*Continued* SuperFractalThing: Arbitrary precision mandelbrot set rendering in Java.

[hapf]

I'm getting 312900 iterations to skip allowing an error <= 0.1% for escape velocity and distance. The region is not 100% identical (no rotation, not exact size). Minit is 313253. That is with 64 coefficients which is chosen as optimal. For 32 I get 300300, for 128 I get also 312900, for 256 312900, and for 1024 312900 as well. Skips are limited to deltas of 100 so this is expected. 64 is optimal since more brings nothing and costs a lot more to compute. And there is corruption in the non reference centers by the way.

[quaz0r]

i have noticed a more empirical way to determine when overskipping is occurring:  less reference points get used.  i have also observed that readily discernible visual differences in the final image do not tend to present immediately after having crossed the threshold into overskip territory.

assuming here for test purposes of course that your system of glitch tolerance is disabled and things are allowed to run their full course down to every last point.  for instance, at claude's test location from this thread, at 1280x720, a correct render for me uses 44 reference points.  once i pass into overskip territory, this number steadily goes down, though as i recall visually obvious deformations do not present until this number gets down into the low 20s or so.

logically this makes sense:  when visually obvious overskipping occurs, in complex julia sorts of areas that would normally require a relatively higher number of references to accurately complete, you instead see simple non-julia sorts of formations take their place.

this is just a free sort of metric here;  if one wanted to perform a more perfectly concrete, more expensive comparison, you could of course compare the escape times and/or distance estimates of each point among different renders of a location.

it would be quite useful to compile a collection of test locations together with empirically-determined skip boundaries for an array of SA term counts.

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19582

a test render of TheRedshiftRider's "Light Years Away" location, inspired by the discussion on series approximation.

1920x1080 @ 6x supersampling for an effective resolution of 11520 x 6480.
min iteration was 313765, with 32 SA terms skipping 290484 iterations.
glitch tolerance disabled for a total of 1438 reference points  

[quaz0r]

i noticed again claude's post about trying just || for the test, which indeed is a pretty good starting point just on its own.  still rather conservative, but at least is consistent in its behavior.  which i think shows that overall these formulas are good to go, except what to use with ||.

I haven't yet tried to implement my version of residual estimation formula. It's on my todo list.

your mention of some mathy thing ive never heard of inspired me to poke around and do some reading..  i got to reading about things like standard deviation, and decided to give that a try.  using the standard deviation of the values from the SA' thing with || seemed potentially promising, but went a little too far.  adjusting || by some constant to make it smaller helps but is still imperfect.  maybe there is some formula which utilizes something like the standard deviation which would be good?

it still seems like there should be a definitive answer as to what formula to use here, instead of just trying random stuff and seeing what happens  

and im still falling back to (max1*max2)/(min1*min2) until someone thinks of something more legitimate.  i dont know why it works as well as it seems to, but its still the best for me so far.

[knighty]

Quazor:
Until now, I haven't had any problem with the locations provided by claude. In some cases the skipped iterations number is optimal. It is sub optimal when there are those substructures that cause blobs and need their own reference.

The test as proposed is not perfect and doesn't give an absolute guarantee on the number of iterations to skip. So far it works well (even if it is sub optimal) and I need to understand first the how and why. There are better and more reliable (and a little bit complicated) ways to do the test but I haven't yet had time to implement them...

The "light years away" location is somehow problematic. When using 100 terms, I get the same glitches as kalles fraktaler at 313'088 skipped iterations. Even the per pixel test doesn't detect correctly the glitches. at 128 terms, nothing is detected (at 64 terms everything is detected). Note that in those cases R is bigger than the last few terms of the SA. These glitches appear when skipping 313'084. At 313'083 skipped iterations there are no SA glitches at all. When using 64 terms, skipping 313'083 iters gives no deformations or SA glitches. At 313'084, there are a lot of deformations that are hopefully detected by the per pixel test. (See pictures below -4x antialiasing-)

With 100 or 128 terms, those glitches don't look like a deformation but as a random noise. They seem to be caused by rounding errors. This is not surprising because there are a lot of terms to add in the computation of the coefficients and the SA itself: the more terms, the more rounding errors. It is possible to account for the rounding errors in the computation of the SA coefficients by adding them to the residual estimation (and using the suitable rounding rule) but that would be too slow. Maybe using Horner method would reduce somehow those rounding errors.

