The Ward-BRDF does not only ignore Fresnel effects, but also a geometric attenuation factor:
The Gaussian distribution has shown up repeatedly in theoretical formulations of reflectance [\(\ldots\)], and it arises from certain minimal assumptions about the statistics of a surface height function. It is usually preceded by a Fresnel coefficient and geometrical attenuation factors, and often by an arbitrary constant. Since the geometric attenuation factors are typically difficult to integrate and tend to counteract the Fresnel factor anyway, we have replaced all of these coefficients with a single normalization factor that simply insures the distribution will integrate easily and predictably over the hemisphere Ward (1992, p. 268).
Thus, in a strict sense, using the Fresnel-BRDF proposed by
Walter et al. (2007) does not isolate the contribution of Fresnel effects
\(F\), but the difference to the Ward-BRDF is also due to the inclusion of the attenuation factor
\(G\). The attenuation factor leads to a darkening of the specular reflection near grazing angles and thus counteracts Fresnel effects, which are reflected in an
increase of specular reflection near grazing angles. This means that an increase of reflection strength near grazing angles above that observed with the Ward-BRDF can only be due to
\(F\).
In the BRDF,
\(F\),
\(G\), and the Beckmann distribution
\(D\) are multiplied with each other. The upper row of
Figure 24 illustrates the effects of
\(F\) and
\(G\).
\(G(\omega _i,\omega _o, m)\) is a function of the incidence direction
\(\omega _i\), the reflection direction
\(\omega _o\) and the local surface normal
\(m\).
\(F(\omega _i,m)\) is a function of
\(\omega _i\) and
\(m\). The function
\(G\) is separable:
\(G(\omega _i,\omega _o, m) \approx G_1(\omega _i,m) G_1(\omega _o,m)\). The top right panel in
Figure 24 shows the rational approximation to
\(G_1\) for three realistic roughness values (compare
Figure 8 and Eq. 27 in
Walter et al., 2007). That is, the attenuation works only near grazing angles. The middle panel in
Figure 24 shows
\(F\) for a typical refractive index of 1.5. Finally, the top left panel shows the combined effect of
\(G\) and
\(F\) for
\(\omega _o = \omega _i\):
\(G\) attenuates
\(F\) near grazing angles and this effect increases with
\(\alpha\).
The middle row of
Figure 24 shows renderings with a patched version of the Mitsuba renderer, in which either the effect of
\(G\) or
\(F\) was selectively discarded (and thus the other factor isolated). To eliminate Fresnel effects,
\(F\) was set to a constant value
\(f = 2(ior-1)^ 2/(ior+1)^2\), i.e., to twice the value for perpendicular incidence with a given refractive index
\(ior\). To eliminate
\(G\), this factor was was set to 1. The renderings are for
\(ior = 1.5\) and
\(\alpha = 0.15\). A comparison of the correct rendering with the restricted renderings, which isolate
\(F\) and
\(G\), respectively, reveals that the effect of
\(G\) is very small. As expected, the reflection strength without attenuation
\(G\) is slightly increased near grazing angles. When only the factor
\(G\) is considered, the result is very similar to that obtained with the Ward-BRDF. The difference in the relative effects of isolating
\(F\) and
\(G\) can also be seen in the difference images that compare the luminance of full and restricted renderings.
Together, these findings indicate that, at least for the parameter values realized in the experiment, the differences between the stimuli obtained with Fresnel- and Ward-BRDF are almost completely due to the inclusion of Fresnel effects.