Lamp and daylight together sit below the locus

A display’s shadows sit on a floor made of its own black and the room’s light reflected off the screen — two parts a dimming panel has no single black found folding into one declared number — and ΔEITP — the colour difference built for high-dynamic-range television — reads those shadows correctly only when it is told that floor, as a guessed veil halves the error first found for the room’s share of it. A warm room tints the black, not the halo found a floor declared in grey leaving a dark picture’s shadows tinted towards a 2,700 K lamp by up to 17.7 ΔEITP units at a tenth of a candela on the screen, and a floor declared in the room’s own colour removing nearly all of it.

A room sensor must read the lamp to a hundred kelvin asked how precisely a display must know its room for that declaration to work, and found, in the moon picture at a tenth of a candela, about fourteen mireds in temperature, twelve per cent in level, and a Duv of 0.003 across the Planckian locus — the line of colours that heated bodies glow, along which a colour temperature is a position. A sensor that reports only a temperature declares a point on that line, and any room whose light is not on it is declared wrongly by its distance off.

The sensor essay’s rooms were all lit by a single 2,700 K radiator, on the locus by construction. An evening room is often lit by a lamp and the last of the daylight together, a mixture whose chromaticity is neither the lamp’s nor the day’s. The prediction was that a lamp-and-daylight mixture sits well above the locus — by more than the 0.003 the census allows — so that declaring its correlated colour temperature tints the moon picture beyond a unit at a tenth of a candela for every mixture, while declaring its true chromaticity stays under a unit throughout.

The mixtures sit below the locus, and the temperature fails only midway #

A 2,700 K lamp mixed with D65 daylight, the daylight from a tenth to three quarters of the light’s luminance, sits below the Planckian locus at every share: a Duv of −0.0020 at a tenth, −0.0041 at a quarter, −0.0048 at a half, −0.0022 at three quarters. Declared as a Planckian lamp at the mixture’s correlated temperature, the moon picture’s shadows at a tenth of a candela are tinted by 0.60 units with a tenth daylight, 1.11 with a quarter, 1.24 with four tenths, 1.16 with a half, 0.95 with six tenths and 0.48 with three quarters. Daylight alone, at D65’s own +0.0032 above the locus, is tinted by 0.67 declared from its temperature, and by under a hundredth declared in grey. Declared in the mixture’s own chromaticity, every share at every level is tinted by no more than the halo’s residue in the single-lamp room.

• The prediction is wrong about where the mixtures sit. They are below the locus, pinkish, not above it.
• It is right that the temperature is not enough, for a quarter to a half daylight at a tenth of a candela, and at more shares in a brighter room.
• The chromaticity is enough every time. Two numbers declared, the mixture’s tint is the single lamp’s.

The three curves span three orders of magnitude, and the axis is logarithmic. The grey declaration falls steadily as daylight replaces the lamp, from 17.7 units with the lamp alone to 1.6 with nine tenths daylight and to under a hundredth with daylight alone, because D65 is the display’s own white and a grey declaration is exactly right for a daylit room. The chromaticity declaration sits at the floor throughout, 0.19 with the lamp alone — the halo’s residue — and falling as daylight rises. The temperature declaration rises from the floor, peaks above a unit between a quarter and a half daylight, falls to 0.16 at nine tenths, and rises again to 0.67 with daylight alone.

Moved to a hundredth of a candela, every tint is a quarter as large, and the temperature declaration never reaches a unit. Moved to one candela, it passes a unit from a tenth daylight to three quarters, peaking at 2.8, and daylight alone declared from its temperature is tinted by 1.39.

Why a mixture sits below the locus #

The prediction reasoned from a familiar fact: daylight lies above the Planckian locus, on its green side, and so does a mixture that is mostly daylight. But a mixture of two lights lies on the straight line between them in a chromaticity plane, and the Planckian locus between 2,700 and 6,500 K curves upward — towards green — in the middle of that range. A straight line between a point on the locus at 2,700 K and a point just above it at 6,500 K passes under the bulge. The mixtures from a tenth to three quarters daylight lie on that chord, at most 0.0049 below the curve near four tenths daylight, and only the mixtures that are nearly all daylight rise past it.

