⚡ electrical engineering

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

This paper introduces Continuous Oklab fallback (COFb), a differentiable, one-parameter chroma-gating method that seamlessly blends OKLCH and linear Oklab interpolation paths to eliminate both inter-hue detours and off-line bows in color gradients without relying on discrete threshold switches.

Original authors: Naoyuki Uchida

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Naoyuki Uchida

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Problem: The "Detour" and the "Confused Compass"

Imagine you are a digital artist trying to draw a smooth line (a gradient) between two colors on a computer screen. You want to go from Blue to Yellow.

In the old way of doing things (using a space called OKLCH), the computer tries to keep the "vibrancy" (chroma) of the colors constant while changing the hue. Think of this like walking around a giant round table.

The Blue-to-Yellow Problem: If you walk from the "Blue" side of the table to the "Yellow" side, the shortest path is a straight line across the middle. But the computer is told to walk around the edge of the table to keep the distance from the center constant.
The Result: Instead of a clean Blue-to-Yellow transition, the path takes a huge detour. It swings out wide, passing through Green and Red on the way. This is the "Inter-hue Cast." It's like trying to drive from New York to Boston, but the GPS forces you to drive all the way to Chicago first just to keep your speedometer reading the same.

There is a second problem:

The "Black" Problem: If you try to fade from Green to Black, the computer gets confused. Black has no color, so it has no "hue" (no direction on the compass). But the computer still tries to guess a direction and swings the path off course, creating a weird curve instead of a straight line. This is the "Achromatic Cast."

The Old Fix: The "On/Off Switch"

Previously, engineers tried to fix the "Black" problem with a simple On/Off Switch.

The Rule: "If one of the colors is Black (or gray), turn off the 'keep the path curved' rule and just draw a straight line. If both colors are colorful, keep the curved rule."
The Flaw: This switch only works when one color is Black. If you are going from Blue to Yellow (both colorful), the switch stays "Off," and the computer still takes that weird detour through Green. The old fix completely ignored the Blue-to-Yellow problem.

The New Solution: The "Dimmer Switch" (COFb)

Naoyuki Uchida proposes a new method called Continuous Oklab fallback (COFb). Instead of a rigid On/Off switch, imagine a Dimmer Switch or a Volume Knob.

How it works: The computer constantly checks how "colorful" (chroma) the path is at every single step.
- High Color (Vibrant): When the colors are bright and strong, the dimmer is turned up. The path follows the curved OKLCH route, keeping the colors vibrant and following the designer's intent.
- Low Color (Fading to Gray/Black): As the colors get closer to gray or black, the dimmer turns down. The computer gently stops trying to keep the curved path and starts blending toward a straight line.
- The Magic: This happens smoothly. It doesn't wait for a color to be perfectly black to switch modes. It starts relaxing the curve the moment the colors get a little dull.

What This Achieves

Fixes the Detour: Because the dimmer turns down when the path gets close to the center (where the Blue-to-Yellow detour happens), the computer stops swinging wildly through Green. It takes a much more direct, natural path.
One Rule for All: You don't need a special "Black" rule anymore. The same dimmer switch handles the Blue-to-Yellow detour and the Green-to-Black confusion automatically.
The Trade-off: To fix the weird curves, the path gets slightly less vibrant (a tiny bit grayer) as it approaches the center. The author admits this is a small price to pay to stop the colors from swinging wildly off-course.

The "Recipe" (The Math)

The author suggests a specific formula for this dimmer switch, similar to a recipe used in biology (Michaelis-Menten kinetics).

The Setting: He recommends a specific "sensitivity" setting (called $\sigma \approx 0.19$ ).
The Result: With this setting, the weird detour for Blue-to-Yellow is cut in half. The path is much straighter and more natural, without needing complex checks to see if a color is "black" or "not black."

Summary

Think of the old method as a traffic light that is either Red (stop curving) or Green (keep curving). It only worked when you hit a specific stop sign (Black).

The new method (COFb) is like adaptive cruise control. It constantly adjusts the car's steering based on how close you are to the center of the road. If you are driving fast on a wide road (vibrant colors), it keeps the lane. As you slow down and approach a narrow bridge (gray/black), it gently steers you straight to avoid hitting the walls, without needing a sudden brake or a hard turn.

This makes digital color gradients look smoother and more natural, especially when mixing bright colors or fading into black, and it works on older computer systems that don't have the fancy new "Black-detecting" software built-in.

Technical Summary: Continuous Oklab Fallback (COFb) for OKLCH Interpolation

1. Problem Statement

OKLCH, the cylindrical form of Ottosson's Oklab color space, is the recommended interpolation space for CSS Color 4 gradients and color-mix(). While its polar parameterization (Lightness, Chroma, Hue) provides intuitive hue arcs, it introduces geometric pathologies near the neutral axis (zero chroma) where the hue angle is undefined. These pathologies manifest as two distinct artifacts:

Inter-hue Color Cast (Type 1): Occurs when both endpoints are chromatic but lie on opposite sides of the neutral axis (e.g., Blue $\to$ Yellow). The constant-radius hue arc forces the interpolation path to detour around the origin, bulging in chroma and sweeping through an unintended hue (e.g., passing through green). Existing remedies fail to address this because both endpoints have defined hues.
Achromatic-Endpoint Cast (Type 2): Occurs when one endpoint is achromatic (e.g., Green $\to$ Black). Although the hue is undefined at the achromatic point, raw OKLCH carries a hue value forward, causing the path to bow away from the straight chromatic-to-neutral line.

