SCS — Module 08 - SCS vs CIELAB in the dark region

SCS vs CIELAB in the dark region

CIELAB uses a piecewise transform whose linear leg flattens discrimination as Y → 0. The Fisher–Bernoulli geodesic used by SCS has a sensitivity 1/√(ℓ(1−ℓ)) that diverges at both ends, which is the correct behavior for threshold discrimination near black. This module lets you see the difference on a dark gradient and on the sensitivity curves.

Dark-region upper bound (Y_max)

5.0%

Number of uniform steps

● CIELAB — L* = 116·f(Y) − 16

Step contrast range

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Step uniformity

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● SCS — d_lum(Y) = 2·arcsin(√Y)

Step contrast range

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Step uniformity

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Perceptual sensitivity dP/dY (log scale) — the slope each metric assigns to a luminance change

CIELAB (piecewise: linear leg for Y < δ³, cube root above) SCS Fisher–Bernoulli: 1/√(Y(1−Y))

Published benchmark. On COMBVD pairs with L* < 25 (n = 174), the Fisher–Bernoulli geodesic reaches r = 0.625 against human DV ratings, compared with r = 0.558 for CIELAB. The gap is structurally explained by the sensitivity curves above: as Y approaches 0, CIELAB's slope is finite (determined by the linear leg f(t) = t/(3δ²) + 4/29), while the Fisher–Bernoulli slope 1/√(Y(1−Y)) diverges — which matches the psychophysical behavior of discrimination near threshold.

Theory

CIELAB's piecewise transform. L* = 116·f(Y/Y_n) − 16 with f(t) = t^(1/3) for t > δ³ (δ = 6/29, so δ³ ≈ 0.00886) and f(t) = t/(3δ²) + 4/29 below. The linear leg exists to prevent the cube-root singularity at 0, but it makes the slope constant in the deep-dark region — meaning CIELAB cannot distinguish luminance differences finer than a fixed step there.

Fisher–Bernoulli geodesic. The SCS metric uses d_lum(Y₁, Y₂) = 2|arcsin(√Y₁) − arcsin(√Y₂)|, the Fisher–Rao geodesic on the Bernoulli family. Its local sensitivity is d/dY[2 arcsin(√Y)] = 1/√(Y(1−Y)). This diverges as Y → 0 or Y → 1, matching the well-known increase in JND sensitivity near the black and white endpoints.

The ladder. The left ladder shows n equal CIELAB L* steps in [0, Y_max], rendered as sRGB. The right ladder shows the same number of equal d_lum steps over the same range. In the dark region, the SCS ladder spreads the darkest patches further apart than CIELAB does, because SCS allocates more perceptual distance there.

Step uniformity. We measure the coefficient of variation of the displayed sRGB luminance between consecutive steps. A lower value means steps are more visually uniform given sRGB output. The sensitivity curves below explain where the two metrics place their perceptual resolution budget.

Source. The numbers 0.625 and 0.558 are Pearson correlations on COMBVD L* < 25 (PT_COLOR.tex §CIELAB vs SCS, reproducible via scripts/delta_e_scs00.py --region dark).