A light's distance from the equal-signal colour sets when it goes

A display that adapts its picture to the room reads the room’s light through an ambient sensor and judges its reading by a dim test picture: a moon at a tenth of a candela per square metre, where a declared white that is off shows as a tint in ΔEITP, and a unit of tint is the line between a reading that serves and one that does not. The census holds fifty-one lights — lamps, LEDs, tubes, mixtures — and a seven-channel sensor that seven channels end the trade and lose the tube and its successors built to read every one of them within that unit. Seven channels carry their noise as lightly as three and a tilt moves every light’s reading alike found it keeping them there through independent noise and a gain tilt.

A dark offset pulls every light toward one colour added the same signal to every channel, as an uncorrected dark current does. Every light’s reading moved straight toward one colour — the one the sensor’s matrix reports for equal signals in every channel, which lies at the blue end of the room’s lights — and the seven-channel sensor held every light within a unit to an offset of two per cent of each light’s strongest channel signal, its worst at 0.94, and lost sixteen at five. The sixteen were the warm ones: every 2,700 K LED, the 2,200 K radiator, two warm mixtures and the 3,000 K triphosphor sibling.

Between two and five per cent the census held no readings, and the offset’s geometry made a prediction possible for each light separately. A light’s reading moves along a straight line toward the equal-signal colour, by an amount that grows with the offset, so a light far from that colour should go sooner. The prediction was that the first light is lost near three per cent and is the census’s warmest, that a light’s crossing offset falls with its distance from the equal-signal colour, and that its tint grows nearly linearly in the offset from its own starting value, so that the crossing is predicted within a fifth from two numbers a light. If the crossings did not follow the distance, the moon picture’s sensitivity to a white’s error would differ by direction.

One light goes at two per cent, twelve together near four #

The seven-channel sensor loses its first light at an offset of 2.12 per cent — the 2,200 K radiator, the census’s warmest light. It holds every other light to 3.5 per cent; at four it has lost four, at four and a half fourteen, at five sixteen, at six seventeen and at eight nineteen. The warmest goes first, as predicted, but a third sooner than predicted.

The count’s shape is the room’s shape, read from its warm end inward. The radiator stands alone at the warm end of the census and goes alone; the next three lights to go are a warm mixture and two of the LEDs, and then ten more LEDs go inside the next half per cent. Above five per cent the count creeps upward one light at a time, as the census’s cooler lights, nearer the equal-signal colour, cross in turn.

Below half a per cent nothing moves #

The 2,200 K radiator’s tint is 0.24 with no offset, 0.24, 0.25 and 0.26 at a tenth, a quarter and a half of a per cent, and then 0.42 at one, 0.94 at two and 1.89 at four. For every light the sensor loses, the tint rises by under a twentieth up to half a per cent and by at least four times as much between one and two per cent. The prediction’s third part fails in form: the tint does not grow linearly from its starting value. It lies flat, then climbs in a nearly straight line.

The flat stretch has a plain cause in the geometry. A light’s tint with no offset is the sensor’s own error on that light, a small vector in chromaticity; the offset adds a second vector pointing toward the equal-signal colour, growing with the offset. Where the second is small beside the first and not parallel to it, the sum’s length barely changes; once the second dominates, the tint grows with it, nearly in proportion. So the tint follows something close to the length of a sum of two vectors, which is flat while one is small and linear once it is large — a hockey stick, not a straight line, and the knee is near half a per cent.

The further from the equal-signal colour, the sooner a light goes #

Among the nineteen lights lost by an eight-per-cent offset, the crossing offset falls with the light’s distance from the equal-signal colour at r = −0.93. The furthest, the 2,200 K radiator at 0.091 in CIE 1960 uv, crosses at 2.1 per cent; the nearest, the 3,500 K triphosphor sibling at 0.035, at 7.9. The second part of the prediction holds.

The relation is not exact, and the residue is informative. At one distance — the twelve 2,700 K LEDs’ readings all lie within 0.0006 of each other, near 0.063 — the crossings spread over half a per cent, so something besides the distance enters. The natural candidates are each light’s own error with no offset, which the pull adds to at some angle, and how its signal is spread across the channels, since the offset is set as a share of the strongest one; this census does not separate them. Distance sets the order across families; something in the recipe sets it within one.

Two readings a light predict where it crosses #

A straight line through each lost light’s tints at one and two per cent predicts its crossing between 0.2 and 7.7 per cent early, never late. The third part of the prediction holds in substance — the crossing from two numbers a light, within a fifth — even though its stated form, linear from the starting value, was wrong. Above the knee the tint is nearly straight, and two readings above the knee find the straight part.

That the error is always early, and grows with the crossing offset, says the climb eases slightly as the offset grows. A sum of two vectors grows a little less than linearly once the larger term turns the sum toward itself, and the furthest-out crossings, extrapolated furthest, show it most. For a sensor designer the asymmetry is the useful part: two readings give a crossing that is safe.

Twelve warm LEDs cross within half a per cent #

The twelve 2,700 K LEDs cross between 3.87 and 4.39 per cent, nine of them within a fifth of a per cent of the middle. They share a chromaticity, and so a distance from the equal-signal colour, and they cross as a group; what spreads them is their recipes, the LEDs pumped at 452 nm with no red phosphor going first and those pumped at 440 nm with a full red going last.

This bunching is what made the count jump from four to fourteen between four and four and a half per cent. It is the same kind of fact each direction falls short by one band’s ratio found in a different census: when many cases share the quantity that decides them, a count over a threshold moves in a lump as the threshold passes them, though nothing about any one case changes abruptly.

