An appearance model sees the per-channel sky turn twice as far

A picture graded on a bright master display has to be shown on dimmer ones, and a tone curve decides how. ITU-R BT.2390’s curve leaves a signal’s dark levels alone and compresses everything above a knee towards the target display’s peak, and the question that matters for colour is what it is applied to. A tone-mapped blue turns where it lands found a pure blue’s behaviour in ΔEITP — where its added-green step turns, how sharp its corner is — set only by the level the blue is delivered at, so that for a pure blue any tone curve is as good as any other that delivers it to the same level.

A sky is not a pure blue. A tone curve per channel turns a sky towards cyan mixed the blue with a share of green and showed it through BT.2390 two ways: applied to each channel’s level separately, and applied to the largest channel with all three scaled by the same factor. Per channel, the bright blue channel is compressed from 1,000 cd/m² towards 100 while the green’s 100 cd/m² sits just above the curve’s knee and loses under a third, so on a 100 cd/m² display a tenth of green arrives as 69 per cent of the blue’s drive, and the mixture’s ICtCp hue turns 23° towards the green. Scaled, the hue moves under three degrees at every share.

Twenty-three degrees is a large turn on ICtCp’s hue circle, but ICtCp’s hue angle is not a measure of how different two colours look. ICtCp is built so that distances in it predict discrimination thresholds at fixed adaptation; its hue angle at a blue may be crowded or stretched against what a viewer sees. A quantiser keeps the blue a spike had already found ΔEITP and an appearance model disagreeing at a display’s blue in opposite directions — ΔEITP sharpening the blue corner that CAM16-UCS blunts nearly to a face — so there was reason to expect the two to read a blue’s hue differently here too. The prediction was that each mixture, converted to CIECAM16 under its own display’s white at that display’s adapting luminance, still shows the per-channel change at a tenth of green as more than twenty hue-quadrature units — a fifth of the way from blue to green — and more than ten ΔEITP units, while the scaled curve’s stays under three of each. If CIECAM16 had shrunk the per-channel change below ten units, the blue’s hue scale would have absorbed the turn, and the per-channel curve’s cost would lie in the green’s brightness more than in its hue.

The turn is half the way to green #

Read in CIECAM16, a blue mixed with a tenth of green sits at a hue quadrature of 312.9 on the master and at 260.9 through the per-channel curve on the 100 cd/m² display: a change of 52 units, more than half the way from unique blue at 300 to unique green at 200. A twentieth of green moves 35 units and a quarter 56; on the 300 cd/m² display 9, 17 and 47. Through the scaled curve no mixture on either display moves by more than 0.19 units.

• The hue half of the prediction holds, at two and a half times the twenty units asked.
• The ΔEITP half fails in the way that matters. Between master and display both curves are more than a hundred units apart from the master on the 100 cd/m² display, and the scaled curve is the further, because ΔEITP measures absolute light.
• The alternative does not arise. The appearance model does not absorb the turn; it magnifies it.

Hue quadrature is CIECAM16’s hue expressed against the four unique hues — red at nought and four hundred, yellow at a hundred, green at two hundred, blue at three hundred — so that equal steps divide each quarter of the circle evenly between the hues observers name. A change of fifty-two is a colour that an observer would describe as about as near to green as to blue, where the master’s was well inside the blues. The per-channel curve has turned a sky into a teal.

The change is largest near a tenth or a quarter of green and smaller at a half. The reason is in the curve: per channel, the green’s share is multiplied most when the green sits near the curve’s knee and the blue far above it. At a half of green both channels are above the knee and compressed together, and the per-channel curve begins to resemble the scaled one. On the 300 cd/m² display the knee is higher, so a twentieth or a tenth of green is compressed less unequally and turns a third as far; a quarter, whose green now sits at the knee, turns almost as far as on the dimmer display.

CIECAM16 counts the blue’s hue finely #

At a tenth of green on the 100 cd/m² display, ICtCp’s hue angle turns 23.4° and CIECAM16’s 57.8°; at a twentieth, 16.9° and 48.9°; on the 300 cd/m² display at a tenth, 9.9° and 29.2°. Every per-channel mixture turns two to four times further in the appearance model’s hue angle than in ICtCp’s.

The prediction expected the opposite risk, and it was reasoning from a misreading. An appearance model flattens the blue by its hue found CAM16-UCS making a display’s blue corner the bluntest of twelve — and found the cause in its hue scale, which at a display’s blue counts one degree of CIELAB hue as three and a half of its own. A unit that counts hue that finely spreads directions apart, which blunts a corner; the same fineness stretches a turn. The earlier essay’s closing section described that finding as compression near the blue. It was the reverse, and the reverse is what this calculation finds: the appearance model’s hue moves faster than ICtCp’s wherever a colour leaves the blue towards the cyans.

