I made my first IFS images back in 1998 in basic, but later on I wrote a simple C program that could render high resolution images, like these ones below, that where in fact computed at 6000x9000 pixels with 2x2 antialias (12000x18000 effective sampling points). One of them made it into the finals of a fractal image contest, and the other one was used as background for a cover of a book, but that's another story.
Click to enlarge
Click to enlarge
Really understanding the maths of IFS is not that easy, but implementing it is amazingly simple. The images are the attractors of the actual IFS. The functions are just linear functions, affine transformation T if you want, that take a point p and transform it into p' with rotation, skew and scale factor A and a translation B:
|that can also be written as|
When applying T to all the points of a plane (an image if you want) this gets expanded or contracted, and that contraction fractor is just the determinant of A (since the transformations are linear), as usual.
For rendering all the IFS images in this site I only used four such transformations, but you can choose any number. We will be applying these transformations to points in the plane, and assuming the overall transformation is contracting, the iterative process will makes the complete plane converge to a given set, the final image. The geometry of this set, the attractor of the IFS, is usually fractal, even if the transformations are linear, but that's out of the scope of this little article.
The iterations happend like this: choose a random point in the plane, one of the four transformations at random, and apply it. This gives a new point. Now, pick again one of the four transformations at random, and apply it again. Repeat, few thousand millions of times.
This will give an "orbit" for that very first starting point we chose. The chaos game theory ensures that if the average contraction factor is less than that one, we will get not a random point cloud neither an exploding orbit, but an well defined shape, the attractor.
Well, for this to be true, the chaos theory and symbolic dynamic mathematics say that you must randomly choose the transformation from the transformation set of the IFS. This way, the orbit densely overlaps the attractor. In the other hand, to allow the chaos game converge faster, it is better not to randomly select a transformation from the transformation set of the IFS, but with a probabilty propotional to the area covered by this transformation over the total area of all the transformations:
To make things more interesting, it is also possible not to only record the attractor, but also the densisty of the attractor. This can be interpreted as the probabilty of the iteration to step at a given point across the orbit. After selecting the probabilities according to the equation above, this density over the attractor should be quite smooth.
Finally, a bit more interesting images can be created if other transformations can be applied before accumulating the orbit on the plane. It is important to note that to keep the chaos game unaltered, this transformations must be done out of the iteration feedback loop. To create these images, a simple rotating transformation was applied, in wich the rotating angle was proportinal to the signed horizontal distance from the point to the origin.
Additionaly, a sinousoidal component was added to this angle, that changed each iteration.
Adjusting the frequency and amplitude of this sinusoidal component, the thickness and intensity of the "arcs" on the picture can be adjusted. Without this variation, the arcs would become just "threads".
The following images bellow were created by randomly choosing the tranformation functions (you can click on them to see a higher resolution version):
I wrote a small application that randomly created thousands of IFS images. To quickly discard those transformations creating disperse attractors, the application only used those random transformations with a contracting coefficent smaller than 1.5. These images where, of course, low-resolution and low quality (low number of iterations). The very first two images in this page were created by choosing from this colection of random images.
For example, to make the first image, I chose one of the random images I liked, and worked on it. It's color was decomposed in the hue and his intensity. As explained, the iterative process does not just show the attractor, but its density. So, for the color intensity, the density of the attractor was choosen. In fact, it is the square root of the density normalized to the maximun/minimun value over the image. The square root was not choosed for any mathematical reason; it was just used to adjust the contrast of the image in the image-synthesis part (where the colors are kept in floating point), to avoid crapy manipulation in Photoshop using 8 bit precision.
The color was a bit more complicated to create. Actually, it stores important information about the dynamics happening on the attractor: each point in the attractor is accessed from another given point of the attractor by the application of one of the transformations used, his "preimage" (if these transformations overlap, there is more than one preimages exist). In other words, each point in the attractor can be identified by the symbolic secuence of transformations used to get that point from the original one. This sequence is called the "address". This address, expressed in the correct base (four in my case, because I selected four transformations), defines a number between 0 and 1. The coloring of this images was done based on that number. Each new point on the orbit generated a color based on his actual address, and the color was accumulated on the plane using a 5%-95% blending with the previously accumulated color. This is how many applications out there produce the Flame images.
This color was multiplied by the previously explained intensity value to get the final color. To get a color based on the address of a point, a sinusoidal function was used for each color component (red, green and blue):
red = 0.4 + 0.25*sin( 0.5 + address )
green = 0.6 + 0.25*sin( 0.9 + address )
blue = 0.7 + 0.25*sin( 1.3 + address )
This color was desaturated in a 30% (mixed with his grey-level version). Finally, a "glowing" postprocess effect was done to enrich the visual quality. This effect is responsible for the light emmission effect on some parts of the image (the brighter ones). To get it, the image was low-pass filtered (a gaussian filter) and added back to the originial image in a 50% proportion.