Graphics Reference
In-Depth Information
0
consisted of a finite collection of point lights, the integral
that defines
M
became a simple sum.
As an approach to solving the rendering equation, this involves many of the
methods described in Section 31.2: The restriction to a few point lights amounts to
solving a subproblem. The truncation of the series amounts to approximating the
solution method rather than the equation. The transport operator,
M
,usedinthe
early days was also restricted: All surfaces were Lambertian, although this was
soon extended to include specular reflections as well.
Improving the algorithm to use
g
Note that since
0
(i.e., including shadows) was a
subject of considerable research effort, with two main approaches: exact visibility
computations, and inexact ones. Exact visibility computations are discussed in
Chapter 36.
A typical inexact approach consists of rendering a scene from the point of
view of the light source to produce a
shadow map:
Each pixel of the shadow map
stores the distance to the surface point closest to the light along a ray from the
light to the surface. Later, when we want to check whether a point
P
is illuminated
by the light, we project
P
onto the shadow map from the light source, and check
whether it is farther from the light source than the distance value stored in the map.
If so, it's occluded by the nearer surface and hence not illuminated. This approach
has many drawbacks, the main one being that a single sample at the center of a
shadow map pixel is used to determine the shadow status of all points that project
to that pixel; when the view direction and lighting direction are approximately
opposite, and the surface normal is nearly perpendicular to both, this can lead to
bad aliasing artifacts (see Figure 31.16).
0
instead of
Figure 31.16: Aliasing produced
by a low-resolution shadow map.
The aliasing on the shadows is
the problem; the stripes on the
cubes themselves arise from a
different problem. (Courtesy of
Fabien Sanglard.)
By the way, the approaches used in the early days of graphics were not, at the
time, seen as approximate solutions to the rendering equation. They were practi-
cal “hacks,” sometimes in the form of applications of specific observations (e.g.,
Lambert's law for reflection from a diffuse surface) to more general situations
than appropriate, and sometimes were approximations to the phenomena that were
observed, without any particular reference to the underlying physics. When you
read older papers, you'll seldom see units like watts or meters; you'll also on rare
occasions notice an extra cosine or a missing one. Be prepared to read carefully
and think hard, and trust your own understanding.
Spherical Harmonics
We've discussed patch-based radiosity, in which the field radiance is approximated
by a piecewise constant function; one can also think of this as an attempt to write
the field radiance in a particular
basis
for a subspace of all possible field-radiance
functions, in this case the basis consisting of functions that are identically one on
some patch
j
, and zero everywhere else. Linear combinations of these functions
are the piecewise constant functions used in radiosity.
A similar approach is to represent the surface radiance at a point (which is a
function on the hemisphere of incoming directions) in some basis for the space
of functions on the sphere. Assuming we limit ourselves to continuous func-
tions, such a basis is provided by spherical harmonics,
h
1
,
h
2
,
...
, which are
the analog, for
S
2
, of the Fourier basis functions
sin(
2
π
nx
)
(
n
=
1, 2,
...
) and
cos(
2
π
nx
)
(
n
=
0, 1, 2,
...
)
on the unit circle. The first few spherical harmonics,
