Mandelbrot Exterior Optimization

[Botond Kósa]

I have modified my monotonicity checking algorithm. First I implemented a monotonicity check for the differential of the values, but combined with the harmonic interpolation it produced quite a few visual glitches. Then I inserted a threshold into the original monotonicity check:

if(Math.max(Math.abs(minDiff),Math.abs(maxDiff))<THRESHOLD) then monotonic=true;

After trying different threshold values, I found that 0.3 has the most savings while still not distorting the visuals. Here are two examples about the effect of this threshold (left: original monotonicity checking; right: original monotonicity checking + threshold).

As you can see, the curves in the middle of monotonic areas have disappeared. This results in savings of up to 30%, depending on the image.

[reesej2]

Wow, that algorithm certainly looks like it's working. Very nice!

[cKleinhuis]

where is the difference to "normal" guessing algorithm ?!
and what the heck are these bands, that do not belong to the set, (mostly blue and curly )

[Timeroot]

@Trifox: The normal guessing algorithm only works if the colors are exactly the same. This works even if they change, and then it interpolates. Guessing does no interpolating.

@Botond:

...the actual pixel values do not follow any analytic function. (If that were the case, it wouldn't be a fractal.)

Actually, they do (I think). Please check out the Riemann mapping theorem: There exists, for any shape in the complex plane, an analytic function mapping the shape to the unit disk. And in the case of Julia sets, the potential can be calculated by taking the inverse of the function and measuring the distance to the disk.

[Botond Kósa]

Actually, they do (I think). Please check out the Riemann mapping theorem: There exists, for any shape in the complex plane, an analytic function mapping the shape to the unit disk. And in the case of Julia sets, the potential can be calculated by taking the inverse of the function and measuring the distance to the disk.

Does this also mean that the actual number of iterations before bailout (and including the fractional part) on a straight line (the boundary of an interpolated area) follows an analytic function?

[Botond Kósa]

and what the heck are these bands, that do not belong to the set, (mostly blue and curly )

Those bands are of course not parts of the set, they simply consist of outer points where the iteration count measured on a horizontal or vertical line has a local minimum or maximum. They are not visible on the final (interpolated) image. Also, the newest version of my algorithm eliminates them completely (see my last post with attachments).