I gave it a very quick try using FractInt, and came up with these PARs which you can use to load and try out.  I also saved the MAP files used to create them if you wish those separately.  (See the images further on down.)

test_01            { ;  (c) Paul N. Lee -- Apr 13, 2007 - 02:15:26
                     ; time=  0:00:00.82 on a PIII-700 at 640x480
                     ; Video=F5 using FractInt 2099.8
  reset=2099 type=mandel passes=t
  center-mag=+0.29124525926602574/-0.01530508041716197/3112.83
  params=0/0 float=y inside=0 colors=www000<253>000
  }

test_02            { ;  (c) Paul N. Lee -- Apr 13, 2007 - 02:15:26
                     ; time=  0:00:00.82 on a PIII-700 at 640x480
                     ; Video=F5 using FractInt 2099.8
  reset=2099 type=mandel passes=t
  center-mag=+0.29124525926602574/-0.01530508041716197/3112.83
  params=0/0 float=y inside=0 colors=000www<253>www
  }

test_03            { ;  (c) Paul N. Lee -- Apr 13, 2007 - 02:24:10
                     ; time=  0:00:00.71 on a PIII-700 at 640x480
                     ; Video=F5 using FractInt 2099.8
  reset=2099 type=mandel passes=t
  center-mag=+0.29130558810737911/-0.01519509048573419/18097.85
  params=0/0 float=y inside=0 colors=www000<253>000
  }

test_04            { ;  (c) Paul N. Lee -- Apr 13, 2007 - 02:24:10
                     ; time=  0:00:00.71 on a PIII-700 at 640x480
                     ; Video=F5 using FractInt 2099.8
  reset=2099 type=mandel passes=t
  center-mag=+0.29130558810737911/-0.01519509048573419/18097.85
  params=0/0 float=y inside=0 colors=000www<253>www
  }

 

 
 
 

They look like inverse Julia images, wherein the iteration uses the inverse of the Julia calculation and the orbits settle on the fractal boundary of the Julia set.

Kerry

They are definitely Mandelbrot set plots. Is the inverse iteration method applicable to Mandelbrot sets?



Duncan C

A quick look at the code in "The Science of Fractal Images" for the "Distance Estimator Method for Mandelbrot set" reveals a matrix with only the values '0' and '1' assigned to it, so they don't seem to use grayscale.

I think it has to do with the prints, a combination of 'more details' caused by the "DEM/M" and a very 'fine' grid of pixels, that way it can look like grey when a black and a white pixel are directly adjacent to eachother (dithering effect).

Regards,
Stijn Wolters.

Hi Duncan,

The key to distance estimation is what I call the "running derivative", here's how the default (Mandelbrot) Dist. Est. formula for Ultrafractal works:

For the entire fractal compute pm1 = p-1 and ip = 1/p where p is the power (estimated rate of divergence)

Before iterating (i.e. per-pixel) initialise a complex variable dz to (0,0)

At the end of each iteration compute the running derivative:
  dz = dz*p*(z^pm1) + 1

After bailout use something like:
  Dist. Est. = (p*log(cabs(z))*cabs(z)/cabs(dz))^ip

For a Julia Set initialise dz to (1,0) and omit the +1 in the iteration calculation.

The above method is a simplification of the original Distance Estimation by Hart for which I don't have a direct reference to hand - I think it's available as a PDF on the net.

Note that both Mandelbrot and Julia versions work better with higher bailouts - typically >65536 or so.

I've taken a first stab at implementing the Distance Estimator Method (DEM) documented in The Science of Fractal Images.

I still can't get the crisply defined shapes that the authors do, but I'm getting closer.

The image below was generated at 1400 x 1400 pixels, then downsampled to 750x750, and contrast adjusted. This improves the look of the image greatly.


Duncan C

David,

I've googled "estimated rate of divergence" and found all kinds of unrelated stuff, by inserting "fractal" into the search, All I came up with is this thread in this forum.

.......  How do I estimate the rate of divergence?

(In particular I'm playing around with 3D Quaternions)

Thanks