A curve on intensity cannot see a blue's brightness

A picture graded on a bright master display has to be shown on dimmer ones, and a tone curve decides how. A tone curve per channel turns a sky towards cyan showed ITU-R BT.2390’s curve doing that two ways to a blue mixed with a little green — applied to each channel separately, or to the largest channel with all three scaled by one factor — and an appearance model sees the per-channel sky turn twice as far read the results in CIECAM16 under each display’s own white. Per channel, a tenth-green blue graded on a 1,000 cd/m² master moves fifty-two hue-quadrature units towards green on a 100 cd/m² display, more than half the way from unique blue to unique green, and its lightness doubles; with the largest channel scaled, hue and lightness hold to a quarter of a unit.

BT.2390 describes a third way. Apply the curve not to any channel but to ICtCp’s intensity I — the PQ-encoded mean of the cone-like L and M signals — and scale the two chroma axes by the smaller of the new intensity over the old and its inverse. That keeps ICtCp’s hue angle exactly, since both chroma axes are multiplied by one number, and it keeps the colour’s chroma in proportion to its intensity. A quantiser keeps the blue a spike found ICtCp treating a display’s blue as its sharpest corner where an appearance model treats it as nearly a face, so ICtCp’s geometry at a blue was not a safe guide to appearance; the question here is narrower, whether its hue lines are. The prediction was that this intensity curve keeps each mixture’s CIECAM16 hue quadrature within three units of the master’s and its lightness within two at every share of green, close to the scaled curve, because ICtCp’s intensity tracks luminance and the chroma scaling holds the ratio of colour to light. If the hue had moved by more than ten units at a tenth of green, ICtCp’s hue axis would not be an appearance hue even when only the intensity is changed.

The hue holds and the colour cannot be shown #

Mapped on intensity, every mixture’s ICtCp hue equals the master’s to a thousandth of a degree, and its CIECAM16 hue quadrature lies within 2.3 units of the master’s on both displays — a tenth of green moves 2.0 units on the 100 cd/m² display and 0.2 on the 300. But every mixture’s drives leave the display’s gamut: on the 100 cd/m² display a pure blue asks for 7.14 times the blue channel’s full drive and a half-green sky 2.13 times; on the 300 cd/m² display, 3.33 and 2.26.

• The hue half of the prediction holds, a little inside the three units asked.
• The lightness half cannot be read on a display. As mapped, a tenth-green sky’s lightness rises from 31.5 to 81.9 against the display’s white, but that is light the display cannot emit.
• The alternative does not arise. ICtCp’s hue and the appearance’s agree to two units when only the intensity is moved; the trouble is elsewhere.

The figure’s four lines separate into two pairs. As mapped, the intensity curve is flat along nought, as the prediction said. Brought back into the gamut by clipping each channel at full drive, it falls to within five units of the per-channel curve, turning a tenth-green sky 47 units towards green. Brought back by scaling all three drives down until the largest is at full, it stays near nought, at 1.8 units. A colour that has to be put on a display has to be put inside its gamut, and the way that is done decides the hue a viewer sees.

Why the curve barely touches a blue #

ICtCp’s intensity is built from the L and M cone-like signals, and a saturated blue drives them weakly: most of its light lands on S. The pure 465 nm blue at full drive on the 1,000 cd/m² master has an intensity equal to the PQ level of 79 cd/m²; a tenth-green sky’s corresponds to 141 cd/m² and a half-green sky’s to 387. BT.2390’s knee, below which it changes nothing, sits at 27.9 cd/m² for a 100 cd/m² display and 161.4 for a 300 cd/m² one. On the brighter display the curve does not touch a pure blue or any sky up to a tenth of green; on the dimmer one it takes the pure blue’s intensity from 79 to 62 cd/m² — a fifth — and a tenth-green sky’s from 141 to 80.

The blue channel itself, read the way the other two curves read it, sits at 1,000 cd/m², the top of the curve, and is compressed by a factor of ten on the 100 cd/m² display. The intensity curve and the channel curves are compressing different quantities: one asks how bright the colour is to the L and M cones, the other how much the blue primary is being driven. For a sky they differ by an order of magnitude, and a display’s limits are set by the drives.

So the intensity curve hands the display a blue nearly as bright, in absolute terms, as the master’s, on a panel with a tenth of the master’s peak. The mapped pure blue asks the 100 cd/m² display’s blue primary for seven times the light it can give. No display can do that, and what happens next is not the tone curve’s decision.