I've also tried the glitch detection formula that was proposed here (I use pmax=32). It works very well but doesn't seem to do better than PauldelBrot's in practice (beside detecting some more glitches in "light years away" location.

Something interesting that happens in the computations of the SA, especially with lots of coefficient: It is very slow at the beginning then accelerate dramatically after the first period of the reference is attained. It is probably because of the underflows and subnormals which are 10's of times slower to compute.

[quaz0r]

hmm, perhaps i didnt understand what the per pixel test was supposed to be.  i thought it was just something you were using for your own tests.  is it something we are actually supposed to use?  like a sort of overskip detection you can perform when initializing each point with the series?  in that case, i guess you would treat these areas like the perturbation glitch areas, and do new reference points inside them?  what is the best way to choose a new reference then, the point with the highest error value?

and so are we saying then that this is the best we can do, more glitch detecting and more reference points?  or there is still hope to figure out a totally proper way to get one golden skip value for the whole image?

[knighty]

I'm using the per pixel test for testing purposes. . In my previous post, the (global) test used for light years away location gives 282752 skipped iterations Wich is far below the max possible: 313083.

The per pixel test is not very expensive so it can be used as a checking option. Having such error metric is already better than nothing.

You are right. It is possible to use the per pixel test to mark the areas detected as overskipped and do a glich like correction. Then the next reference point is around the max error points. But the goal is still to find a good way to get the near optimal skip value over all the rendered area.

The current test works, and goes beyond one period of the reference, because it ignores the zeroes of the derivative of the SA that are inside some area around the reference. When we reach one period P, at P+1 there is always a point inside the minibrot where the derivative vanishes. That point remains until the next period. At 2P+1, we will get 3 point where the derivative is 0. Then 7 at 3P+1 ... so we have 2ⁿ-1 zero derivative points inside the minibrot between n*P+1 and (n+1)*P iterations. This means that the test should not be:
  max abs term - every thing else
but (more conservative but guaranteed assuming the reference dominanat and is inside a minibrot):
  max abs term corresponding to a power of two - every thing else
Or maybe (not sure!):
  sum of abs terms corresponding to a power of two - every thing else.

The obvious disadvantage is that using a reference that is not inside the dominant minibrot will "underskip" a lot. Also the number of skipped iterations will be less than log2(m)*P (where m is the number of coefficients of the SA and P the period of the reference). In order to go further, finer and more expensive methods have to be used. This is not so dramatic because the expensive test doesn't have to be performed at every iteration.

[stardust4ever]

Looking at the green areas in the image two posts up, I see a lot of distortion. What should be circular formations have literally become crescent moons. Is the whole green area significantly perturbed?

[quaz0r]

indeed, the green area is the "this is your brain on drugs" area.  the goal is to keep our mandelbrots drug-free.

[stardust4ever]

indeed, the green area is the "this is your brain on drugs" area.  the goal is to keep our mandelbrots drug-free.

LOL! I've had people from video gaming forums (retro gaming is my other hobby) watch some of my YT fractal vids, and they said watching 15 minutes of continuous colorful zoom action was like taking an acid trip!

[TheRedshiftRider]

indeed, the green area is the "this is your brain on drugs" area.  the goal is to keep our mandelbrots drug-free.

LOL! I've had people from video gaming forums (retro gaming is my other hobby) watch some of my YT fractal vids, and they said watching 15 minutes of continuous colorful zoom action was like taking an acid trip!

Same here, I showed one of my videos to my classmates. They asked which drugs I was using.

[quaz0r]

yes if you like fractals you should definitely give psychedelic drugs a try sometime 

[TheRedshiftRider]

yes if you like fractals you should definitely give psychedelic drugs a try sometime  

I would imagine it could nice but I suppose opinions differ greatly. I am not sure what to think about it.
Alright, let's return to the original topic.

I really like the way my location is used here. The different palettes look nice on this location.

[quaz0r]

yes, i need to look at your zooming technique again sometime.  its really great how you can get that much going on at such a shallow depth!

[TheRedshiftRider]

yes, i need to look at your zooming technique again sometime.  its really great how you can get that much going on at such a shallow depth!

Thank you. I actually have not figured out the math behind it so I still do not know how it works. Help and interest are always appreciated.

[stardust4ever]

yes if you like fractals you should definitely give psychedelic drugs a try sometime 

If they legalize mushrooms (not the culinary type) or THC in my state, I'll give it a try!