# Scientists discover a 100-year-old math error, changing how humans see color

- Researchers identify a key math error in the theory of color perception.
- The theory was developed by famous scientists including Erwin Schrödinger.
- Correcting the error could lead to major improvements in numerous industries.

Scientists have corrected a significant math error in a theory used to describe human color perception for over 100 years. The mathematical paradigm containing the error was first developed by legends in math and physics – Bernhard Riemann, Hermann von Helmholtz, and the Nobel Prize winner Erwin Schrödinger.

Correcting this mistake in the modeling that is commonly utilized to describe how our eyes can tell one color from another could have important ramifications, leading to improvements in televisions, other devices reliant on image and video processing, the textile and paint manufacturers, as well as visualizations of scientific data.

The new paper, published in the Proceedings of the *National Academy of Science*, is the work of the lead author and computer scientist Roxana Bujack and a research team from the *Los Alamos National Lab,* who blended psychology, biology, and mathematics for their study.

In a press release, Bujack, who creates scientific visualizations at Los Alamos National Laboratory, called the current mathematical models used for color perceptions incorrect and requiring a “paradigm shift.”

## A surprise finding

Being able to accurately model human color perception has a tremendous impact on automating image processing, computer graphics, and visualization. Bujack’s team first set out to develop algorithms that would automatically enhance color maps used in data visualization to make it easier to read them.

To come up with a concrete mathematical model of perceived color space, red, green, and blue are plotted in 3D space. That's because these colors are registered most strongly by light-detecting cones in our retinas. These are also the colors that blend together into images of an RGB computer screen.

The team was working on algorithms that would automatically improve color maps used in data visualization, making them easier to understand and interpret.

What the team was surprised to find is that they were the first ones to realize the established practice of applying “Riemannian geometry” to the 3D space didn’t work.

Riemannian geometry is unlike the Euclidian geometry you may be familiar with from school, but as Bujack explained, it “allows generalizing straight lines to curved surfaces.”

Bujack and her team showed that using Riemannian geometry actually results in overestimating how large color differences are perceived.

This happens due to the effect of “diminishing returns” where “large color differences are perceived as less than the sum of small differences,” the scientists wrote in the study.

In other words, a big difference in color is perceived to be less than the sum of small differences in color that lie between two widely separated shades. The researchers demonstrated that this effect cannot be accounted for in a Riemannian geometry.

### What’s next?

When reached by Interesting Engineering (IE) for comment, Bujack explained that it’s hard to know why the error in modeling devised by giants in her field persisted for so long without correction.

“If I had to guess,” shared Bujack, “I would say that maybe the color researchers thought about the Riemannian (curved) space somehow as the ‘opposite’ of the Euclidean (straight) space and ignored that it is a pretty regulated construct itself.”

When asked what kind of geometry their team might use to describe perceptual color space going forward, Bujack said that they are investigating what that looks like.

“If we are lucky, a Riemannian space with a scaling function could do the trick, but more experiments are needed to see if that works,” she added.

Bujack also thinks that “a path-connected metric space would be a good model.”

“But of course, you have to allow for some perceptual 'noise' like in the theory of Thurstone. Without the stochastical component, it would violate the most fundamental metric property: the identity of indiscernibles, i.e., that zero is only returned if both inputs are identical. You can present two very close colors to an observer between which he will not see a difference even though they are not 100% identical,” the scientist explained.

## Potential tech improvements

The scientists believe their work will eventually result in improvements to visualization technologies, including televisions and monitors. But, as Bujack explained to IE, it will take a while to get there.

“Most experimental data on color perception is on very small differences because we thought we could add them up to get the large ones,” she said, adding, “Now we know that there is a lot of work needed to map out the large distances.”

What this leads to is that the scientists would have to “generalize the existing algorithms to run on that space.” And only when that’s achieved will we start seeing more accurate measures of color difference and improvements in almost every kind of image processing technique.

Bujack provided one example: “If we can compute the perceived difference between two images perfectly mathematically, we can adjust the compression rate of videos for streaming to be exactly "this far" off from the ground truth for an observer and save bandwidth.”

The research article "The non-Riemannian nature of perceptual color space" was first published in the April issue of the scientific journal *PNAS*.

**Study Abstract:**

The scientific community generally agrees on the theory, introduced by Riemann and furthered by Helmholtz and Schrödinger, that perceived color space is not Euclidean but rather, a three-dimensional Riemannian space. We show that the principle of diminishing returns applies to human color perception. This means that large color differences cannot be derived by adding a series of small steps, and therefore, perceptual color space cannot be described by a Riemannian geometry. This finding is inconsistent with the current approaches to modeling perceptual color space. Therefore, the assumed shape of color space requires a paradigm shift. Consequences of this apply to color metrics that are currently used in image and video processing, color mapping, and the paint and textile industries. These metrics are valid only for small differences. Rethinking them outside of a Riemannian setting could provide a path to extending them to large differences. This finding further hints at the existence of a second-order Weber–Fechner law describing perceived differences.