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"""Palette-constrained dithering.
Every routine takes the working image in CIELAB plus a per-pixel table of the
palette indices that pixel is *allowed* to use (because the VIC-II only lets a
given screen cell show a small set of colours), and returns an (H,W) image of
chosen palette indices (0..15). Because the allowed set is per-pixel, error that
diffuses across a cell boundary is automatically re-clamped to the neighbour
cell's own colours -- exactly the constraint real C64 hardware imposes.
"""
from __future__ import annotations
import numpy as np
DITHER_MODES = ["bayer", "bluenoise", "yliluoma", "floyd", "atkinson", "stucki",
"jarvis", "sierra", "sierra_lite", "burkes", "riemersma",
"ostromoukhov", "none"]
def _bayer_int(n: int) -> np.ndarray:
"""Integer Bayer matrix with values 0..n*n-1 (n a power of two)."""
if n == 1:
return np.array([[0]])
s = _bayer_int(n // 2)
return np.block([
[4 * s + 0, 4 * s + 2],
[4 * s + 3, 4 * s + 1],
])
def bayer_matrix(n: int) -> np.ndarray:
"""Normalised (0..1) Bayer threshold matrix of size n x n (n power of two).
Thresholds are centred at (i + 0.5) / n^2 so they span (0,1) symmetrically --
normalising only once (not at every recursion level, which would compress the
range and collapse the dither toward a single threshold)."""
return (_bayer_int(n) + 0.5) / (n * n)
def _gather_colors(palette_lab: np.ndarray, allowed: np.ndarray) -> np.ndarray:
# allowed: (H,W,K) palette indices -> (H,W,K,3) Lab
return palette_lab[allowed]
def _ordered_core(img_lab, allowed, palette_lab, thr_full, strength=1.0):
"""Ordered dithering between each pixel's two best colours, using a supplied
(H,W) threshold map (Bayer, blue-noise, ...). For every pixel we take its
nearest and second-nearest allowed colour, project the pixel onto the segment
between them, and the threshold decides which of the two to emit -- smooth
blends without ever leaving the cell's legal colour set."""
H, W, _ = img_lab.shape
colors = _gather_colors(palette_lab, allowed) # (H,W,K,3)
d = np.sum((img_lab[:, :, None, :] - colors) ** 2, axis=-1) # (H,W,K)
i1 = np.argmin(d, axis=-1)
d2 = np.array(d)
np.put_along_axis(d2, i1[..., None], np.inf, axis=-1)
i2 = np.argmin(d2, axis=-1)
yy, xx = np.indices((H, W))
c1 = colors[yy, xx, i1] # (H,W,3)
c2 = colors[yy, xx, i2]
seg = c2 - c1
seg_len2 = np.sum(seg * seg, axis=-1) + 1e-9
t = np.sum((img_lab - c1) * seg, axis=-1) / seg_len2 # projection 0..1
t = np.clip(t * strength, 0.0, 1.0)
chosen = np.where(t > thr_full, i2, i1)
return np.take_along_axis(allowed, chosen[..., None], axis=-1)[..., 0]
def quantize_ordered(img_lab, allowed, palette_lab, strength=1.0, n=8):
"""Ordered (Bayer) dithering."""
H, W, _ = img_lab.shape
yy, xx = np.indices((H, W))
thr = bayer_matrix(n)
return _ordered_core(img_lab, allowed, palette_lab, thr[yy % n, xx % n], strength)
_BLUENOISE = {} # cached void-and-cluster matrices, keyed by size
def bluenoise_matrix(n: int = 64, sigma: float = 1.9) -> np.ndarray:
"""Tileable n x n blue-noise threshold matrix (0..1), via void-and-cluster
(Ulichney). Like Bayer but with no regular cross-hatch grid -- the dot
pattern is isotropic and organic, so gradients look far cleaner."""
