The model has a hue shift it was never given

Make a monochromatic light brighter and its hue changes. Yellow-greens and reds move towards yellow, blue-greens and violets towards blue, and three wavelengths — commonly cited at about 474, 506 and 571 nanometres — do not move at all. The effect is named for Bezold and Brücke, it has been in the literature since the 1870s, and it is not among the things CIECAM16 was fitted to.

The model takes a luminance and compresses its signals nonlinearly, so it is entitled to an opinion. It turns out to have one.

The right shape, in nearly the right places, at a fraction of the size #

CIECAM16 predicts a Bezold–Brücke shift nobody put into it, with invariant wavelengths near the measured ones and a magnitude far below what is reported.

• The shift is real and structured: positive at the short end, negative through the greens, positive again through the yellows and reds, crossing zero four times.
• The model’s invariant wavelengths are 459.0, 495.2, 502.4 and 569.6 nanometres, against the cited 474, 506 and 571. The yellow one agrees to 1.4 nanometres.
• The size is small: a thirtyfold rise in luminance moves the hue angle by at most 2.2 degrees, which is a median of 0.27 nanometres of equivalent wavelength and at most 3.5.
• Brightening the room is not the same experiment as brightening the patch: a thousandfold rise in the room’s adapting luminance moves the same hues by at most 0.88 degrees, and at several wavelengths in the other direction.

What the model was asked and what it answered #

The test is as plain as it sounds. Take the matching functions at a wavelength, scale them to a relative luminance of 2, and put the result through CIECAM16 in a room of 100 candelas a square metre. Do it again at a luminance of 60. Subtract the two hue angles.

Nothing in that asks the model for a Bezold–Brücke shift, and nothing in the model’s construction mentions one. The hue angle comes from the ratio of two opponent signals, those signals come from the adapted cone responses, and the adaptation includes a hyperbolic compression whose argument is the absolute signal level. Because the compression acts on each channel separately and the channels are unequally excited, brightening the stimulus changes the ratio — and the ratio is the hue. The effect is a side effect of a nonlinearity introduced for a different purpose.

That is worth stating carefully, because it is the interesting part. A model that reproduces an effect it was fitted to has demonstrated arithmetic. A model that reproduces an effect nobody gave it has demonstrated that the mechanism it encodes is at least the right kind of mechanism.

Where its invariants fall #

The crossings are the model’s own invariant hues, found by bisection on the shift.

They come out at 459.0, 495.2, 502.4 and 569.6 nanometres. The reported values are 474, 506 and 571. The agreement at the long end is close to exact — 569.6 against 571 — and the green one is bracketed by a pair of crossings a few nanometres apart, at 495 and 502, where the measured value of 506 sits just outside. The short-wavelength one is the worst: the model crosses at 459 where the literature reports 474, fifteen nanometres away.

Two of the three, then, are where they should be, and the model produces a fourth crossing the literature does not report — the doubled green one, which is an artefact of the shift being very nearly zero over a whole stretch of the locus rather than crossing cleanly.

The size is the problem #

The shift in degrees is hard to compare with anything, because the literature reports the effect as a change of wavelength: how far along the spectrum a dim stimulus would have to move to match the hue of a bright one.

Converted, the model’s effect is a median of 0.27 nanometres and a maximum of 3.5 for a thirtyfold change in luminance. The Bezold–Brücke effect as reported is larger than that — commonly quoted in the range of several to tens of nanometres for comparable luminance ratios. The effect has not been re-measured here and cannot be: this is a computation, and the comparison is against numbers quoted from elsewhere, which is the one kind of number worth leaning on least.

What can be said without leaning on anybody’s quoted magnitude is internal. The model’s own hue scale has structure at a much larger scale than its Bezold–Brücke shift. The hue scale has four corners found that the quadrature’s slope jumps by a factor of 1.74 at green and runs 4.1 times faster near yellow than near blue — so the scale a user reads hue in is piecewise, and the largest hue-angle movement this effect produces, 2.2 degrees, is well under the width of the irregularities in the scale it would be read on.

