Reference Tables: Color Spaces and HDR

Chris Lomont, with Claude Fable 5.2

What this article is. Lookup tables for the articles on color and HDR: RGB spaces and matrices, transfer functions, Y′CbCr variants, chroma subsampling, code points, HDR formats, pixel formats and stops.

Related articles. Expert notes: Color and HDR Notes . General-audience summary: Color and HDR Notes Summary . Error guide: Best Practices and Gotchas .

Contents

• 1. RGB spaces
• 2. Transfer functions
• 3. Luma/chroma forms (“YUV flavors”)
• 4. H.273 / CICP code points (the common ones)
• 5. HDR formats
• 6. Pixel formats for HDR data
• 7. Stops, decades, ratios, bits
• 8. Color float candidates (from §8 of the expert notes)
• 9. Perceptual spaces and difference metrics
• 10. Formulas to implement

Matrices and derived numbers are computed from the published primaries [Code]; check against the standard before relying on the last digit.

[Code]: private test code written for these articles; not published.

1. RGB spaces

White points: D65 = (0.3127, 0.3290); D50 = (0.3457, 0.3585); D60 ≈ (0.32168, 0.33767); DCI = (0.314, 0.351).

“P3” is three different things: DCI-P3 (greenish white, γ 2.6, 48 cd/m²), P3-D65 (same primaries, D65, usually γ 2.6 or PQ in mastering) and Display P3 (D65, sRGB curve).

Linear conversion matrices (D65 to D65, no adaptation needed)

BT.709 → BT.2020:

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 0.6274  0.3293  0.0433
 0.0691  0.9195  0.0114
 0.0164  0.0880  0.8956

BT.2020 → BT.709:

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 1.6605 -0.5876 -0.0728
-0.1246  1.1329 -0.0083
-0.0182 -0.1006  1.1187

Display P3 → BT.709:

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 1.2249 -0.2249  0.0000
-0.0421  1.0421  0.0000
-0.0196 -0.0786  1.0983

Apply to linear values only. Negative results are real out-of-gamut colors, not errors.

Luminance weights (the Y row)

2. Transfer functions

Signal levels worth memorizing

Camera log curves (scene-referred)

3. Luma/chroma forms (“YUV flavors”)

Range

Narrow-range scaling between bit depths is a shift (×4 from 8 to 10 bits), not a rescale to the full code range.

Chroma subsampling

Common pixel-format names: NV12 (8-bit 4:2:0, interleaved chroma), I420/YV12 (8-bit 4:2:0 planar), P010 (10-bit 4:2:0 in the high bits of 16), yuv420p10le (10-bit 4:2:0 in the low bits of 16), UYVY/YUY2 (8-bit 4:2:2 packed), v210 (10-bit 4:2:2, three samples per 32-bit word), Y410 (10-bit 4:4:4 + 2-bit alpha).

Chroma siting: center (JPEG, MPEG-1); left (MPEG-2, H.264, HEVC default); top-left (BT.2020/BT.2100 UHD).

4. H.273 / CICP code points (the common ones)

Typical triples: sRGB still image 1-13-0 full; HD video 1-1-1 narrow; HDR10 9-16-9 narrow; HLG 9-18-9 narrow; Display P3 image 12-13-0 full. Code 2 means “unspecified”, and decoders then guess.

5. HDR formats

6. Pixel formats for HDR data

7. Stops, decades, ratios, bits

Sensor makers quote dB (20 log₁₀ of the electron ratio): 84 dB is 14 stops.

The same signal under different definitions (stops)

Typical luminance ranges

8. Color float candidates (from §8 of the expert notes)

Threshold: the Barten-model implementation of the expert notes (see the caveat at Figure 10 of the expert notes).

9. Perceptual spaces and difference metrics

10. Formulas to implement

Constants are exact unless marked. Sources: BT.2100 for PQ, HLG and ICtCp; BT.2124 for the color difference; CIE 015 for CIELAB; BT.2446 for the tone curve; the Ultra HDR and Adobe gain-map specifications. The Oklab constants are from its author’s publication and were not rechecked here.