if n in _BLUENOISE:
return _BLUENOISE[n]
rng = np.random.default_rng(12345)
# toroidal Gaussian centred at (0,0); rolling it centres it on any pixel
ax = np.minimum(np.arange(n), n - np.arange(n))
g = np.exp(-(ax[:, None] ** 2 + ax[None, :] ** 2) / (2 * sigma ** 2))
def roll(p):
r, c = divmod(p, n)
return np.roll(np.roll(g, r, axis=0), c, axis=1)
binary = np.zeros((n, n), bool)
ones = max(1, (n * n) // 10)
for p in rng.choice(n * n, ones, replace=False):
binary.flat[p] = True
e = np.zeros((n, n))
for p in np.flatnonzero(binary):
e += roll(p)
# phase 1: even out the initial pattern (move tightest cluster -> largest void)
while True:
tc = int(np.argmax(np.where(binary.flat, e.flat, -np.inf)))
binary.flat[tc] = False; e -= roll(tc)
lv = int(np.argmin(np.where(binary.flat, np.inf, e.flat)))
binary.flat[lv] = True; e += roll(lv)
if lv == tc:
break
rank = np.full(n * n, -1, np.int64)
proto = binary.copy(); ep = e.copy()
# phase 2a: remove tightest clusters, ranking downward
b = proto.copy(); e = ep.copy(); cnt = int(b.sum())
for r in range(cnt - 1, -1, -1):
tc = int(np.argmax(np.where(b.flat, e.flat, -np.inf)))
b.flat[tc] = False; e -= roll(tc); rank[tc] = r
# phase 2b: fill largest voids, ranking upward
b = proto.copy(); e = ep.copy()
for r in range(cnt, n * n):
lv = int(np.argmin(np.where(b.flat, np.inf, e.flat)))
b.flat[lv] = True; e += roll(lv); rank[lv] = r
m = (rank.reshape(n, n) + 0.5) / (n * n)
_BLUENOISE[n] = m
return m
def quantize_bluenoise(img_lab, allowed, palette_lab, n=64):
"""Ordered dithering with a blue-noise mask instead of Bayer (no grid)."""
H, W, _ = img_lab.shape
yy, xx = np.indices((H, W))
thr = bluenoise_matrix(n)
return _ordered_core(img_lab, allowed, palette_lab, thr[yy % n, xx % n])
def _quantize_diffusion(img_lab, allowed, palette_lab, kernel, divisor):
"""Generic serpentine error-diffusion constrained to per-pixel allowed sets."""
H, W, _ = img_lab.shape
work = img_lab.astype(np.float64).copy()
out = np.zeros((H, W), dtype=np.int64)
pal = palette_lab
for y in range(H):
cols = range(W) if (y % 2 == 0) else range(W - 1, -1, -1)
flip = 1 if (y % 2 == 0) else -1
for x in cols:
allow = allowed[y, x]
cand = pal[allow]
diff = cand - work[y, x]
k = int(allow[np.argmin(np.sum(diff * diff, axis=-1))])
out[y, x] = k
err = work[y, x] - pal[k]
for dx, dy, w in kernel:
nx, ny = x + dx * flip, y + dy
if 0 <= nx < W and 0 <= ny < H:
work[ny, nx] += err * (w / divisor)
return out
# (dx, dy, weight) relative to current pixel, assuming left-to-right scan.
_FLOYD = [(1, 0, 7), (-1, 1, 3), (0, 1, 5), (1, 1, 1)]
_ATKINSON = [(1, 0, 1), (2, 0, 1), (-1, 1, 1), (0, 1, 1), (1, 1, 1), (0, 2, 1)]
# Larger kernels spread error further -> smoother gradients (best for grayscale).
_STUCKI = [(1, 0, 8), (2, 0, 4),
(-2, 1, 2), (-1, 1, 4), (0, 1, 8), (1, 1, 4), (2, 1, 2),
(-2, 2, 1), (-1, 2, 2), (0, 2, 4), (1, 2, 2), (2, 2, 1)]
_JARVIS = [(1, 0, 7), (2, 0, 5),
(-2, 1, 3), (-1, 1, 5), (0, 1, 7), (1, 1, 5), (2, 1, 3),
(-2, 2, 1), (-1, 2, 3), (0, 2, 5), (1, 2, 3), (2, 2, 1)]
# Sierra-3: like Jarvis but lighter third row -> a touch less smearing.
_SIERRA = [(1, 0, 5), (2, 0, 3),
(-2, 1, 2), (-1, 1, 4), (0, 1, 5), (1, 1, 4), (2, 1, 2),
(-1, 2, 2), (0, 2, 3), (1, 2, 2)]
# Sierra-Lite: tiny 3-tap kernel -> crisp, fast, minimal bleed across cells.
_SIERRA_LITE = [(1, 0, 2), (-1, 1, 1), (0, 1, 1)]
