Erwin Schrödinger’s Math Error

 ( Send contributions to mbrill@datacolor.com ) 

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 Schrödinger 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] https://discover.lanl.gov/news/0810-color-perception .

[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

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!

 

References:

[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 johnthemathguy.blogspot.com. 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.”

https://www.techtarget.com/whatis/definition/unobtainium#:~:text=Unobtainium%20is%20a%20term%20used,other%20than%20not%20actually%20existing.

 

The Oldest Hack in Color Engineering

  (Send contributions to mbrill@datacolor.com)

 

When I was at MIT, I learned what it meant to call something “a hack.” The term referred to a patch in software (or hardware) that was ad hoc but clever and fixed a problem. There are many other definitions of “hack” (the noun), but I don’t use those here. To me, “hack” is not necessarily pejorative, but descriptive.

 

In color engineering there have been many hacks, originating even before the term “color engineering” was popular. Two rather good ones are Cal McCamy’s approximation for correlated color temperature of a light given its chromaticity [1] and the CIELAB L* function as an approximate inversion of a fifth-degree polynomial that characterized the Munsell Renotation System lightness scale [2].

 

Just for fun, I now ask you: What is the oldest hack in color engineering? One could nominate Newton’s representation of color as a closed circle. Newton must have been aware that the purples are not elementary colors that he could see with his prisms. But Newton and others may have adopted the circle as an instructive idealization, and that is not a hack.

 

To get the ball rolling (I expect hundreds of reader responses), I nominate the object-color tristimulus value (X, Y or Z, for any illuminant and observer you like). This quantity is defined as a specific ratio (see ASTM E-308, any edition). The numerator is the wavelength integral of the product of illuminant power density, reflectance and color-matching function (generally x-bar, y-bar or z-bar). The denominator is the wavelength integral of the product of the illuminant power density and the function y-bar. There’s also a factor of 100, but that doesn’t matter.

 

What problem does this definition cleverly solve? In a single stroke, it renders the object-color tristimulus value dimensionless and independent of the absolute light intensity, as is its cousin – the emissive-mode tristimulus value. (This latter fact emerges from the grounding of emissive-mode tristimulus values on the emissive-mode color-matching experiments that underlie basic colorimetry. In a short paper I just submitted to Color Research and Application, I explain how the titration in a color match leads to cancelation of all dimensions.) Creating the object-mode ratio makes the tristimulus values from reflected lights appear comparable to the values from emitted lights and emissive displays. Certainly color-matching equivalence classes of reflected lights are not disturbed by the ratio. But, as many color-management experts have warned, we should be careful in asserting exact colorimetric color reproduction between emissive and reflective media. For one thing, only the reflective tristimulus space has an unambiguous white point.

 

How is the object-color tristimulus value a hack? It is ad hoc, solves the units problem, and is clever enough to survive generations of standards bodies like the ASTM and the CIE. The ad hoc quality manifests when we observe that the object-color tristimulus values have a distinct and asymmetric dependence on Y wrought by the denominator, yet that asymmetry was not based on color matching (as should befit a true tristimulus value). Furthermore, as the current object-color tristimulus value is a ratio between two spectrum integrals, it will have a weird illuminant-invariance such as I have found for Von Kries adapted tristimulus values [3] and band ratios [4]. If the illuminant is restricted to a finite linear function space, changing its coefficients will not alter the object-color tristimulus value if the reflectance is outside a forbidden function subspace. There’s no room for such games in the emissive-mode tristimulus values.

 

So that is why I consider the object-color tristimulus value as a hack. My only remaining question is, how old is it? Wyszecki and Stiles [5] point to the dawn of the 1931 CIE system of colorimetry. I have not investigated further, but it’s entirely possible that this hack precedes not only the term “color engineering,” but also the name “tristimulus value” itself. I’ll leave that subject to serious historians in our ranks.

 

[1] McCamy CS, Color Res Appl 17 (1992), 142-144 (with erratum in Color Res Appl 18 (1993), 150.

[2] Newhall SM, Nickerson D and Judd DB, Final report of the OSA subcommittee on spacing of the Munsell colors, J Opt Soc Am 33 (1943), 385-418.

[3] Brill MH, Minimal Von-Kries illuminant invariance, Color Res Appl 33 (2008), 320-323.

