Tuesday, November 27, 2018

Visualizing Four-Dimensional Colorimetry

This essay is a return to the now-popular topic of tetrachromacy (four-color vision [1]), but with a geometric flavor that responds to a challenge by Jan Koenderink:

 “I hold the view that it is not possible to understand human color vision completely without having appreciated the tetrachromatic (or polychromatic of any order) embedding. I consider it to be somewhat of a scandal that the literature has so little to offer there.” [2, p. 200]

Since the publication of Koenderink’s book, as if on cue, researchers confirmed one example of a human tetrachromat [3,4].  So it is timely to start to answer Koenderink’s challenge, comparing trichromacy with tetrachromacy and including some observations that he himself made.

In three color dimensions (e.g., CIE space), it is common to project out one of the dimensions, to produce a 2D chromaticity space.  The physical-light domain in this space is a planar region delimited by the spectrum locus and the line of purples: i.e., the familiar horseshoe diagram.  Inside the diagram, let’s denote a white point W and an arbitrary color A.  If you draw a line from A through W, any points encountered thereafter are complementary to A.  If you draw the line in the other direction from W through A, you will eventually meet the spectrum locus or the line of purples. If the meeting is with the spectrum locus, the meeting point is called A’s dominant wavelength. If the meeting is with the line of purples, then A has no dominant wavelength.

In four color dimensions, the chromaticity domain has 3 dimensions—still accessible to our spatial visualization, perhaps with some difficulty. The spectrum locus is still a curve, but it is a space curve that spans all three chromaticity dimensions---like a bent wire hanger.  Now imagine the wire hanger shrink-wrapped by a plastic sheet.* Every point within the shrink-wrap can represent a physical light. Now, as in CIE space, denote a white point W and an arbitrary point A within the shrink-wrap.  Again you can draw a line between A and W.  Extending on the W side, the points are legitimate complements of A: they add in certain proportions to give W. Because of the abundance of shrink-wrap area relative to wire-hanger area (theoretically zero), it will come as no surprise that the line will likely end at a shrink-wrap point and not at a wire-hanger point: light A probably has no spectral complement.  Extending the line on the A side, one similarly encounters shrink wrap and not wire hanger. That denies the existence of a dominant wavelength for the vast majority of colors. The selection of possible outcomes is the same for 3D as for 4D colorimetry, but the odds for each outcome are staggeringly different.

You can also use the picture of wire hanger and shrink wrap in understanding the optimal color reflectances in tetrachromatic color spaces.

In trichromacy, it has been shown many times that the optimal reflectances (on the exterior of the object-color solid in tristimulus space) have values 1 or 0 at each wavelength, with at most two transitions between 0 and 1.  To derive the number of transitions [5] one can notionally slice the chromaticity space with a line and define the optimal reflectance transitions as the points on the spectrum locus that were impinged by the slice.  The optimality follows from the argument that 1’s inhabit all of the curve’s wavelengths on one side of the slicing line and 0’s occupy the other side---you can’t do better than that.  In trichromatic space, by the way, the number of transitions is two, if the spectrum locus is convex (which it mostly is).

One can use the same trick in tetrachromatic space, but now one is slicing a 3D chromaticity space with a plane. The number of intersections of the slicing plane with the spectrum locus (wire hanger) is the number of 1-0 transition wavelengths.   It now remains to find the tetrachromat’s analogue to convexity of the spectrum locus and use it to minimize the maximum number of crossovers.   One could begin by asserting that the spectrum-locus curve must span the three dimensions of the chromaticity space, and posit as an axiom that no plane can cross the spectrum locus more than 3 times. 

In performing this exercise, I am realizing that, whereas one can use either chromaticity or tristimulus space to visualize basic colorimetry for trichromats, the chromaticity domain is essential for visualizing tetrachromatic relations.
*I intend shrink-wrap as a metaphor for the boundary of the 3D convex hull of the spectrum locus. Real shrink-wrap will sometimes incur concavity, so my metaphor is imperfect.
[1] Jennings C. All the colors we cannot see: tetrachromacy in humans. ISCC News Issue 482 (Spring 2018), pp. 13-14.

