The sky a dim display can show best is the master's own

A sky graded on a 1,000 cd/m² master has to be shown on displays a tenth and a third as bright, and the last few essays have followed one route there: ITU-R BT.2390’s tone curve applied to ICtCp’s intensity, which keeps ICtCp’s hue by construction. A curve on intensity cannot see a blue’s brightness found that a saturated blue sits under the curve’s knee and comes out asking for two to seven times the blue drive a display has. Lowering only the chroma turns the sky white and its hue with it brought those colours back by their ICtCp chroma alone, holding intensity and hue. ICtCp’s hue and intensity were kept to a hundredth; the CIECAM16 hue moved eight to seventeen units toward green, the lightness rose 24 to 54 units over the scaled return, and every sky lay 13 to 19 CAM16-UCS units from the master, three to five times as far as scaling the three drives until the largest is full.

Two things went wrong in that return, and they are separable. ICtCp’s constant-hue line from a saturated blue toward white drifts toward green in CIECAM16, so holding ICtCp’s hue did not hold the hue a viewer sees. And the intensity the curve chose belongs to a saturated colour; held while the saturation goes, it makes a pale one. A return along CIECAM16’s own constant-hue line removes the first by construction and keeps the second.

The prediction was that a return built by CIECAM16’s inverse — at the master’s hue, at the lightness the intensity curve delivers, with the chroma reduced until every drive lies between nought and one — keeps the hue within half a unit, needs a deeper chroma cut than ICtCp’s return because CIECAM16’s hue line from a blue runs nearer the display’s blue edge, and still lies at least twice the scaled return’s distance from the master on the 100 cd/m² display. If it landed within the scaled return’s distance, the fault would have been ICtCp’s hue line alone.

The hue is held, and more chroma with it #

Returned along CIECAM16’s hue at the intensity curve’s lightness, every mixture’s hue quadrature is the master’s within a hundredth of a unit, on both displays. That part is by construction, and the construction works.

At every mixture on both displays the CIECAM16 return keeps more chroma than ICtCp’s: a pure blue 46.0 against 37.4 on the 100 cd/m² display and 75.6 against 71.8 on the 300. The prediction had it the other way round. One reading, consistent with the numbers and not separately tested, is that the drift itself is the reason: ICtCp’s hue line from the blue leans toward green as it lightens, the display’s gamut at those lightnesses reaches less far in that direction than along the blue’s own appearance hue, and so a colour following the true hue stays nearer the blue edge of the gamut and keeps more of its colour before a drive reaches its limit.

Holding the true hue saves one or two units of seventeen #

On the 100 cd/m² display the CIECAM16 return lies 15.7 to 17.3 CAM16-UCS units from the master, against 16.8 to 18.9 for ICtCp’s return and 4.5 to 5.6 for the scaled return. Every mixture lies more than two and a half times as far from the master as scaling leaves it, and the third part of the prediction holds.

So the hue drift, which was the striking failure of the essay before, is the smaller part of what went wrong. Holding CIECAM16’s hue instead of ICtCp’s brings each mixture between 0.8 and 2.3 units nearer the master, under a fifth of the distance it had. The return along the true hue is still a pale sky: its lightness is the one the intensity curve chose, and that is the larger error.

A pure blue has a CIECAM16 lightness of 21.4 on the master; mapped by the intensity curve it has 66.8 on the 100 cd/m² display and 41.3 on the 300. The curve passes a saturated blue’s intensity almost unchanged because it lies under the knee, and on a display a tenth as bright, with the viewer adapted to that display’s white, an unchanged absolute intensity is a much larger share of white. The curve was built to keep things under the knee where they are in absolute terms; for a viewer whose adaptation follows the display, keeping them there makes them up to three times as light.

The master’s own drives are the nearest the display can come #

The obvious next return holds the master’s appearance itself — its lightness, chroma and hue, read under its own viewing conditions — and rebuilds it under each display’s. If it fits in the gamut, nothing nearer is possible in CIECAM16’s terms.

The master’s appearance, rebuilt under each display’s viewing conditions, needs drives within 0.008 of the master’s own at every mixture on both displays — a pure blue at 0.001, 0.001, 1.003 on the 100 cd/m² display, a half-green mixture at 0.006, 0.502, 0.992. It fits, to within three thousandths of full drive, and it is the master’s drives.

