Scientists Discover 100-Year-Old Math Mistake That Changes How Humans See Colors

Scientists Discover 100-Year-Old Math Mistake That Changes How Humans See Colors

The new article, published in the Proceedings of the National Academy of Sciencesis the work of lead author and computer scientist Roxana Bujack and a research team from Los Alamos National Laboratory, who mixed psychology, biology and mathematics for their study.

In a press release, Bujack, who creates science visualizations at Los Alamos National Laboratory, called the current mathematical models used for color perception incorrect and in need of a “paradigm shift.”

A surprise discovery

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

To provide a concrete mathematical model of the perceived color space, red, green and blue are plotted in 3D space. This is because these colors are registered most strongly by the light-detecting cones in our retinas. These are also the colors that blend in images on an RGB computer screen.

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

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

Riemannian geometry is different from the Euclidean geometry you might know from school, but as Bujack explained, it “allows you to generalize straight lines to curved surfaces.”

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

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

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

And after?

When contacted by Interesting Engineering (IE) for comment, Bujack explained that it is unclear why the modeling error engineered by the giants in her field has persisted for so long without correction.

“If I had to guess,” Bujack shared, “I’d say that maybe color researchers thought of Riemannian (curve) space somehow as ‘the opposite’ of the Euclidean (straight) space and have ignored that it is a fairly regulated construction itself.

When asked what kind of geometry their team might use to describe perceptual color space in the future, Bujack said they are investigating what it looks like.

“If we’re lucky, a Riemannian space with a ladder function might do the trick, but more experiments are needed to see if it works,” she added.

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

“But of course you have to take into account some perceptual ‘noise’ like in Thurstone’s theory. Without the stochastic component, this would violate the most fundamental metric property: the identity of indistinguishables, i.e. zero is only returned if the two inputs are identical. We can present an observer with two very similar colors between which he will not see a difference even if they are not 100% identical,” explains the scientist.

Potential technological improvements

Scientists believe their work will eventually lead to improvements in viewing 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 about very small differences because we thought we could add them together to get the bigger ones,” she said, adding, “Now we know there’s lots of work to do to map the large distances. ”

This leads scientists to “generalize existing algorithms to run on this space”. And it’s not until that’s achieved that we’ll begin to see more accurate color difference measurements and improvements in nearly every type of image processing technique.

Bujack provided an example: “If we can perfectly mathematically calculate the perceived difference between two images, we can adjust the compression rate of videos for streaming so that it is exactly ‘as far’ from the ground truth for an observer and save bandwidth”.

The research paper “The Non-Riemannian Nature of Perceptual Color Space” was first published in the April issue of the scientific journal PNAS.

Summary of the study:

The scientific community generally agrees on the theory, introduced by Riemann and extended 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 color differences cannot be derived by adding a series of small steps and therefore the perceptual color space cannot be described by Riemannian geometry. This finding is incompatible with current approaches to perceptual color space modeling. Therefore, the assumed shape of the color space requires a paradigm shift. The implications 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 only valid for small differences. Rethinking them outside of a Riemannian framework could provide a way to extend them to large differences. This finding suggests the existence of a second-order Weber-Fechner law describing perceived differences.

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