A team led by Los Alamos scientist Roxana Bujack used geometry to build a mathematical definition of color perception based on hue, saturation, and lightness. Their results, presented at a visualization science conference, formalize Schrödinger's model of color and show that these familiar color qualities are built into the structure of color perception itself.

"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."

Completing Schrödinger's Color Puzzle

By defining these perceptual attributes more rigorously, the researchers have supplied a missing piece in Schrödinger's long standing vision for a closed mathematical model of color. The goal was to define hue, saturation, and lightness using only the geometric property of highest color similarity.

Human color vision is based on three types of cone cells, which are centered around red, blue, and green. That gives color spaces three dimensions, allowing scientists to organize and compare colors mathematically.

In the 19th century, mathematician Bernhard Riemann proposed that perceptual color spaces are not flat or straight, but curved. In the 1920s, Schrödinger built on that idea by defining hue, saturation, and lightness within a Riemannian model of color perception, using a metric that describes how people perceive color differences.

Fixing a Century Old Mathematical Gap

Schrödinger's definitions have shaped color science for roughly 100 years. But while the Los Alamos team was developing algorithms for scientific visualization, they found that the mathematics behind the model had important weaknesses.

The biggest problem involved the neutral axis, the line of grays that runs from black to white. Schrödinger's definitions of hue, saturation, and lightness depend on where a color sits in relation to that axis, yet he never formally defined the axis itself.

That omission created a serious gap. Without a precise definition of the neutral axis, the entire construction was formally incomplete. The team's most important advance was finding a way to define the neutral axis using only the geometry of the color metric.

To accomplish that, the researchers had to move beyond the traditional Riemannian model. That shift represents a major mathematical advance for visualization science.

A Better Model of How Colors Change

The team also corrected two other important issues in the older framework.

One involved the Bezold- Brücke effect, a phenomenon in which changing light intensity can make a color appear to shift in hue. The researchers addressed this by using the shortest path in their geometric model of color perception rather than relying on a simple straight line.

They also used the shortest path in a non-Riemannian space to account for diminishing returns in color perception, another effect that had not been fully captured by the older approach.

Why Color Perception Matters

The research was presented at the Eurographics Conference on Visualization and builds on a broader Los Alamos project on color perception. That project also produced a groundbreaking 2022 paper in the Proceedings of the National Academy of Sciences.

A more precise model of color perception could have wide value in fields that depend on accurate color, including photography, video, visualization, and related technologies. It could also improve the way scientists create and interpret visual data.

Scientific visualization plays an important role in helping researchers understand complex information. Better color models can support more effective analysis across many areas, including national security sciences.

The team's work now provides a foundation for future color modeling in non-Riemannian space.

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.