"Popular Brot" of a Buddhabrot

[tit_toinou]

Hi there!
If you compute first a Buddhabrot (+1 in every slots of the matrix (the discretization of the complex plane) of the orbit of a point that do not belong to the Mandelbrot Set), and then compute a Mandelbrot where you have a variable = 0 at the beginning, and you add to it the value of the Buddhabrot matrix slot for every point in the orbit of an escaping point (and at the end dividing the variable by the number of iterations so that you compute an arithmetic mean)... You have the Popular Brot!
The point is that computing a Buddhabrot is expensive and it is difficult to zoom into in.
But once a Buddhabrot is computed, the Popular Brot is as fast as the Classic Mandelbrot Set and so it is easy to zoom in !
And it apparently features "Buddha" like in the original Buddhabrot !

The Buddhabrot :


The Popular Brot Associated with :

[kram1032]

So what you're saying is, you take a binary map of the buddhabrot set (white if rays stay inside the set, black if not) and then take that map to iterate the normal Mandelbrot set in only those places?
If you do the same thing with the anti-buddhabrot, you'll essentially get an efficient map of interesting points to plot. All that's black in the anti-buddhabrot will escape anyway, so no need to plot those regions. It's a good upper bound. Much better than the typical circle of radius 2.
However, doing this with the normal buddha-brot might lead to a different interesting picture?
How do zoomed-in regions look like? Just like those you'd expect from the MSet, or do omitted sections change the outcome in an interesting manner?

[Alef]

I found that zooming in buddhabrot is painfully slow, but by accident I found that you can scale your buddhabrot, what effectively works as zoom in. And scaling is somewhat faster than zooming in. Actualy I copy pasted some code. Maybe with little bitt of experimenting someone can find best way for this.

[tit_toinou]

So what you're saying is, you take a binary map of the buddhabrot set (white if rays stay inside the set, black if not) and then take that map to iterate the normal Mandelbrot set in only those places?

Mmmh... No (or I don't understand what you mean).
Here is the algorithm for one value of c :

Code:

c is given
i = 0
s = 0.0
z = c
while( i < IterationMax && abs(z)^2 < 4 )
{
  z = z^2 + c
  s += Buddhabrot[zToScreenY(z)][zToScreenX(z)]
  i++
}

if( i == IterationMax )
{
  pixel is white
}
else
{
  pixel grayscale is s/i / Max(s/i for all is in the image)
}

The normalization operation (last line) must be done afterwards of course.

I hope I made myself clear   .

I found that zooming in buddhabrot is painfully slow, but by accident I found that you can scale your buddhabrot, what effectively works as zoom in. And scaling is somewhat faster than zooming in. Actualy I copy pasted some code. Maybe with little bitt of experimenting someone can find best way for this.

What do you mean by "scaling" ? You are computing a bigger image ?


I uploaded the image. We can cleary see cloudy features of buddhabrot and of course discontinuities.
Finding a way to remove them would be cool.
And I don't know how to color the interior of the mandelbrot set since their colors would depend on the iterationMax value with this algorithm.
And the Popular Brot image gets better with a fully cooked buddhabrot.

How to interpret the Popular Brot image : the more black is a point, the more its orbits went through "popular" points i.e. frequently visited areas of the orbits of escaping points.

[Alef]

What do you mean by "scaling" ? You are computing a bigger image ?

No, the same image resolution, but a bigger fractal;) Result is like zooming in.

It looks too jagged. Maybe try smoothed iterations (some log formula) or some smooth formula, who works in insides and outsides such as exponent smoothing and larger bailout.