Fractional Brownian Motion noise

Math essence

fbm(x) = sum_{k=0..N-1} G^k * noise(2^k * x + offset_k) where noise(.) is a smoothly-interpolated value noise built from a 32-bit integer hash. Hurst exponent H in [0,1] determines G = exp2(-H); the canonical 'yellow noise' corresponds to H = 1 (G = 0.5). Optional domain offset slides the field across the canvas to simulate animation.

Parameters

Reference

First 60 lines of reference.py. Full source: library_external/hodgin_iquilezles/FBMNoise/reference.py

from __future__ import annotations

from typing import ClassVar

import numpy as np


def _hash1(ix: np.ndarray, iy: np.ndarray, seed: int) -> np.ndarray:
    ix = ix.astype(np.uint64, copy=False)
    iy = iy.astype(np.uint64, copy=False)
    h = (ix * np.uint64(0x27D4EB2F165667C5)) ^ (
        iy * np.uint64(0x165667B19E3779F9)
    ) ^ (np.uint64(np.int64(seed) & 0xFFFFFFFF) * np.uint64(0x9E3779B97F4A7C15))
    h &= np.uint64(0xFFFFFFFFFFFFFFFF)
    h ^= h >> np.uint64(33)
    h *= np.uint64(0xFF51AFD7ED558CCD)
    h &= np.uint64(0xFFFFFFFFFFFFFFFF)
    h ^= h >> np.uint64(33)
    h *= np.uint64(0xC4CEB9FE1A85EC53)
    h &= np.uint64(0xFFFFFFFFFFFFFFFF)
    h ^= h >> np.uint64(33)
    out = (h & np.uint64(0xFFFFFFFF)).astype(np.float64) / 4294967296.0
    return out.astype(np.float32)


class FBMNoise:
    """Fractional Brownian Motion noise — Inigo Quilez (iquilezles.org).

    Quilez's article *fbm* (https://iquilezles.org/articles/fbm/) describes
    the canonical fBM accumulation:

        float fbm( in vec2 x, in float H ) {
            float G = exp2(-H);
            float f = 1.0, a = 1.0, t = 0.0;
            for ( int i = 0; i < numOctaves; i++ ) {
                t += a * noise( f * x );
                f *= 2.0;
                a *= G;
            }
            return t;
        }

    With ``G = 0.5`` (H=1) we get the popular "yellow noise" / isotropic
    fBM used for terrain, clouds, bark and many natural textures.  We use
    a 2D value-noise base (Quilez's hash22 → dot/rand) and expose both
    ``H`` and the number of octaves.
    """

    family: ClassVar[str] = "hodgin_iquilezles"
    sketch: ClassVar[str] = "fbmnoise"

    def __init__(
        self,
        dim: int = 360,
        scale: float = 4.0,
        octaves: int = 6,
        hurst: float = 1.0,
        contrast: float = 1.1,
        offset_x: float = 0.0,
        offset_y: float = 0.0,