That has a practical edge. A grading tool that checks a tone-mapped sky’s hue in ICtCp and accepts a few degrees will pass changes that a viewer reads as two or three times as large. ICtCp’s hue angle is not wrong — it is a coordinate of a space designed for thresholds — but it is not the hue a viewer names.

The sky gets twice as light #

Per channel on the 100 cd/m² display, a blue mixed with a twentieth of green goes from a CIECAM16 lightness of 26.8 on the master to 54.9; with a tenth, from 31.5 to 66.3; with a quarter, from 42.7 to 75.9. Through the scaled curve every mixture keeps its lightness within 0.25. Read against each display’s own white, the per-channel sky is twice as light as the master’s.

The green carries most of a mixture’s luminance — a tenth of the 530 nm green carries more than the whole 465 nm blue does — so multiplying its share by seven, as the per-channel curve does at a tenth, multiplies the mixture’s luminance relative to the display’s white by 4.4, and lightness follows luminance as roughly its cube root and the surround’s exponent. So the per-channel cost is not only a hue. Brightness and the green sharpen the blue almost independently separated a display’s blue corner’s dependence on brightness from its dependence on the green primary; here a tone curve moves both at once, and the appearance model reads the move as a pale teal standing in for a mid blue.

Chroma tells less. The per-channel mixtures’ CIECAM16 chroma stays within about ten units of the master’s — 85.3 to 90.6 at a tenth of green — because the green raises the colour’s lightness and its saturation against the white falls in proportion. The sky is not more vivid; it is lighter and greener.

Absolute light ranks the curves backwards #

On the 100 cd/m² display a pure blue is 152 ΔEITP units from the master’s whichever curve shows it; with green in the mixture the scaled curve’s distance rises to 156–168 and the per-channel curve’s falls to 114–134. On the 300 cd/m² display the scaled curve’s distances run 83 to 91 and the per-channel curve’s 61 to 65. For every mixture with any green in it, ΔEITP between master and display calls the per-channel curve the closer.

That is the second half of the prediction’s failure, and the more instructive. ΔEITP is a distance in an absolute space: its intensity axis is the PQ-encoded luminance itself, so a colour shown at a tenth of its graded luminance is far from the graded one whatever its hue. Any tone curve that maps a 1,000 cd/m² master to a 100 cd/m² display moves every bright colour by a hundred units or more, and the per-channel curve, by keeping the dim green near its graded level while the blue falls, keeps more of the master’s absolute light. In absolute light it is the more faithful curve; in appearance it is the less. The prediction asked for both readings to agree, and they disagree by sign.

The disagreement is not a flaw in either measure. ΔEITP was built to measure how distinguishable two colours are when both are on the same display under the same adaptation. Master and target are two displays a viewer adapts to separately, and the right comparison between them is between appearances, each under its own white.

Read in appearance, the scaled sky is the closer #

The scaled curve’s CAM16-UCS distance from the master is 4.1 on the 100 cd/m² display and 2.8 on the 300 for a pure blue, and stays within a quarter of a unit of those at every share of green; the per-channel curve’s rises to 16.1 at a twentieth of green and 17.6 at a tenth on the dimmer display, and to 14.5 at a quarter on the brighter. Per channel, every mixture with green in it sits three to six times further from the master in appearance than the scaled curve leaves it.

The scaled curve’s residual four units are worth reading. They are not a hue or a lightness — those it keeps to a fifth of a unit — but colourfulness: CIECAM16’s colourfulness depends on the absolute luminance, so the same colour on a display a tenth as bright looks less colourful, the Hunt effect. A pure blue loses it on either curve, and no tone curve that dims the display can avoid it. Everything beyond those four units is the per-channel curve’s own.

Where the mixtures land #

On the master, a blue mixed with a twentieth of green sits at a hue quadrature of 316.9, a tenth at 312.9, a quarter at 302.9 and a half at 277.5. Per channel on the 100 cd/m² display they land at 281.6, 260.9, 246.8 and 243.4. A tenth of green arrives further towards green than a half was graded, and every mixture through a quarter has crossed below unique blue, which on the master only the half had.

