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Author Topic: Mandelbrot Set Distance Estimator Rendering Problem  (Read 1576 times)
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aleph0
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Posts: 27



« on: April 29, 2014, 10:29:35 PM »

I've noticed a problem which, surprisingly, seems to affect all Ultra Fractal outside colouring formulae that use the running derivative approach from the Mandelbrot Set distance estimation algorithm (DEM/M) documented on page 196 of the Science of Fractal Images (SOFI).

See the three sample images below which were all produced using the Distance Estimator formula from Standard.ucl.

Render of the largest bud attached to the top of the main cardioid. The filaments on the left side (in red box) are much narrower than those on the right hand side (blue box). My understanding is that the filaments on both sides of the bud should be of similar width and should definitely not exhibit the dramatic differences shown in the image. A nicely balanced rendering is shown, for example, on page XII of the Beauty of Fractals.



Render of the largest minibrot to the North of the main cardioid. Some filaments to West and South are rendered too narrow.



Render of the area East of the largest minibrot on the real axis. The filament overlying the real axis is broken in parts.



Many deep zooms featuring visible minibrots exhibit similar imbalances in filament width around the minibrot. For images where the minibrots are too small to be seen and the main Mandelbrot Set body is not visible, filament rendering seems to be OK, i.e. filament width is broadly consistent across the image.

I'm seeing the same problem with distance estimator formulae that I've written myself. I also use the running derivative approach (without the overflow test, which as far as I'm aware is not often used and may only have been included as an optimisation).

The three sample images were all rendered using the one-pass linear drawing method. However, the same effects are visible for all three drawing methods, so the problem is definitely not a side effect of drawing method.

I'm fairly sure this is not a result of an issue with the DEM/M algorithm itself, as it has been successfully used in many programs. None of the DE images in the various books written by Peitgen and colleagues exhibits the problem. It's also absent in Fractal Works for Mac, which definitely uses the running derivative. See the two beautiful minibrot renderings in this post which show no sign of filament width irregularities:
- http://www.fractalforums.com/mandelbrot-and-julia-set/how-are-bandw-images-from-'beauty-of-fractals'-created/msg4290/#msg4290

Has anyone else observed this problem when working with distance estimation formulae based on the running derivative in Ultra Fractal? Are there any known workarounds or solutions?
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Syntopia
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« Reply #1 on: April 29, 2014, 10:59:49 PM »

Seems weird. I have implemented a Mandelbrot with Distance Estimated in Fragmentarium, and here the image is completely symmetric. See attached.


* mandel.png (93.78 KB, 700x570 - viewed 503 times.)
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aleph0
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Posts: 27



« Reply #2 on: April 29, 2014, 11:24:29 PM »

Seems weird. I have implemented a Mandelbrot with Distance Estimated in Fragmentarium, and here the image is completely symmetric. See attached.

Exactly, that's how these images are meant to look, so that's why I'm thinking this is specific to UF rather than the algorithm.

Your image has some thinning of the filaments as you head North, but that's to be expected. That effect is clearly visible too in the image on page XII of the Beauty of Fractals (though overall the filaments are tuned wider in that image).
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xenodreambuie
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« Reply #3 on: April 30, 2014, 12:31:20 AM »

I've noticed a problem which, surprisingly, seems to affect all Ultra Fractal outside colouring formulae that use the running derivative approach from the Mandelbrot Set distance estimation algorithm (DEM/M) documented on page 196 of the Science of Fractal Images (SOFI).


The problem is most likely that the derivative needs to be calculated before the new z, not after it. One of the symptoms of doing it the wrong way is fuzzy patches.
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Regards, Garth
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David Makin
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« Reply #4 on: April 30, 2014, 12:46:12 AM »

The calculation is only an approximation and several people have calculated it different ways.
The most important thing to me is that to be correct the Mandy and Julia versions need to be different.

Here's a Minibrot using a "correct" version of DE:

Fractal1 {
fractal:
  title="Fractal1" width=928 height=696 layers=1
  credits="Dave Makin;4/29/2014;D J Makin;12/20/2011;Dave Makin;10/5/2\
  011;David Makin;6/15/2008;Frederik Slijkerman;7/23/2002"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=-0.156807372395/1.03204270755 magn=100.08595
formula:
  maxiter=100 percheck=off filename="Standard.ufm" entry="Mandelbrot"
  p_start=0/0 p_start3=0.0 p_ang=0.0 p_coord3=0.0 p_xmod=2.0
  p_ymod=2.0 p_power=2/0 p_bailout=128 p_smallbail=1E-13
inside:
  transfer=none
outside:
  density=1024 transfer=linear repeat=no filename="mmf3.ucl"
  entry="MMF3-DistanceRotation" p_smoothd=yes p_smoothr=no
  p_bailout=128.0 p_power=2.0 p_julia=no p_method=Distance
  p_dscale=1.0 p_rscale=1.0 p_distance=Cabs p_addreal=2.0
  p_addimag=2.0 p_addimag1=0.5 p_addreal1=0.5
gradient:
  smooth=yes rotation=1 index=0 color=16777215 index=45 color=6555392
  index=399 color=0
opacity:
  smooth=no index=0 opacity=255
}

Now look at the change in the following Julia carefully when the "Julia switch" parameter is enabled....

Fractal2 {
fractal:
  title="Fractal2" width=928 height=696 layers=1
  credits="Dave Makin;4/29/2014"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=0/0 magn=1
formula:
  maxiter=100 percheck=off filename="Standard.ufm" entry="Julia"
  p_seed=-0.5948275862/0.4827586207 p_power=2/0 p_bailout=128
  p_smallbail=1E-13
inside:
  transfer=none
outside:
  density=64 transfer=linear repeat=no filename="mmf3.ucl"
  entry="MMF3-DistanceRotation" p_smoothd=yes p_smoothr=no
  p_bailout=128.0 p_power=2.0 p_julia=no p_method=Distance
  p_dscale=1.0 p_rscale=1.0 p_distance=Cabs p_addreal=2.0
  p_addimag=2.0 p_addimag1=0.5 p_addreal1=0.5
gradient:
  smooth=yes rotation=1 index=0 color=16777215 index=45 color=6555392
  index=399 color=0
opacity:
  smooth=no index=0 opacity=255
}
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aleph0
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Posts: 27



« Reply #5 on: April 30, 2014, 01:08:45 AM »

Thank you Garth and David, that works nicely.

I will update my custom formulae to use this approach.
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