel- and Ward-BRDFs, respectively. In total, there were three different blobs (blob1, blob2, and blob3 shown in this order in the last three columns) rendered in two orientations under two different illuminations and with varying reflection strength. A stimulus in one illumination (first column) was compared with objects shown in the other illumination. The shape was either identical (second column), identical but viewed from another angle (rotation about the vertical axis), or different with two degrees of dissimilarity; here the fourth column shows the more similar blob, the fifth the more dissimilar one. All stimuli within each row have the same reflection strength, i.e., the four stimuli on the right represent “perfect constancy” matches to the standard depicted in the leftmost column.
Figure 21
 
Some of the stimuli used in Experiment 4. Upper and lower rows show the same stimuli rendered with Fresnel- and Ward-BRDFs, respectively. In total, there were three different blobs (blob1, blob2, and blob3 shown in this order in the last three columns) rendered in two orientations under two different illuminations and with varying reflection strength. A stimulus in one illumination (first column) was compared with objects shown in the other illumination. The shape was either identical (second column), identical but viewed from another angle (rotation about the vertical axis), or different with two degrees of dissimilarity; here the fourth column shows the more similar blob, the fifth the more dissimilar one. All stimuli within each row have the same reflection strength, i.e., the four stimuli on the right represent “perfect constancy” matches to the standard depicted in the leftmost column.
Figure 22
 
Mean reflection strength errors for all standard values depending on the BRDF (blue vs. red), the standard illuminations (top vs. bottom), and the degree of shape dissimilarity (left to right in the order of increasing dissimilarity; see text for the exact definition). A reflection strength error of zero means perfect gloss constancy. The shaded error regions correspond to ±1 SEM. The dashed vertical lines enclose the standard values from 0.3 to 0.63 for which the error made with different BRDFs can best be compared. Please note that the sample sizes differ in the four dissimilarity conditions. The proportions from zero to three are 1, 1, 4/3, 1/2.
Figure 22
 
Mean reflection strength errors for all standard values depending on the BRDF (blue vs. red), the standard illuminations (top vs. bottom), and the degree of shape dissimilarity (left to right in the order of increasing dissimilarity; see text for the exact definition). A reflection strength error of zero means perfect gloss constancy. The shaded error regions correspond to ±1 SEM. The dashed vertical lines enclose the standard values from 0.3 to 0.63 for which the error made with different BRDFs can best be compared. Please note that the sample sizes differ in the four dissimilarity conditions. The proportions from zero to three are 1, 1, 4/3, 1/2.
Figure 23
 
The standard deviation of the reflection strength errors in Experiment 4 for each subject. The values in the upper row are computed for all standard values, those in the lower row only for the standard values between 0.3 and 0.63 that were similar in both BRDFs.
Figure 23
 
The standard deviation of the reflection strength errors in Experiment 4 for each subject. The values in the upper row are computed for all standard values, those in the lower row only for the standard values between 0.3 and 0.63 that were similar in both BRDFs.
Figure 24
 
The effect of the illumination on gloss constancy depending on BRDF and standard value. The data points are absolute values of the mean differences between the reflection strength errors made with different standard illuminations under otherwise identical conditions. For additional information, see the caption of Figure 22.
Figure 24
 
The effect of the illumination on gloss constancy depending on BRDF and standard value. The data points are absolute values of the mean differences between the reflection strength errors made with different standard illuminations under otherwise identical conditions. For additional information, see the caption of Figure 22.
Figure 25
 
The effect of shape on gloss constancy. The data points show the mean reflection strength errors depending on the standard values. They correspond to the means of the values given in Figure 22 across the two standard illuminations. For additional information, see the caption of Figure 22.
Figure 25
 
The effect of shape on gloss constancy. The data points show the mean reflection strength errors depending on the standard values. They correspond to the means of the values given in Figure 22 across the two standard illuminations. For additional information, see the caption of Figure 22.
Figure 26
 
Relative intensity of the mirror image of a homogeneous diffuse global illumination according to Fresnel's equations, shown for a sphere and the four blobs depicted in the four rightmost columns of Figure 21. The Fresnel effects are computed for a refractive index of 1.45 and displayed with a gamma of 1.8.
Figure 26
 
