The following code is very slow, with Mathematica 13.2 on a Mac Silicon M2 Pro:

domains = 1500; (* Number of domains in 3D *)

randomDomains = Table[{RandomReal[], RandomReal[], RandomInteger[{1, 2}]}, {n, 1, Ndomains}];

ImageDomaines = MedianFilter[
    InterpolationOrder -> 0,
    Frame -> False,
    ColorFunction -> "Rainbow",
    PlotRangePadding -> 0,
    ImageSize -> {1024, 1024}], 0],

Why is it so slow? How can we speed it up, and get curved random patterns?

Here's a preview of what this code is doing:

enter image description here

  • $\begingroup$ Why do you make three-dimensional domains if you're only interested in a two-dimensional image? $\endgroup$ Jul 27, 2023 at 2:04
  • $\begingroup$ @DavidG.Stork, it's just the simplest way I found to make a random pattern like this. $\endgroup$
    – Cham
    Jul 27, 2023 at 2:05
  • $\begingroup$ Try it in TWO D. $\endgroup$ Jul 27, 2023 at 2:26
  • $\begingroup$ ListContourPlot[RandomReal[1, {35, 35}], InterpolationOrder -> 1, Contours -> 1, ContourShading -> {Red, Purple}, Frame -> False] could be a starting point. $\endgroup$
    – Syed
    Jul 27, 2023 at 3:05
  • 1
    $\begingroup$ @DavidG.Stork It is impossible to draw such 2D image without any extra data(for example 3D data) or extra restrict function( for example a distance function define on 2D) $\endgroup$
    – cvgmt
    Jul 27, 2023 at 5:11

2 Answers 2


IMHO, InterpolationOrder->0 should use Voronoi cells. So you can do it this way:

Ndomains = 1500;
x = RandomReal[{0, 1}, {Ndomains, 2}];
cols = RandomChoice[
    Directive[FaceForm[ColorData["Rainbow"][0]], EdgeForm[]], 
    Directive[FaceForm[ColorData["Rainbow"][1]], EdgeForm[]]
M = VoronoiMesh[x, 
  MeshCellStyle -> Thread[Thread[{2, Range[Ndomains]}] -> cols], 
  PlotRange -> {{0, 1}, {0, 1}}];
img = MedianFilter[Rasterize[M, ImageSize -> {1024, 1024}], 6]

enter image description here

  • $\begingroup$ This answer doesn't give the same kind of output. It's not similar to the example I gave in my question. $\endgroup$
    – Cham
    Jul 27, 2023 at 13:01
  • $\begingroup$ Better? Note that ListDensityPlot fails at sampling the edges and corners of the domains correcly. VoronoiMesh does much better which is why edges and corners appear sharper than in your plot. $\endgroup$ Jul 27, 2023 at 13:33
  • $\begingroup$ Yes, it looks better. But the shapes are having too much straight borders. This is the issue. My slow code could smooth the shapes and give curved borders, but it's very slow. $\endgroup$
    – Cham
    Jul 27, 2023 at 13:44
  • 2
    $\begingroup$ Hmm, the Rasterize may be a good trick, despite the ugly crude borders. Try this in your code: MedianFilter[Rasterize[M, ImageSize -> {1024, 1024}], 6] $\endgroup$
    – Cham
    Jul 27, 2023 at 13:57
  • 1
    $\begingroup$ Hm. Maybe you are interested in shapes that arise during the Cahn-Hillard flow (see the animation in my post)? This flow models two phases that slowly demix. Think of small grease drops on a soup (water) for example. $\endgroup$ Jul 27, 2023 at 13:57

Based on @Syed:

In 0.264381 seconds on a Mac laptop:

ListContourPlot[RandomReal[1, {70, 70}], InterpolationOrder -> 1, 
 Contours -> 1, ContourShading -> {Red, Purple}, Frame -> False]

enter image description here

  • $\begingroup$ This method works, but apparently it gives many vertical and horizontal shapes. It's not as "natural looking" as the slow method of my question. We can feel a horizontal-vertical grid behind the structures. $\endgroup$
    – Cham
    Jul 27, 2023 at 13:04
  • $\begingroup$ @Cham: I don't necessarily agree that it "gives many vertical and horizontal shapes"; it may just be that human brains are great at recognizing patterns, even when they don't exist. You could test this out by creating an image "blindly", making a randomly rotated version, cropping the rotated version and the original, and seeing if you can tell the difference. $\endgroup$ Jul 27, 2023 at 17:43
  • 1
    $\begingroup$ Also, I did try running Fourier analysis on this image file and didn't see any particular evidence of correlations along the axes vs. at angles to the vertices; the transform looked pretty rotationally symmetric. $\endgroup$ Jul 27, 2023 at 17:44
  • $\begingroup$ Yep. I agree............. $\endgroup$ Jul 27, 2023 at 17:47
  • $\begingroup$ Well, yet there's something wrong with the output. There are some ugly artifacts and spikes... $\endgroup$
    – Cham
    Jul 27, 2023 at 17:54

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