[2] Koenderink J. Color for the Sciences. Cambridge, MA: MIT Press, 2010, Section 5.10.3.

[3] Jordan G, Deeb S, Bosten J, Mollon JD. The dimensionality of color vision in carriers of anomalous trichromacy. J of Vision 10 (2010), p. 12.

[4] Jameson KA, Winkler AD, Goldfarb K. Art, interpersonal comparisons of color experience, and potential tetrachromacy. Invited proceedings paper for the 2016 IS&T International Symposium on electronic Imaging (EI 2016). Technical session on Human Vision and Electronic Imaging.

 [5] West G, Brill MH, Conditions under which Schrödinger object colors are optimal, J. Opt. Soc. Am. 73, 1223-1225 (1983).

Michael H. Brill

Thursday, August 30, 2018

The YouTube Theory of Colour Vision

When writing for a general audience, vision scientists often refer to the long-, middle- and short-wavelength cone classes (L, M and S) as red, green and blue cones respectively. While this simplification may seem harmless it unfortunately has been the starting point for a cascade of misunderstandings about human colour vision. To begin with it reinforces the assumption that hues are properties residing in wavelengths of light, and then understandably leads to the assumption that the three cone types individually detect red, green and blue hues/wavelengths. Together these assumptions lead to the conclusion commonly encountered in discussions of colour vision on social media that we “only really see three colours”. In turn this conclusion has teamed up with the homunculus fallacy to spawn a model of colour vision in which the cone cells send hue signals directly to an observing brain. When the brain receives a combination of cone signals that could be produced by a “real” colour in the spectrum it “thinks it sees” that colour. This model bears little resemblance to current science but has achieved the status of orthodoxy on several online platforms.

Expositions of this model can be found in four YouTube videos recorded as having from nearly half a million to more than 18 million views: “This Is Not Yellow” (Michael Stevens, September 2012 [1]), “How We See Color” (Colm Kelleher, TED-Ed, January 2013 [2]), “Colour Mixing: The Mystery of Magenta” (Steve Mould, The Royal Institution of Great Britain, February 2013 [3]) and “Does This Look White to You?” (Dianna Cowern, October 2015 [4]). Kelleher’s version is typical: “There are three kinds of cone cells that roughly correspond to the colors red, green, and blue. When you see a color, each cone sends its own distinct signal to your brain. For example, suppose that yellow light, that is real yellow light, with a yellow frequency, is shining on your eye. You don't have a cone specifically for detecting yellow, but yellow is kind of close to green and also kind of close to red, so both the red and green cones get activated, and each sends a signal to your brain saying so.”

Based on the assumptions that hues reside in the wavelengths of the spectrum and that the function of colour vision is to detect these hues/wavelengths of monochromatic light, when we “think we see” yellow while looking at a mixture of long and middle wavelengths our brain is being “tricked” or ”lied to”; this mixture of wavelengths is only “fake yellow” [1].

In each video the verbal explanation implies that the “red and green cones” respond most strongly to “red and green wavelengths” respectively, and that there is no cone responding most strongly to “yellow wavelengths”, even when an accompanying diagram shows correctly that the “real yellow” wavelength lies near the peak of the “red cone” response [1,4]. Stevens adds the additional misconception that bright yellow objects such as lemons reflect only “real yellow” wavelengths (they in fact reflect strongly most of the long and middle wavelengths of the spectrum).

Mould [4] adds the novel idea that when the brain receives a combination of “red cone” and “blue cone” signals that could not be produced by a single wavelength it “makes up” a colour, magenta. The view that magenta alone is a “made up colour”, which is now also popular on the internet, reflects the assumption that the spectral hues are not “made up” because they “exist” in the wavelengths of the spectrum.

