Chroma-gated, differentiable OKLCH interpolation: Continuous Oklab fallback for color-cast reduction

Abstract: OKLCH -- the cylindrical (lightness, chroma, hue) form of Ottosson's Oklab color space -- is the interpolation space recommended by CSS Color 4 for gradients and color-mix(), and it is now broadly deployed. Its polar parameterization, however, casts color near the neutral axis in two ways: (1) an inter-hue detour between two chromatic endpoints that sweeps through an unintended hue (blue to yellow visibly passing through green), and (2) an off-line bow when one endpoint is achromatic. Existing remedies are uniformly two-valued -- a threshold switch that fires only at an achromatic endpoint -- so they address only (2); on chromatic pairs every one of them reduces to raw OKLCH, leaving the (1) inter-hue cast untreated. We introduce Continuous Oklab fallback (COFb), a one-parameter, differentiable chroma gate $w(C)=C<sup>n/(C<sup>n+σ<sup>n)$ that continuously blends the OKLCH path toward the linear Oklab path as chroma falls. A single gate reduces the (1) cast that the two-valued family leaves untreated and unifies the handling of (1) and (2) without any endpoint test. We characterize a cast-hue trade-off frontier, adopt a default ($n=1$, the rational Michaelis-Menten form; $σ\approx0.19$ for a typical sRGB palette, from a normalization-independent cast-half criterion), and verify the gate's properties symbolically. At the default, COFb halves the inter-hue path detour (mean lateral deviation -49.5%, chroma-weighted hue excursion -35.5%). We also state the method's limits: on (2) alone the two-valued switch remains better, and like any Cartesian blend COFb does not preserve chroma. In deployment, COFb runs entirely in plain Oklab (a,b) to sRGB, so it serves as a fallback that delivers the same cast-reduced gradients where modern CSS color interpolation (color-mix(in oklch) and the like) is unavailable -- older engines, image and video pipelines, or GPU shaders.

• The paper introduces Continuous Oklab Fallback (COFb), which blends raw OKLCH with linear Oklab using a differentiable chroma gate and reduces mean inter-hue lateral deviation by 49.5% at σ≈0.19.
• COFb addresses both inter-hue and achromatic-endpoint casts without branching, reducing chroma-weighted hue excursion from 20.3° to 13.1° for chromatic pairs while retaining a continuous, GPU-friendly formulation.
• The method does not fully outperform hard achromatic switching or preserve chroma, and its optimal σ depends on gamut and content, leaving perceptual validation and adaptive scaling as open questions.

The problem: two casts from one singularity

OKLCH, the cylindrical (lightness, chroma, hue) form of Ottosson's Oklab space, is the interpolation space recommended by CSS Color 4 for gradients and color-mix(), and it is deployed across major browsers and CSS tooling. Its polar parameterization is singular on the neutral axis: at zero chroma the hue $h=\operatorname{atan2}(b,a)$ is undefined, and near the axis small displacements in $(a,b)$ produce arbitrarily large hue changes. Uchida's paper separates the resulting artifacts into two named cases:

• (1) Inter-hue cast: both endpoints chromatic but on opposite sides of the neutral axis (e.g., blue → yellow). The constant-radius hue arc detours around the origin, bulging in chroma and sweeping through an unintended hue—blue → yellow visibly passes through green.
• (2) Achromatic-endpoint cast: one endpoint achromatic (e.g., green → black). Its hue is genuinely undefined, yet a value is carried forward, bowing the path off the chromatic-to-neutral line.

The paper's central observation is that all existing remedies—the CSS missing-component mechanism (issue #6107) and ColorAide's is_achromatic() threshold—are two-valued: a hard switch that fires only when an endpoint is judged achromatic. On chromatic pairs every such method reduces exactly to raw OKLCH, so artifact (1) is a blind spot of the entire existing family.

Method: Continuous Oklab fallback

COFb computes both the raw OKLCH path and the straight linear-Oklab segment in the $(a,b)$ plane, then blends them point-by-point with a chroma gate evaluated at the interpolated chroma $C(t)$ along the OKLCH path:

$$w(C)=\frac{C^{n}}{C^{n}+\sigma^{n}}, \qquad a(t)=w\,a_{\text{oklch}}(t)+(1-w)\,a_{\text{lin}}(t),$$

with lightness interpolated linearly and shared. Because the gate reads the path's own chroma at each $t$, it engages wherever chroma is low—at an achromatic endpoint and along an inter-hue detour alike—with no endpoint test or branching. As $\sigma\to 0$ the family recovers raw OKLCH; as $\sigma$ grows it collapses to linear Oklab, so COFb spans the two standard choices continuously where the existing fallback switches discretely between them.