Every sky asks for more blue than the display has #

On the 100 cd/m² display the intensity mapping asks the blue channel for 7.14 times full drive for a pure blue, 5.78 for a twentieth of green, 4.84 for a tenth, 3.25 for a quarter and 2.13 for a half; on the 300 cd/m² display for 3.33 at every share up to a tenth, 3.07 at a quarter and 2.26 at a half. The more green a sky carries, the less it overshoots, because green raises the intensity the curve sees and so brings more of the colour into the range the curve compresses. On the 300 cd/m² display the overshoot is flat at 3.33 — exactly the ratio of the master’s peak to the display’s — for every sky the curve leaves unchanged, since those colours arrive at the display with their master’s absolute light.

The overshoot is largest exactly where a display’s blue matters most. The green primary decides how sharp the blue is and brightness and the green sharpen the blue almost independently found a display’s most saturated blues the colours whose difference metric changes fastest; a broad green sharpens the blue, a broad blue dulls it found the blue’s corner sensitive even to a primary’s spectral width. The intensity curve sends those colours to the gamut’s boundary and beyond it by a factor of several, so whatever the display does at its boundary is done to its most sensitive colours.

A grader working on the master sees none of this. The master has the light to show every one of these skies, and the mapped colours’ ICtCp — hue intact, chroma reduced a little, intensity lowered — looks like a careful tone map in any ICtCp plot. Only the conversion to the target display’s drives shows that the colour cannot exist there.

Clipping restores the per-channel turn #

Clipped channel by channel at full drive, a tenth-green sky’s green arrives at 65 per cent of the blue’s drive on the 100 cd/m² display, against the per-channel curve’s 69 and the 13 per cent the intensity mapping asked for; a quarter of green arrives at full green beside full blue. The blue is held at full drive while the green, which asked for little more than the display has, keeps nearly all of it, and the share the master graded is multiplied by six.

That is the per-channel curve’s mechanism arrived at by a different road. The per-channel curve compresses the bright blue channel heavily and the dim green lightly; clipping compresses the blue to full drive and the green not at all. On the 300 cd/m² display the two are identical for every sky up to a tenth of green — the clipped intensity curve’s hue changes are −8.7 and −16.7 units, the per-channel curve’s −8.7 and −16.7 — because there the intensity curve leaves those colours exactly as graded, and clipping a graded colour channel by channel is what the per-channel curve does above its knee.

Clipped, a tenth-green sky’s lightness goes from 31.5 to 65.3, close to the per-channel curve’s 66.3; scaled into the gamut, to 34.5, three units above the master’s. The scaled return is the gentler one: dividing every drive by the largest keeps the drives’ proportions, and with them the hue and nearly the lightness the intensity mapping had chosen.

Scaled back, the intensity curve becomes the scaled curve #

The scaled return deserves a closer look, because it is not quite the scaled curve. The scaled curve applies BT.2390 to the largest channel’s luminance and multiplies all three by the same ratio; the intensity curve with a scaled return applies BT.2390 to the intensity, scales the chroma, and then divides all three drives by whatever is needed to bring the largest to full. On the 300 cd/m² display, where the intensity curve leaves skies up to a tenth of green untouched, the scaled return gives exactly the scaled curve’s colours — the same hue, the same lightness, the same CAM16-UCS distance of 2.7. On the 100 cd/m² display the two differ a little: the intensity curve’s chroma scaling, 0.89 at a tenth of green, desaturates the colour before the return, and the return then carries that desaturation down with it.

At a tenth of green on the 100 cd/m² display the CAM16-UCS distance from the master is 17.6 through the per-channel curve, 4.0 through the scaled curve, 17.0 through the intensity curve clipped and 4.9 scaled back into the gamut; at a half of green, 11.2, 3.9, 11.6 and 5.6. The four curves form two pairs at every share with green in it: the two that clip land within a unit of each other, the two that scale within two. A pure blue is the exception that proves the mechanism — the intensity curve’s chroma scaling adds a little green and red to it, which clipping the blue then keeps, so the clipped pure blue lies 8.4 units from the master where the per-channel curve leaves a pure blue at 4.1. The intensity curve’s chroma scaling adds about one unit of appearance difference at a tenth of green and two at a half — the cost of desaturating the sky by a tenth to a fifth for a reason the display then overrides.

