Graphics Reference
In-Depth Information
much high-frequency information in it. While this loose statement may be cor-
rect, it's not the case that the smoothed lighting of smoothed objects produces the
same result as taking full-detail lighting of the full-detail model and smoothing the
results. The “smoothing” operation (filtering out high frequencies) does not com-
mute with the “product” operation in the rendering equation (another instance of
the Noncommutativity principle). Then again, it's the best we've got at present,
and it's an approach that forms the basis for many techniques.
Before leaving this high-level discussion, we have two more observations. The
first is that, in thinking about graphics, we tend to think about the kinds of mod-
els being used to represent the world, and it's easy to confuse the “nature of the
world” with the “nature of models representing it.” As three points on the con-
tinuum of model classes, consider (a) tessellation-independent models, like the
implicit surfaces used in making a simple ray tracer, (b) collections of triangles
and/or “primitives” like cubes, spheres, cones, spline patches, etc., combined with
unions, intersections, etc., to describe shapes, and (c) “object-based” graphics,
in which the world is populated by objects, each one modeled in its own way
and each one supporting various operations like “Where does this ray intersect
you?” and “Give me a simplified representation of yourself.” If objects are them-
selves represented as meshes (as they often are), it's very natural to ask, “What's
a simplified representation of this object?” to do level-of-detail computations at a
per-object level. The problem with this approach is that the result of simplification
depends on our description of the world, not the world itself
. For instance, if we
have a sphere that's represented by an icosahedral mesh, it's natural to simplify
it by replacing that mesh with an octahedron or tetrahedron. But if we happen
to have created that same shape with 20 objects, each of them a single triangle,
then there's no possible simplification: Each triangle is as simple as can be. To
give another example, if we have two irregularly shaped objects partly overlap-
ping each other (see Figure 25.18), and we simplify each of them to remove fine
detail, the gap between them can remain as a small detail, and hence a source of
aliasing when we sample the scene. This happens in practice when we model a
city as many buildings: Even when all the buildings are simplified to cuboids, the
spaces between them may be rectangular gaps that are so small, in screen space,
that they produce aliases.
Figure 25.18: Two irregularly
shaped objects overlap; when we
simplify each one, getting rid of
small details, the space
between
the objects remains as a small
detail, unrecognized by our sim-
plification process.
One approach to level of detail is therefore to consider the simplification of
a whole scene at a time. Perbet and Cani [PC01] took this approach in modeling
prairies: Grass near the camera was modeled as individual blades; slightly more
distant grass was represented by flexible vertical panels on which the blades of
grass were “painted.” And very distant grass was represented by textures applied
to large horizontal planar polygons. The intermediate representation—a textured
polygon that faces the viewer—is one instance of a
billboard.
Numerous permuta-
tions of this billboard idea have been used over the years, from representing trees
by an intersecting pair of billboards (which looks decent, except when viewed
from above), to increasingly complex combinations such as the representation of
clouds by multiple semi-transparent billboards [DDSD03], to the representation of
crowd characters by billboard assemblies with time-varying textures [KDC
+
08].
One very natural approach to level of detail is to represent the world (or
an object) as a union of spheres. Simplification is natural: One replaces sev-
eral small spheres with a larger sphere that encloses (partially or completely) the
small ones. Such representations are easy to translate and rotate, and spheres are
simple enough that they're amenable to lots of algorithmic tricks that make the
