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Author Topic: (Question) What could these artifacts be on my Lyapunov fractal renders?  (Read 194 times)

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Offline algorev

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(Question) What could these artifacts be on my Lyapunov fractal renders?
« on: July 30, 2021, 09:54:50 PM »
I'm writing a fractal renderer in Haskell and i'm implementing lyapunov fractals. Using "standard" coloring (shades of yellow for negative lyapunov exponents, blue for positive and black for 0), i get this image for sequence "AB" in the range [2;4] x [2;4]:
(attachment 1)

I chose these parameters because they're those of the wikipedia sample image:


Note that the image is mirrored because the coordinates are not in the same order in the program. This however should not effect what bugs me: these 'spots' all over the image. Is there an error in my implementation? These blue marks everywhere that are not in the reference image seem very regular, but the fact that, well, they're not in the reference image bugs me.

Here is my code to find the value for one pixel:

Code: [Select]
lyapunov :: Int -> Int -> Maybe Double
lyapunov x y = let string = "AB"
     r n = if (string !! (n `rem` (length string))) == 'A' then x else y
     iterLyapunov i n result = if n < maxIter then
        let i' = (r n) * i * (1 - i)
      ln a = (log a) / (log 2.71)
      result' = (+) result $ ln $ Prelude.abs $ (r n) * (1 - 2*i')
      in iterLyapunov i' (n + 1) result'
else result
in return $ iterLyapunov 0.5 0 0

What could the source of those spots be?

Linkback: https://fractalforums.org/index.php?topic=4352.0

Offline marcm200

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Re: What could these artifacts be on my Lyapunov fractal renders?
« Reply #1 on: July 30, 2021, 10:19:08 PM »
Such spots are very often seen in Lyapunov images. Usually they are reduced by using a larger value for the skipping iterations before one actually computes the lyapunov exponent. I'm not familiar with Haskell, but from the code I gather you're computing the Lyapunov exponent right from the start. I usually take 50 skipping and 100 calculating iterations for a quick look at an image, and 5000/10000 for the actual image.

Also the reference image might be super-sampled and/or scaled down.

Offline algorev

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Re: What could these artifacts be on my Lyapunov fractal renders?
« Reply #2 on: July 30, 2021, 10:39:22 PM »
Such spots are very often seen in Lyapunov images. Usually they are reduced by using a larger value for the skipping iterations before one actually computes the lyapunov exponent. I'm not familiar with Haskell, but from the code I gather you're computing the Lyapunov exponent right from the start. I usually take 50 skipping and 100 calculating iterations for a quick look at an image, and 5000/10000 for the actual image.

Also the reference image might be super-sampled and/or scaled down.

Those are very good tips, thanks a lot! :D
And yup, the reference image is supersampled, according to the description. I didn't think it was important.
I have a question, though: why skip iterations? If i understand you correctly, it removes these artifacts and improves the quality of the image, but how does it work? I'm puzzled.

Offline marcm200

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Re: What could these artifacts be on my Lyapunov fractal renders?
« Reply #3 on: July 30, 2021, 11:06:38 PM »
The "problem" with pretty much all fractal-type systems, is, that, what happens at the beginning of an actually infinite iterational scheme, is often not descriptive of the long-time limit behaviour. And that's happening with the perturbed Lyapunov exponent from Mario Markus' algorithm as well. If I recall correctly, he states in his 90's article something like "let the x-value settle in". It's similar to the statement "for sufficiently large N" in math papers or in general with convergent series.

The actual value of how many skipping iterations one performs, is, however, (at least afaik) just a guess till you're satisfied with the quality of the image. In my images, I could not get rid of those artifacts in full, one example being the UFO image, where vasyan did a great job removing those spots:

vasyan's: https://fractalforums.org/index.php?action=gallery;sa=view;id=2388
my version (with spots): https://fractalforums.org/index.php?action=gallery;sa=view;id=1960


Offline algorev

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Re: What could these artifacts be on my Lyapunov fractal renders?
« Reply #4 on: July 30, 2021, 11:16:23 PM »
That makes a lot of sense. Thanks for the explanation!
I've implemented your suggestion and you're absolutely right, my renders are way nicer now!


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