Helmlab: A Two-Space Family of Analytical, Data-Driven Color Spaces for UI Design Systems

Gorkem Yildiz

cs.GRFeb 26, 2026

Helmlab: A Two-Space Family of Analytical, Data-Driven Color Spaces for UI Design Systems

Authors: Gorkem Yildiz

Organizations: Independent Researcher

Abstract

We present Helmlab, a family of two purpose-built color spaces for UI design systems sharing a common 11-stage analytical structure: MetricSpace, a 72-parameter space optimized for color-difference prediction, and GenSpace, a 44-parameter space optimized for gradient and palette generation. The forward transform maps CIE XYZ to a perceptually-organized Lab representation through learned matrices, per-channel power compression, Fourier hue correction, and embedded Helmholtz-Kohlrausch lightness adjustment. A post-pipeline neutral correction holds gray-axis chroma below 1e-5 on a 21-step ramp, and a rigid rotation of the chromatic plane improves hue-angle alignment without affecting the distance metric (which is invariant under isometries). On COMBVD (3,813 color pairs), MetricSpace v21 achieves STRESS 22.48, a 23 percent reduction from CIEDE2000 (29.20). On the held-out MacAdam 1974 dataset it scores 19.51 (CIEDE2000: 22.13; CAM16-UCS leads at 18.71). On a self-collected 3,552-judgement screen-condition set it scores 23.26 vs 62.54 for CIEDE2000. On academic He et al. 2022 (82 3D-printed pairs) MetricSpace scores 35.9 vs CIEDE2000 32.6, a regression we own. Averaging the three primary datasets, MetricSpace scores 21.75 vs the next-best baseline CIECAM02-UCS at 35.98. GenSpace v0.11.1 trades distance accuracy for generation quality: on a 90-metric, 3,038-pair gradient/palette benchmark across sRGB, P3, and Rec.2020, it wins 65 of 90 vs OKLab. The transform is invertible with round-trip errors below 1e-13. Production implementations ship on PyPI, npm, Color.js (PR 722, merged), and as a PostCSS plugin.

Figures & tables

Figure 1: The Helmlab MetricSpace forward transform pipeline. Blue: linear operations; yellow: nonlinear corrections; green: structural guarantees (NC, rotation). The 72 jointly-optimized parameters are distributed across the eleven stages as shown.

Figure 2: stress on COMBVD (3,813 pairs). Lower is better. MetricSpace v21 is bottom-row.

Method	combvd	MacAdam	HumFB	Avg
MetricSpace v21	22.48	19.51	23.26	21.75
CIE94	33.37	19.78	59.75	37.63
CIEDE2000	29.20	22.13	62.54	37.96
DIN99	35.57	23.31	56.44	38.44
$J_{z}a_{z}b_{z}$	41.92	24.14	61.74	42.60
CIE Lab ( $\Delta E^{*}_{76}$ )	42.86	24.53	62.32	43.24

Table 1: stress across three datasets. Lower is better. HumFB = 3,552 self-collected screen-condition judgements from 71+ observers, see caveat in Section 6.1 . Avg is the unweighted three-dataset mean. He et al. 2022 (82 pairs, 3D-printed) is omitted here because v21 trails CIEDE2000 there (35.9 vs 32.6); see Section 6.1 . All baselines use Euclidean distance in their native Cartesian coordinates with per-pair Bradford CAT to D65 where appropriate.

Figure 3: Predicted vs observed color differences on COMBVD for MetricSpace v21.

Figure 4: Cross-dataset validation on held-out data.

Figure 5: Left: neutral ramp step uniformity. Right: achromatic chroma leakage. Helmlab ’s NC drives

C<10^{-5}

for grays on the 21-step ramp.

Property	Value
Achromatic max $C$ (NC on)	$<10^{-5}$
Hue RMS (6 sRGB primaries)	26.4°
Hue max error	53.4°
Round-trip (sRGB)	$\sim$ 7 $\times$ 10 -15
Round-trip (random XYZ)	$\sim$ 1 $\times$ 10 -13
Munsell CV (3,590 iso-hue pairs)	32.0%

Table 2: Generation quality metrics for MetricSpace v21.

Configuration	COMBVD stress	$\Delta$
Full MetricSpace v21	22.48	—
Euclidean distance only	$\sim$ 27.6	$+5.1$
No H-K embedding	$\sim$ 26.5	$+4.0$
No hue correction (Stage 4)	$\sim$ 37.9	$+15.4$
No dark-L compression	$\sim$ 22.7	$+0.2$
Rotation $\varphi=-28.2$ °	22.48	0.00

Table 3: Frozen ablation on v21: effect of disabling components without re-optimisation.

