Graphics Reference
In-Depth Information
If T is a sampling interval, then 1/T is called the
sampling frequency
and 1/(2T) is
called the
Nyquist limit
. The Whittaker-Shannon Theorem says that if a function is
sampled less often than its Nyquist limit, then a complete recovery is impossible. One
says that the function is
undersampled
in that case. Undersampling leads to a phe-
nomenon referred to as
aliasing
, where fake frequencies or patterns appear that were
not in the original object. The two-dimensional situation is similar, but in practice
one must sample a lot more because of limitations of available reconstruction
algorithms.
Now in the discussion above, it was assumed that we were taking an infinite
number of samples, something that we obviously cannot do in practice. What happens
if we only take a finite number of samples? Mathematically, this corresponds to where
we multiply the sampled result by a function that vanishes outside a finite interval.
The main result is that it is in general
impossible
to faithfully reconstruct a function
that has only been sampled over a finite range. To put it in another way, no function
that is nonzero over only a finite interval can be band-limited and conversely, any
band-limited function is nonzero over an unbounded set.
The practical consequences of the theory sketched above can be seen in lots of
places. Aliasing is most apparent along edges, near small objects, along skinny high-
lights, and in textured regions. Ad hoc schemes for dealing with the problem may be
disappointing because of the human visual system's extreme sensitivity to edge dis-
continuities (
vernier acuity
). Aliasing is also a problem in animation. The best-known
example of temporal aliasing is the case of the wagon wheel appearing to reverse its
direction of motion as it spins faster and faster. Other examples are small objects flash-
ing off and on the screen, slightly larger objects appearing to change shape and size
randomly, and simple horizontal lines jumping from one raster line to another as they
move vertically. See Figure 2.12. This happens because objects fall sometimes on and
sometimes between sampled points.
Jaggies do not seem to appear in television because the signal generated by a tel-
evision camera, which is sampled only in the vertical direction, is already band-limited
before sampling. A slightly out of focus television camera will extract image samples
that can be successfully reconstructed on the home television set. People working in
computer graphics usually have no control over the reconstruction process. This is
part of the display hardware. In practice, antialiasing techniques are imbedded in
algorithms (like line-drawing or visible surface determination algorithms). The
approaches distinguish between the case of drawing isolated lines, lines that come
from borders of polygons, and the interior of polygons.
There are essentially two methods used to lessen the aliasing problem. Intuitively
speaking, one method treats pixels as having area and the other involves sampling
at a higher rate. The obvious approach to the aliasing problem where one simply
Figure 2.12.
Objects appearing, disappearing,
changing size.
