Graphics Reference
In-Depth Information
→
n
→
→
r
h
θ
h
→
θ
r
i
θ
i
x
β
y
Figure 8.11
Coordinate system for Ward's BRDF model. The specular component depends on the
half-angle θ
h
, which is the angle the vector
h
halfway between the incoming and outgoing
directions makes with the surface normal. (After [Ward 92].)
not as accurate. Also, the dynamic range of the measured BRDF is limited by the
camera.
In the Ward model, the Fresnel reflectance and local occlusion terms are re-
placed with a single Gaussian function. The primary reason for this, according
to Ward himself, is that the measurement apparatus was not capable of captur-
ing measurements at angles low enough to accurately record these phenomena.
Omitting these effects undermines the physical basis of the model; however, for
many surfaces the increased reflectance due to Fresnel reflection is partially coun-
teracted by the effects of self-occlusion. In any case, it makes the model simpler.
The Ward model represents the ratio of reflected light by a Gaussian function of
the tangent of the half-angle
between the surface normal and the half-vector
h
θ
(
Figure 8.11
)
:
e
−
tan
2
θ
h
/
k
2
1
√
cos
.
(8.5)
k
2
θ
i
cos
θ
r
4
π
The coefficient
k
, which represents the falloff rate of the Gaussian function (the
standard deviation, in statistics parlance), corresponds to the roughness of the sur-
face and can be fit to the measured data (
Figure 8.12
)
.
7
Equation (8.5) models isotropic reflection: rotating the incoming and outgoing
vectors about the surface normal does not change BRDF value. In fact, the model
depends only on the angle
θ
h
the half-vector makes with the surface normal. Ward
extended the model to anisotropic reflection by adding a dependence on the angle
β
the half-vector makes with the coordinate axes (
Figure 8.11
)
. The anisotropic
7
Arne Dur showed that omitting the square root in in Equation 8.5 provides a better normalization.