[4] Brill MH, "Can color-space transformation improve color computations other than von Kries?" in:  Human Vision, Visual Processing, and Digital Display IV, J. P. Allebach and B. E. Rogowitz, Editors, Proc. SPIE 1913,485-494 (1993).

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

 

 

Michael H. Brill

Datacolor

 

 

My Big Win in Vegas!

  (Send contributions to mbrill@datacolor.com)

When I find myself in a casino, it’s always because of a professional meeting. My reaction to these casino visits is perhaps predictable.

 

In 1979, I attended the Illuminating Engineering Society meeting in Atlantic City. There I dropped a coin in a slot machine, and then became so absorbed in a conversation with Bill Thornton that I was oblivious when a red carpet was unrolled behind me and Sammy Davis, Jr. walked from one end to the other.

 

In 1985, I attended the quadrennial AIC meeting in Monte Carlo. There I sacrificed another coin to a one-armed bandit, and then became so absorbed in a conversation with Claude Pelissier that I didn’t see the sights all around me on the “topless” beach.

 

In 1992, a field trip after an Acoustical Society meeting in Salt Lake City lodged me at an inexpensive casino/hotel in Mesquite, NV. A somewhat larger sacrifice went to a slot machine there. I was so impressed by the casino’s use of light and color to disorient patrons so they would gamble, that I presented a summary of the ruse at an ISCC Interest Group III panel discussion (see ISCC News Issue 340). Nobody told me what glamorous opportunity I missed in Mesquite, but I must have missed something because I got lost on the way back to my table from the restaurant salad bar due to the evil genius of the casino architects.

 

In 2005, I returned to Nevada, this time to an ASTM E12 meeting in Reno. By now my obligatory slot sacrifice swelled to five dollars. I’m reasonably sure I didn’t miss any glamorous sights, and I was aware that, in all my experiences with casinos, I hadn’t retrieved a penny from the slot machines.

 

But then, in September 2021, I visited Las Vegas and had my Big Win. The meeting this time (American Urology Association) was not mine but my partner Karen’s. After she made travel arrangements with me in tow, the AUA thought it prudent to make the meeting virtual. Karen was determined not to change our plans, so we checked into the Venetian, the conference hotel. (It was on the opposite end of the price scale from the hotel I’d stayed at in Mesquite in 1992.) Karen attended the virtual meeting from our hotel room. To me it was a bit eerie, like a séance with ghosts from an alternate universe. Meanwhile, I explored the hotel/casino complex. True to my experience in Mesquite, I became repeatedly lost on a grand scale. One small but effective confusion was to call two of the three hotels in the complex “Venetian” and “Venezia.” The architecture and a sinuous indoor canal spoke “Venice,” but straight paths and right angles were rare, and the site map had little visual correspondence to the site itself. Even employees of some of the concessions could not describe how to get to other places under the same roof.

 

Later, we embarked on an evening tour on a double-decker bus, and I had an opportunity to meet my obligatory slot machine. The tour guide gave us an hour in the environs of the Golden Nugget Casino. Stepping inside, I found that the only acceptable way to sit down for an hour was to gamble. Karen slid a $20 bill into a machine, and we managed to keep busy for more than a half hour. Her goal was to get as much run-time as possible out of that investment. She found that strategy was great fun when she visited Las Vegas with her mother several years earlier. When I slid my $20 bill into the same machine, I had a different goal. After a while I had won almost ten dollars, whereupon I decided to cash out. A little slip of paper appeared, and I grabbed it and asked the nearest bystanders where I could find the cashier prior to the imminent departure of our bus. Peering with cow-like eyes, they told me they didn’t know. How could anyone gamble without a plan for even the first two minutes that would ensue if they actually won? It was sad.

 

Out of time, I returned to the bus and read the paper slip from the machine: Void after 30 days! I’d have to come back to the Golden Nugget later in my trip. But I never did.

 

So that was how I scored my Big Win in Vegas. Well, maybe a Bigger Win was to have made the trip despite some significant health issues and with the overarching Covid risk. Some people thought I was crazy to do it. Maybe it’s good to indulge such craziness just once amid the larger-than-life habitués of Vegas.

 

Michael H. Brill

Datacolor