A sensor calibrated on lamps cannot read an LED

A display’s shadows sit on a floor made of its own black and the room’s light reflected off the screen, and ΔEITP — the colour difference built for high-dynamic-range television — reads those shadows correctly only when the display is told what that floor is. 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 many units at a tenth of a candela on the screen, and a floor declared in the room’s colour removing nearly all of it.

Two essays since have asked what declaring the room’s colour requires. A room sensor must read the lamp to a hundred kelvin found that, in the moon picture at a tenth of a candela, a sensor must read a warm lamp to about fourteen mireds and to a Duv of 0.003 off the Planckian locus. Lamp and daylight together sit below the locus found that a room lit by a lamp and the last of the daylight cannot be declared from its colour temperature at all — its light sits off the locus, below it — but can be declared from its chromaticity, which keeps the tint at the halo’s own small residue for every mixture.

That last result assumed the chromaticity was known exactly. A display learns it from an ambient-light sensor: a chip a few millimetres across behind the bezel, with three or four broad colour filters over photodiodes and a matrix that turns their signals into an estimate of the room light’s XYZ. The prediction was that a three-channel sensor calibrated on a 2,700 K and a 6,500 K source reads the lamp-and-daylight mixtures within a Duv of 0.001, because they are mixtures of its two calibration lights, and misreads phosphor-converted LEDs by 0.003 to 0.006, tinting the moon picture beyond a unit under them. The alternative was that the mixtures are misread too, in which case the calibration must span more than two lights.

Smooth lights read; spiky ones do not #

A modelled sensor of three Gaussian channels — 450 nm and 80 wide, 540 and 90, 610 and 90 — whose matrix is the least-squares copy of the CIE 1931 functions that reads a 2,700 K and a 6,500 K Planckian lamp exactly, misreads the lamp-and-daylight mixtures by up to 0.0023 in the CIE 1960 plane, rising with the daylight and passing 0.001 at half; at a tenth of a candela on the screen no mixture’s tint reaches a unit, the worst 0.54. It misreads Planckian lamps from 2,200 to 8,000 K by 0.0010 to 0.0033. It misreads every one of thirty white LEDs built onto the locus by between 0.0073 and 0.0172 — above the predicted ceiling of 0.006 for every one — and every one tints the moon picture by between 1.66 and 3.60 units. Three fluorescent tubes are misread by 0.009 to 0.016. Red phosphor lowers an LED’s error at every pump and temperature, from 0.013–0.017 with none to 0.007–0.013 with the most.

• The prediction for the mixtures nearly holds. They are not quite mixtures of the calibration lights, because daylight is not a 6,500 K lamp, and the error grows with the daylight; but it stays inside what the shadows tolerate.
• The prediction for the LEDs is three times too kind. Every LED fails, and fails by more than a unit and a half.
• The divide is the spectrum’s shape, not its colour. Every smooth light is read within 0.0035; every light with a narrow emitter is misread by more than 0.007.

The forty-seven lights fall into two bands with nothing between them. The lower band is every Planckian lamp and every mixture, from 2,200 to 8,000 K, all under 0.0035; the upper is every LED and every fluorescent tube, from 0.0073 up. At the same correlated colour temperature, 4,000 K, a Planckian lamp is misread by 0.0020 and the twelve LEDs by 0.0073 to 0.0172. Nothing about the light’s colour separates the bands: an LED built onto the locus at 4,000 K has the lamp’s chromaticity to a millionth, and the sensor reports them as different lights.

What three broad channels can copy #

A sensor reads a light’s chromaticity correctly for every spectrum only if some combination of its channels reproduces the three colour-matching functions exactly — the Luther condition, which a filter maker approaches and never meets. Three Gaussian channels cannot meet it. The copy of ȳ is close: the 540 nm channel is nearly the right shape. The copies of x̄ and z̄ share a channel, the 450 nm one, and x̄’s copy carries a broad lift between 450 and 550 nm where the standard function has a trough, and a long-wavelength lobe that is wider and lower than the standard’s. The copy of z̄ is too wide on both sides.

For a smooth spectrum those errors largely integrate away. A Planckian lamp’s power changes slowly across the visible, so a copy that is too high in one place and too low in the next sums to nearly the right answer, and the two calibration lamps fix the sums exactly at two temperatures. A spiky spectrum puts its power where the copy is wrong and nowhere else to cancel it. A white LED is a narrow blue line from its pump and a broad yellow hump from its converter, with a trough between them at about 480 nm; a fluorescent tube is a handful of mercury lines and narrow phosphor peaks — the two shapes a lamp is not a blackbody found a colour temperature saying nothing about. Each delivers most of its blue in a band twenty nanometres wide, and the sensor’s copy of x̄ and z̄ is wrong by different amounts there than across the smooth lamp it was pinned against.

