Improved Noise — Ken Perlin (SIGGRAPH 2002, faithful reimplementation)

Math essence

For query (x, y): X = floor(x) & 255, Y = floor(y) & 255, xf = x - floor(x), yf = y - floor(y), u = fade(xf), v = fade(yf). A = p[X] + Y, B = p[X+1] + Y, AA = p[A], AB = p[A+1], BA = p[B], BB = p[B+1]. n_{ij} = grad(hash, xf-i, yf-j) where grad selects one of 16 cube-edge directions from h & 15. Result = lerp(v, lerp(u, n00, n10), lerp(u, n01, n11)). For fBM we sum amp*lac^k * perlin(lac^k * x) over k octaves.

Parameters

Reference

First 60 lines of reference.py. Full source: library_external/shader_legends/KenPerlinNoisePure/reference.py

from __future__ import annotations

from typing import ClassVar

import numpy as np


_PERMUTATION = (
    151, 160, 137, 91, 90, 15, 131, 13, 201, 95, 96, 53, 194, 233, 7, 225,
    140, 36, 103, 30, 69, 142, 8, 99, 37, 240, 21, 10, 23, 190, 6, 148,
    247, 120, 234, 75, 0, 26, 197, 62, 94, 252, 219, 203, 117, 35, 11, 32,
    57, 177, 33, 88, 237, 149, 56, 87, 174, 20, 125, 136, 171, 168, 68, 175,
    74, 165, 71, 134, 139, 48, 27, 166, 77, 146, 158, 231, 83, 111, 229, 122,
    60, 211, 133, 230, 220, 105, 92, 41, 55, 46, 245, 40, 244, 102, 143, 54,
    65, 25, 63, 161, 1, 216, 80, 73, 209, 76, 132, 187, 208, 89, 18, 169, 200,
    196, 135, 130, 116, 188, 159, 86, 164, 100, 109, 198, 173, 186, 3, 64, 52,
    217, 226, 250, 124, 123, 5, 202, 38, 147, 118, 126, 255, 82, 85, 212, 207,
    206, 59, 227, 47, 16, 58, 17, 182, 189, 28, 42, 223, 183, 170, 213, 119,
    248, 152, 2, 44, 154, 163, 70, 221, 153, 101, 155, 167, 43, 172, 9, 129,
    22, 39, 253, 19, 98, 108, 110, 79, 113, 224, 232, 178, 185, 112, 104, 218,
    246, 97, 228, 251, 34, 242, 193, 238, 210, 144, 12, 191, 179, 162, 241,
    81, 51, 145, 235, 249, 14, 239, 107, 49, 192, 214, 31, 181, 199, 106, 157,
    184, 84, 204, 176, 115, 121, 50, 45, 127, 4, 150, 254, 138, 236, 205, 93,
    222, 114, 67, 29, 24, 72, 243, 141, 128, 195, 78, 66, 215, 61, 156, 180,
)


def _make_perm(seed: int) -> np.ndarray:
    base = np.array(_PERMUTATION, dtype=np.int32)
    rng = np.random.default_rng(seed)
    perm = base.copy()
    rng.shuffle(perm)
    full = np.zeros(512, dtype=np.int32)
    full[:256] = perm
    full[256:] = perm
    return full


def _fade(t: np.ndarray) -> np.ndarray:
    """Ken Perlin's improved fade curve: 6t^5 - 15t^4 + 10t^3."""
    return t * t * t * (t * (t * 6.0 - 15.0) + 10.0)


def _lerp(t: np.ndarray, a: np.ndarray, b: np.ndarray) -> np.ndarray:
    return a + t * (b - a)


def _grad(hash_idx: np.ndarray, x: np.ndarray, y: np.ndarray) -> np.ndarray:
    """Perlin's grad for 2D noise (z=0), derived from the lo 4 bits.

    The original 3D grad uses the lo 4 bits of the hash to pick one
    of 16 gradient directions. For z=0, this collapses to:
        h & 1  → sign of u
        h & 2  → sign of v
        h & 4  → u = x or y
        h & 8  → v = x or y
    """
    h = hash_idx & 15
    u_x = (h & 1) == 0
    u_y = ~u_x