Two matrices fail under a warm white

The blue’s hue turn was lost with a matrix traced CIECAM16’s small Abney turn at blue — four degrees where the constant-hue data Oklab was fitted to say sixteen — to the cone space its compression works in. CIECAM02 adapted in one space, CAT02, and compressed in another, Hunt–Pointer–Estévez, a close approximation to the cones; CIECAM16 does both in one sharpened space, CAT16. Compressing in Hunt–Pointer–Estévez instead brought the blue’s turn back to fifteen degrees and the whole census of display colours to within three of Oklab’s.

The merger had a reason, and the essay named it. Converting between two cone spaces sends some colours to negative signals, and a power-law compression cannot take a negative number; CIECAM02 carried a sign through the compression to keep the arithmetic going, and the negative cases were among its known faults. A front end that adapts in CAT16 and compresses in Hunt–Pointer–Estévez would restore that trouble somewhere. The prediction was that the failures are confined to monochromatic light below about 450 nm and a thin shell of the most saturated violets, well outside any display or print gamut — so that such a model would keep the blue’s Abney turn for every colour anyone reproduces and lose only colours nobody does. If the failures reached into the display gamut instead, the single matrix was a real trade, and the blue’s turn is its price.

The white decides where it fails #

Where the two-matrix front end fails depends on the white it is adapted to, and neither white does what the prediction said. Adapted to the display’s own white, the failures are at the other end of the spectrum: spectral lights from 561 to 700 nm send their short-wave signal below nought, by at most four thousandths of their luminance, and nothing on sRGB’s boundary fails; P3’s and BT.2020’s red corners fail by the same four thousandths. Adapted to illuminant A — a tungsten white — the failures are where the prediction put them and much larger than it thought: every spectral light from 380 to 474 nm sends its long-wave signal negative, by up to 6.7 times its luminance at the violet end; a point at sRGB’s own blue corner fails; BT.2020’s blue corner fails by 0.45. And there the blue’s restored Abney turn becomes a swing of fifty degrees.

• Under daylight the trade is almost free. The two-matrix front end restores the blue’s turn, and its negative signals are thousandths, at colours no display reaches except at its reddest corner.
• Under a warm white it is not. The adaptation that brings a tungsten white to neutral multiplies the blue end of the spectrum by far more than the conversion’s small negative coefficients can absorb.
• The failure reaches the colours people reproduce, including sRGB’s blue, once the white is warm.
• The single matrix was a trade after all, and what it traded the blue’s turn for was stability under warm adaptation.

The two lines are the whole result. Adapted to the display white, the smallest signal stays above nought across the violet and blue and grazes below it past 560 nm, too little to see at this scale. Adapted to illuminant A, it plunges below nought at the violet end — the figure is cut at −2 — and climbs back through nought at 474 nm. The same conversion, the same colours, and a different white.

Adapting in one space and compressing in another #

The front end is CIECAM16’s, changed as the earlier essay changed it. A colour’s tristimulus values are taken into CAT16 space and multiplied, channel by channel, by the gains CIECAM16’s own adaptation rule gives for the white and the viewing conditions: an average surround, an adapting luminance of 100 candelas a square metre, and the model’s degree of adaptation for those, 0.94. The adapted signals are taken back to XYZ and on into Hunt–Pointer–Estévez space, where the compression would act. The question at each colour is whether any of the three Hunt–Pointer–Estévez signals is below nought.

Adaptation is a set of gains, and gains are linear, so a colour’s signs depend only on its chromaticity. Doubling a colour’s luminance doubles all three signals. The census therefore runs over chromaticities: the spectrum locus at every nanometre from 380 to 700, the purple line joining its ends, the edges of sRGB, Display P3 and BT.2020, and the chromaticities of every block reflectance on the site’s five-nanometre grid under the adapting white’s own spectrum — reflectance one across a band and nothing outside it, or the reverse, which are the boundary of the object-colour solid at every luminance. Each is tried under two whites: the display white the Abney census used, and CIE illuminant A.

Under the display white, the failures are red and tiny #

Adapted to the display white, no violet or blue fails. The failing stretch is the long-wave end of the locus, from 561 to 700 nm, and the signal that fails is the short-wave one. Hunt–Pointer–Estévez’s short-wave signal is simply Z; CAT16’s third row gives long-wave light a tiny negative weight, and after adaptation and conversion back, a pure red’s Z comes out a few thousandths below nought. The deepest failure anywhere on the locus is 0.0037 of the light’s luminance, and among the optimal object colours the same. Of the display boundaries, sRGB’s never fails; P3’s and BT.2020’s fail at their red corners, by that same four thousandths.

