Path: utzoo!utgpu!jarvis.csri.toronto.edu!cs.utexas.edu!sun-barr!newstop!sun!coherent!dplatt
From: dplatt@coherent.com (Dave Platt)
Newsgroups: comp.graphics
Subject: Re: Excluding Mandelbrot set
Message-ID: <40446@improper.coherent.com>
Date: 27 Nov 89 19:00:23 GMT
References: <3544@quanta.eng.ohio-state.edu> <480003@hpsad.HP.COM> <19885@pasteur.Berkeley.EDU> <1989Nov25.173416.8870@isy.liu.se>
Reply-To: dplatt@improper.UUCP (Dave Platt)
Organization: Coherent Thought Inc., Palo Alto CA
Lines: 74


In article <1989Nov25.173416.8870@isy.liu.se> pell@isy.liu.se (P{r Emanuelsson) writes:
>Jeff Anton:
>>In one of the books about fractals there is a system of determining
>>how far away a given point is from any point in the Mandelbrot set.
>>From that you could write a program which could eliminate pixels from the
>>set until it can eliminate no more leaveing pixels in the Mandelbrot set.
>>[...]
>>I think the system is not really better than traditional approach and
>>is more complicated.
>
>Perhaps not, but it's a different way! And it's fun to watch too!
>
>The main drawback as I see it is that the algorithm only generates B/W images.

This isn't strictly true... although the algorithm is used only for B/W
images in "The Science of Fractal Images".

It's true that the disk-marking algorithm does not generate the nice,
smooth dwell-bands that we're used to seeing with the standard
raster-scan algorithm.  The overlapping disks do look rather jarring if
displayed in black&white, or with colors that contrast significantly.

However, it's possible to extend the disk-marking algorithm a bit, and
to generate some _very_ nice colored images from it.  There are two
aspects to this extension:

[1] When iterating points, mark them in one of the following categories:
    
    -  "In M".
    -  "On the border".  Not in M, but calculated distance to the nearest
       point in M is less than a suitable fraction of a pixel.  I've
       found that border widths ranging from .3 to .001 pixels work
       well, depending on the complexity of the image.
    -  "Dwell disk".  Not in M, calculated distance > border limit.
       Mark all points in the disk with the dwell count for the point at
       the center.  [Fisher simply marks them "not in M" and doesn't
       save the dwell count]

[2] When displaying points, show points "in M" in black, points "on the
    border" in a contrasting bright color, and the points in the
    dwell-disks in a series of smoothly-shaded colors.

The calculation process in step [1] takes no longer than the algorithm
as described by Yuval Fisher... it simply uses a slightly more complex
scheme for marking points in the dwell-disks.  The display process in
[2] is the critical one... you must choose colors for the dwell-disks
that blend so smoothly that the borders between the disks aren't obvious
to the eye.

If done correctly, this process generates a very striking image... a
lacy, highly-detailed tracery stands out against a smoothly-shifting
background of colors.

>The main advantage is speed - especially when you are not interested about
>details in the current magnification, but want to move on to more interesting
>details.

The other advantage is that the border-of-M algorithm produces a
very-high-detail image, which _really_ stands out against the
background.

I could probably mail out a small image calculated in this way... my
program can write 24-bit RGB TIFF files.  If there's somebody out there
who'd be willing to accept such an image and then make it available for
anonymous FTP, please drop me a line... our site isn't on the Internet,
so I can't do this myself.  If somebody else would be willing to convert
it to GIF or a similar, popular format, please speak up!

-- 
Dave Platt                                             VOICE: (415) 493-8805
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