That is a property of how the appearance model treats a display whose viewer adapts to it, and it is worth seeing how little of the display’s own character enters. The primaries are the same on master and display here, so the drives mean the same colours; on a display with a different blue, a green’s long tail sharpens the blue; its short tail does not and its neighbours found the blue corner moving with the primaries, and the master’s appearance would need different drives — but it would still be the master’s appearance, rebuilt, that set the floor, and the rebuild is the operation a display’s colour management already performs for colours that fit. Tone mapping, for colours under the knee and a viewer adapted to the display, has nothing to add to it beyond the overall level. CIECAM16’s lightness and chroma are relative to the adapted white, and the census sets each display’s adapting luminance at a fifth of its peak, so a colour at the same relative drives on a dimmer display has nearly the same lightness and chroma as on the master. What changes is the colourfulness, which depends on absolute luminance; that is the Hunt effect, and it is the residue.

The master’s own drives lie 4.02 to 4.14 CAM16-UCS units from the master on the 100 cd/m² display, and 2.70 to 2.77 on the 300, whatever the mixture. The distance barely depends on the green share at all, which is what a loss of colourfulness that scales every colour alike should look like. An appearance model sees the per-channel sky turn twice as far had found a pure blue, where every curve agrees, lying between two and five units from the master and called it the dimmer display’s loss of colourfulness alone. That is very nearly a floor. Matching the master’s lightness, chroma and hue exactly leaves the colourfulness short, and a colour a little different in lightness and chroma can make up a fraction of that in CAM16-UCS — but only a fraction, as the curves below show.

Most of the miss is the lightness the curve held #

Of the ICtCp return’s 18.9 units for a pure blue on the 100 cd/m² display, holding CIECAM16’s hue removes 2.2, showing the master’s own lightness and chroma removes 12.6 more, and 4.1 remain — the display’s dimness. At every mixture the middle bar is the largest, by a factor of five or more over the hue’s.

The decomposition is the answer the run of essays on this curve has been reaching for. The intensity curve’s failure on saturated skies was never mainly its hue; it was that it set the sky’s lightness by an absolute intensity, for a viewer who reads lightness relative to the display. Every return that kept that intensity inherited the failure, and the return that discards it — scaling all three drives — lands close to the master’s own drives, because the master’s own drives are what a viewer adapted to the dimmer display reads as the master’s sky.

On the 300 cd/m² display the CIECAM16 return lies 10.5 to 14.4 units from the master, the scaled return 2.69 to 3.13, and the master’s own drives 2.70 to 2.77. On the brighter display the scaled return is already at the floor for every mixture but the half-green one, where the curve’s compression of the green channel near the top costs it four tenths of a unit.

Every curve tried so far, against the floor #

With a floor in hand, every tone curve these essays have tried can be placed on one scale. A tone-mapped blue turns where it lands and a tone curve per channel turns a sky towards cyan applied BT.2390 to each channel; the essay that read those curves in CIECAM16 applied it to the largest channel with the others scaled; the last three essays applied it to ICtCp’s intensity and returned the result three ways.

On the 100 cd/m² display, a tenth-green sky lies 17.6 CAM16-UCS units from the master under the per-channel curve, 4.03 under the scaled curve, 18.5 returned by ICtCp chroma, 17.3 returned along CIECAM16’s hue, 4.94 scaled into the gamut after the intensity curve, and 4.07 shown as the master’s own drives. On the 300 cd/m² display the same six are 11.7, 2.69, 13.8, 12.5, 2.69 and 2.73. The scaled curve sits on the floor at every mixture, within 0.13 of it and on its near side by up to that much, because dimming every channel by the same factor is very nearly what showing the master’s drives on a dimmer display does.

So the curves fall into two groups, and the line between them is whether the curve changes the ratios between the channels. The per-channel curve compresses the strong blue channel more than the weak green one and turns the sky; the intensity curve keeps the ratios in ICtCp but asks for an intensity the gamut cannot hold, and every way back changes them. The scaled curve and the master’s drives keep the ratios, and they are within a few tenths of each other and of the floor. The gamut shrinks in the dark found what a display can show depending on its level; what it can best show of a saturated sky depends on its level only through the colourfulness, and the ratios carry the rest.

