Wavelet noise

Wavelet noise is an alternative to Perlin noise which reduces the problems of aliasing and detail loss that are encountered when Perlin noise is summed into a fractal.

Algorithm detail
The basic algorithm for 2-dimensional wavelet noise is as follows: This results in an image that contains all the information that cannot be represented at half-scale. From here, $$N$$ can be used similarly to Perlin noise to create fractal patterns.
 * Create an image, $$R$$, filled with uniform white noise.
 * Downsample $$R$$ to half-size to create $$R^\downarrow$$, then upsample it back up to full size to create $$R^{\downarrow\uparrow}$$.
 * Subtract $$R^{\downarrow\uparrow}$$ from $$R$$ to create the end result, $$N$$.