Reverse engineering a fractal image?

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Offline marcm200

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« on: February 06, 2019, 11:48:02 AM »
I was wondering whether one could retrieve formula information from a calculated image of, say, a black and white Julia set. Since it only has a finite number of points with finite precision, there's hardly much information present except for the shape.

Is this information sufficient to deduce one c value (might not be the single one existing) that would deliver said Julia set when computed? Or is escape iteration information necessary?

And what about other types of fractals?

Does anyone know of literature/articles dealing with that topic? Or has experience in doing so?

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/reverse-engineering-a-fractal-image/2598/


Offline marcm200

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« Reply #2 on: February 07, 2019, 10:13:47 AM »
Thanks a lot! Seems one needs a lot of general information about the overall structure of the Mandelbrot set, just the image and the general iteration formula doesn't seem enough.

Would be interesting to see how the Julia sets behave with finite resolution and how many different shapes there are. I might take a closer look into that.

Offline claude

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« Reply #3 on: February 07, 2019, 06:11:53 PM »
Julia sets are invariant under both forward and backward iteration.  So perhaps you could phrase it as an optimization problem.  Assuming you don't know the coordinates of the image, but that it is just uniformly scaled and rotated and translated, that requires solving for 3 complex variables a b c:

\[ I = \frac{f_c(a I + b) - b}{a} \] where I is the image of the Julia set

As you have more than 3 pixels (I hope), the system is overconstrained so least squares optimization might work.  For other transformations (eg perspective, for a photo of a poster of a Julia set), the process is similar but there may be more variables.

Maybe the cost function to optimize could be based on distance fields (per pixel distance to Julia set, scaled by derivative of f_c transform at each pixel), such that perfect alignment would minimize the cost?  Not sure exactly how it would work, but maybe it's possible.

Offline marcm200

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« Reply #4 on: February 07, 2019, 07:52:16 PM »
Thanks for the in-depth answer! Now I know where to start.