The YouTube explanations make no explicit reference to cone opponency, the process by which the cone responses are compared with each other beginning in the retina (rather than proceeding directly to the brain). Nor do they mention the important higher-level process of hue opponency [5], by which perceptions of hue are generated in a yet poorly understood way as combinations of red/green and yellow/blue hue components. It is in fact difficult to see how the concept of hue opponency could be grafted without causing great confusion onto an explanation that already invokes red, green and blue signals arising at the level of the cones.

In explaining colour vision science, it is important to make it clear that hues do not reside in wavelengths of light and that at the level of cone responses and cone opponency our vision detects not hues but variations in the balance of the long-, middle- and short-wavelength components of light across the visual field. Hues should not enter the narrative until the higher-level stage of hue opponency, in which detected variations in the spectral composition of light and in the spectral reflectance of objects evoke red/green and yellow/blue hue-opponent perceptions.

1 https://www.youtube.com/watch?v=R3unPcJDbCc
2 https://www.youtube.com/watch?v=l8_fZPHasdo; https://ed.ted.com/lessons/how-we-see-color-colm-kelleher
3 https://www.youtube.com/watch?v=iPPYGJjKVco
4 https://www.youtube.com/watch?v=uNOKWoDtbSk
5 Wuerger, S., & Xiao, K. (2015). Color Vision, Opponent Theory. In R. Luo (Ed.), Encyclopedia of Color Science and Technology (pp. 413-418). Springer. doi:10.1007/978-3-642-27851-8_92-1

David J. C. Briggs (National Art School and Julian Ashton Art School, Sydney, Australia)
Dr. David Briggs has taught courses in colour for painters for twenty years and is the author of the website www.huevaluechroma.com. A recording of his recent ISCC webinar “The New Anatomy of Colour” is available through the ISCC website (https://iscc.org/SeminarSeries).

Friday, April 27, 2018

Psychology and the Big Adventure of the Elastic Yardstick

A psychologist looks to color science to resolve old measurement quandaries.

 A news flash:  psychology is hard if you don’t have the right tools.  Psychology has a significant problem with measurement.  Call its problem what you will: a crossing of ideas, a confluence of concepts, a tired old mistake, or even a methodological thought disorder (the last phrase is from Michell 1997, p. 374). Psychology has had a perennial problem with measurement since the mid-1800s (if not before) when Gustav Fechner announced his law to relate psychological qualities to physical magnitudes.  The heart of the problem is that measurement in psychology is taken to be the same activity as quantitative measurement in physics.  Measurement in physics involves knowing how to measure distances and knowing how to measure time, among other notions; it is clear that we cannot extend the same activities of measurement to psychology without some conceptual upheaval in our understanding of measurement.  "It is as if I were to say, 'You surely know what "It’s 5 o’clock here" means; so you also know what "It’s 5 o’clock on the sun" means.' " (Wittgenstein, 1953/2009, § 350, p.118e)  We do not, at least not without a lot more work and a few conventions about astrophysics.  The one activity (in psychology) is just not the same as the other (in physics).  More than that: it is meaningless to begin to describe how the two activities are the same or different, before we think harder about measurement and psychology.

One can begin in psychology by honoring Fechner and repealing his law, repeating a phrase from Stevens (1957).  We are not rid of the problem of measurement in psychology merely by repealing Fechner’s law, though.  Stevens himself caused as much trouble by introducing another procedure he called ‘magnitude estimation’: a procedure of attaching number words to stimulus magnitudes.  The very act of attaching numbers to perceptible magnitudes was supposed to constitute measurement, somehow.  One can’t just attach numbers to situations and expect the procedure to stand as measurement: such an attitude trivializes psychology in a parody of physics. “The proposition that one conversation is ten times as boring as another is neither true nor false, but is simply a string of words to which no sense may be attached.” (von Kries in Niall, 1995).  Stevens only created trouble by introducing one more procedure unworthy of being called measurement.  So let us honor Stevens and repeal his law of magnitude estimation in turn.
Is there hope left for measurement in psychology ?  Measurement continues to be a problem all over psychology today, but hope remains.  We do know what constitutes effective measurement.  The formal or mathematical conditions for quantitative measurement are well-known (as in Krantz, Luce, Suppes, and Tversky, 1971).  Problems of measurement do matter, if a coherent description of color space matters in colorimetry – as one example.   What can be done to resolve issues of the application of measurement in psychology?  We can begin by recognizing that measurement is something that may be possible in psychology, or else it may fail to obtain.  There may be no measurement in most domains of psychology: we just do not know when measurement makes sense. 