# Burkes: two-row Stucki relative -> smooth gradients, cheaper than Stucki.
_BURKES = [(1, 0, 8), (2, 0, 4),
(-2, 1, 2), (-1, 1, 4), (0, 1, 8), (1, 1, 4), (2, 1, 2)]
def quantize_floyd(img_lab, allowed, palette_lab):
return _quantize_diffusion(img_lab, allowed, palette_lab, _FLOYD, 16)
def quantize_atkinson(img_lab, allowed, palette_lab):
return _quantize_diffusion(img_lab, allowed, palette_lab, _ATKINSON, 8)
def quantize_stucki(img_lab, allowed, palette_lab):
return _quantize_diffusion(img_lab, allowed, palette_lab, _STUCKI, 42)
def quantize_jarvis(img_lab, allowed, palette_lab):
return _quantize_diffusion(img_lab, allowed, palette_lab, _JARVIS, 48)
def quantize_sierra(img_lab, allowed, palette_lab):
return _quantize_diffusion(img_lab, allowed, palette_lab, _SIERRA, 32)
def quantize_sierra_lite(img_lab, allowed, palette_lab):
return _quantize_diffusion(img_lab, allowed, palette_lab, _SIERRA_LITE, 4)
def quantize_burkes(img_lab, allowed, palette_lab):
return _quantize_diffusion(img_lab, allowed, palette_lab, _BURKES, 32)
def _hilbert_path(W: int, H: int):
"""Sequence of (x,y) covering a WxH grid along a Hilbert space-filling curve
(points outside the grid are skipped). Used by Riemersma dithering so the
1-D error trail stays spatially compact in 2-D."""
order = 1
while (1 << order) < max(W, H):
order += 1
side = 1 << order
pts = []
for d in range(side * side):
rx = ry = 0
t = d
x = y = 0
s = 1
while s < side:
rx = 1 & (t // 2)
ry = 1 & (t ^ rx)
if ry == 0: # rotate quadrant
if rx == 1:
x = s - 1 - x
y = s - 1 - y
x, y = y, x
x += s * rx
y += s * ry
t //= 4
s <<= 1
if x < W and y < H:
pts.append((x, y))
return pts
def quantize_riemersma(img_lab, allowed, palette_lab, qlen=16, ratio=1.0 / 16):
"""Riemersma dithering: error diffusion along a Hilbert curve with a decaying
memory of recent errors (geometric weights). Spreads error in every
direction without the directional grain of raster diffusion."""
H, W, _ = img_lab.shape
pal = palette_lab
out = np.zeros((H, W), dtype=np.int64)
# geometric weights, most-recent first, summing to 1 (error-conserving)
w = ratio ** (np.arange(qlen) / max(1, qlen - 1))
w = w / w.sum()
hist = np.zeros((qlen, 3), dtype=np.float64)
for x, y in _hilbert_path(W, H):
target = img_lab[y, x] + w @ hist
allow = allowed[y, x]
cand = pal[allow]
diff = cand - target
k = int(allow[np.argmin(np.sum(diff * diff, axis=-1))])
out[y, x] = k
hist[1:] = hist[:-1]
hist[0] = img_lab[y, x] - pal[k]
return out
# Ostromoukhov variable-coefficient error diffusion ("A Simple and Efficient
# Error-Diffusion Algorithm", SIGGRAPH 2001). The three forward coefficients
# (right, below-left, below) vary with the *tone* being quantised, which breaks
# up the worm artefacts of fixed-kernel diffusion. Table given at breakpoints
# (luminance 0..255) and linearly interpolated; symmetric about 127.5.
_OSTRO_KNOTS = np.array([
(0, 13, 0, 5), (1, 13, 0, 5), (2, 21, 0, 10), (3, 7, 0, 4),
(4, 8, 0, 5), (10, 47, 3, 28), (15, 23, 3, 13), (16, 15, 3, 11),
(32, 43, 7, 36), (64, 21, 8, 21), (96, 39, 7, 35), (112, 19, 7, 17),
(127, 38, 8, 35),
], dtype=np.float64)
def _ostro_coeffs():
"""Build the 256x3 (normalised) Ostromoukhov coefficient table by
interpolating the knot table and mirroring it about the midpoint."""