A negative signal of four thousandths of a colour’s luminance passes through the compression as a tiny negative response, the sign carried through as CIECAM02 carried it. It moves a hue by nothing measurable. This is the two-matrix front end’s failure under daylight: real, and irrelevant to every colour a display shows.

Under illuminant A, the failures are blue and deep #

Adapted to a tungsten white, the picture turns round. Every spectral light from 380 to 474 nm sends its long-wave signal negative, and the failing stretch reaches across the blue into the blue-green. The depth grows towards the violet: a light at 410 nm comes out with a long-wave signal of −6.7 times its own luminance.

The mechanism is the adaptation itself. A tungsten white has little blue, so bringing it to neutral multiplies CAT16’s short-wave channel several times over; a blue light’s large short-wave signal becomes very large, and the conversion back to XYZ and on to Hunt–Pointer–Estévez carries some of it into the long-wave signal with a negative coefficient. Under the display white the short-wave gain is near one, and the same small coefficient does nothing.

Warm failures reach the colours people reproduce #

The prediction’s test was whether the failures stay outside any display or print gamut. Adapted to illuminant A they do not. A point on sRGB’s boundary at its blue corner fails, by a hundredth of its luminance; one point of P3’s does the same; fifteen of BT.2020’s six hundred sampled boundary points fail, by up to 0.45 of their luminance. Among the optimal object colours, 611 block reflectances fail, the narrowest by several times their luminance. Adapted to the display white, by contrast, the same boundaries fail at most by four thousandths and sRGB’s not at all.

So the answer to the prediction’s question is conditional. For a viewer adapted to daylight, the two-matrix front end loses only colours nobody reproduces, as predicted. For a viewer adapted to a warm room — the ordinary case for a print under household light, and the one appearance models are most needed for — it fails at the display’s own blue.

The restored turn swings under a warm white #

The two-matrix front end was wanted for its hue behaviour, so the census’s last question is what the failures do to it. Each of the display’s twenty-four most saturated colours is mixed with the adapting white down to a fifth of its purity, and its hue angle’s change is read in CIECAM16’s own front end and in the two-matrix version.

Adapted to the display white, the two-matrix front end turns the blue 14.4 degrees where one matrix turns it 3.6 — the restored turn, close to the earlier essay’s 15.6 and to Oklab’s 16.3. Everywhere else the two front ends agree within a few degrees. Adapted to illuminant A, the two-matrix front end turns the same blue 50.4 degrees, and one matrix 5.8. On the way to a fifth of its purity the blue meets a negative long-wave signal, and near nought the compression’s power law is steep: a signal crossing zero turns the hue angle sharply. The neighbouring blues, at 210 and 270 degrees of HSV hue, turn 21 and 17 degrees in two matrices under A, against 11 and nought in one.

A turn of fifty degrees takes a blue most of the way to violet on its way to white. Nothing in the constant-hue data suggests adapting to tungsten does that to a display blue, and the one-matrix model’s six degrees is the more believable. Under a warm white the two-matrix front end’s Abney turn is an artefact of its failure, not a better prediction.

The blue is where every part of the model is most strained #

That the failure lands at blue is no accident of this census; the blue is where CIECAM16’s front end is under the most strain from every direction. The blue resists through the model’s denominator found the model’s colourfulness at blue set by a cone-signal sum that its opponent signals are divided by, and white drains the blue last in every model found every model agreeing that white takes colourfulness from blue most slowly. Both are statements about a signal much larger in one channel than the others. Adaptation to a warm white makes that one channel larger still, and a conversion that was harmless for balanced signals is no longer harmless for this one.

The compression then does the rest. A power law with an exponent of 0.42 is steep near nought and flat far from it, so a signal that approaches nought from above — as the display blue’s long-wave signal does on its way to white under a tungsten adaptation — swings the compressed response, and the hue angle computed from it, far more than the same change in a signal well away from nought. The model has a hue shift it was never given found the compression producing a hue shift with luminance that the model was never fitted to; here it produces a hue shift with purity, of fifty degrees, for the same reason. Near nought, a compressive response amplifies whatever reaches it, and the two-matrix front end delivers something to nought under a warm white.