All gate properties are verified symbolically with SymPy: $w(0^+)=0$, $w(\sigma)=1/2$, monotonicity, analyticity on $(a,b)$0, and the $(a,b)$1-dependent endpoint slope $(a,b)$2 for $(a,b)$3, $(a,b)$4, $(a,b)$5. The default is $(a,b)$6 (the rational Michaelis–Menten form $(a,b)$7), chosen for three reasons: finite endpoint slope ($(a,b)$8 up to the neutral axis), a GPU-friendly rational form with closed-form inverse $(a,b)$9, and consistency with the author's Oklch+ toolset (Uchida, 3 Jun 2026). The alternative $(a,b)$0 (the Naka–Rushton exponent fit to COMBVD data in Oklch+) differs only at the error level, so adopting $(a,b)$1 carries no trade-off cost.

Evaluation

Metrics are computed directly in the Oklab $(a,b)$2 plane over five endpoint pairs split into the two groups: lateral deviation from the chord (primary metric), excess chroma, and chroma-weighted hue excursion. Three findings stand out.

On chromatic pairs, all existing methods are numerically identical to raw OKLCH—their achromatic test never trips—and share its full cast (e.g., Blue→Yellow: lateral deviation 0.1768, hue excursion 25.42°). COFb is the only method compared that acts here, halving the group-mean lateral deviation from 0.1378 to 0.0697 (−49.5%) and reducing chroma-weighted hue excursion by 35.5%. Excess chroma nearly coincides with lateral deviation on these pairs, confirming the detour is essentially a pure radial bulge.

Method	(1) lat. dev.	(1) hue exc.	(2) lat. dev.	(2) hue exc.
Raw OKLCH	0.1378	20.3°	0.0821	40.0°
NaN-inherit / Two-valued	0.1378	20.3°	0.0000	0.0°
COFb ($(a,b)$3, $(a,b)$4)	0.0697	13.1°	0.0313	11.2°

On achromatic pairs, the two-valued methods remain superior in isolation, driving lateral deviation to exactly zero; COFb improves substantially over raw OKLCH (0.0821 → 0.0313) but does not reach zero. The paper states this plainly rather than claiming uniform superiority: COFb's contribution is unification of (1) and (2) under one differentiable expression, not dominance on either case alone.

A parameter sweep exposes a cast–hue trade-off frontier whose ideal corner (full cast removal plus full hue control) is unreachable. The default $(a,b)$5 comes from a normalization-independent cast-half criterion (cast = half the raw baseline); a hue-priority operating point sits at $(a,b)$6. A frontier-knee estimate ($(a,b)$7) is consistent but grid-dependent and not counted as independent corroboration. A cross-check confirms the gate-based hue-retention proxy tracks actual paths almost perfectly (Pearson $(a,b)$8 against path-based retention), so the trade-off is not a proxy artifact.

Limitations and open questions

The paper is explicit about boundaries. First, artifact (2) taken alone is better served by the existing switch, which reaches exactly zero lateral deviation; closing that residual gap within a continuous formulation remains open. Second, like any Cartesian blend, COFb does not preserve chroma—the path grays slightly near the neutral axis—and a chroma-preserving variant would require a non-Cartesian construction outside this work's scope. Third, no perceptual optimality is claimed: $(a,b)$9 is anchored to a geometric criterion for a moderate-chroma sRGB-typical palette, and because $C(t)$0 is a threshold in chroma units, more saturated content or wider gamuts shift the optimum upward; content-adaptive $C(t)$1 scaling is left uncharacterized. Fourth, while the gate is $C(t)$2 in chroma at the neutral axis for $C(t)$3, the time-derivative behavior of the rendered path at an achromatic endpoint (where $C(t)$4 combines with the gate response) has not been exhaustively characterized. Finally, whether the geometric cast reduction corresponds to observer-judged improvement awaits a perceptual study, and step-size uniformity along the path lies on a separate evaluation axis not addressed here.

Conclusion

This paper identifies a genuine gap in deployed color-interpolation practice: the two-valued achromatic fallbacks specified for CSS leave the inter-hue cast entirely untreated, since they reduce to raw OKLCH whenever both endpoints are chromatic. Continuous Oklab fallback closes that gap with a single differentiable chroma gate that halves the inter-hue detour at its recommended default ($C(t)$5, $C(t)$6), unifies both artifacts without branching, and runs entirely in plain Oklab-to-sRGB arithmetic—making it usable as a fallback in engines, pipelines, and shaders lacking native OKLCH interpolation. Its costs—a residual achromatic gap versus the two-valued switch and slight near-axis graying—are stated alongside the results, positioning COFb as a continuous extension of missing-component handling rather than a replacement for it.