if "tab" in _BLUENOISE: # reuse the module cache dict
return _BLUENOISE["tab"]
xs = _OSTRO_KNOTS[:, 0]
c = np.empty((256, 3))
half = np.arange(128)
for j in range(3):
c[:128, j] = np.interp(half, xs, _OSTRO_KNOTS[:, j + 1])
c[128:] = c[:128][::-1] # symmetric for the upper half
c /= c.sum(axis=1, keepdims=True)
_BLUENOISE["tab"] = c
return c
def quantize_ostromoukhov(img_lab, allowed, palette_lab):
H, W, _ = img_lab.shape
work = img_lab.astype(np.float64).copy()
out = np.zeros((H, W), dtype=np.int64)
pal = palette_lab
coeffs = _ostro_coeffs()
for y in range(H):
ltr = (y % 2 == 0)
cols = range(W) if ltr else range(W - 1, -1, -1)
flip = 1 if ltr else -1
for x in cols:
allow = allowed[y, x]
cand = pal[allow]
diff = cand - work[y, x]
k = int(allow[np.argmin(np.sum(diff * diff, axis=-1))])
out[y, x] = k
err = work[y, x] - pal[k]
tone = int(np.clip(work[y, x, 0] * 2.55, 0, 255)) # L* (0..100) -> 0..255
cr, cdl, cd = coeffs[tone]
for dx, dy, wgt in ((1, 0, cr), (-1, 1, cdl), (0, 1, cd)):
nx, ny = x + dx * flip, y + dy
if 0 <= nx < W and 0 <= ny < H:
work[ny, nx] += err * wgt
return out
def quantize_yliluoma(img_lab, allowed, palette_lab, plan=16, n=8):
"""Yliluoma-style ordered dithering. For each pixel we greedily build a
'mixing plan' -- a list of `plan` palette entries (with repeats) from its
allowed set whose running average best matches the target -- then index into
the plan, sorted by lightness, with the Bayer threshold. Mixes more than two
colours per cell, so flat-palette platforms get far richer apparent colour."""
H, W, _ = img_lab.shape
colors = _gather_colors(palette_lab, allowed) # (H,W,K,3)
yy, xx = np.indices((H, W))
running = np.zeros((H, W, 3)) # sum of chosen Lab
chosen = np.empty((H, W, plan), dtype=np.int64) # local index into allowed
for t in range(plan):
# average if we appended each candidate next
avg = (running[:, :, None, :] + colors) / (t + 1)
d = np.sum((avg - img_lab[:, :, None, :]) ** 2, axis=-1) # (H,W,K)
pick = np.argmin(d, axis=-1) # (H,W)
chosen[:, :, t] = pick
running = running + colors[yy, xx, pick]
# sort each pixel's plan by the lightness (L*) of its colours
plan_L = colors[yy[..., None], xx[..., None], chosen, 0] # (H,W,plan)
order = np.argsort(plan_L, axis=-1)
sorted_plan = np.take_along_axis(chosen, order, axis=-1)
thr = bayer_matrix(n)
idx = np.clip((thr[yy % n, xx % n] * plan).astype(np.int64), 0, plan - 1)
local = np.take_along_axis(sorted_plan, idx[..., None], axis=-1)[..., 0]
return np.take_along_axis(allowed, local[..., None], axis=-1)[..., 0]
def quantize_none(img_lab, allowed, palette_lab):
colors = _gather_colors(palette_lab, allowed)
d = np.sum((img_lab[:, :, None, :] - colors) ** 2, axis=-1)
i1 = np.argmin(d, axis=-1)
return np.take_along_axis(allowed, i1[..., None], axis=-1)[..., 0]
def quantize(img_lab, allowed, palette_lab, mode="bayer"):
if mode == "bayer":
return quantize_ordered(img_lab, allowed, palette_lab)
if mode == "bluenoise":
return quantize_bluenoise(img_lab, allowed, palette_lab)
if mode == "yliluoma":
return quantize_yliluoma(img_lab, allowed, palette_lab)
if mode == "floyd":
return quantize_floyd(img_lab, allowed, palette_lab)
if mode == "atkinson":
return quantize_atkinson(img_lab, allowed, palette_lab)
if mode == "stucki":
return quantize_stucki(img_lab, allowed, palette_lab)
if mode == "jarvis":
return quantize_jarvis(img_lab, allowed, palette_lab)
if mode == "sierra":
return quantize_sierra(img_lab, allowed, palette_lab)
if mode == "sierra_lite":
return quantize_sierra_lite(img_lab, allowed, palette_lab)
if mode == "burkes":
return quantize_burkes(img_lab, allowed, palette_lab)
if mode == "riemersma":
return quantize_riemersma(img_lab, allowed, palette_lab)
if mode == "ostromoukhov":
return quantize_ostromoukhov(img_lab, allowed, palette_lab)
return quantize_none(img_lab, allowed, palette_lab)