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7.5. BRDF Theory 241
Equation 7.38 only applies to optically flat surfaces. A generalized
version that can be used to compute a reciprocal, energy-conserving dif-
fuse term to match any specular term was proposed by Ashikhmin and
Shirley [42]:
f
diff
(l, v)=k
norm
(1 − R
spec
(l))(1 − R
spec
(v))ρ, (7.39)
where k
norm
is a constant computed to ensure energy conservation. Given a
specular BRDF term, using Equation 7.39 to derive a matching diffuse term
is not trivial since in general R
spec
() does not have a closed form. Compu-
tation of the normalization constant k
norm
can also be an involved process
(BRDF term normalization is further discussed in Section 7.6). The deriva-
tion of the diffuse term of the Ashikhmin-Shirley BRDF [42] is an example
of the application of Equation 7.39. Kelemen and Szirmay-Kalos [640] also
derived a diffuse term for their BRDF using Equation 7.39; rather than
attempting to derive a closed form for R
spec
(), their implementation used
precomputed lookup tables.
Theoretical considerations aside, the simple Lambertian term shown in
Equation 7.36 is the most commonly used diffuse term in practice.
7.5.5 Microgeometry
The previous sections discussed material properties relating to the compo-
sition or internal structure of the object, with the surface assumed to be
optically flat. Most real surfaces exhibit some roughness or structure that
affects how light reflects from them. Surface detail modeled by a BRDF is
microscale—smaller than the visible scale or, in other words, smaller than a
single pixel. Larger details are typically modeled with textures or geometry,
not with BRDFs. As we discussed in Section 5.3, whether surface struc-
tures are considered to be microscale depends on the scale of observation
as much as on the size of the structures themselves.
Since such microgeometry is too small to be seen directly, its effect
is expressed statistically in the way light scatters from the surface. The
microscale structures or irregularities are assumed to be significantly larger
than visible light wavelengths (approximately 400 to 800 nanometers). This
is because structures have no effect on the light if they are much smaller
than visible light wavelengths. If the size is on the same order as the
wavelength, various wave optics effects come into play [44, 515, 1213, 1345,
1346]. Surface detail on this scale is both difficult to model and relatively
rare in rendered scenes, so it will not be further discussed here.
The most important visual effect of the microgeometry is due to the
fact that many surface normals are present at each visible surface point,
rather than a single macroscopic surface normal. Since the reflected light
direction is dependent on the surface normal (recall Equation 7.30), this
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242 7. Advanced Shading
Figure 7.23. Light reflecting from two surfaces: a slightly rough surface on the left and
a considerably rougher surface on the right.
causes every incoming light ray to be reflected into many directions; see
Figure 7.23.
Since we cannot see that the microgeometry surface normals and their
directions are somewhat random, it makes sense to model them statisti-
cally, as a distribution. For most surfaces, the distribution of microgeome-
try surface normals is a continuous distribution with a strong peak at the
macroscopic surface normal. The “tightness” of this distribution is deter-
mined by surface smoothness. On the right side of Figure 7.23, we see light
reflecting from a surface that is rougher (and thus has its microgeometry
normals clustered less tightly around the overall surface normal) than the
surface on the left side of the figure. Note that the wider distribution of
normals causes the reflected light to “spread” in a wider range of directions.
Figure 7.24. Gradual transition from visible detail to microscale. The sequence of images
goes top row left to right, then bottom row left to right. The surface shape and lighting
are constant; only the scale of the surface detail changes.
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7.5. BRDF Theory 243
The visible effect of increasing microscale roughness is greater blurring
of reflected environmental detail. In the case of small, bright light sources,
this blurring results in broader and dimmer specular highlights (highlights
from rougher surfaces are dimmer because the light energy is spread into a
wider cone of directions). This can be seen in the photographs in Figure 5.8
on page 106.
Figure 7.24 shows how visible reflectance results from the aggregate
reflections of the individual microscale surface details. The series of im-
ages shows a curved surface lit by a single light, with bumps that steadily
decrease in scale until in the last image the bumps are all smaller than
a single pixel. It can be seen that statistical patterns in the many small
highlights eventually become details in the shape of the resulting aggregate
highlight. For example, the relative sparsity of individual bump highlights
in the periphery becomes the relative darkness of the aggregate highlight
away from its center.
For most surfaces, the distribution of the microscale surface normals is
isotropic—rotationally symmetrical, or lacking any inherent directionality.