Quantitative structure is a contingent matter for colorimetry as it is for psychology generally – not a law at all, not by Fechner and not by Stevens and not by anyone.  I venture to say the quantitative nature of color space has not been demonstrated in full: we still do not know if measurement works within colorimetry in the same way it does for other, physical magnitudes.  In a profoundly ironic twist though, measurement in color space has a far better chance of working than the application of quantitative measurement in a geometry of ‘visual space’.  Color theory has a better legacy: the pioneers of modern color theory were acutely aware of problems of measurement as they advanced the notion of color space, and a ‘line element’ for color space.  In contrast my bet is that ordinary measurement is meaningless for visual shape, that is, under what has been called ‘the geometry of visual space’.  But that conclusion follows from a long and abstract argument which I leave for another day (though see Suppes, 1991, p.48).


Krantz, D.H., Luce, R.D., Suppes, P. & Tversky, A. (1971).  Foundations of measurement. vol.1.  Academic Press.  https://doi.org/10.1016/b978-0-12-425403-9.50007-2

Michell, J. (1997).  Quantitative science and the definition of measurement in psychology.  British Journal of Psychology, 88(3), 355 – 383.  https://doi.org/10.1111/j.2044-8295.1997.tb02641.x

Niall, K.K. (1995).  Conventions of measurement in psychophysics: von Kries on the so-called psychophysical law.  Spatial Vision, 9(3), 275 – 305.  https://doi.org/10.1163/156856895x00016

Stevens, S.S. (1957).  On the psychophysical law.  Psychological Review, 64(3), 153 – 181.  https://doi.org/10.1037/h0046162

Suppes, P. (1991).  The principle of invariance with special reference to perception.  In: J.-P. Doignon & J.-C. Falmagne, Eds.  Mathematical psychology: current developments.  New York: Springer, pp. 35 – 53. https://doi.org/10.1007/978-1-4613-9728-1_2

Wittgenstein, L. (1953).  Philosophical investigations / Philosophische Untersuchungen.  Revised 4th edition, 2009.  Wiley-Blackwell.

Keith K. Niall
Keith lives in Toronto, Canada.  He is translator and editor of Erwin Schrödinger’s Color Theory, and he is looking forward to writing a book about vision in his own voice. His email address is Keith.Niall@drdc-rddc.gc.ca.

Monday, February 12, 2018

We'll Always Have Parrots

My significant other and I have two parrots, an African grey and a yellow-naped Amazon. They are 24 years old and very feisty--i.e., apt to bite fingers and toes.  They will also be around long after we have departed this Earth, for they live to be at least 55 or 60.  So that is why I have said "we'll always have parrots" in "parroty" of Rick in Casablanca.

And the parrots will always have colors: Yellow and green (with a white eye ring) for the Amazon, and various lightnesses of gray and red for the African grey.  But the colors may vary according to the aqueous environment: A grey feather immersed in water will stay grey but darken slightly. Orange, yellow,  and red will also hold their color in water. But green (and blue, I am told) will change. In particular, green turns brown when immersed in water.  Clearly there are at least two mechanisms for the color: diffraction/iridescence for colors that change on immersion in water (change of refractive environment), and conventional pigment reflectance for colors that don't change on exposure to water. For more on bird-feather color, see 

Many experiments are possible, including immersion of the whole bird. Sometimes I imagine I understand how Edgar Allan Poe could have been a bit freaked out by his raven because it was likely to outlive him. But, as Ilsa heard at that immortal moment in Casablanca, it is more likely that "we'll always have parrots."

Left to right: Alex and Poobah

Michael H. Brill