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A Metric for Three-Dimensional Color Discrimination Derived from V1 Population Fisher Information

Michael Menke

Abstract

We derive a Riemannian metric on three-dimensional color space from the Fisher information of neural population codes in the visual pathway. Photoreceptor adaptation, retinal opponent channels, and cortical population encoding each map onto a geometric construction, producing a metric tensor whose components correspond to measurable neural quantities. The resulting 17-parameter model is fitted jointly to four independent threshold datasets: MacAdam's (1942) chromaticity ellipses, the Koenderink et al. (2026) three-dimensional ellipsoids, Wright's (1941) wavelength discrimination function, and the Huang et al. (2012) threshold color difference ellipses, covering 96 independently measured discrimination conditions across varied chromaticities and luminances. The joint fit achieves STRESS of 23.9 on MacAdam, 20.8 on Koenderink et al., 30.1 on Wright, and 30.8 on Huang et al.

A Metric for Three-Dimensional Color Discrimination Derived from V1 Population Fisher Information

Abstract

We derive a Riemannian metric on three-dimensional color space from the Fisher information of neural population codes in the visual pathway. Photoreceptor adaptation, retinal opponent channels, and cortical population encoding each map onto a geometric construction, producing a metric tensor whose components correspond to measurable neural quantities. The resulting 17-parameter model is fitted jointly to four independent threshold datasets: MacAdam's (1942) chromaticity ellipses, the Koenderink et al. (2026) three-dimensional ellipsoids, Wright's (1941) wavelength discrimination function, and the Huang et al. (2012) threshold color difference ellipses, covering 96 independently measured discrimination conditions across varied chromaticities and luminances. The joint fit achieves STRESS of 23.9 on MacAdam, 20.8 on Koenderink et al., 30.1 on Wright, and 30.8 on Huang et al.

Paper Structure

This paper contains 27 sections, 2 theorems, 48 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Photoreceptor Adaptation
  3. Cone excitations
  4. Von Kries adaptation
  5. Opponent Channel Formation
  6. Parvocellular coordinate
  7. Koniocellular coordinate
  8. Cortical Population Fisher Information
  9. Fisher information for a single Poisson neuron
  10. From Fisher information to discrimination ellipsoids
  11. Directional tuning and the rank-1 structure
  12. Population additivity
  13. Three V1 sub-populations
  14. Luminance scaling
  15. Cross-channel normalization
  16. ...and 12 more sections

Key Result

Proposition 3.1

The coordinate $z$ depends only on chromaticity, not on luminance. That is, changing $Y$ at fixed $(x,y)$ leaves $z$ unchanged.

Figures (3)

  • Figure 1: Huang et al. (2012) threshold ellipses: observed (blue, solid) versus predicted (red, dashed) at $5\times$ magnification in CIE $(x,y)$ coordinates, after transforming the original CIELAB ellipses.
  • Figure 2: Koenderink et al. (2026) discrimination ellipsoids projected onto three pairs of sRGB coordinates: observed (blue, solid) versus predicted (red, dashed) at $2.5\times$ magnification. A mostly non-overlapping subset of ellipsoids is shown in each panel (number indicated in subtitle); the STRESS-optimal scale factor is computed from all 35 ellipsoids. Centre dots are coloured by sRGB value. STRESS = 20.8.
  • Figure 3: MacAdam (1942) discrimination ellipses: observed (blue, solid) versus predicted (red, dashed) at $10\times$ magnification. The predicted ellipses use the STRESS-optimal global scale factor $F$. The model captures ellipse orientations and aspect ratios across the chromaticity diagram. STRESS = 23.9.

Theorems & Definitions (5)

  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 4.1: Positive definiteness
  • proof