Function: f(x,y) = Abs(Sin(x^2 + y^2))
All of the images are 720 pixels in width and 240 pixels in height,
with the "real world" origin at the upper left corner. One pixel
represents a "real world" square area of 0.05 by 0.05 units, meaning
that each image covers an x range of 0 (left) to 36 (right) and a
y range of 0 (top) to 12 (bottom). The value of the function being
evaluated is always in the range 0 to 1, so the value was scaled to
the range 0 to 255 so that 256-gray-level images could be created,
0 being black and 255 being white, with 254 levels of gray in-between.
The images should be viewed in a mode which allows 256 levels of gray to be
shown with no further color quantization or dithering, since such
degradations would introduce other artifacts into the images. If a
rainbow-like color pattern appears in the images, it is probably due to the
computer monitor being pushed to its resolution limits.
The first image was created by sampling the function once at the
center of each pixel. Note that the aliasing is severe in this case.
The concentric circle structure at the right upper corner looks just
as valid as the structure at the upper left, although the upper right
structure is an artifact of the sampling process and is not actually
part of the original signal.
The second image was created by numerically computing the average value
of the function over the area of each pixel by evaluating the integral
of the function over that area and dividing by the pixel area. This
method, although computationally expensive, creates the smoothest
possible sampling. Note that the aliasing is still present, but the
amplitude of it is significantly reduced. In fact, the amplitude of
the aliasing becomes less as distance from the origin increases, unlike
in the first case, where one sample in the center of each pixel was
taken, where the aliasing occurs with the same magnitude at periodic
intervals throughout the sampled image.
The third image was created by sampling the function once within the
area of each pixel, but the location within each pixel was randomly
chosen. This, in a way, emulates the way film with a random distribution
of grains, would capture an image. It is not a perfect emulation because
there is the condition that exactly one sample must lie within each pixel
area, where grains randomly distributed about a surface are generally not
subject to that distribution. That is why a slight bit of aliasing is still
visible, but the aliasing has been practically reduced to barely-noticeable
levels, at the expense of having noise where aliasing artifacts were produced
using the other sampling methods. In some applications, the noise may be
more desirable than the aliasing caused by the other sampling methods.
Evans A Criswell (firstname.lastname@example.org)