Limitations of Existing Remedies: Current solutions (e.g., CSS missing-component handling, ColorAide's is_achromatic threshold) are uniformly two-valued. They employ a hard threshold switch that only triggers when an endpoint is judged achromatic. Consequently, they successfully address Type 2 artifacts but reduce to raw OKLCH for Type 1 artifacts (chromatic pairs), leaving the inter-hue cast entirely untreated.

2. Methodology: Continuous Oklab Fallback (COFb)

The paper introduces Continuous Oklab fallback (COFb), a one-parameter, differentiable interpolation method that unifies the handling of both artifact types without endpoint testing or branching.

Core Construction

COFb operates in the Cartesian $(a, b)$ plane of Oklab. It blends two reference paths point-by-point based on the interpolated chroma $C(t)$ :

Raw OKLCH Path: $(a_{lch}, b_{lch})$ , following the polar arc.
Linear Oklab Path: $(a_{lab}, b_{lab})$ , a straight line segment (cast-free, but loses hue control).

The blending is governed by a chroma-dependent gate function $w(C)$ :
$\begin{aligned} a(t) &= w(C(t)) \cdot a_{lch}(t) + (1 - w(C(t))) \cdot a_{lab}(t) \\ b(t) &= w(C(t)) \cdot b_{lch}(t) + (1 - w(C(t))) \cdot b_{lab}(t) \\ L(t) &= (1-t)L_0 + tL_1 \end{aligned}$

The Gate Function

The gate uses a Naka–Rushton / Michaelis–Menten saturation function:
$w(C) = \frac{C^n}{C^n + \sigma^n}$

High Chroma ( $C \to \infty$ ): $w \to 1$ , the path follows raw OKLCH (full hue control).
Low Chroma ( $C \to 0$ ): $w \to 0$ , the path relaxes toward the linear Oklab segment (eliminating the cast).
Differentiability: The gate is differentiable for $C > 0$ . The paper selects $n=1$ (Michaelis–Menten form) to ensure $C^1$ continuity at the neutral axis ( $C=0$ ), avoiding the vertical tangent present when $n < 1$ .

Parameter Selection

$\sigma$ (Threshold): Represents the chroma value at which the gate is half-open. The paper adopts a default of $\sigma \approx 0.19$ , derived from a normalization-independent "cast-half" criterion (reducing the inter-hue detour by 50%).
$n$ (Exponent): The default is $n=1$ . An alternative $n=0.87$ (fit to COMBVD data in prior work) is noted, but the paper finds the optimum $\sigma$ practically insensitive to $n$ within the tested range, making $n=1$ preferable for its rational, GPU-friendly form.

3. Key Contributions

Addressing the Inter-hue Blind Spot: COFb is the only method evaluated that reduces the Type 1 (inter-hue) cast. Existing two-valued methods are numerically identical to raw OKLCH for chromatic pairs.
Unified Continuous Handling: A single differentiable expression replaces the special-cased, two-valued logic. It acts continuously wherever chroma is low—whether at an achromatic endpoint or along the origin-skirting detour of an inter-hue arc.
Cast–Hue Trade-off Characterization: The paper maps the frontier between cast removal and hue retention. It establishes that removing more cast necessarily relaxes hue control.
Symbolic Verification: The properties of the gate (monotonicity, smoothness, endpoint behavior) are verified symbolically using computer algebra systems.

4. Evaluation Results

Experiments were conducted on five endpoint pairs (three inter-hue, two achromatic-endpoint) using metrics for lateral deviation (distance from the straight line), excess chroma, and hue excursion.

Inter-hue Pairs (Type 1):
- Raw OKLCH and all existing two-valued methods showed identical high deviation (e.g., mean lateral deviation $\approx 0.1378$ ).
- COFb (at default $\sigma \approx 0.19$ ) reduced the mean lateral deviation by 49.5% (to $\approx 0.0697$ ) and the chroma-weighted hue excursion by 35.5%.
- The reduction effectively halves the "detour" around the origin.
Achromatic-Endpoint Pairs (Type 2):
- Existing two-valued methods drive lateral deviation to exactly 0 (perfect linear Oklab).
- COFb significantly improves upon raw OKLCH (reducing mean deviation from $\approx 0.0821$ to $\approx 0.0313$ ) but does not reach zero.
- Claim: The paper explicitly states that for Type 2 cases in isolation, the existing two-valued switch remains superior. COFb's advantage is the unification of both cases into a single continuous mechanism.
Trade-off: The default $\sigma \approx 0.19$ represents a balance where the inter-hue cast is halved while retaining approximately 43–66% of the hue control depending on the metric used.

5. Significance and Claims

The paper positions COFb as a geometric, behavior-level construction rather than a claim of perceptual optimality. Its significance lies in:

Implementation Flexibility: COFb runs entirely in plain Oklab $(a, b) \to sRGB$ . It does not require native OKLCH interpolation support in the target environment. This makes it a viable fallback for older engines, image/video pipelines, and GPU shaders where modern CSS color interpolation is unavailable.
Unification: It transforms the existing discrete, (2)-only fallback logic into a continuous, differentiable family that also resolves the previously untreated (1) inter-hue cast.
Modest Scope: The authors explicitly acknowledge limitations:
- It does not preserve chroma (the path grays slightly near the neutral axis).
- It does not match the zero-deviation performance of two-valued switches on achromatic endpoints alone.
- The default $\sigma$ is content-dependent (optimized for a typical sRGB palette) and not a universal constant.

In summary, COFb offers a practical, mathematically continuous solution to the "blind spot" in current OKLCH interpolation strategies, reducing visible color casts in gradients without requiring changes to the CSS specification.