What the Dichromat's Eye Tells the Trichromat's Brain

At last February’s SPIE Electronic Imaging Symposium in Burlingame, CA, I posed the question, “Can trichromats really know what dichromats see?” I did this as a mini-paper in a session called “The Dark Side of Color” in which the goal is to ask controversial questions but not to answer them. I will re-express my argument here, because Hue Angles is also a place for controversial questions.

One would think trichromats do know what dichromats see, judging from several algorithms and software applications that simulate the appearance to a dichromat of any given trichromatic image [1-4]. But my SPIE talk challenged that presumption.

We do know what sets of tristimuli are matches for each kind of dichromat (protanope, deuteranope, or tritanope according to which cone system the dichromat lacks). The confusion loci define parallel lines through tristimulus space. In the central-perspective view that is chromaticity space, the parallel lines converge to a vanishing point, called a co-punctal point, that is different for each kind of dichromat.

Although useful, confusion loci and copunctal points don’t define how to map the appearance of a dichromatic color on the appearance from a trichromatic space. It is not even necessary for the dichromatic appearance of a light to match the trichromatic appearance of one of the lights on a confusion locus.  So how can one make the map?

Formally, the problem is as follows: Find a light Q that looks the same to a trichromat as a light P looks to a particular dichromat. If you could find a light Q for each light P, you could present such lights Q in an image as a simulation of the dichromat’s perception.

One is helped in this task by the theoretical assumption that a dichromat’s visual system is the same as a trichromat’s, except the dichromat lacks one kind of cones. One replaces the missing-cone input channel with one of the non-missing cone types. At a post-receptoral stage in the model (say, at an opponent-color stage), the theory chooses a channel and then either nulls it or substitutes one of the remaining channels. There are many color-vision models (ranging from Guth’s ATD model to CIECAM02), and the above algorithm allows several choices for implementation in each model. 

Having chosen a color-vision model and a way to “dichromatize” it, Capilla et al. [2]  predict Q’s from P’s by borrowing the asymmetric-matching idea from chromatic-adaptation studies.  They call their approach the “corresponding pair” idea: Map XYZ of light P to the dichromat’s model output, and then send that model output through the inverse of the trichromat’s model to arrive at Q.

Using this model structure, Capilla et al. compare images transformed using different color-vision models and “dichromatizing” options. The results are quite diverse, showing the impact of the choice that remains even after the assumed simplifications. Only experiment can decide which choice (if any) is right.  What experiment shall that be?

Even the existence of an experiment is problematic. On one level, my SPIE question devolves to the classic philosophical conundrum of my not being able to know if I see the same blue that you do.  The situation is saved to some extent by the existence of unilateral dichromats. There the appearance matches between the dichromatic eye and the trichromatic eye promise to be a legitimate “Rosetta Stone.” Indeed, unilateral dichromats depose the naïve model of a dichromat’s color always having the appearance of one of its confusion aliases in trichromatic vision. But to be trustworthy, color-appearance matches must be made cetera paribus---that is, all other variables being equal. The spatial context of a scene always affects the appearance of a color in that scene, and the contexts themselves cannot be equated between a dichromat and a trichromat. You would have to ask the unilateral dichromat to match all the colors in all the possible scenes in your universe to be sure that you had a good simulation. An additional complication is that unilateral dichromats are so rare that we cannot be too fussy about assuring that the trichromatic eye is really “normal.” Finally, the colors dichromats see can be as unstable as Gestalt effects like the Necker cube. One dichromat I know reported the following experience: At a distance, red roses look achromatic to him. When he gets nearer to the roses, they suddenly look red. Then, they stay red as he backs away from them, reverting to their previous achromatic appearance only when he looks away from them and back.

In my SPIE talk, I concluded that, if you still want to predict and simulate what dichromats see, you have truly passed…to the Dark Side of Color. Do you agree?

1. H. Brettel, F. Vienot, and J. D. Mollon, Computerized simulation of color appearance for dichromats, J. Opt. Soc. Am. A 14, 2647-2655 (1997).
2. P. Capilla, M. A. Diez-Ajenjo, M. J. Luque and J Malo, Corresponding-pair procedure: a new approach to simulation of dichromatic color perception. J. Opt. Soc. Am A 21, 176-186 (2004).
3. H. Kotera, A study on spectral response for dichromatic vision, Proc. 19th IS&T Color & Imaging Conference, pp. 8-13 (2011).
4. http://www.vischeck.com/daltonize/ 

Michael H. Brill
Datacolor

Solution to Cryptogram

And the answer [1]  is…

"Within your lifetime will, perhaps,
As souvenirs from distant suns
Be carried back to earth some maps
Of planets and you'll find that one's
So hard to color that you've got
To use five crayons. Maybe, not."

The poet was Marlow Sholander. He was my freshman calculus professor at Western Reserve University (before it united with Case Institute of Technology). I don't know when he wrote it, or why. He was known for chain smoking and for phrases like, "There are no Gausses in this class"---proved by lofting epsilons and deltas over our heads. But he said not a word about the four-color-map theorem. It was only a conjecture and not a theorem when I knew Sholander.  The proof would come in 1976 and be published in 1977 [2,3]. Even then, the proof was questioned because it required a computer. In fact, it was the first major theorem that was proved using a computer.

For new initiates: The four-color map theorem says that, no matter how you carve up a plane into connected (contiguous) areas, to assure that no two abutting regions have the same color, you don’t need more than four colors. “Abutting” means sharing a boundary of at least two points, so, e.g., Arizona and Colorado (which share only one point) could have the same color on a U.S. map.

You won’t find the theorem bandied about by geographers. The maps are entirely in the minds of mathematicians, e.g. the following from Wikipedia (http://en.wikipedia.org/wiki/Four_color_theorem):

Why spend a career trying to prove (or disprove) something about four-color maps? To put it abstractly, I think it allows you to hold (and maybe control) certainty in the palm of your hand.  The intoxication of knowing exactly what could not have come from a distant planet---no matter how far away---is the essence of Sholander’s poem.

I would not have guessed he had it in him, and it was not he who got me excited about what was then the four-color conjecture.  Years before, my tiny sixth-grade class trooped across the soccer field to Brentwood High School, invited to partake in a flight of fancy led by a 12th-grade prodigy. This prodigy inundated us with fun and challenges from constructing flexagons and polyhedra to reading Fantasia Mathematica---which contains a story about an impossible five-color map. We made three visits after school, as I recall.
The others in my class returned and made flexagons. I spent every boring class moment for the next six years trying to disprove various “easy” truths like the four-color-map problem. And math classes didn’t have boring moments anymore.

The name of the prodigy? Jef Raskin, who started the MacIntosh project at Apple Computer. The rest is history, as Wikipedia will attest (in a different article). Some years after our visits, I chanced to meet him again, and he said he’d outgrown the childish pursuits he had started me on.

How strange that it was the dry, hierarchy-obsessed professor who carried the wonder to distant planets through his poem!

Michael H. Brill
Datacolor

1. The only solver was Paul Centore.
2. Appel, Kenneth; Haken, Wolfgang (1977), "Every Planar Map is Four Colorable Part I. Discharging", Illinois Journal of Mathematics 21: 429–490
3. Appel, Kenneth; Haken, Wolfgang; Koch, John (1977), "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567