September 9th 2009

Non-attenuated pattern piling

The "pattern piling" technique, used in all my work since around 2002, consists, starting from a pattern covering the plane, in generating rescaled copies of it and merging these copies together. Usually, the small copies of the pattern are attenuated. Indeed, the smallest copies are too small to be resolved on low resolution images and would produce a random noise which would hide the large scale structures if they were not attenuated. The attenuation is controlled by a parameter, traditionally called beta (B), whose effect can be visualized here. The smaller B is, the less attenuated small copies of the pattern are, B=0 corresponding to no attenuation at all.

As a consequence of attenuation, the resulting image is not scale invariant. At high magnification, the small copies of the pattern which produce the visible structure are attenuated, and the latter look washed out. An instance of this phenomenon is obvious in my older images, for instance this one, in which the smallest visible triangular pattern is barely noticeable.

During all the years that I've spent exploring this technique, the mean value of B decreased steadily. Back in 2002, 2003 or 2004, I often used values of B between 0.7 and 1. In 2008, B was still between 0.5 and 1, with occasional texture layers using lower beta value. My 2009 works often use values of B around 0.3. Recently, I tried B=0.1 in a few images, like the one below.

20090825-1. A very low value of B is used on a smooth pattern, in conjuction with a large magnification step (here 15). See also 20090815-1, 20090816-1, 20090822-1, 20090824-1, 20090826-1, 20090827-1, 20090905-1 and 20090906-1 for a similar technique.

One can wonder if it is possible to make an image in which the smaller copies of the pattern are not attenuated and which does not look like random noise. Such an image would have the very elegant property of being scale invariant: the patterns would not fade out when zooming on them. It turns out that it is much more easy that I thought to create such images, provided one is a little bit careful when choosing the pattern to be piled. The image below uses randomly distributed squares. The density of squares is very low, so that the small scale patterns do not interfere significantly with the large scale pattern.

20090830-1, using B=0. Note the scale invariance of the pattern as you zoom.

On the contrary, this image uses a smooth pattern, but a rather large magnification step (8).

20090907-1, using B=0.

The only trouble with these non-attenuated images is that they look rather different before and after anti-aliasing For instance, here is 20090907-1 without anti-aliasing, as it appears in Ultra Fractal.

20090907-1 without anti-aliasing.

I had several candidate works which looked great as raw images, but turned out to be boring when rendered. Some practice is still needed to be able to foresee the result of anti-aliasing.