So a lamp-and-daylight room is pinkish against the locus, not greenish, and a temperature-only declaration puts its white slightly too green. The chord crosses back above the locus at about 86 per cent daylight, between three quarters (Duv −0.0022) and nine tenths (+0.0008), where the mixture’s correlated temperature is near 5,600 K; above that the mixture is greenish, as daylight is, and below it pinkish. The sign of a temperature-only declaration’s error therefore flips across an evening: too pink while daylight still dominates the room, too green once the lamp takes over, and the tint’s direction flips with it. A display that corrected its shadows by a fixed offset learned in one part of the evening would make the other part worse. The size of the miss is set by how far the chord sags below the curve, and the sag is largest where the two lights contribute equally, which is why the temperature declaration fails in the middle of the range and not at its ends.

Read as a Duv, the mixtures make a shallow U below the axis: −0.0020 at a tenth, −0.0041 at a quarter, −0.0049 at four tenths, the deepest, −0.0048 at a half, −0.0022 at three quarters, +0.0008 at nine tenths, and +0.0032 for daylight alone. Their correlated temperatures run from 2,889 K at a tenth daylight to 3,976 at a half and 5,036 at three quarters. The sensor census’s tolerance, a Duv of about ±0.003 at a tenth of a candela, is crossed from about a sixth daylight to about two thirds, and the tint curve crosses a unit a little inside those, at about a fifth and three fifths. The Duv alone predicts the failure, and a sensor that reports a Duv beside its temperature reports what the declaration needs.

The distance off the locus is the tint #

A temperature declaration gets a mixture’s white wrong by one thing only: the declared white sits on the locus at the mixture’s correlated temperature, and the true white sits off it by the mixture’s Duv. If that distance is what tints the shadows, the tint should be proportional to it. At a hundredth of a candela it is, almost exactly: between 57 and 61 units of tint per unit of Duv for every mixture from a tenth daylight to all. In brighter rooms the points still fall close to straight lines through the origin, but the warm mixtures’ lines are steeper than the cool ones’ — 300 units of tint per unit of Duv for a tenth daylight at a tenth of a candela and 209 for daylight alone; 739 and 435 at one candela.

The proportion is the practical result. A display that knows the room’s light level and the Duv of its light can predict the tint a temperature-only declaration would leave, without computing the picture: at a tenth of a candela a Duv of about 0.004 costs a unit, and the mixtures’ worst, 0.0049, costs 1.24. That is the sensor census’s tolerance of 0.003, met from the other direction; a sensor’s Duv error and a mixture’s own Duv are the same kind of error to the shadows, and they add.

The falling slope with more daylight means a cool room forgives a given Duv a little more than a warm one. Why it falls the census does not take apart; it reports the slope, which is what a display needs in order to use the rule.

Daylight alone is off the locus too #

A daylit room needs no declaration at all: D65 is the display’s own white, and a grey declaration is exact. A sensor that reads a daylit room’s temperature, 6,505 K, and declares the Planckian lamp at that temperature misplaces the white by daylight’s own Duv of +0.0032, tinting the moon picture’s shadows by 0.18 units at a hundredth of a candela, 0.67 at a tenth and 1.39 at one. The most common bright room a display sits in is the one a temperature-only declaration gets wrong by the most at one candela, and the one in which no declaration was needed.

That is the general version of the mixture’s problem. The CIE’s daylights lie on their own curve, above the Planckian locus, and lamps of other kinds — fluorescent tubes, LEDs — lie wherever their phosphors put them. A declaration needs the light’s chromaticity, and a sensor that converts a measured chromaticity to a temperature and back throws away exactly the coordinate that separates daylight from a heated body. Nor does a viewer adapted to the room discount what remains: adapting to a warm room deepens the shadows’ tint found adaptation to a warm lamp deepening a dark picture’s tint rather than removing it.

Every level, declared from the temperature #

The tint scales with the room’s light, as the sensor census found, and the three curves are the same shape at three heights; the one-unit line they are judged against does not move with the room, a choice a tolerance has no light level examined. At a hundredth of a candela the temperature declaration’s worst tint is 0.29 units; at a tenth, 1.24; at one candela, 2.82 — each at four tenths daylight, where the mixture sits furthest below the locus. A dim room forgives the mixture entirely; a dim room with a lamp on forgives it at the ends of the range; a lit room forgives it only when the light is all lamp or nearly all daylight.

Brighter pictures forgive the mixture #

The moon picture is the census’s darkest, and the earlier census found every picture’s tolerance growing with how much of the screen is bright — partly because a bright picture puts light into the viewer’s eye, the veil the eye’s own glare hides most of the halo found hiding a floor’s misreadings near the light. The mixtures follow the same order. At a tenth of a candela, with four tenths daylight, the temperature declaration tints the moon picture by 1.24 units, the subtitle picture by 0.63 and the window picture by 0.18; at one candela, by 2.82, 2.43 and 1.30. A dim room with a lamp on and some daylight left fails the dark film scene and passes the evening news; a brightly lit room fails all three.

