I should have made clear that in the context of the DEM calculation my mentioning of the estimated degree of divergence only really applies to formulas like z^n+c - the generic DEM calculation is actually:

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*f'(z) + 1.0

After bailout use something like:
  Dist. Est. = (p*log(cabs(z))*cabs(z)/cabs(dz))^ip
(where ip is 1/estimated divergence which could be calculated here using the method I mentioned for smooth iteration in my previous post)

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

Where f(z) is the iterated formula and f'(z) is its derivative.

The above should work in Quaternion form as well as Complex.

i may have linked this elsewhere, but i'm pretty sure you'll find this interesting (the power of the gpu) and useful ("zealously commented" source code): http://graphics.cs.uiuc.edu/svn/kcrane/web/project_qjulia.html

Is the value your calculating above the DEM or the value of p?

ah whoops, trust me to go simplifying the wrong equation done that way too often in exams as well... careless!

Ok....  now I'm lost   

I can calulate the value of p by pixel after the iterations are complete with...
p =  log(cabs(final z))/log(cabs(penultimate z));

pm1 = p-1;
ip    = 1/p;

DistEst =  (p*log(cabs(z))*cabs(z)/cabs(dz))^ip;

dz is initialized to [0,0];

At each iteration dz = dz*p*(z^pm1) + 1;

What I can't figure out is what value to use for p to iterate dz inside the iteration loop?

Hi all,

Sorry - I obviously confused everyone by mentioning that you can calculate the degree of divergence - it can only be calculated at the end i.e.
The calculation for the degree of divergence,

p = log(cabs(final z))/log(cabs(penultimate z))

Only works for the bailout (last) iteration where you're using final z and penultimate z, itr does not work with z and zold on any earlier iterations.

For z^2+c you should use p=2, pm1=1 and ip = 0.5 all fixed for the whole fractal, that holds for any z^p + c fractal eg. for z^3+c then p=3, pm1=2 and ip=0.3333333

For a general fractal z = f(z) then on each iteration you calculate:

dz = dz*f'(z) + 1.0

where f'(z) is the derivative of f(z)

and in the final calculation after bailout the ip value needs to be the inverse of the estimated divergence - again probably best using a fixed value for the whole fractal.

Greetings and welcome to this particular Forum !!     

One can never know too much.     

I'm finally generating DEM (Distance Estimate Method) based images that I'm happy with. It turns out that my code had a couple of bugs in it.

The new code runs reasonably fast, even with all the extra math to compute distance estimates.

The algorithm in S of F I ("The Science of Fractal Images" uses a sharp cut-off from white to black when pixels get close enough to the set, but I find this makes for jagged looking plots. I've implemented a non-linear color gradient that works pretty well.

For pixels where their DE value is  threshold>=DE>=0, I scale the value of DE to 1>=DE>=0, then do the calculation:
    gradient_value = 1 - (1-DE)^n,     ("^n" means to the n power. I wish I could use superscripts!)

"n" is an adjustment factor that lets me change the shape of the gradient curve.

The value "gradient_value" determines the color of the pixel. If it's 0, the pixel is colored at the starting color (white, for a B&W plot.) If "gradient_value" is 1, the pixel gets the end color (e.g. black)

For values of n>1, this gives a rapid change in color for pixels that are far from the set, and a slower change in value as DE approaches 0.

For values of n<1, it gives a slow color change for pixels far from the set, and rapid change close to the set. For n=1, I get a linear color gradient.

The non-linear gradient lets me use the DE value to anti-alias my plots. By adjusting my threshold value and my adjustment value, I can get good looking results for a variety of images.

Here is my latest version of the sample image from "The Beauty of Fractals":



And here are a couple of other images where I really like the lacy effect I get from DEM (the distance estimator method):







My app stores the DE value of each pixels, and does the color assignments at display time. This means that you can tweak the color values and get different looks without having to re-calculate the fractal.


Duncan C