A new study corrects an important error in the 3D mathematical space

Math error!


For over 100 y, the scientific community has adhered to a paradigm introduced by Riemann and furthered by Helmholtz and Schrodinger, where perceptual color space is a three-dimensional Riemannian space. This implies that the distance between two colors is the length of the shortest path that connects them.

A new study corrects an important error in the 3D mathematical space. It could potentially boost scientific data visualizations, improve TVs and recalibrate the textile and paint industries.

Roxana Bujack, a computer scientist with a background in mathematics who creates scientific visualizations at Los Alamos National Laboratory, said, “Our research shows that the current mathematical model of how the eye perceives color differences is incorrect. That model was suggested by Bernhard Riemann and developed by Hermann von Helmholtz and Erwin Schrödinger—all giants in mathematics and physics—and proving one of them wrong is pretty much the dream of a scientist.”

Automation of image processing, computer graphics, and visualization activities is possible by modeling human color perception.

Bujack said, “Our original idea was to develop algorithms to improve color maps for data visualization automatically, to make them easier to understand and interpret.”

Scientists were surprised to discover- they were the first to determine the longstanding application of Riemannian geometry. This allows generalizing straight lines to curved surfaces, which didn’t work.

A detailed mathematical model of the seen color space is required to develop industry standards. Early attempts made use of the familiar geometry taught in many high schools, Euclidean spaces; more sophisticated models used Riemannian geometry. The models plot red, green, and blue in the 3D space. The colors that blend to generate all the images on your RGB computer screen are those that are most strongly detected by the light-detecting cones in our retinas.

In the study, which blends psychology, biology, and mathematics, scientists discovered that using Riemannian geometry overestimates the perception of significant color differences.

That’s because people perceive a big difference in color to be less than the sum you would get if you added up small color differences that lie between two widely separated shades.

Bujack said“We didn’t expect this, and we don’t know the exact geometry of this new color space yet. We might be able to think of it normally but with an added dampening or weighing function that pulls long distances in, making them shorter. But we can’t prove it yet.”

Journal Reference:

  1. Roxana Bujack et al, The non-Riemannian nature of perceptual color space, Proceedings of the National Academy of Sciences (2022). DOI: 10.1073/pnas.2119753119
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