So the question a display maker has to answer is not whether a temperature-only sensor is good enough on average, but whether it is good enough for the darkest content the display will show in the brightest mixed room it will sit in. For the moon picture at a tenth of a candela, every mixture from about a fifth daylight to three fifths is outside the unit, and many living rooms pass through that mixture at dusk.

What a display maker should take from this #

A room sensor that will be used to declare a display’s floor should report a chromaticity, or a temperature and a Duv, not a temperature alone. With the chromaticity, every mixture of a warm lamp and daylight is declared as well as a single lamp is; with the temperature alone, a dim room with a lamp on and a little daylight left is tinted beyond a unit, and a brightly daylit room is tinted by more than a unit for no reason at all. The input nobody declared argued that every input a colour pipeline assumes should be stated; here the input must be stated in the right coordinates as well.

The prediction got the arithmetic of mixtures wrong in an instructive way. It treated daylight’s position above the locus as the fact that would carry over to a mixture, when the fact that carries over is the shape of the locus between the two lights. Mixtures of two whites sit below the locus wherever it bulges between them, and every evening room is a mixture.

How the mixtures were built and read #

The panel, pictures and declarations are the warm-room census’s: a 1,000 cd/m² local-dimming panel of 32 × 18 zones, the moon picture on the middle diffuser, its two floors declared and the eye’s veil added. The room’s light is a 2,700 K Planckian radiator and the CIE D65 daylight spectrum added in the proportions that give daylight a share from nought to all of the luminance, reflected off the screen at 0.01, 0.1 or 1 cd/m². Each mixture’s correlated colour temperature and Duv are read in the CIE 1960 plane against the Planckian locus.

The floor is declared three ways: in D65 grey at the room’s level; in the colour of a Planckian radiator at the mixture’s correlated temperature, at the room’s level; and in the mixture’s own chromaticity. The tint is the chroma part of ΔEITP between a grey on the declared floor and the same grey on the true one, the worst over every zone and dark grey. With no daylight, the temperature and chromaticity declarations are the sensor census’s exact reading.

What this does not reach #

Two lights. A real evening room may add a television’s own glow, a second lamp of another temperature, or daylight already tinted by the sky’s colour at sunset; each adds a point off both ends of the chord.

Equal geometry. The census adds the two lights before they reach the screen. A window and a lamp light a screen from different directions, and its reflection may favour one.

The moon picture. The darkest picture is the most demanding; the sensor census found the bright window picture tolerating six times as much.

Still open: whether a sensor’s own chromaticity error undoes the declaration #

A sensor that reports a chromaticity reports it with an error, and the census here declares the mixture’s chromaticity exactly. Small ambient sensors read light through three or four broad filters, and their chromaticity errors are largest exactly where lamps and daylight differ most, in the blue.

The calculation is a modelled three-channel sensor with filters like a small ambient-light sensor’s — broad Gaussian channels near 450, 540 and 610 nm — calibrated on a 2,700 K and a 6,500 K source, reading each mixture and each of a census of white LEDs, its chromaticity error turned into a declaration and the tint read as here. The prediction is that a three-channel sensor reads the lamp-and-daylight mixtures within a Duv of 0.001, because they are mixtures of its two calibration lights, and misreads phosphor LEDs by 0.003 to 0.006, tinting the moon picture beyond a unit at a tenth of a candela under the LEDs. If the mixtures are misread too, a sensor’s calibration must span more than two lights before its chromaticity can be declared. A sensor calibrated on lamps cannot read an LED makes that calculation.

The carried fact and the carrying fact #

The habit here is about which property of the parts carries over to a whole.

The prediction knew two facts about the parts: the lamp is on the locus and daylight is above it. It carried the second to the mixture, reasoning that a mixture with daylight in it inherits daylight’s offset. The fact that actually governs a mixture’s position is a property of neither part: the shape of the curve between them, against which a straight-line mixture sags.

The failure mode is to predict a mixture’s property from its components’ properties when the property is defined against a curve. The test is to place the mixture where mixing puts it — on the chord — and measure it against the curve there. Here that put every evening room below the locus and turned a prediction about greenish mixtures into a finding about pinkish ones.