Are fractal image generators one-way functions?

Question

Is it hard to calculate the coordinates and zoom factor that was used to generate a fractal image of, say, the Mandelbrot set? If you know the rest of the parameters, like how many iterations where used, in generating the image.

Answer 1

Sketch of an approach commonly used in image processing / computer vision.

Say we want to do a systematic computational approach ( not using the algebraic properties of the fractal ) to do this and we know the mapping between color and iteration count. To be time efficient we would want to use clues from the fast-to-compute points first, i.e. the points escaping the set after a low number of iterations.

1. Start finding sets of points of fast divergence in the image we are searching for. For most well-behaved fractals these will be connected sets.
2. Analyze the shapes of their boundaries. This can be done in multiple resolutions to give a hint of where to zoom in to produce a fit.
3. Start at a zoomed-out state, try and match the descriptors we have built from our zoomed-in-image. At first it will probably suffice to use stuff like mean iteration count over areas to find reasonable areas to search in. But as we go further in matching shape will get more and more important.