PQ (ST 2084), exact rationals

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m1 = 2610/16384        = 0.1593017578125
m2 = 2523/4096 * 128   = 78.84375
c1 = 3424/4096         = 0.8359375      (= c3 - c2 + 1)
c2 = 2413/4096 * 32    = 18.8515625
c3 = 2392/4096 * 32    = 18.6875
EOTF:     Y = 10000 * ( max(E^(1/m2) - c1, 0) / (c2 - c3 * E^(1/m2)) )^(1/m1)     cd/m2
Inverse:  E = ( (c1 + c2 * y^m1) / (1 + c3 * y^m1) )^m2,   y = Y/10000

Landmarks: signal 0.5 is 92.25 cd/m²; 0.5807 is 203; 0.75 is 983; 0.9 is 3906.

HLG (BT.2100)

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a = 0.17883277,  b = 1 - 4a = 0.28466892,  c = 0.5 - a*ln(4a) = 0.55991073
OETF:     E' = sqrt(3E)              for E <= 1/12
          E' = a*ln(12E - b) + c     otherwise            (E scene light in [0,1])
Inverse:  E  = E'^2 / 3              for E' <= 1/2
          E  = (exp((E' - c)/a) + b) / 12   otherwise
OOTF:     Ys = 0.2627 R + 0.6780 G + 0.0593 B;   Rd = Lw * Ys^(gamma - 1) * R   (same for G, B)
gamma:    1.2 + 0.42*log10(Lw/1000)          for 400 <= Lw <= 2000
          1.2 * 1.111^log2(Lw/1000)          outside that range
Black:    beta = sqrt(3 * (Lb/Lw)^(1/gamma));  use max(0, (1 - beta)*E' + beta) as the signal
75% signal as display light:  Lw * 0.26496^gamma

ICtCp and ΔE ITP

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From linear BT.2020 RGB, 1.0 = 10000 cd/m2:
L = (1688 R + 2146 G +  262 B) / 4096
M = ( 683 R + 2951 G +  462 B) / 4096
S = (  99 R +  309 G + 3688 B) / 4096
L', M', S' = PQ inverse EOTF of L, M, S          (HLG OETF for the HLG variant)
I  = 0.5 L' + 0.5 M'
PQ:   Ct = ( 6610 L' - 13613 M' + 7003 S') / 4096     Cp = (17933 L' - 17390 M' - 543 S') / 4096
HLG:  Ct = ( 3625 L' -  7465 M' + 3840 S') / 4096     Cp = ( 9500 L' -  9212 M' - 288 S') / 4096

Delta E ITP = 720 * sqrt( dI^2 + (0.5 dCt)^2 + dCp^2 )      always from the PQ form, on display light

About 1 is one just-noticeable difference in the most critical viewing state. BT.2124 suggests that a tolerance below 3 may be acceptable for a reference display. Older papers scale the same quantity differently; check before comparing numbers.

CIELAB, exact constants

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f(t) = t^(1/3)                 for t > (6/29)^3 = 216/24389
f(t) = (841/108) t + 4/29      otherwise
L = 116 f(Y/Yn) - 16;   a = 500 [f(X/Xn) - f(Y/Yn)];   b = 200 [f(Y/Yn) - f(Z/Zn)]

For display light, $Y_n$ is the diffuse white (203 cd/m² by BT.2408), not the peak.