It is worth being plain about what this does not say. BT.2390 exists for colours above its knee, which a dimmer display cannot show at their master luminance and must compress; for those, the master’s drives are not available and the choice of where to apply the curve matters in a way this census of saturated skies, all under the knee, cannot test. Brightness and the green sharpen the blue almost independently found how a display’s level and its green shape the blue corner these skies sit in; a bright sky, nearer white and above the knee, would sit somewhere else entirely.

What a grading engineer should take from this #

For colours under the knee, show the master’s drives. On these skies a viewer adapted to the dimmer display reads the master’s relative drives as the master’s appearance, to within the colourfulness the display’s lower luminance must cost. A tone curve that leaves such colours at their absolute intensity makes them lighter for that viewer, and every return from there inherits it.

Separate the hue error from the lightness error before fixing either. The chroma return’s hue drift was the most visible symptom and a fifth of the cost. A return along the true hue fixed the symptom and left four fifths of the distance in place.

Know the floor. The master’s own appearance, rebuilt on the target display, sets a distance no tone mapping beats by more than a tenth of a unit here; measuring every curve against that floor rather than against nought says how much of a curve’s error is the curve’s.

How the colours were computed #

The display’s primaries are at 630, 530 and 465 nm; a blue at full drive with a share of green from nought to a half is graded on a 1,000 cd/m² master and mapped by BT.2390 on ICtCp intensity, its chroma axes scaled, to displays of 100 and 300 cd/m². Each display is read in CIECAM16 under its own D65 white, adapting luminance a fifth of its peak, background 20, average surround; the master under its own. The CIECAM16 return builds colours by the model’s inverse at the master’s hue angle and the mapped colour’s lightness, reducing chroma by bisection until every drive lies between nought and one; the master’s own appearance is the inverse at the master’s lightness, chroma and hue under each display’s conditions. Every colour is compared with the master in CAM16-UCS.

What this does not reach #

The floor depends on the adapting luminance. At a fifth of each display’s peak the viewer’s adaptation scales with the display, which is what makes the master’s drives the master’s appearance; a viewer adapted to a fixed room level, not to the display, would set a different floor and might favour a different return.

Saturated skies only. Colours above the knee, which the curve exists to compress, are not in this census; for them the master’s drives do not fit, and the question of which return keeps their appearance is a different one.

One appearance model. The decomposition into hue, lightness and dimness is CIECAM16’s; another model would divide the same distance differently.

Still open: whether a viewer adapted to the room changes the best return #

Every reading here adapts the viewer to the display — its adapting luminance a fifth of its peak — and under that adaptation the master’s relative drives are the master’s appearance. A display viewed in a lit room, with the viewer adapted partly to the room, reads a dim display’s colours as dimmer than its own white makes them.

The calculation is every return here, and the master’s own appearance rebuilt, with each display’s adapting luminance held at the master’s 200 cd/m² instead of a fifth of the display’s peak, read in CAM16-UCS against the master as here. The prediction is that the master’s own appearance then needs drives above full on the 100 cd/m² display for every mixture, because a fixed adaptation asks a dim display for the master’s absolute colourfulness, so that it no longer fits, and that the intensity curve’s lightness, which keeps absolute intensity, comes within twice the scaled return’s distance instead of four times. If the scaled return stays nearest under a fixed adaptation, the drives’ relative scale is the right one whatever the viewer adapts to.

The curve’s choice that was never about hue #

The habit here is about finding which part of an error is which before mending any of it.

The ICtCp return failed in a way that looked like a hue problem, because the hue drift was the large, visible, nameable change. Mending the hue removed the drift exactly and left the sky as far from the master as before, to within two units. Splitting the distance against a floor — the master’s own appearance rebuilt on the dimmer display, which turned out to be the master’s own drives — put four fifths of it on the lightness the curve chose, a choice that had been made before any return was applied.

The failure mode is to fix the most visible component of an error and read the remaining distance as the next most visible. The test is to build the best achievable result, measure every candidate against it, and divide each candidate’s distance from the truth into the parts that each change removes.