Read as a grading decision, the figure is stark. A colourist who graded a sky as a blue just short of unique blue, with a tenth of green to warm it towards cyan, will find it on the dimmer display as a teal past where a half-green turquoise was graded. Nothing in the master’s grading decided that; the per-channel curve did. A tone curve per channel turns a sky towards cyan put the turn at 23° of ICtCp hue, which reads as a moderate shift. In the hue a viewer names, it is a change of category.

What a display engineer should take from this #

Never tone-map a picture’s channels separately where hue matters. A green’s long-wavelength tail sharpens the blue and its neighbours found a display’s blue sensitive to small changes in the green primary’s spectrum; a per-channel tone curve changes the green’s share of a blue by sevenfold, which dwarfs any primary’s tail. BT.2390 applied to the largest channel with the three scaled together keeps every mixture’s CIECAM16 hue within a fifth of a unit and its lightness within a quarter, at the cost only of the dimmer display’s loss of colourfulness.

Check a tone curve’s hue in an appearance model, not in ICtCp. At a display’s blue, CIECAM16’s hue moves two to four times as far as ICtCp’s for the same change of colour, and the appearance model’s is the one a viewer reports.

Do not compare master and target in an absolute colour difference. ΔEITP between a 1,000 and a 100 cd/m² rendering is dominated by the dimming and ranks a hue-shifting curve above a hue-keeping one. Compare appearances, each under its own white.

How the colours were read #

The display has red, green and blue primaries at the spectral locus at 630, 530 and 465 nm and a D65 white, built by the census’s primary matrix — the 530 nm green one of the greens the green primary decides how sharp the blue is set against the blue, and the 465 nm blue near the middle of the range over which that essay found the blue corner turning. A mixture is the blue at full drive plus a share of the green — nought, a twentieth, a tenth, a quarter or a half — graded on a 1,000 cd/m² master; the target displays peak at 100 and 300 cd/m². BT.2390’s curve is applied either to each channel’s luminance separately or to the largest channel’s, the other two scaled by the same factor. Each colour’s XYZ, relative to its own display’s white at a hundred, is converted to CIECAM16 with that white, an adapting luminance a fifth of the display’s peak, a background of twenty and an average surround; hue quadrature, hue angle, lightness, chroma and colourfulness are the model’s, and the CAM16-UCS distance is the model’s own uniform-space formula. ICtCp is computed from each colour’s absolute luminance through the PQ response, and ΔEITP is the Euclidean distance between ICtCp triples with the chroma axes scaled as the standard defines.

What this does not reach #

Other appearance models and viewing conditions. CIECAM16 with an adapting luminance of a fifth of the peak and an average surround is one choice; a dark surround, typical of a cinema, lowers the model’s lightness contrast and would change the lightness figures more than the hue.

Other tone curves. BT.2390 is one curve; its knee placement drives the per-channel distortion, and a curve with a lower knee distorts more.

Real pictures. A sky is a gradient of mixtures; every point of it turns by a different amount, and how a viewer reads a gradient that changes category across its width is not modelled.

Still open: whether a curve on intensity keeps the sky #

The scaled curve keeps the appearance by applying the tone curve to one number — the largest channel — and scaling the rest by the same factor. BT.2390 also describes a second way that keeps hue by construction: apply the curve to ICtCp’s intensity and scale the two chroma axes by the ratio of new intensity to old.

The calculation is each mixture shown on the 100 and 300 cd/m² displays through BT.2390 applied to ICtCp intensity with the chroma axes scaled in proportion, read in CIECAM16 as here beside the scaled and per-channel curves. The prediction is that the intensity curve keeps hue quadrature within three units and lightness within two of the master’s at every share, close to the scaled curve, because ICtCp’s intensity tracks luminance and the chroma scaling holds the ratio of colour to light. If its hue moves by more than ten units at a tenth of green, ICtCp’s hue axis is not an appearance hue even when nothing but the intensity is changed, and a tone curve that keeps ICtCp’s hue does not keep the viewer’s.

A distance that measured the wrong comparison #

The habit here is about choosing a difference measure for a comparison across conditions.

The prediction asked one distance, ΔEITP, to grade a tone curve between two displays, and it ranked the curves backwards. ΔEITP had not failed at its job; it was asked a different question from the one it answers. It measures how far apart two colours are under one adaptation, and the master and the target are seen under two. An appearance model, which takes each colour against its own white, ranked them the way a viewer would.

The failure mode is to compare colours seen under different conditions with a difference formula built for colours seen under the same one. The test is to ask what the viewer is adapted to when each colour is seen, and to measure the difference in a space that holds that adaptation — or, where none does, to say that the number is a distance in light, not in appearance.