Which way the readings go wrong #

The directions say as much as the sizes. The mixtures are misread straight up the v axis, by up to 0.0023 with daylight alone, and not along u at all: daylight differs from the 6,500 K lamp the sensor was pinned on in its blue and violet, where D65 carries the absorption structure of the atmosphere and more violet than a Planckian lamp, and the sensor’s copies of the colour-matching functions are wrong by different amounts in that region than across the smooth lamp they were pinned against. The Planckian lamps between and beyond the calibration temperatures are misread by a little in both directions, small arrows pointing nowhere in particular.

The LEDs are all misread the same way, towards lower u and higher v — towards green, off the locus they sit on. The arrows are long and nearly parallel, whatever the LED’s temperature. A display reading its room from this sensor would declare an LED-lit room as greener than it is, by about a hundredth in the 1960 plane, and carry that error into the shadows of a dark picture along the green–magenta direction, wherever the room’s light makes up most of the floor.

That consistency is a clue the next essay can use. An error that points the same way for thirty LEDs of three pumps, four red-phosphor strengths and three temperatures is not thirty errors; it is nearly one, scaled. A calibration that knew the room was LED-lit could take most of it out, which is the alternative the prediction named in another form — the calibration spanning more than two lights.

What the reading’s error costs the shadows #

The tint follows the reading’s error almost linearly, about two hundred ΔEITP units per unit of 1960 chromaticity at a tenth of a candela, at a rank correlation of 0.98 across all forty-seven lights. The tint crosses a unit near an error of 0.0045, a little beyond the 0.003 off the locus the room-sensor essay found a temperature declaration tolerating, and every smooth light falls short of it while every spiky light passes it. The least-wrong LED, 465 nm with the most red phosphor at 4,000 K, still tints the moon picture by 1.66 units; the worst, a 440 nm LED with no red phosphor at 4,000 K, by 3.60.

That ordering has one regularity worth taking: red phosphor helps at every pump and every temperature, each step from none to the strongest lowering the error, from 0.0134–0.0172 with none to 0.0073–0.0128 with the most. A red phosphor fills the long end of the spectrum with a broad band, so an LED with more of it is a little less spiky — its power spread more evenly across the region where the sensor’s copy of x̄ is wrong. It is the smooth-against-spiky divide again, inside one family.

The mixtures, against their truth #

The mixtures are the case the earlier essay solved and the one the prediction expected the sensor to keep solved. At a tenth of a candela the true chromaticity’s tint falls from 0.19 with the lamp alone to nothing with daylight alone — the floor declared exactly leaves only the halo’s residue, and daylight leaves none. The sensor’s reading runs the other way, from 0.19 to 0.54, because its error grows with the daylight. At a tenth of a candela that is still under a unit at every share. At one candela — a brighter room, or a picture a viewer has turned down — the reading’s tint passes a unit from about three quarters daylight, 1.13 with daylight alone, where the true chromaticity’s tint is nothing.

So the prediction was right to expect the mixtures to be the easy case and wrong about why. A mixture is not a mixture of the calibration lights unless its daylight is a Planckian lamp, and there is no D65 lamp: the daylight standard is a measured spectrum with the atmosphere’s absorption in it, and a sensor calibrated on a radiator at 6,500 K reads it the way it reads any other light it was not shown — well, here, because daylight is smooth.

A brighter room multiplies everything #

The room’s light on the screen sets how much of the shadows’ floor is the room, and so how much a wrong declaration of it costs. Each light’s tint rises about fourfold for each tenfold of room light. At a hundredth of a candela — a nearly dark room — the LED’s misreading costs under a unit, 0.91 for the 452 nm LED at 4,000 K; at a tenth it costs 3.55, and at one, 7.7. The triphosphor tube follows it a quarter lower. Daylight and the Planckian lamp sit a factor of six to eight below at every level; daylight passes a unit only at one candela, and the lamp not at all.

The level is the display’s to know as well as the colour, and a room sensor must read the lamp to a hundred kelvin found it tolerating twelve per cent of error in level against far less in colour. Here the colour error alone, at every LED, outweighs what twelve per cent of level would cost.

What a display maker should take from this #

A three-channel sensor calibrated on incandescent-like sources is not a chromaticity sensor in an LED-lit room. It reads a lamp, the evening’s mixture of lamp and daylight, and daylight itself well enough to keep a dark picture’s shadows under a unit at a tenth of a candela. It misreads every white LED the census holds by enough to tint them by two to four units, in a direction — towards green — that a viewer would see as magenta shadows.