The hue scale has four corners is the reminder that CIECAM16 reports hue as a quadrature between four unique hues, so an angle of fifty degrees is not merely a number. Between the model’s unique blue, at 237.5 degrees, and its unique red, at 380.1, fifty degrees is more than a third of the way, and a reader of the hue quadrature would be told the blue had moved a third of the way to purple.

What the merger bought #

The cones an appearance model uses found CAT16’s axes to be fitted to corresponding colours rather than measured on receptors, committing to a confusion point far from the measured one; Hunt–Pointer–Estévez is the more receptor-like of the two spaces. The earlier essay found that compressing in it gives the right Abney turn at blue. This census adds the other side of the ledger: a front end that adapts in one space and compresses in another is stable only when the adaptation gains are near one, and under a warm white they are not.

CIECAM16’s single space was therefore a trade with a real price paid at a real place. It gave up the blue’s hue behaviour under daylight — white turns a hue, and the models part at blue — to keep the model’s arithmetic sound under warm light, where a two-space model sends the display’s own blue to a negative signal. A model that keeps both would have to adapt and compress in the same receptor-like space, and the earlier essay found that adapting in Hunt–Pointer–Estévez gives up the adaptation performance CAT02 and CAT16 were sharpened for.

How the census was taken #

The conversions are the standard ones: XYZ to CAT16 by CIECAM16’s matrix, XYZ to Hunt–Pointer–Estévez by CIECAM02’s. The gains are CIECAM16’s — D times the white’s luminance over the white’s CAT16 signal, plus one minus D — with D from an average surround at an adapting luminance of 100 cd/m². The display white is sRGB’s at a luminance of 100; illuminant A is the CIE table’s, scaled to the same luminance. A colour’s signals are divided by its own luminance so that they read as fractions of it; a negative signal is any below nought. The locus runs at every nanometre from 380 to 700 through the 1931 observer, the purple line at forty-one points, each display’s triangle at two hundred points an edge, and the optimal colours at every block of the 5 nm grid from 380 to 780 in both senses, under the white’s own spectrum. The Abney turn is the change in CIECAM16’s hue angle, computed from the compressed signals with CIECAM16’s opponent rows and the sign carried through the compression, from each HSV-saturated sRGB colour to a mixture with the adapting white at the same luminance holding a fifth of its purity.

What this leaves out #

The compression carries the sign, as CIECAM02 did. Another treatment — clipping a negative signal to nought, or a compression that is defined below nought — would give a different hue for the failing colours, and none of them would make the failure go away.

Two whites. A white between daylight and tungsten would put the failing stretch between the two found here, and the first white at which sRGB’s blue fails is not located.

The hue is the only output read. CIECAM16’s colourfulness and lightness also pass through the compressed signals, and a compression goes below the floor is a reminder that what happens near nought in a compression is where a model’s quantities stop meaning what they say.

Still open: the warmest white a two-matrix front end tolerates #

The failure at sRGB’s blue appears somewhere between the display white and illuminant A. Where it appears is the working limit of a two-space front end: below it, a model could compress in Hunt–Pointer–Estévez for its hue behaviour and trust the result.

The calculation is this census along the daylight and Planckian loci from 6500 K down to 2856 K, with the correlated colour temperature at which sRGB’s boundary first fails, and the Abney turn of the display blue at each. The prediction is that sRGB’s blue first fails near 3500 K — a warm-white LED’s temperature — and that the blue’s two-matrix turn stays within five degrees of its daylight value down to about 4000 K before rising steeply, so that a two-space model is sound for daylight and neutral rooms and unsound for the warm ones. If sRGB fails only very close to 2856 K, the practical range of a two-matrix front end is wider than CIECAM16’s merger assumed, and the blue’s turn could have been kept for most viewing.

A failure is a property of the conditions as well as the colours #

The habit is about stating where a model fails.

The prediction located the two-matrix front end’s failures by colour — monochromatic light below 450 nm and a thin shell of violets — and the census found that description true under one white and false under the other. The same conversion fails at the red end under daylight and at the blue end under tungsten, by thousandths in one case and several times the luminance in the other, because what fails is not a colour but a colour under an adaptation. A model’s failure region is a joint statement about its inputs and its viewing conditions.

The failure mode is to test a model’s limits under the one condition it is usually shown in. The test is to vary the condition the model was built to handle — here the adapting white — and see whether the limits move.