Oklab

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From linear sRGB:
l = 0.4122214708 r + 0.5363325363 g + 0.0514459929 b
m = 0.2119034982 r + 0.6806995451 g + 0.1073969566 b
s = 0.0883024619 r + 0.2817188376 g + 0.6299787005 b
l', m', s' = cube roots
L = 0.2104542553 l' + 0.7936177850 m' - 0.0040720468 s'
a = 1.9779984951 l' - 2.4285922050 m' + 0.4505937099 s'
b = 0.0259040371 l' + 0.7827717662 m' - 0.8086757660 s'

Luma and chroma, general form

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Y'  = KR R' + (1 - KR - KB) G' + KB B'
Cb  = 0.5 (B' - Y') / (1 - KB)
Cr  = 0.5 (R' - Y') / (1 - KR)
          KR       KB
BT.601    0.299    0.114
BT.709    0.2126   0.0722
BT.2020   0.2627   0.0593
Narrow range, n bits:  Y code = (219 Y' + 16) * 2^(n-8);   C code = (224 C + 128) * 2^(n-8)
Full range,   n bits:  Y code = (2^n - 1) Y';              C code = (2^n - 1) C + 2^(n-1)

For other primaries, KR and KB are the red and blue entries of the Y row of that space’s RGB-to-XYZ matrix.

Integer arithmetic that is exact

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Float to 8-bit:         i = round(255 f), after clamping f to [0,1] in floating point
8-bit to float:         f = i / 255
a*b/255, rounded:       t = a*b + 128;   (t + (t >> 8)) >> 8
16-bit to 8-bit:        (v*255 + 32895) >> 16
8-bit to 16-bit:        v * 257
n to m bits, full range:   round(v * (2^m - 1) / (2^n - 1))
n to m bits, narrow range: v << (m - n)

Tone curve with an exact inverse (BT.2446 Method C, luminance in cd/m²)

This is the default parameter set printed in BT.2446-1, for HLG display light. Use the constants as printed; recomputing $k_2$ and $k_4$ from the rounded $k_1$ and $k_3$ gives slightly different values.

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Y_sdr = k1 * Y_hdr                              for Y_hdr < Y_ip = 58.5 / k1
Y_sdr = k2 * ln(Y_hdr / Y_ip - k3) + k4         otherwise
k1 = 0.83802,  k2 = 15.09968,  k3 = 0.74204,  k4 = 78.99439

Gain maps

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Make:   G = log2( (HDR + k_hdr) / (SDR + k_sdr) )        linear light, same primaries
        g = clamp( (G - Gmin) / (Gmax - Gmin), 0, 1 )^gamma;   code = floor(255 g + 0.5)
Apply:  g = code/255 (after resampling);   G = Gmin + (Gmax - Gmin) * g^(1/gamma)
        H = log2( display peak / display SDR white )
        W = clamp( (H - CapMin) / (CapMax - CapMin), 0, 1 )
        HDR = (SDR + k_sdr) * 2^(G * W) - k_hdr
Defaults: k_sdr = k_hdr = 1/64

Quantization and stops

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Stops between two luminances:        log2(L2 / L1)
Relative step of a power-law code:   dL/L = (gamma / N) * L^(-1/gamma)      N codes over [0,1], L relative
Codes per stop at code v, gamma g:   v * (1 - 2^(-1/g))
Worst blend error in a power-law space, black over white at coverage f:   f^(1/g) - f
      peak at f = g^(-g/(g-1)):   g = 2.2 gives f = 0.236, error 0.283 (codes 60 against 132)

Lookup-table conventions

• A “17-point” table for 8-bit data has two incompatible meanings: nodes at 0, 16, 32 and so on up to 256, where white (255) is not a node, or 17 nodes spread evenly from 0 to 255. Mixing them is a scale error of 256/255.
• In .cube files red varies fastest. In the Common LUT Format blue varies fastest. Converting without reordering swaps red and blue.
• A uniform grid on linear input spends its first cell on the darkest fifth of the lightness scale. Put a log or PQ shaper in front.
• Tetrahedral interpolation always runs from the cell’s black corner to its white corner, so grays are interpolated only from gray nodes. For fractional position (r, g, b) with r ≥ g ≥ b the weights are 1 − r, r − g, g − b and b on corners 000, 100, 110 and 111; the other five orderings follow by permutation.
• Reported table sizes for errors under one just-noticeable difference: about 33 per side for SDR with tetrahedral interpolation, and 55 to 85 per side for 12-bit PQ output depending on content. Sizes depend strongly on the test images, especially dark saturated ones.