Declaring the room’s chromaticity is only as good as the sensor’s worst light. The earlier essay’s conclusion, that the chromaticity declaration works for every mixture, still holds in principle and holds in practice for the lights the sensor was built for. In an LED-lit room it carries the sensor’s error straight into the shadows, two to four units of it, with nothing in the display to say so.

The fix belongs in the calibration, and the arrows say what it should know. The LEDs’ errors point one way, so a sensor that could recognise an LED’s spectrum — as radiators and LEDs are one family to the sensor found a different, narrow-channel sensor partly can — could apply a correction for it. A phone’s sensor and its camera were found sorting lamps into smooth and structured without a mistake, from the difference between two readings; a single three-channel sensor has no second reading to compare, and its only recourse is a calibration fitted to the lights it will meet.

How the sensor was built and read #

The sensor’s channels are Gaussian sensitivities on the 5 nm grid, centred at 450, 540 and 610 nm with full widths at half height of 80, 90 and 90 nm — a modelled ambient-light sensor, not any one product’s. Its matrix from channel signals to XYZ is fitted row by row: each row is the combination of channels closest in least squares to one CIE 1931 colour-matching function, constrained so that the 2,700 K and 6,500 K Planckian lamps’ estimated XYZ equal their true XYZ exactly. Those two are read to a part in ten million million.

The lights are the nine mixtures of a 2,700 K Planckian lamp and D65 the earlier essay used; the thirty white LEDs LEDs built on the locus still fall off the line built onto the locus — pumps at 440, 452 and 465 nm, four strengths of red phosphor, converters solved to put each white within a millionth of the locus at 2,700, 4,000 or 6,500 K; Planckian lamps at 2,200, 3,000, 4,000, 5,000 and 8,000 K; and triphosphor, halophosphate and broadband fluorescent tubes. Each light’s error is the distance in the CIE 1960 plane between its chromaticity and the sensor’s. The tint is the earlier census’s: the moon picture on a 1,000 cd/m² local-dimming panel of 32 × 18 zones, the worst ΔEITP chroma difference over every zone and dark grey between the shadows as they are and as rendered with the floor declared in the sensor’s chromaticity, at the room’s level.

What this does not reach #

One sensor. The channels’ positions and widths are a modelled choice. A sensor with four channels, or narrower ones, or channels placed to copy the colour-matching functions more closely, would draw both bands differently; the divide between smooth and spiky lights is the mechanism, and its size is this sensor’s.

A viewer who adapts. Every tint here is a colour difference on the screen. Adapting to a warm room deepens the shadows’ tint found the viewer’s own adaptation to the room changing how far such a tint looks wrong; that is not computed for an LED-lit room.

A sensor that sees the screen. A real ambient-light sensor behind a bezel sees the room through a window and sometimes the display’s own light reflected back; neither is modelled.

A sensor without noise. Every reading here is exact arithmetic on the channel signals. A real sensor at a tenth of a candela of room light reflected off the screen is reading a few lux, where photon and dark noise add their own scatter.

Still open: whether a calibration fitted to the room’s lights rescues the LEDs #

The LEDs’ errors are large and nearly parallel, which says the sensor’s matrix is wrong for them in one consistent way. A matrix fitted on more than two lights — a lamp, daylight, and LEDs at two or three temperatures — would trade a little accuracy on the lamps for a great deal on the LEDs, and a matrix fitted in least squares to every light in a census would spread its error over all of them.

The calculation is this sensor recalibrated three ways: exactly on the 2,700 K lamp, D65 and one 4,000 K LED; in least squares on the lamp, daylight and six LEDs; and in least squares on every light in the census, each LED and tube left out in turn and read by the matrix fitted without it. The prediction is that the least-squares matrix brings every LED under 0.004 and under a unit at a tenth of a candela, while the Planckian lamps rise towards 0.003; and that the three-light matrix rescues the LEDs near the one it was shown and fails the rest. If no matrix brings the LEDs and the lamps under a unit together, three broad channels cannot serve a room that might be lit by either, and the sensor needs a fourth channel, not a better calibration. No calibration reads the lamps and the LEDs together makes that calculation.

The light it was shown #

The habit here is about what a calibration certifies.

A sensor calibrated on two lamps reads those two lamps perfectly, and the prediction reasoned outward from them: lights that are mixtures of the calibration lights should read well, lights that are not should read less well. The second half was right in kind and wrong by a factor of three, and the first half was right for a reason it did not state.

The failure mode is to judge a light’s distance from the calibration by its colour. An LED at 4,000 K sits exactly between the two calibration lamps in chromaticity and is the worst-read light in the census. The calibration’s reach is set by the spectra it was shown, and what a sensor certifies is that it reads lights shaped like them. The test is to read lights of every shape the room might hold — here, five families — and to see which property of the light the error follows.