Relative intensity of the mirror image of a homogeneous diffuse global illumination according to Fresnel's equations, shown for a sphere and the four blobs depicted in the four rightmost columns of Figure 21. The Fresnel effects are computed for a refractive index of 1.45 and displayed with a gamma of 1.8.
Figure 27
 
Combining a given “Fresnel reflection pattern” with correct and incorrect diffuse reflections leads to qualitatively different gloss impressions. Panels B and C show the same specular reflections computed according to Fresnel's equations combined with either the correct diffuse component shown in panel A or with an isolated diffuse component shown in panel D, which is incompatible (see also Figure 3). Although the image of the incorrect combination in panel C is everywhere brighter, the quality of the gloss impression seems actually slightly reduced, i.e., the material appears both more inhomogeneous and less realistic (with respect to the interpretation as an opaque glossy surface).
Figure 27
 
Combining a given “Fresnel reflection pattern” with correct and incorrect diffuse reflections leads to qualitatively different gloss impressions. Panels B and C show the same specular reflections computed according to Fresnel's equations combined with either the correct diffuse component shown in panel A or with an isolated diffuse component shown in panel D, which is incompatible (see also Figure 3). Although the image of the incorrect combination in panel C is everywhere brighter, the quality of the gloss impression seems actually slightly reduced, i.e., the material appears both more inhomogeneous and less realistic (with respect to the interpretation as an opaque glossy surface).
Figure A1
 
Main effects of the six context factors on the brightness similarity ratings for both Ward BRDFs. The error bars correspond to ±2 SEM.
Figure A1
 
Main effects of the six context factors on the brightness similarity ratings for both Ward BRDFs. The error bars correspond to ±2 SEM.
Figure A2
 
Comparison stimuli for the rating of gloss strength in Experiment 3. Only the images for integer rating values are shown, where s = 0 was the smallest and s = 9 the largest scale value. Intermediate values were also possible.
Figure A2
 
Comparison stimuli for the rating of gloss strength in Experiment 3. Only the images for integer rating values are shown, where s = 0 was the smallest and s = 9 the largest scale value. Intermediate values were also possible.
Figure A3
 
Ranges of specular reflection strength for Ward- and Fresnel-BRDFs and corresponding ranges in gloss strength. Upper row: An equidistant subset of the 51 ρs values ranging from 0 to 0.28 used for the Ward-BRDF in Experiment 4. Middle row: An equidistant subset of the 51 ior values ranging from one to two used for the Fresnel-BRDF in Experiment 4. Lower row: Values for ρs ranging from 0 to 0.17 match the gloss range shown in the middle row more closely than the range shown in the upper row.
Figure A3
 
Ranges of specular reflection strength for Ward- and Fresnel-BRDFs and corresponding ranges in gloss strength. Upper row: An equidistant subset of the 51 ρs values ranging from 0 to 0.28 used for the Ward-BRDF in Experiment 4. Middle row: An equidistant subset of the 51 ior values ranging from one to two used for the Fresnel-BRDF in Experiment 4. Lower row: Values for ρs ranging from 0 to 0.17 match the gloss range shown in the middle row more closely than the range shown in the upper row.
Table A1
 
Results of a seven-way ANOVA for the reflection strength settings in Experiment 1. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A1
 
Results of a seven-way ANOVA for the reflection strength settings in Experiment 1. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A2
 
Results of a seven-way ANOVA for the gloss similarity ratings. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A2
 
Results of a seven-way ANOVA for the gloss similarity ratings. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A3
 
Results of a seven-way ANOVA for the brightness similarity rating for the match in Experiment 1. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A3
 
Results of a seven-way ANOVA for the brightness similarity rating for the match in Experiment 1. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A4
 
Results of a seven-way Anova for the gloss strength ratings from Experiment 3. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A4
 
Results of a seven-way Anova for the gloss strength ratings from Experiment 3. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A5
 
Results of a seven-way ANOVA for the gloss quality ratings from Experiment 3. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
Table A5
 
Results of a seven-way ANOVA for the gloss quality ratings from Experiment 3. Notes: Shown are all main effects and larger interaction effects. The results are given in descending order of mean sq.
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