Seamless tiling Perlin noise

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I'm looking for good references on how to implement seamless, tiling Perlin noise. So far, everything I've found on google has been useless. As I understand it, there are two approaches to tile Perlin noise. 1. Interpolating N noises and weighting them. This is the approach described in The Perlin noise math FAQ. In 2D the formula is
Ftileable(x, y) = (
F(x, y) * (w - x) * (h - y) +
F(x - w, y) * (x) * (h - y) +
F(x - w, y - h) * (x) * (y) +
F(x, y - h) * (w - x) * (y)
) / (wh)

The problem seems to be the final division by (w * h). For low values of w and h (ex.: w = h = 1.0) this leads to over-saturation of values: the function no longer returns values in the [-1;+1] range, but easily in the [-3;+3] range. For large values of w and h (ex.: w = h = 128.0), the values are under-saturated; values start to converge to 0. This happens even faster in 3D due to the cubic term. 2. The modulo approach. The idea here is that you can tile in noise space over a period that is a power of 2 by inserting modulo operations when calculating the indices for the permutation table. See this thread on gamedev.net for an explanation. It works well when the period is indeed 2^n, but this restricts the fractal to have a lacunarity of 2.0, which severely limits the kind of fractals that you can generate. The second approach looks hopeless to me, but I'd like to know if anybody found a solution to the saturation problem in the first one.. Y.

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How about just calculating F(fmod(x,..), fmod(y,..) ) and bilinear interpolating it?

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So I ended up with a tiling method for improved noise that works on any INTEGER location.

I got the idea when I realized that, as implemented, it tiles ever 256 units anyway (Perlin noise is not actually infinite, though I'm sure you probably already knew that).

Here's a dev journal post containing the solution I came to.

A similar method works with worley (voronoi) noise as well, though I didn't write the code in my journal post.

QUICK EDIT: Being able to tile at any integral coordinate definitely is way less restrictive on the lacunarity, though you do have to be a little bit creative with your fBm implementation (to ensure all of the tile points are at integral locations).

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Thank you. I finally solved my problem.

Modifying the code to make it tiling at integer locations is easy, but it generates a lot of artifacts/bad patterns when the tiling factor decreases to 1. And of course, integrating it to a fractal is harder.

I worked on the first approach a bit more, and realized that the maths formula above required its input to be in the tiling range. In other words, I had to perform a:

x = fmod(x, w)
y = fmod(y, h)

Before calling the function.

Now everything works well, and I can tile Perlin noise at any frequency, even lower than 1 if needed.

Y.

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