Saturday, February 4, 2023

The Seven Pillars of SI Wisdom

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Early metrology is sometimes exemplified by the King’s foot, a convention established in the 12th Century by Henry I of England. The most obvious problem with the King’s foot was the need to recalibrate when the regime changed.  The underlying problem was more subtle. Any anthropocentric metric—by which I mean a metric based on human attributes or performance--- is both statistically and conceptually more fragile than one based on universal constants of nature.


Nowadays (  we start with seven precisely related constants of nature and derive from them seven reference units which are said to be all we’ll ever need.   The constants are cesium hyperfine frequency ΔνCs , Planck’s constant h, the speed of light in vacuum c, the elementary charge e, Boltzmann’s constant k, Avogadro’s number NA , and the luminous efficacy of a defined visible radiation K.  The basic units are mass (kilogram), distance (meter), time (second), amount of matter (mole), electric current (Ampere), temperature (kelvin degree), and luminous intensity (candela). The whole system, called SI, comprises what we might call the seven pillars of SI wisdom.  The system seems to have no anthropocentrism.   


But wait! The seventh constant and the seventh unit are not like the others. This seventh pillar depends not only on humanity in general, but on particular observers whose flicker sensitivity and brightness data the CIE aggregated to define the 1924 luminous efficiency function V) in visible wavelength λ .


The history of the candela in Wyszecki and Stiles (Color Science, 2nd ed, Wiley 1982; pp. 254-255) is quite educational.  Rather than use the whole 1924 V(λ) curve (which was to be obsoleted and conditionalized a lot in the next century), standards bodies defined the candela with only two human-related numbers: the peak wavelength (555 nm) of V(λ) and the watt-to-lumen ratio (1/683).  Interestingly, the SI does not define the candela for any light other than monochromatic at 555 nm, so, for example, I cannot ask SI what the candela count is for a given wattage of light at 460 nm.  This illustrates that any reduction of the candela’s dependence on human vision decreases the universality of SI.


The candela didn’t enter the SI system uncontested.  A sign of the struggle was that for many years the US National Bureau of Standards (NBS) divorced itself from all human factors including metrology of vision and other senses.  When NBS deflected responsibility for calibration of color-measurement instruments, the need for such calibration was satisfied by private companies such as Hemmendinger Color Lab.  Fortunately, NBS (now NIST) takes on metrology of a more human sort, so they’re helping to manage the seventh pillar.


We’ve come a long way in the standardization of fundamental constants and their units.  But one of the seven basic units of SI, the candela, is tied to a human-based standard.  Even in the newest refinement of SI, a vestige of the King’s foot remains!


That the SI metrologists felt forced to include a human-vision metric in one of its seven pillars reminds us of the importance of vision in our understanding of the universe. A question to ponder: Of all the five senses, why was vision salient? 


Michael H. Brill

Retired Color Scientist




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