Lowering only the chroma turns the sky white and its hue with it

A high-dynamic-range master holds a sky graded on a 1,000 cd/m² monitor, and the same sky must be shown on a display a tenth or a third as bright. The standard way to make the trip is a tone curve — ITU-R BT.2390’s — that compresses everything above a knee, and the question that has run through these essays is where it should be applied. A tone-mapped blue turns where it lands and a tone curve per channel turns a sky towards cyan applied it to each of the display’s three channels separately, which compresses the strong blue channel more than the weak green one and turns a sky toward cyan; an appearance model sees the per-channel sky turn twice as far read that turn in CIECAM16 — 52 units of hue quadrature for a blue with a tenth of green in it — and found the curve applied to the largest channel with the others scaled keeping hue and lightness within a quarter of a unit.

BT.2390 describes a third way, the curve applied to ICtCp’s intensity with the two chroma axes scaled to match, which keeps ICtCp’s hue by construction. A curve on intensity cannot see a blue’s brightness found that it does keep the hue — within 2.3 quadrature units in CIECAM16 — but delivers colours no display can show. A saturated blue has a low ICtCp intensity for its luminance, so it sits under the curve’s knee and passes almost unchanged, and every sky then asks for two to seven times the blue drive the display has. Clipped per channel, it turned as the per-channel curve did; scaled into the gamut, it landed near the scaled curve. The gamut return, not the curve, decided the appearance.

That leaves the return that the intensity curve seems to ask for. The mapped colour has the right ICtCp hue and the intensity the curve chose; it fails only because a blue that saturated cannot be that intense. The prediction was that holding the intensity and the hue and reducing only the chroma until every drive lies between nought and one keeps the CIECAM16 hue within three quadrature units, raises the lightness by ten or more over the scaled return — the intensity curve’s own lightness, now displayable — and halves the chroma, so that the sky arrives as a paler blue of the right hue. If the hue moved by more than five units, ICtCp’s hue lines would not be constant-hue lines in appearance once chroma changes, and every return would have to be judged in an appearance model.

ICtCp’s hue and intensity are kept #

For every mixture, from a pure blue to one with half as much green as blue, on both displays, the chroma-only return keeps the mapped ICtCp intensity within a thousandth of itself and the master’s ICtCp hue within a hundredth of a degree. The return does exactly what it was defined to do: the colour moves only along the line from itself toward the achromatic axis at its intensity, in ICtCp.

What it costs to get there is the first surprise.

On the 100 cd/m² display the two chroma axes must be cut to 0.179 of their mapped value for a pure blue and to 0.033 for the half-green mixture; on the 300 cd/m² display to 0.503 and 0.095. The prediction had imagined a halving. The actual cut is to between a third and a thirtieth.

The reason is the knee itself. A saturated blue’s intensity passes the curve nearly untouched because it is low — ICtCp’s intensity weighs a blue lightly — but low for a blue is high for the display’s gamut at that hue. At the intensity the curve leaves it, the only colours the display can make are near the achromatic axis, where all three primaries contribute. The curve chose an intensity, and the display can show that intensity only as a nearly white colour.

The sky is lit far past the master #

A pure blue has a CIECAM16 lightness of 21.4 on the master and 22.7 scaled into the 100 cd/m² display’s gamut; returned by its chroma alone it has 75.9 on that display and 46.2 on the 300 cd/m² display. With half as much green as blue, 98.4 and 94.9. Over every mixture the chroma-only return lies 24 to 54 units of lightness above the scaled return, against the ten the prediction asked.

So the half of the prediction that expected the intensity curve’s lightness to become displayable was right in sign and wrong by a factor of three to five, and the reason is the same as the chroma’s. The intensity the curve keeps is the intensity of the master’s colour, and when the colour’s chroma is removed at that intensity, what remains is a light colour, because a near-achromatic colour of that ICtCp intensity is light. A deep blue sky is dark because it is saturated; take the saturation away and hold the intensity, and the sky is pale.

The chroma return keeps under half a sky’s colour #

On the 100 cd/m² display a pure blue keeps 41 per cent of the master’s CIECAM16 chroma and a half-green mixture 8 per cent; on the 300 cd/m² display, 79 and 18 per cent. The scaled return keeps 92 to 99 per cent of the master’s chroma on the dimmer display and 98 to 100 on the brighter.

