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image processing - Generating animations of clouds with Mathematica



I'd like to generate some visually-pleasing animations of clouds, fog or smoke with Mathematica. My idea of "visually-pleasing" is along the lines of one of the images on the Wikipedia article for random Perlin noise.


enter image description here



Image description: "Perlin noise rescaled and added into itself to create fractal noise."



Based on the example MATLAB code found here, I wrote the following function in Mathematica:


perlin3D[n_, t_, r_] := Module[{s, w, i, d},
  s = ConstantArray[0., {t, n, n}];
  w = n;
  i = 0;

  While[w > 3,
   i++;
   d = GaussianFilter[RandomReal[{0, 1}, {t, n, n}], r*i];
   s = s + i*d;
   w = w - Ceiling[w/2 - 1];
   ];
  s = (s - Min@s)/(Max@s - Min@s)
  ]

The results are OK, but not as good as I'd like. It's not as smooth as the example image above, nor is the image contrast as strong.



(* Generate 100 frames of 128*128 pixels *)
res = perlin3D[128, 100, 4];
imgres = Image@# &/@ res;
ListAnimate[imgres, 16]

enter image description here


How can I improve the quality of the generation using Mathematica, and is there anyway to speed it up for larger and/or longer animations?


Update


The contrast can be improved a little, as pointed out by N.J.Evans in a comment, by removing the first and last few frames before scaling, namely s = s[[r*i ;; -r*i]]. However it's still not as "fog-like" as the Wikipedia example.


enter image description here




Answer



This is a 2D Gaussian random field with a $1/k^2$ spectrum and linear dispersion $\omega \propto k$. I clip the field to positive values and square root it to give an edge to the "clouds".


n = 256;
k2 = Outer[Plus, #, #] &[RotateRight[N@Range[-n, n - 1, 2]/n, n/2]^2];

spectrum = With[{d := RandomReal[NormalDistribution[], {n, n}]},
   (1/n) (d + I d)/(0.000001 + k2)]; 
spectrum[[1, 1]] *= 0;

im[p_] := Clip[Re[InverseFourier[spectrum Exp[I p]]], {0, ∞}]^0.5


p0 = p = Sqrt[k2];

Dynamic @ Image @ im[p0 += p]

enter image description here


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