spotty interior distance estimate for Julia sets

[claude]

I'm having issues with interior distance estimate for Julia sets with an attracting cycle - getting spots at the attracting cycle and its pre-images

I asked a question on stack exchange, with relevant equations, images, and some experiments with complex-valued distance estimates:
http://math.stackexchange.com/questions/1153052/interior-distance-estimate-for-julia-sets-getting-rid-of-spots

Any ideas to solve the spots?  Or should I give up on interior distance estimate and use edge-detection on Fatou basins?

[xenodreambuie]

I also get spots and don't know if there's a good way to eliminate them. But as far as I know the spots should be sharp (single pixel or sub-pixel) and not fading in like the true set boundary, hence much less annoying.

[Adam Majewski]

Do you have similar problem when using both external and internal DEM on parameter plane for drawing boundary of Mandelbrot set ?

Here is the code for Mandelbrot set
http://www.moleculardensity.net/buddhabrot/appendix/2

where I see no spots 

[xenodreambuie]

Do you have similar problem when using both external and internal DEM on parameter plane for drawing boundary of Mandelbrot set ?

For myself, I don't do internal DE for Mandelbrots so I haven't tried. The disc removal method is probably immune to the spots but not so good if one is using both smooth DE and potential for various coloring on a per sample basis.

Claude, the reason for your spots being fuzzy is using a pixel size of 2. Try it with between 0.5 and 1 for much better results. Then your exclusion method might work better and you could reduce the threshold too.  However, the spots themselves seem to be an artifact of the DE formula for points that converge fastest (?) to the critical points. Could you be explicit about the formula you use for the phase difference of the complex DE for validity? Then perhaps someone can find an efficient modification of the DE formula that works.

[claude]

(oops, wrote this a while ago but forgot to hit post)
dwell bands get closer together as you approach both the Julia set (red, dwell increasing) and also the attracting cycle (green, dwell decreasing) - possible issues with de related to self-intersecting equipotentials?  I have a vague "intuition" of de as "dwell band spacing"...


Adam: no spots with interior de for Mandelbrot set (I guess because the hyperbolic components are much more nicely behaved (can be conformally mapped to a disc) than a Fatou component (fractal boundary).

xenodreambuie:  phase difference between a and b is arg(a / b)  (handles 2pi wrapping better than arg(a) - arg(b)). so i compute complex de for each pixel, then the gradient (a complex number) of its (real) magnitude using central differences (except at the image boundary):

Code:

gradient[i][j] = (abs(de[i+1][j]) - abs(de[i-1][j]))/2 + I * (abs(de[i][j+1]) - abs(de[i][j-1]))/2

[Adam Majewski]

such image can be made with :
* level set of attraction time  https://commons.wikimedia.org/wiki/File:ILSMJ.png
* internal distance estimation
* Koenigs coordinate  https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Koenigs_coordinate

What are the differences ?