However, some surfaces have microscale structure that is anisotropic.This
results in anisotropic surface normal distributions, and directional blurring
of reflections and highlights (see Figure 7.25).
Some surfaces have highly structured microgeometry, resulting in inter-
esting microscale normal distributions and surface appearance. Fabrics are
a commonly encountered example—the unique appearance of velvet and
satin is due to the structure of their microgeometry [41].
Although multiple surface normals are the primary effect of microgeom-
etry on reflectance, other effects can also be important. Shadowing refers to
occlusion of the light source by microscale surface detail, as shown on the
Figure 7.25. On the left, an anisotropic surface (brushed metal). Note the directional
blurring of reflections. On the right, a photomicrograph showing a similar surface. Note
the directionality of the detail. (Photomicrograph courtesy of the Program of Computer
Graphics, Cornell University.)
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244 7. Advanced Shading
Figure 7.26. Geometrical effects of microscale structure. On the left, the black dashed
arrows indicate an area that is shadowed (occluded from the light) by other microge-
ometry. In the center, the red dashed arrows indicate an area that is masked (occluded
from view) by other microgeometry. On the right, interreflection of light between the
microscale structures is shown.
left side of Figure 7.26. Masking (visibility occlusion of microscale surface
detail) is shown in the center of the figure.
If there is a correlation between the microgeometry height and the sur-
face normal, then shadowing and masking can effectively change the normal
distribution. For example, imagine a surface where the raised parts have
been smoothed by weathering or other processes, and the lower parts re-
main rough. At glancing angles, the lower parts of the surface will tend
to be shadowed or masked, resulting in an effectively smoother surface.
See Figure 7.27. In addition, the effective size of the surface irregulari-
ties is reduced as the incoming angle θ
i
increases. At extremely glanc-
Figure 7.27. Microgeometry with a strong correlation between height and surface normal
(raised areas are smooth, lower areas are rough). In the top image, the surface is
illuminated from an angle close to the macroscopic surface normal. At this angle, the
rough pits are accessible to many of the incoming light rays, and many of them are
scattered in different directions. In the bottom image, the surface is illuminated from
a glancing angle. Shadowing blocks most of the pits, so few light rays hit them, and
most rays are reflected from the smooth parts of the surface. In this case, the apparent
roughness depends strongly on the angle of illumination.
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7.5. BRDF Theory 245
ing angles, this can decrease the effective size of the irregularities to be
shorter than the light’s wavelength, making the surface optically smooth.
These two effects combine with the Fresnel effect to make surfaces appear
highly reflective and mirror-like as the viewing and lighting angles approach
90
◦
[44, 1345, 1346]. Confirm this for yourself: Put a sheet of non-shiny
paper nearly edge on to your eye, looking off it toward a computer screen.
Hold a pen with a dark surface along the far edge, so that it points out
from the paper. At a very shallow angle you will see a reflection of the pen
in the paper. The angle has to be extremely close to 90
◦
to see the effect.
Light that was occluded by microscale surface detail does not disappear—
it is reflected, possibly onto other microgeometry. Light may undergo mul-
tiple bounces in this way before it reaches the eye. Such interreflections
are shown on the right side of Figure 7.26. Since the light is being atten-
uated by the Fresnel reflectance at each bounce, interreflections tend not
to be noticeable in insulators. In metals, multiple-bounce reflection is the
source of any visible diffuse reflection (since metals lack subsurface scatter-
ing). Note that multiple-bounce reflections from colored metals are more
deeply colored than the primary reflection, since they are the result of light
interacting with the surface multiple times.
So far we have discussed the effects of microgeometry on specular, or
surface reflectance. In certain cases, microscale surface detail can affect
body reflectance, as well. If the scale of the microgeometry is large relative
to the scale of subsurface scattering (the distances between the entry and
exit points of subsurface-scattered light), then shadowing and masking can
cause a retro-reflection effect, where light is preferentially reflected back
toward the incoming direction. This is because shadowing and masking
will occlude lit areas when the viewing and lighting directions differ greatly
(see Figure 7.28).
Figure 7.28. Retro-reflection due to microscale roughness. Both sides show a rough
surface with low Fresnel reflectance and high scattering albedo (so body reflectance is
visually important). On the left, the viewing and lighting directions are similar. The
parts of the microgeometry that are brightly lit are also the ones that are highly visible,
leading to a bright appearance. On the right, the viewing and lighting directions differ
greatly. In this case, the brightly lit areas are occluded from view and the visible areas
are shadowed. This leads to a darker appearance.
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