Graphics Reference
In-Depth Information
matrix
C
represents the ordinary specular reflection. With
C
x
=
1,
C
becomes the identity matrix and the effective specular direction is just the incom-
ing direction ω
i
. In this case, the lobe models
retroreflection
, which is a property
of some surfaces that reflect a significant amount of light back into the direction
from which it came.
8
The values of
C
x
,
C
y
,and
C
z
cannot be arbitrarily chosen.
For example, an isotropic surface requires that
C
x
C
y
=
C
z
=
=
C
y
. Lafortune and his coau-
thors suggested replacing the diagonal matrix with a more general 3
×
3matrixto
model anisotropic reflection.
9
Figure 8.15(a) shows a series of images rendered from different viewpoints
using the traditional cosine lobe model. In these images, the specular reflection
on the surface of the desk disappears as the viewing direction gets more parallel
to the surface. In reality, the opposite effect is observed: surfaces usually look
more reflective at grazing angles. Figure 8.15(b) shows the same series rendered
using the Lafortune model; the reflection toward grazing is more plausible.
The Lafortune BRDF model has been successful in representing many com-
mon types of reflection, including directionally diffuse reflection, general off-
specular reflection, and retroreflection. Because the isotropic model specifies each
lobe as a function of four parameters:
C
x
,
C
y
,
C
z
, and the exponent
n
; only three
parameters are needed for an isotropic lobe, as
C
x
C
y
. The authors found that
fitting these parameters to measured BRDF data obtained at the Cornell Light
Measurement Laboratory resulted in a good match. One advantage of the mul-
tilobe nature of the Lafortune model is that more nodes can be added to better
approximate the underlying data.
10
The simplicity of the Lafortune model makes
it suitable for hardware rendering. The paper “Efficient Rendering of Spatial bidi-
rectional Reflectance Distribution Functions” by David K. Mcallister, Anselmo
Lastra, and Wolfgang Heidrich [Mcallister et al. 02] presents a GPU-based tech-
nique that stores the diagonal matrix elements and the exponent of each Lafortune
lobe in the RGBA channels of a texture map texel. The textures then represent
=
8
Retroreflection can be observed in a number of real surfaces. The surface of the moon is an
example; without retroreflection, the edges of a full moon would appear darker than the center. This
absence of “limb darkening” causes the full moon to look like a disc instead of a sphere. Another
example of retroreflection is in the paint used in some road signs and lane markings. This special
paint is impregnated with highly reflective tiny glass beads, which produce retroreflection even at
very low angles of incidence. As a result, distant road markings become visible to a driver from
retroreflection of the car's own headlights.
9
Some justification of the general matrix approach is given in the paper “Applications of Irradiance
Tensors to the Simulation of Non-Lambertian Phenomena, 1995” by James Arvo [Arvo 95].
10
The simplicity of the Lafortune model is also a drawback: the small number of parameters limits
the flexibility of the model. Subsequent research has shown that although the model fits measured
data well in the plane of incidence, the lobes do not get sufficiently narrow in the azimuthal direction
near grazing reflection [Ngan et al. 04, Stark et al. 05]. As a result, no sum of Lafortune lobes can
accurately represent certain types of reflection.
