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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.
 
 
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