Sub-dataset	$N$	v21	CIEDE2000
BFD-P(C)	200	29.08	29.08
BFD-P(D65)	2,028	21.54	24.09
BFD-P(M)	548	21.75	35.23
LEEDS	307	21.84	19.25
RIT-DuPont	312	21.90	19.47
WITT	418	30.93	30.22

Table 4: stress on individual combvd sub-datasets (v21 production checkpoint).

Figure 6: GenSpace v0.11.1 vs OKLab — per-category breakdown across all 90 ColorBench metrics. Headline tally is 65W / 9L / 16T ( Table 5 ). Categories ordered by total metric count.

Category	GenSpace	OKLab	Ties
Gamut geometry	24	0	3
Application	9	0	3
Gradient quality	7	3	1
Independent †	6	1	0
Perceptual	5	0	0
Structural	4	2	2

Table 5: GenSpace v0.11.1 vs OKLab — head-to-head on ColorBench. † Independent = Hung-Berns, Ebner-Fairchild, Pointer-gamut metrics (held out from optimizer). Visual breakdown in Figure 6 .

Figure 7: sRGB gamut in Helmlab $a$ – $b$ plane at three L levels. Points colored by their sRGB values.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Stage	Parameter	Value	Description
$\mathbf{M}_{1}$ (9)	Row 0	[ $-$ 0.7213, $-$ 0.4534, $-$ 0.1929]	XYZ $\to$ cone-like
	Row 1	[ $-$ 0.7882, $-$ 1.7952, $-$ 0.0876]
	Row 2	[ $-$ 0.0918, $-$ 0.4577, $-$ 1.2922]
$\gamma$ (3)	$\gamma_{0},\gamma_{1},\gamma_{2}$	0.4723, 0.5149, 0.5113	Signed power compression
$\mathbf{M}_{2}$ (9)	Row 0	[ $-$ 0.2636, $-$ 0.4168, $-$ 0.4927]	Compressed $\to$ Lab
	Row 1	[ $-$ 1.8898, $-$ 3.1212, $-$ 1.0422]

Appendix

Table 6: MetricSpace v21 parameters (

\varphi=-28.2

°). Trained on combvd + He 2022 + MacAdam 1974 with sub-dataset-balanced loss ( Equation 16 ).

Stage	Value
$\mathbf{M}_{1}$	3 $\times$ 3, see JSON (Bradford CAT pre-baked)
Transfer	depcubic, $\alpha=0.021$
$\mathbf{M}_{2}$	3 $\times$ 3, see JSON
Chroma power	0.978
PW $L$ correction	19-point dense, see JSON
Enrichment	L-gated hue rotation

Appendix

Table 7: GenSpace v0.11.1 parameters.

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Jun 3, 2026J/K move · Enter open · S save

Oklch+: A Three-Parameter Extension of Oklab for Improved Color Difference Prediction

Naoyuki Uchida

Independent Researcher

Open paper AskPDF arXiv

Oklab and its cylindrical representation Oklch are widely adopted in interpolation and design workflows as perceptually motivated color spaces, but their color difference prediction accuracy falls short of CIEDE2000. We propose Oklch+, a three-parameter extension of Oklab comprising a power transformation on the L-axis and a Naka-Rushton compression on the C-axis, with Euclidean distance computed in the resulting transformed Oklab coordinates. The Naka-Rushton function is bounded in [0,1], reflecting the saturating nature of chroma sensitivity at high colorimetric values. Evaluated on COMBVD -- 3,813 suprathreshold color difference pairs spanning six independent experimental datasets -- Oklch+ achieves STRESS = 29.09, closely matching CIEDE2000 (29.13; difference = 0.04), using only three parameters optimized against color difference data compared to approximately 17 for CIEDE2000. Cross-validation on a held-out BFD-P D65 subset (2,028 pairs) confirms generalization (STRESS = 26.14), with Oklch+ substantially outperforming Oklab (51.45) and achieving STRESS comparable to CIEDE2000 (24.12) on the held-out set. Improvement over Oklab (47.35) is confirmed across all six COMBVD sub-datasets. Because Oklch+ defines a coordinate system in which Euclidean distance approximates perceptual distance, linear interpolation in the transformed space offers substantially improved perceptual uniformity relative to Oklab. Current evaluation is limited to the sRGB-centered COMBVD dataset; validation in high-chroma regions with empirical observer-rated discrimination data remains future work.

Abstract

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