Problem
I am looking for pointers to existing work or what approach to take to help me solve the following problem of colored fractal curve matching and how to define some metrics and generally how to approach this subject to gain as much information as possible from analyzing these curves. I am at a start of research into those structures and any pointers would be much welcome.
Let's assume a very simple line fractal, I will use the von Koch curve as example. The curve is not uniformly in the same stage though. Some of the lines have been expanded to next stages and some have not. Further each line can has a color for each of its stages. When a line is expanded to 4 lines, the original color is the E(n-1) stage color and the new colors are the E(n) stage colors for each subdivided line. I use color for a multi-dimensional attribute, in this case it can be decomposed to R,G,B values and a closeness of two colors can be judged either by a composite metric or by a set of three metrics along each of the R,G,B dimensions.
I have two (or more) of such curves and I am trying to judge:
- How close they are to each other?
- How to find a set of differences?
- How to find a common base curve at which they started to diverge?
- How to take into account divergence only in color and ignore it if the color divergence is under a certain value for the metrics.
I will be converging a set of these curves to one member of the set:
- How to find the effort for a perfect convergence by reducing all curves to a common base with the convergence target and building them up.
- How to find the effort for imperfect convergence within some metric for color differences.
- Same for a weighted metric - earlier stages color differences have more value than later stages.
PS: The real curve I am interested in is slightly more complicated, where each line can be split into a sequence of equidistant lines before being iterated on with the von Koch transformation, but each member of the sequence has the same color.
Essentially I can reduce some very complex problems down to something isomorphic to this type of curve and then match those curves to gain understanding of the more complex issue.
Representation
Index i for each part of the curve could be drawn as follows. Let's say 0 represents unexpanded line. Numbers 1 to 4 represent the 1st to 4th part of the curve expansion (start,up,down,end). On order 1 curve: (start=0.1,up=0.2,down=0.3,end=0.4). On order 2 curve: (start,start=0.11, start,up=0.12,start,down=0.13,start,end=0.14,up,start=0.21, etc..). A curve could be represented by a set of indexes like: L1 = {0.11,0.12,0.13,0.14,0.2,0.3,0.41,0.42,0.43,0.441,0.442,0.443,0.444}, which would mean that the curve is expanded to order 1 everywhere, start and end part are expanded to order 2 and end of end part is expanded to order 3. You can define a boolean function E to express if index is expanded or not. For the above set E(L1,0)=1; E(L1,0.1)=1; E(L1,0.11)=0; E(L1,0.111)=0; etc. A color is represented by a function C defined for each curve L and index i where E(L,i) = 1. C(L1,0) is defined, but C(L1,0.111) is not.