How it grows with the light #

The shift’s dependence on the luminance ratio is the part an experiment could check most cheaply, so it is worth stating as a prediction.

Taken at 440 nanometres, the model’s shift is 0.29 degrees for a threefold rise, 0.78 for tenfold, 1.49 for thirtyfold and 2.65 for a hundredfold. Fitted through the ends, that is 1.55 degrees per decade of luminance — but the fit is not the curve: the straight line through the ends predicts 1.10 degrees at tenfold where the model gives 0.78, so the growth is slower than a decade rule near the bottom and faster near the top.

Every wavelength behaves the same way with its own sign and size: at 480 nanometres the four numbers are −0.16, −0.43, −0.82 and −1.49; at 590 they are 0.12, 0.32, 0.62 and 1.15. The shape is common and the magnitude is a property of where on the locus the stimulus sits — which is what an invariant hue is, written as a curve instead of a point.

That is a testable statement. If the effect measured on people is proportional to the logarithm of luminance, the model’s slight curvature is wrong; if it saturates at high luminance, the model’s acceleration is wrong in the other direction.

Two ways to add light #

There is a second experiment hiding in the first, and the model distinguishes them.

Brightening the patch acts through the response compression alone: the same room, a stronger signal, a different place on the curve. Brightening the room acts through the degree of adaptation and the induction factors as well, and the model computes those from the adapting luminance before the compression is applied.

The two are not the same size — at most 2.24 degrees against at most 0.88 — and at several wavelengths they are not the same sign. That matters for how the effect would be tested: an experiment that brightens a stimulus inside a fixed surround and one that brightens the whole scene are asking the model two different questions, and it answers them differently. A viewing condition is an argument is the standing statement of why: the room is not a refinement of the stimulus, it is a separate input, and moving one is not moving the other.

What this says about the model #

Three readings, in increasing order of how much they ask.

As a check, it passes. A model built from cone responses, an adaptation and a compression reproduces the sign pattern and roughly the locations of an effect it was never shown. That is evidence the compression is in about the right place in the chain.

As a prediction, it underperforms. If the reported magnitudes are right, the model’s hue shift is smaller than the effect by an order of magnitude, which means anybody using CIECAM16 to predict hue at an unusual luminance is getting a number that moves in the right direction and not far enough.

And as a caution, it is one more prediction made with no clock and no observer in it. The model has no clock found an appearance prediction three and a half units from the settled one half a second after a light changes, and a Bezold–Brücke experiment is exactly the kind where an observer is shown two luminances in succession — so the measured effect includes an adaptation transient the model cannot represent and the computed one does not. Which way that biases the comparison is not obvious, and it is one more reason the size disagreement is not straightforwardly the model’s fault.

The remaining reading: the model reproduces the Hunt effect on colourfulness at a magnitude the field takes seriously, and the hue shift from the same compression at a magnitude it would not. That one coordinate is the right size and another is not is a fact about the fit rather than about the mechanism, and it is the kind of fact that only appears when a model is asked something nobody fitted it to. A patch of a stated size on a stated background would move both, which is a patch is not a scene’s subject and a third way to add light to the same question.

The other coordinate, for comparison #

Hue is the coordinate this essay is about, and it is the one the model handles worst. The other two are worth naming for scale.

Colourfulness rises with luminance in the model, and brighter looks more colourful took that seriously enough to check what it does to apparent contrast. Lightness rises too, and there is no brown light followed the same stimulus from a lightness of 152 to 16 as the surround was raised — a range of nearly ten to one on one coordinate, against a hue movement here of two degrees out of three hundred and sixty.

So within one model, the same compression produces a large effect on two coordinates and a small one on the third. Where the model’s curve does not matter found the nonlinearity’s average behaviour to be nearly linear over a population’s spread and strongly nonlinear over a room’s range of light; this essay finds its effect on hue to be present, correctly shaped and small. Both are statements about the same curve, asked different questions.