A tone-mapped blue turns where it lands found a pure blue’s corner in ΔEITP set by the level it is delivered at and nothing else. The same principle reaches further here. What the viewer sees is set by the drives the display finally shows, and of the two decisions between the master and those drives — the tone curve and the return to the gamut — the second can undo the first entirely.

What a display engineer should take from this #

A tone curve on ICtCp intensity is not a complete tone map for saturated colours. A saturated blue’s intensity sits far below its blue channel’s level, so the curve leaves it bright and the display must bring it back into the gamut. The mapping’s hue is only as good as that return.

Bring colours back into the gamut by scaling, not by clipping. A violet primary is no spike to the quantiser found a display’s blue corner in ΔEITP sharpest with the blue primary at 485 nm and blunter at every step towards 440 — the deep blues are where a display’s colour behaviour changes fastest with its primaries, and they are also the colours a channel clip distorts most, because their blue channel is the one that overflows. Clipping a too-bright blue channel by channel reproduces the per-channel curve’s turn towards cyan almost exactly; scaling all three drives by the largest keeps the hue the intensity mapping chose.

For a sky, the largest-channel scaled curve is simpler and slightly better. It never leaves the gamut, so it needs no return, and it lands a unit or two nearer the master’s appearance than the intensity curve scaled back.

How the colours were mapped and read #

The display has primaries at the spectral locus at 630, 530 and 465 nm and a D65 white. Each mixture is the blue at full drive plus a share of the green, graded on a 1,000 cd/m² master. Its ICtCp is computed from its absolute XYZ through the BT.2020 matrix, the fixed LMS matrix and the PQ response; BT.2390’s curve, written for luminance, is applied to the intensity by taking the intensity’s PQ level as a luminance, mapping it, and encoding the result again; the two chroma axes are multiplied by the smaller of the new intensity over the old and its inverse. The mapped ICtCp is inverted through the same steps to the display’s drives, which reproduce it to a millionth. A clipped colour has each drive limited to the range nought to one; a scaled one has all three divided by the largest. Every colour is read in CIECAM16 relative to its display’s own white, with an adapting luminance a fifth of the peak, a background of twenty and an average surround, as the other two curves were.

What this does not reach #

Better gamut returns. Clipping and scaling are the two simplest; a return that holds ICtCp’s intensity and compresses only its chroma would land somewhere else again, and is the natural next question.

Other colours. A blue is the extreme case of a colour whose intensity is far below its brightest channel; a yellow or a red is nearer, and the intensity curve compresses those more fully.

BT.2390’s own refinements. The recommendation’s full method adds a black-level lift and a saturation-dependent adjustment; this applies the core curve and its chroma scaling only.

Still open: whether a chroma-only return keeps both #

The intensity curve’s colours have the hue the master graded and an intensity the display could show if its chroma were lower; they fail only because a saturated blue at that intensity lies outside the gamut. A return that holds the intensity and the hue and reduces only the chroma until the colour fits would keep the lightness the curve chose and the hue it kept.

The calculation is each mixture mapped by BT.2390 on ICtCp intensity, then brought into the 100 and 300 cd/m² displays’ gamuts by scaling its two chroma axes alone, at fixed intensity, until every drive lies between nought and one, read in CIECAM16 as here. The prediction is that the hue stays within three quadrature units and the lightness rises by ten or more — the intensity curve’s own lightness, now displayable — while the chroma falls by half, so that the sky arrives as a paler blue of the right hue. If the chroma-only return moves the hue by more than five units, ICtCp’s hue lines are not constant-hue lines in appearance once chroma changes, and every return must be judged in an appearance model.

Lowering only the chroma turns the sky white and its hue with it made that calculation: ICtCp’s hue and intensity are kept, the chroma must fall to a fifth or less, the sky goes pale and its hue moves eight to seventeen units toward green.

A decision taken after the one being judged #

The habit here is about what a stage in a pipeline can be credited with.

The intensity curve was judged on its own colours and passed: its hue held. But its colours were not ones a display could show, and the step that brought them back — not part of the curve, not part of the question — decided the final hue entirely, reproducing the very turn the curve was chosen to avoid when that step was a clip, and the curve’s own hue when it was a scale.

The failure mode is to judge a stage of a colour pipeline by its own output when a later stage, chosen separately, can rewrite that output wholesale. The test is to carry every candidate through to what the display actually emits before comparing them, and to report which stage made the difference.