## Thursday, November 3, 2022

### Erwin Schrödinger’s Math Error

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In mid-August, an article from Los Alamos National Laboratory (LANL) News was brought to my attention [1]. The title was provocative: “Math error: A new study overturns 100-year-old understanding of color perception.” The error—made by Erwin Schdinger in 1920 but actually going back to Bernhard Riemann in 1854—was to model color perception as a 3D curved space (called a Riemannian space) in which distance along special curves, called geodesics, represents perceived color difference. The article from LANL News called it a math error—exhilarating to discover among the works of the greats after more than a century. The article cited a research paper in the Proceedings of the National Academy of Sciences (PNAS) and was based on work at LANL [2]. The LANL authors were declaring that their work should inspire a paradigm shift in color science.

I was curious enough to get the PNAS paper. Unsurprisingly, what they called the “math error” was a counterfactual assumption and not a mistake in the algebra. Further, LANL had not proposed an alternative model, and a paradigm shift requires a new as well as an old paradigm. So there’s no paradigm shift yet.

To understand more requires a bit more about Riemannian space. Picture the surface of a sphere. Draw a point on the sphere, and a little circle around the point. On a Euclidean plane the ratio of the circumference to the diameter of that circle would be π, but on the sphere it is less than π (Figure 1), because the surface of the sphere is a Riemannian space of 2 dimensions. A geodesic between two points A and B on a sphere is the big circle on the sphere that is in the plane containing the sphere’s center. The arc of shortest distance d(AB) between A and B on the sphere is on that big circle (Figure 2).

Figure 1 – The distance from a point to the circle, if constrained to be on the sphere, is always greater than the straight-line distance through the interior of the sphere.

Figure 2 – The black arc shows the shortest path between two points

Given this background, here is the logic of the paper:  In a Riemannian space, if 3 points A, B, C are on a geodesic with B between A and C, then their distances d have additivity: d(AB) + d(BC) = d(AC). To test this additivity, the authors first assumed that the neutral colors comprise a geodesic in Riemannian color space, Then they showed experimentally that, for widely separated neutral colors, d(AB) + d(BC) is greater than d(AC). Therefore, colors can’t form a Riemannian space.

The person who brought the LANL News article to my attention wanted to know if the PNAS research paper would lead to a paradigm shift with industrial color implications. A glance at the history of such errors is enough to make a fair prediction.

Consider the basic laws of color matching—Grassmann’s laws. The error there is in the assumption that matches are transitive: If A matches B and B matches C, then A must match C. But in the real world, “A matches B” means “A is within a just-noticeable color difference of B.” So Grassmann was wrong. Yet the New York Stock Exchange was unaffected. In fact, I am not aware of any effect on color-matching protocols.

Next, consider the Euclidean color space (a special case of Riemannian in which the geodesics are straight lines and distance is the square root of the sum of squares of coordinate differences). Euclidean color spaces have existed for more than 150 years. The earliest may be Helmholtz’s color space (which was Euclidean in log RGB coordinates). A typical one is CIELAB (in nonlinear coordinates relative to XYZ). The latest may be DIN99o, the current version of the German standard DIN99, which was published in 2018. Euclidean spaces persist even though their incorrectness was noted by Ludwig Silberstein in 1943 [3].

For CIELAB, the lack of perceptual uniformity was not totally ignored. In response to the long-obvious fact that CIELAB’s Euclidean distance doesn’t track color differences, the standards bodies detoured around the impressive and difficult Riemannian alternative and took a new approach: they built color-difference models using the underlying CIELAB coordinates, but wrote color differences using creative combinations of CIELAB quantities. Examples are CMC, CIE94, and CIE2000—all applicable for small color differences. (Large color differences were left to a few adventurers.) But none of this activity could be called a paradigm shift.

In view of these examples, let’s look at the history of Riemannian color spaces that are not Euclidean. The earliest may be Schroedinger’s in 1920 (missed being Euclidean by “tha…at much”), and the latest may be in a September article [4] which describes the post-CIELAB Riemannian choice as the road rarely taken. That observation in itself denies Riemannian color space the status of “the current scientific paradigm” as asserted in the PNAS paper. The vaunted paradigm shift, then, has neither a “before” nor an “after” paradigm.

And the LANL result has an ironic twist: Its principle of diminishing returns teaches us that we should care less rather than more about evaluating very large color differences, because large color differences matter less than we had assumed. “Relax…it’s no big deal,” it says.

So a paradigm shift is not imminent. But the buzz generated by the LANL article might give The Dress a run for its money!

Thank you to John Seymour for creating the illustrations for this article.

[2] Bujack R, Teti E, Miller J, Caffrey EJ, and Turton TL, The non-Riemannian nature of perceptual color space. Proc Nat Acad Sci, vol 119, No. 18 (2022)

[3] Silberstein L, Investigations on the intrinsic properties of the color domain. J Opt Soc Am 33 (1943), 385-418.

[4] Candry P,  De Visschere P, Neyts K. Line element for the perception of color. Optics Express30(20) 36307-36331(2022).

Michael H. Brill

Datacolor