Solving inverse problem related to Iterated function systems? I generated a Barnsley's fern fractal using details in this link with the aid of MATLAB. My doubts are as follows : 


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*How do we justify the shape generated from those equations?

*Is it possible to find the equations back from just the raw generated data points (the reverse problem)? 


It would be useful for me if someone can suggest any books or links which address these matters?
 A: I'm not sure what you mean by "justify the shape generated from those equations".  An iterated function system, or IFS, is a non-empty list of functions $(f_i)_{i=1}^m$ mapping a a complete metric space (often $\mathbb R^n$) to itself.  If each of those functions is a contraction, then one can prove that there is a unique, non-empty compact set $E$ with the property that 
$$E = \bigcup_{i=1}^m f_i(E).$$
The set $E$ is called the invariant set associated with the IFS and sets generated in this fashion are often called self-similar.
The proof is constructive in nature and we can find the set $E$ by iterating the IFS.  That is, given a compact set $S$, define a transformation $F$ which returns another compact set by
$$F(S) = \bigcup_{i=1}^m f_i(S).$$
If we then let $S_0$ be an arbitrary, non-empty compact set and let $S_n = F(S_{n-1})$, we recursively generate a sequence of compact sets that converges to the invariant set $E$.
Note that an IFS determines a unique self-similar set - not the other way around.  Given a self-similar set, there is (by definition) some IFS that generates that set - but there can and will be many such IFSs.  The problem of finding such an IFS is the inverse problem of which you speak.
More generally, given a non-empty, compact set $S$ in $\mathbb R^2$, we might try to find an IFS whose invariant set is "close" to $S$, even if $S$ is not self-similar.  This is the basis of fractal image compression.  The books by Barnsley are certainly the best references for this


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*Fractal Image Compression

*Fractals Everywhere
