Spatial Anti-aliasing - An Example of An Image With Extreme Pseudo-random Aliasing

An Example of An Image With Extreme Pseudo-random Aliasing

Because fractals have unlimited detail and no noise other than arithmetic roundoff error, they illustrate aliasing more clearly than do photographs or other measured data. The escape times, which are converted to colors at the exact centers of the pixels, go to infinity at the border of the set, so colors from centers near borders are unpredictable, due to aliasing. This example has edges in about half of its pixels, so it shows much aliasing. The first image is uploaded at its original sampling rate. (Since most modern software anti-aliases, one may have to download the full-size version to see all of the aliasing.) The second image is calculated at five times the sampling rate and down-sampled with anti-aliasing. Assuming that one would really like something like the average color over each pixel, this one is getting closer. It is clearly more orderly than the first.

In order to properly compare these images, viewing them at full-scale is necessary.

• 1. As calculated with the program "MandelZot"

• 2. Anti-aliased by blurring and down-sampling by a factor of five

• 3. Edge points interpolated, then anti-aliased and down-sampled

• 4. An enhancement of the points removed from the previous image

• 5. Down-sampled, again, without anti-aliasing

It happens that, in this case, there is additional information that can be used. By re-calculating with the distance estimator, points were identified that are very close to the edge of the set, so that unusually fine detail is aliased in from the rapidly changing escape times near the edge of the set. The colors derived from these calculated points have been identified as unusually unrepresentative of their pixels. Those points were replaced, in the third image, by interpolating the points around them. This reduces the noisiness of the image but has the side effect of brightening the colors. So this image is not exactly the same that would be obtained with an even larger set of calculated points. To show what was discarded, the rejected points, blended into a grey background, are shown in the fourth image.

Finally, "Budding Turbines" is so regular that systematic (Moiré) aliasing can clearly be seen near the main "turbine axis" when it is downsized by taking the nearest pixel. The aliasing in the first image appears random because it comes from all levels of detail, below the pixel size. When the lower level aliasing is suppressed, to make the third image and then that is down-sampled once more, without anti-aliasing, to make the fifth image, the order on the scale of the third image appears as systematic aliasing in the fifth image.

The best anti-aliasing and down-sampling method here depends on one's point of view. When fitting the most data into a limited array of pixels, as in the fifth image, sinc function anti-aliasing would seem appropriate. In obtaining the second and third images, the main objective is to filter out aliasing "noise", so a rotationally symmetrical function may be more appropriate.

Pure down-sampling of an image has the following effect (viewing at full-scale is recommended):

• 1. A picture of a particular spiral feature of the Mandelbrot set.

• 2. 4 samples per pixel.

• 3. 25 samples per pixel.

• 4. 400 samples per pixel.