Image:Julia set (highres 01).jpg
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Julia set, a fractal. C = [0.285, 0.01].
From a self-written tool I named "Julia dream", (after the Pink Floyd tune carrying the same title).
[edit] The story
[edit] Sampling
Normally the escape-time function creates a quite pixly image when the fractal pattern is complex. This is from sampling only the center coordinate of the pixel to be written. That is a single point with no area at all. But the pixel has got the area of one pixel =). If one samples a different point inside the area of the pixel, then it may, (or may not) escape the system earlier or later, (use more or less iterations) than the center-point would. If the pattern is very complex then the sampling of the center-points gives more or less random pixels and that is a problem. The best way I found to prevent this was to use a modified supersample and then use a 3x3 grid. It functions like this:
Divide the pixel to render in 3x3 sub-pixels. Iterate them all and save the results. After all nine sub-points are examined then compare the results and discharge all but the one that used the least number of iterations and use that for the coloursation of the pixel.
On the mailing-list "sci.Fractals" I got this question from some other fractalist: "why do that rather than compute the average color of the subpixels?"
This was my answer to him: "Just because this make the chance bigger two adjecting points will get nearby colours. This makes a sharper, (less blury) image than the average. Think about a Julia spiral that has got spirals, spirals ... Then two nearby points in one of the smaller spirals can have a very big difference in used iterations, (this makes a pixly image if super-sample is not used). The average will still make a pixly image - just less pixly. If one uses least number of iterations, then (maybe?) one of the points is not to deep into the spiral. 2x2 sampling is not enough but 3x3 is pretty good. 4x4 makes a slight better image still but if one considers the time it takes to calculate it - then it is not worth it (16 times slower than normally).
[edit] The colours
The colour-palette used has got one entry for every poissble number of used iterations. If the function uses 16384 (2^14) iterations then the palette has got the same number of colours. To edit a palette like that "by hand" is a bit of work to do and that is nothing you like to do for every image you create. So you will end up using the same palette over and over again and that's boring. I solved this by using sinus-curves. The function randomises three frequences, one for red, green and blue levels respective. And it also randomises the start phase for each of the curves. It also have got functions for frequecy modulations, fading from dark to light and the reversed and some more "goodies". From this I can select a new, hopefully good, palette from the push of a button. While the image is being rendered or after it is done. If it was not good?, then push again and again ...
See also:
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Another highres image from a slight diffrent coordinate.
All fractals are from self-written tools - linear IFS images from my: "2D IFS Studio", non-linear (rev Julia) from "3D RJIFS", (that also writes 2D rev Julias), escape time Z² fractals from "Julia dream", the rest are from small "hacks". See also: Solkoll & Solkoll 3D
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