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.


[1] .

[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


Tuesday, August 30, 2022

Response to Hue Angles ISCC 498


In the Hue Angles column Spring 2022 issue of the ISCC newsletter, Michael Brill offered a challenge for all his fellow chromo-historians: What is the oldest hack in color engineering? Dr. Brill’s nomination was the normalization of the tristimulus values against the illuminant.


I accept his challenge, and submit not one, but two related hacks, just to distinguish my entry from the hundreds of others who responded to the challenge.


I contend that the tristimulus functions are themselves a hack, well, actually two hacks. And since you can’t normalize tristimulus values until you have tristimulus values, I claim that my hacks are slightly older than the normalization hack proposed by Dr. Brill. 


To explain my proposed hacks, I need to give some thrilling backstory. There is a common misconception that the tristimulus functions (also referred to as the Standard Observer) that we know and love were developed to create color metrics that mimicked how we see color. Nope. Not true. There is another common misconception that the tristimulus functions were the best guess in 1931 as to the spectral response of the human eye. Sorry. That’s not true either. The backstory will explain both of those misconceptions.


Color measurement in the 1920s was considerably different than it is today. Today, we punch a button and a set of color coordinates comes out. In the 1920s, measurements were performed by a device called a tristimulus light mixing colorimeter[1]. The user would meticulously adjust the intensities of a red light source, a green light source, and a blue light source to match a sample. This was a time consuming and painstaking task and the human being was an integral part of the color measurement device. The settings of the three light sources were used as a proxy for the measured value of the color [3].

Conceptual drawing of a tristimulus light mixing colorimeter

The Such devices were difficult to use and had poor reproducibility. Someone came up with the clever idea that a spectrophotometer and gobs of arithmetic could be used to emulate a tristimulus colorimeter. I will tentatively say that this person was Deane Judd. At the very least, he was thinking about this in 1930 [4]. 


The aforementioned “gobs of arithmetic” required the creation of gobs of standardized data in the form of color matching functions. These color matching functions answered the question of “how would a hypothetical user adjust the hypothetical knobs of a hypothetical tristimulus colorimeter to match (for example) 530 nm light?”


Two scientists from England, John Guild and W. David Wright, independently took on the task of creating said gobs of data with the help of a total of 17 volunteers between them. Guild and Wright chose different versions of red, green and blue light sources, so their color matching functions were different. But once the correction was made for this difference, their data agreed reasonably well. The averaged data was massaged a bit and standardized in 1931 by the committee that is now known as the CIE. They called it the Standard Observer.


The idea of using a spectrophotometer and gobs of arithmetic to emulate a tristimulus colorimeter was clever. It was also ad hoc, that is, it served as a quick fix to a problem that was at hand. But it did not address the more general need for a way to emulate the human visual system. Hence, it qualifies as a hack. 


The committee now known as the CIE needed to decide which tristimulus colorimeter to emulate. Virtually any choice of red, green and blue lights would suffice. Golly gee, they could even substitute an exotic violet for the blue or a subtle chartreuse for the green. They decided to go even wilder and standardize on lights made from unobtanium[2]. Good golly gosh and gee willickers, since it’s all just computation anyway, who says the choice of standard stimuli for a standard tristimulus colorimeter even has to be physically possible?!??!? 


Being the wild and crazy guys they were, their choice of lights for the primary lights emitted negative amounts of light at certain wavelengths. Why? Their particular choice minimized the gobs of arithmetic needed to determine the tristimulus values that stood as proxy for color measurements. This is my second proposal for a primordial color engineering hack. It was clever and solved an immediate problem – hand calculation of colorimetric values. But once again, the bigger issue of emulating the eyeball went by the wayside.


The tristimulus functions [5]


I propose Seymour’s Rule of Hackery: Every good hack has an equal and opposite unforeseen consequence. As you would expect, there were unforeseen consequences that the 1931 committee did not foresee when they decided to emulate a long-since extinct color measuring device in a way that would save precious microseconds of computing time on any cell phone. Tune in to the next ISCC newsletter and you will see the unforeseen!



[1] Wyszecki G and Stiles WS, Color Science, 1st ed. New York: Wiley, 1967, p. 279.

[2] Smith and Guild, The CIE colorimetric standards and their use, Trans. Opt. Soc. 33, 102 (1931-32), p. 89

[3] Seymour, John, Why does the a* axis point toward magenta instead of red?, Color Research and Application, 23 July, 2020

[4] Judd, Deane B., Reduction of Data on Mixture of Color Stimuli, Bureau of Standards Journal of Research, Vol 4 (1930)

[5] Judd, Deane B., The 1931 I. C. I. Standard Observer and Coordinate System for Colorimetry, JOSA Vol. 23, Oct. 1933


This article was written by John Seymour, who has to his credit several earlier pieces in the ISCC News and has (more famously) a blog at Due to his immeasurable modesty, John has not signed his name, but I reveal his identity here. MHB

[1] Some of you may recognize the word tristimulus. The prefix tri- means three, and stimulus refers to the three flavors of lights in the device. A bit of foreshadowing for the perceptive reader.

[2] “A common observation about unobtainium is that it meets all requirements perfectly, other than not actually existing.”,other%20than%20not%20actually%20existing.