Scientists discover a 100-year-old math error that changes the way humans see color

the new paperpublished in the Chronicle National Academy of Sciencesis the work of lead author and computer scientist Roxana Bojak and a research team of Los Alamos National Laboratory, who mixed psychology, biology and mathematics for their studies.

in press releaseBujack, who created scientific visualizations at Los Alamos National Laboratory, described current mathematical models used for color perception as incorrect and requiring a “paradigm shift.”


The ability to accurately model human color perception has a tremendous impact on the automation of image processing, computer graphics, and visualization. Bujack’s team first set out to develop algorithms that would automatically optimize the color maps used to visualize data to make it easier to read.

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

The team has been working on algorithms that automatically improve the color maps used to visualize data, making them easier to understand and interpret.

What surprised the team discovered was that they were the first to realize that the established practice of applying “Riemannian geometry” to 3D space did not work.

Riemannian geometry is different from Euclidean geometry that you may be familiar with from school, but as Bojak explained, “it allows for the generalization of straight lines over curved surfaces.”

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

This is caused by the effect of diminishing returns, the scientists wrote, in which “large differences in color are perceived as less than the sum of small differences.” In the study.

In other words, a large difference in color is viewed as less than the sum of the small differences in color that lie between two widely spaced shades. The researchers showed that this effect could not be explained in Riemannian geometry.

What’s Next?

When reached out to Interesting Engineering (IE) for comment, Bujack explained that it’s hard to see why the error in modeling created by the giants in their field has persisted for so long uncorrected.

Bujack shared: “If I had to guess, I would say that perhaps color researchers thought of Riemannian (curved) space in some way as ‘the opposite’ of Euclidean (straight) space and ignored that it is a very structured construct in and of itself.”

When asked what kind of geometry their team might use to describe the perceptual color space going forward, Bujack said they’re studying what that might look like.

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

Bujack also believes that “a metric space connected to a track would be a good model.”

“But of course, you have to allow some perceptual ‘noise’ as in Thurston’s theorem. Without the random component, it violates a basic scaling property: the identity of the indistinguishable elements, i.e. zero is returned only if both inputs are identical. You can render two very close colors Any observer would not see any difference between them even though they are not 100% identical,” the scientist explained.

Possible technical improvements

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

“Most of the experimental data on color perception is based on very small differences because we thought we could collect them to get the big ones,” she said, adding, “We now know that there is a lot of work required to map large distances.”

What this leads to is that scientists will have to “generalize existing algorithms to run on that space.” Only when that is achieved will we begin to see more accurate measurements of color difference and improvements in almost every type of image processing technology.

Bujack gave one example: “If we can calculate the observed difference between two images quite mathematically, we can adjust the compression rate of the broadcast videos to be exactly ‘away’ from the ground truth of the 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 summary:

The scientific community generally agrees with the theory, advanced by Riemann and reinforced by Helmholtz and Schrödinger, that the 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 differences in color cannot be derived by adding a series of small steps, and therefore, the perceptual color space cannot be described by Riemann geometry. This finding is incompatible with current methods of perceptual color space modeling. Therefore, the assumed shape of the color space requires a paradigm shift. The consequences of this apply to color scales currently used in the image and video processing, color mapping, paint and textile industries. These scales are only valid for small differences. Rethinking it outside of the Riemann environment could provide a path to extend it to significant differences. This result indicates the existence of a second-order Weber-Fechner law describing perceived differences.