That inverts the comparison the prediction set up. It was supposed that the scaled return dimmed the sky to keep its colour and the chroma return would keep its light at the cost of half its colour. The scaled return in fact keeps both the master’s lightness and its chroma — what it gives up is the master’s absolute luminance, which a dimmer display cannot have anyway. The chroma return keeps the absolute intensity and gives up the colour. For a sky seen on a display a tenth as bright as the master, the first is what a viewer adapted to that display recognises as the same sky.

Holding ICtCp’s hue still turns the sky #

Returned by its chroma alone, every mixture’s CIECAM16 hue quadrature moves toward green, by 16.8, 14.4, 12.6, 17.0 and 9.2 units on the 100 cd/m² display and by 8 to 12 on the 300. Scaled into the gamut, the same mixtures move by 1.3 to 2.2 units on the dimmer display and by 0.6 or less on the brighter. The prediction’s alternative holds: the hue moves by more than five units at every mixture.

ICtCp was designed so that its hue angle stays close to perceived hue along lines of constant hue, and it does far better than older spaces near blue. But a line of constant ICtCp hue running from a saturated blue toward white is not a line of constant hue for a CIECAM16 observer: moving along it, the colour drifts toward green. The drift is the Abney effect, by which a blue desaturated toward white appears to change hue, and no single angle in a space built on fixed matrices can follow it exactly. A return that moves a colour a long way along such a line inherits the whole of the drift.

A pure blue, with no green to turn it, moves 16.8 units returned by its chroma on the 100 cd/m² display and 11.1 on the 300; scaled into the gamut, 1.27 and 0.03. That rules out the green as the cause. The turn is the return’s, and it grows with how far the return moves the colour: on the dimmer display the chroma is cut to under a fifth and the turn is the largest; on the brighter it is cut to a half and the turn is two thirds as large.

By total distance the scaled return is three to five times nearer #

On the 100 cd/m² display the chroma-only return lies 16.8 to 18.9 CAM16-UCS units from the master and the scaled return 4.5 to 5.6; on the 300 cd/m² display 12.8 to 15.4 against 2.7 to 3.1. Every mixture, on both displays, is three to five times nearer the master scaled into the gamut than returned by its chroma.

That settles which return the intensity curve should use, and it is the one that makes the intensity curve unnecessary, at least for skies as saturated as these. Scaling all three drives until the largest is at full drive keeps hue, lightness and chroma within a few units of the master; on these skies it lands within a few units of lightness of the colour the curve applied to the largest channel delivers. The intensity curve kept ICtCp’s hue by construction and lost the appearance in the return; the scaled curve never asked for a colour outside the gamut.

Three returns, three ways to be wrong #

The intensity curve’s colours are now known under three returns, and each fails the master in its own direction, which is the most useful way to remember them.

Clipped channel by channel, a tenth-green sky on the 100 cd/m² display keeps a lightness of 65 against the master’s 32 and turns 47 units toward green; returned by its chroma it reaches 88 and turns 13; scaled into the gamut it keeps 35 and turns under two. Clipping keeps the blue channel at full and lets the green and red rise to whatever the mapped colour asked, so it keeps the most chroma of the two failing returns and turns the most: it is the per-channel curve’s turn arriving by another road, as the essay before found. The chroma return turns less but discards the colour. Scaling keeps everything a viewer adapted to the dimmer display would compare, because it changes only the overall level.

So the intensity curve’s promise — the master’s hue held, its brightness compressed sensibly — can be kept on a gamut-limited display by exactly one of the returns, and that return is the one that throws away the curve’s choice of intensity. The return is doing the tone mapping. The gamut shrinks in the dark found a display’s reachable colours losing a fifth of their reach when the room is turned down, which is the same lesson from the viewer’s side: what a display can show depends on its level, and a curve that decides a colour’s level without asking whether the display can make that colour at that level has decided only half of it.