What was computed, and how #

A monochromatic stimulus is the 1931 matching functions evaluated at a wavelength, interpolated linearly between the five-nanometre grid’s bands, and scaled to a stated relative luminance. The room is the model’s default average surround at an adapting luminance of 100 candelas a square metre, with the white at D65; only the stimulus’s luminance changes between the two readings.

The hue shift is the difference of the two hue angles, wrapped into ±180 degrees. The invariant wavelengths are the sign changes of that shift, refined by bisection to a fortieth of a nanometre. The equivalent wavelength is the shift divided by the model’s own hue-angle slope at that wavelength, and is reported only where that slope exceeds half a degree per nanometre — past about 620 nanometres every red has nearly the same hue, so the conversion there divides by nothing and returns arbitrarily large numbers.

The room comparison holds the stimulus at a relative luminance of 20 and moves the adapting luminance from 1 to 1000 candelas a square metre.

Two things are deliberately not varied. The surround stays average throughout, so the induction factors move only through the adapting luminance and not through a change of room type; and the white stays D65, so nothing in the comparison is a chromatic adaptation. Both of those would move hue as well, and mixing them into a Bezold–Brücke measurement is how the effect gets confused with the effects around it — which is also why the classical experiments are run in the dark with one stimulus at a time.

Where the measurement stops #

The comparison with the reported invariant wavelengths is a comparison with quoted values. The reported numbers vary between studies by several nanometres, they were measured on small numbers of observers, and they depend on the luminance ratio used — so “the model is 1.4 nanometres out at the yellow point” is a statement about one quoted value rather than about the effect.

The stimulus is monochromatic, which is where the effect is classically measured and is the least representative thing anybody looks at. Whether the model’s shift behaves the same way on broadband colours is the same calculation on different spectra, and is not done here.

The invariant hues are also read off a model whose hue angle is not the scale anybody reports hue in. An appearance is not always a stimulus is the standing caution about the model’s coordinates being freer than the things they describe, and a crossing of the hue angle is not quite a crossing of the quadrature — the two agree about where zero is, since a zero shift is a zero shift in any monotone scale, but not about how large a non-zero one is.

And the model is used outside the range it was fitted in. A relative luminance of 2 in a room of 100 is a dark patch, and CIECAM16’s compression is least tested at the bottom of its range — which the appearance model has no straight piece found to matter for a different reason, the price of a deviation near black rising without limit.

Still open: the size, measured rather than quoted #

The experiment that would settle the interesting half is a hue-matching one: a dim monochromatic stimulus matched in hue against a bright one by moving its wavelength, at a stated luminance ratio and in a stated room, on enough observers to have a spread. That produces the equivalent wavelength shift directly and in the unit this essay has had to convert into.

Run at two luminance ratios, it would also say whether the effect is linear in the logarithm of luminance. The model’s own answer is nearly but not quite: 0.29, 0.78, 1.49 and 2.65 degrees at 440 nanometres for ratios of three, ten, thirty and a hundred, which is a curve that steepens rather than a line. An experiment at four ratios would separate the two, and a model whose compression is in the right place but at the wrong strength should get the curvature right while missing the scale. The prediction to test is specific and modest: the model says the shift is under half a nanometre through most of the middle of the spectrum, and if the measurement says five, the compression is in the right place and at the wrong strength.

A model’s unasked-for predictions #

The habit is about what to do with the parts of a model nobody fitted.

A model fitted to one set of data makes predictions about everything else its arithmetic touches, and those predictions are free evidence about its mechanism — better evidence, in a sense, than its residual on the data it was fitted to, which only says the fit converged. A model that reproduces an unrelated effect at roughly the right place has a mechanism doing work; one that reproduces it at the wrong magnitude has a mechanism in the right place and a constant in the wrong one; one that does not reproduce it at all has an effect living somewhere the model has no representation for.

The move is to find an effect the model was not shown, compute what it says about it, and report the sign, the location and the size separately — because a model can get the first two right and the third wrong, which is exactly what happened here, and reporting a single verdict would have lost the distinction.

The failure mode is to test a model only on what it was fitted to. That measures the fit. The unasked-for prediction measures the model.