"What we conclude is that these color qualities don't emerge from additional external constructs such as cultural or learned experiences but reflect the intrinsic properties of the color metric itself," Bujack said. "This metric geometrically encodes the perceived color distance -- that is, how different two colors appear to an observer."

By firmly defining these perceptual features, the researchers supply a crucial missing component that helps fulfill Schrödinger's original goal of creating a self-contained model. In that vision, hue, saturation and lightness would be determined entirely by geometry and the principle of greatest color similarity.

The Geometry Behind Hue, Saturation and Lightness

Human color vision depends on three types of cone cells in the eye, sensitive to red, blue and green light. Because of this, scientists represent color in three dimensions known as color spaces. In the 19th century, mathematician Bernhard Riemann proposed that perceptual spaces could be curved rather than flat. Building on that idea in the 1920s, Schrödinger described hue, saturation and lightness using a mathematical measurement system within this curved framework.

For decades, Schrödinger's definitions shaped scientific understanding of color. However, while developing algorithms for scientific visualization, the Los Alamos team discovered weaknesses in the mathematical foundation of the model. Those gaps opened the door to refining and strengthening the theory.

Defining the Neutral Axis and Fixing Color Theory

A key issue centered on the neutral axis, the line of gray tones that runs from black to white. Schrödinger's definitions rely on how colors are positioned relative to this axis, yet he never mathematically defined it. Without that definition, the structure of the model lacks formal grounding: Without a defined neutral axis, the construction is formally undefined.

One of the team's most important achievements was establishing the neutral axis purely from the geometry of the color metric. Accomplishing this required moving beyond the traditional Riemannian framework, marking a significant advance in the mathematics used for visualization science.

The researchers also corrected two additional problems. They addressed the Bezold- Brücke effect, in which increasing brightness can make a color appear to shift in hue. Instead of assuming colors change along straight lines, they calculated the shortest path within the geometric space. The same shortest-path approach in a non-Riemannian space helped account for diminishing returns in color perception, where increasing differences between colors become less noticeable over time.

Advancing Visualization Science and Real-World Applications

The work, presented at the Eurographics Conference on Visualization, represents the culmination of a broader color perception project that also produced a landmark 2022 paper in the Proceedings of the National Academy of Sciences.

Accurate models of color perception are vital for visualization science, which supports fields ranging from photography and video to advanced data analysis. Clear and reliable color modeling improves how scientists interpret complex datasets and build simulations, including those used in national security research. By establishing a stronger mathematical basis for color in non-Riemannian space, the team has laid the foundation for future advances in visualization technology.

Funding: This work was supported by the Laboratory Directed Research and Development program at Los Alamos and by the National Nuclear Security Administration's Advanced Simulation and Computing program.