Several earlier essays dwelt on the blue corner itself. A broad green sharpens the blue; a broad blue dulls it and brightness and the green sharpen the blue almost independently found how the display’s primaries shape the corner where a saturated sky lives. A green’s long tail sharpens the blue; its short tail does not traced that sharpness to the green primary’s spectrum. Every tone curve here meets the corner where it is sharp, and the gamut narrows fast there as a colour lightens; the natural reading is that this is why the chroma return has to cut so far, and that a display whose blue corner was blunter would need less of a cut and would turn its sky less. Neither is computed here, and the corner’s shape is the variable a next calculation on another display would have to move.

What a grading engineer should take from this #

A hue held in one space is not held in another once chroma moves far. ICtCp’s hue is a good guide to appearance for small chroma changes and a poor one for large ones near blue, where a return toward white turns the hue toward green by up to seventeen quadrature units.

The intensity a curve keeps belongs to the colour that had it. A saturated blue’s intensity is low because it is saturated; held while the saturation goes, it becomes the intensity of a pale colour, and the sky turns white rather than paler blue.

For saturated skies, return by scaling the drives. On these mixtures the scaled return keeps the master’s appearance to within a few CAM16-UCS units on both displays; clipping and lowering only the chroma each move it by much more, in different ways.

How the colours were computed #

The display’s primaries are at 630, 530 and 465 nm. A blue at full drive mixed with a share of green from nought to a half is graded on a 1,000 cd/m² master, converted through the display’s matrix and Rec. 2020’s to LMS and through PQ to ICtCp; BT.2390’s curve is applied to the intensity, from the master’s peak to the display’s, and the two chroma axes scaled by the smaller of the new intensity over the old and its inverse. The chroma-only return multiplies both chroma axes by one factor, found by bisection, until every drive lies between nought and one; the colour is converted back through PQ and the matrices. Each colour is read in CIECAM16 under the display’s own D65 white with an adapting luminance of a fifth of its peak, a background of 20 and an average surround, and compared with the master read under its own; CAM16-UCS distance is the appearance model’s uniform-space difference.

What this does not reach #

One blue primary and one green. The return’s turn depends on where a desaturation line from this blue runs through CIECAM16; a display whose blue sat at another wavelength would turn by another amount.

One appearance model. The turn is CIECAM16’s reading of a desaturation along ICtCp’s hue; whether other appearance models read it alike is not computed.

Returns at fixed intensity only. A return that lowers the intensity and the chroma together — between the scaled return and this one — is not tried.

Still open: whether a return along CIECAM16’s hue keeps the sky #

The chroma return failed because ICtCp’s constant-hue line from a saturated blue drifts toward green in CIECAM16. A return that lowers chroma along CIECAM16’s own constant-hue line — holding the master’s hue quadrature and the curve’s lightness, and reducing colourfulness until the colour fits — would not drift by construction.

The calculation is each mixture mapped on intensity, then returned into the display’s gamut along CIECAM16’s constant-hue line at the lightness the intensity curve delivers, reducing chroma by bisection until every drive lies between nought and one, read in CAM16-UCS beside the scaled return. The prediction is that the hue is kept within half a unit by construction and the chroma falls further than ICtCp’s return needed, because CIECAM16’s hue line from a blue runs nearer the display’s blue edge, while the distance from the master stays at least twice the scaled return’s on the 100 cd/m² display, because the lightness the intensity curve chose is still a pale colour’s. If the CIECAM16 return lands within the scaled return’s distance, the fault was ICtCp’s hue line alone and not the intensity it held.

The sky a dim display can show best is the master’s own made that calculation: the true hue is held and more chroma kept, but the sky stays as far from the master, because the lightness the curve held is four fifths of the miss.

A hue held in the wrong space #

The habit here is about where an invariant lives.

The intensity curve was designed to hold ICtCp’s hue, and the chroma return extended that design: hold the hue and the intensity, give up only chroma. Every number the design promised was kept, to a hundredth. And the sky came out pale, lit past its master by fifty units of lightness and turned toward green by seventeen, because the quantities held were held in a space where they do not mean what a viewer means by them once a colour moves far.

The failure mode is to preserve an invariant exactly in the space it was defined in and assume the appearance is preserved with it. The test is to read the result in the appearance model the invariant was meant to approximate, at the distances the operation actually moves the colour.