### 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 (criswell@itsc.uah.edu)

2001/01/30