Fractal parameters estimation thru contour analysis - Part I

Introduction
I really love fractals theory. Nothing fascinates me as the amazing structures generated by a chaotic processes that describe (hyperbolic) fractals.
It's fairly easy to draw a fractal image:

• The input
• A small set of very easy function (linear transformations)
• A probability to choose one of the aforementioned functions.
• The process
• Choose randomly a starting point.
• Choose randomly one of the above functions according to the above defined probability.
• Map the starting point using the above selected function.
• iterate as long as you want :)
• Admire the meaning of the life!
• just plot the points

Wherever you are, whatever decision you make, despite the chaos, the overall picture will be always the same. More you will move, more clear the picture will be!

When I approached for the first time fractals,  after the reading of the great Barnley's book (Fractals Everywhere) I opened the laptop and I wrote the routine to draw my first IFS!
...By mistake I also plotted the lines that join the points:

...It's quite a messy, isn't it?
Sometimes it's better to give up to connect the points, just relax and wait until everything will be clear. If we focus on every single detail we risk to lose what really counts!

What is a fractal (Iterated Fractal System)?
There are many formal definitions to describe an IFS, in my opinion the most effective describe a fractal as the fixed point of the Markov process (mentioned above).
The process converges to the fractal :)

The powerful of the fractal
In literature there are plenty articles in which fractals are used to solve complex problems (financial analysis, stock exchange preditictions, description of biological processes,...).
Also in Computer Science fractals fond a good collocation, think about the image compression: the ivy leaf image takes around 25MB of space, all the information to obtain it thru the IFS takes less 1KB!!
The problem 
Given the ivy leaf image, what are the parameters of the Markov process to generate it?
A bit more formally what we have to estimate is:

• A set of functions: $\{ \tau_1, \cdots,\tau_N \}$, where $\tau_i(p_0)=A_1 \cdot p_0 +Q_1$, where:

$ A= \begin{bmatrix}\alpha & \beta\\ \gamma & \delta \end{bmatrix} Q= \begin{bmatrix} \epsilon\\ \zeta \end{bmatrix} $

• We need also the estimate the probability to choose each $\tau_j$

The combination of the collage theorem and the box counting approach is the most common technique to solve the problem.
When I was student I approached the problem from a different angle. I have to say that the results obtained were partials but I still think that something more can be done :)
Before to start we need one more notion: the contractive maps (have a look at Banach Theorem)
Under certains conditions the iteration  $\tau_j$ led to a fixed point:

Example of contractive map applied to an ellipsoid.
It converges to a fixed pointed.

First conjecture:

• An IFS is characterised by a fixed point that lies on its convex hull.
• From a fixed point that lies on the border of the IFS, the iterations of the contractive maps that generate the fractal return the convex hull of the fractal.

The ivy leaf IFS is generated by 4 contractive maps. Each color describes the map used to generate the point.

The above animated gif shows the meaning of the conjecture.

An experimental evidence about the fact that at least one fixed point lies on the convex hull of the fractal can be obtained changing the params of the maps:

The light blue points depict the fixed point of the maps used to plot the fractals.

Despite the changes in the maps, the fixed point on top of the leaf still lays on the convex hull of the fractal.

In the next post I'll show you a nice recursive algorithm I found to obtain different levels of convex hull for an IFS.

Stay tuned

Cristian.