At eight per cent the warm half of the room is lost #

At an eight-per-cent offset the nineteen lights lost all lie at a CIE 1960 u of 0.237 and above; the thirty-two kept all at 0.235 and below. The split is clean: on this axis alone, which runs from the coolest daylight to the warmest radiator, every light on the warm side of 0.236 is lost and every light on the cool side kept.

A room sensor must read the lamp to a hundred kelvin found the warm end of the room the hardest for this picture to forgive, because a warm white declared wrong tints a dim moon visibly. The offset compounds that: it attacks the warm end hardest, because the warm end is furthest from where the matrix sends an equal-signal input. A sensor that must work in dim, warm rooms — the evening the moon picture models — meets its worst offset error and its least forgiving lights together.

Three disturbances, three shapes #

The seven-channel sensor has now been disturbed three ways, and each left a different shape on the room’s fifty-one lights. It is worth setting them side by side, because the shape says what a correction must do.

Independent noise of one per cent on every channel scattered each light’s reading about its true value and lost none; a gain tilt of two per cent at the ends of the range moved every light by nearly the same vector and lost none; a dark offset of two per cent drew every light toward the equal-signal colour and lost none, and at 2.1 per cent lost the first. The three are similar in size where they begin to matter, and opposite in structure. Noise has no direction and averages away over repeated readings. A tilt has one direction for the whole room, which a single reference reading of any known light measures and removes. An offset has one point for the whole room, which a single dark reading measures and removes — and which no reference light can, because a light’s reading under an offset depends on how far it sits from the point.

Relatives, not twins, teach a sensor a tube found the census’s hardest light to fit, a triphosphor tube, read correctly only once its relatives were in the fitting set; under an offset, the tube itself is never lost to eight per cent, and of its siblings only the two warmest go — the 3,000 K sibling at 4.5 per cent and the 3,500 K one last of all, at 7.9 — because the family’s chromaticities sit nearer the cool end, toward the equal-signal colour. The lights that were hardest to calibrate and the lights that are hardest to protect are different lights, and a sensor’s error budget has to name both.

The warm end’s exposure has a history in this census too. A sensor calibrated on lamps cannot read an LED found warm LEDs the first casualty of a fit that had not seen them, and lamp and daylight together sit below the locus found warm-and-cool mixtures leaving the locus where a sensor’s declared white must still be right. Under an offset the warm LEDs go together and early again — this time not because the fit lacked them but because the matrix sends an equal-signal input to the cool end, and they are the lights furthest from it.

So the order in which a room’s lights fail is not one property of the sensor, fixed by its channels and its fit, but one per disturbance, and the dark offset’s order is the simplest of the three to state: by distance from one point, warmest first.

What a sensor designer should take from this #

Budget a dark offset by the warmest light the sensor must read. The first loss is the light furthest from the equal-signal colour; on this sensor that is a 2,200 K radiator at 2.1 per cent of its strongest channel. Every cooler light allows more.

Expect nothing below half a per cent. The offset’s pull is lost in each light’s own error until it grows past it; a sensor whose dark current stays under half a per cent of its weakest-room signal needs no dark subtraction for this picture.

Two readings find the margin. A light’s tint at two offsets above the knee predicts where it crosses, early and within a twelfth; a calibration that measures the tint at one and two per cent knows the crossing without searching for it.

How the crossings were read #

The census holds fifty-one lights; the sensor has seven Gaussian channels between 445 and 650 nm; each light is read by the matrix fitted by least squares to the other fifty. The offset adds to every channel the same amount, a stated share of that light’s largest channel signal, at nought, a tenth, a quarter and a half of a per cent, one, two, every half per cent to five, six and eight. A light’s tint is the moon picture’s ΔEITP error when the display declares the reported white; its crossing is found by straight lines between readings; its distance is from its offset-free reading to the equal-signal colour, the matrix fitted on all fifty-one applied to a vector of ones, in CIE 1960 uv. The prediction from two readings extends the straight line through the tints at one and two per cent to a unit.

What this does not reach #

The offset is a share of each light’s own strongest channel. Converting the crossings into room illuminance needs the sensor’s dark current in absolute terms and each light’s signal at each level, which the census does not model.

One sensor. The equal-signal colour, and so the order, are this seven-channel matrix’s; a sensor fitted to a different set of lights would send an equal-signal input elsewhere.

Equal offsets in every channel. A channel-by-channel dark current is a different vector and a different colour, and would reorder the crossings.

Still open: whether placing the equal-signal colour among the lights moves the first loss out #

The first light lost is the one furthest from the colour the matrix reports for equal signals, and that colour lies at the blue end of the census because nothing in the matrix’s fit asks it to lie anywhere. A matrix fitted with one more constraint — that a vector of equal signals reports the centre of the room’s lights — would put it in the middle.

The calculation is the seven-channel sensor’s matrix fitted to the fifty lights with an added row requiring equal signals to report the census’s mean chromaticity, read with each light left out as here, at every offset from nought to eight per cent. The prediction is that the constrained matrix costs no light at no offset, keeps every light within a unit to at least three per cent, and loses its first light — now the census’s coolest or its warmest, whichever is further from the centre — at least half a per cent later than the 2.1 per cent here. If the constraint costs lights with no offset, the equal-signal colour is a property the sensor’s channels fix and the fit cannot move without misreading the room.

A curve that was a sum #

The habit here is about the shape of an error built from two parts.

The prediction expected each light’s tint to grow in a straight line from where it started, because the offset’s pull grows in proportion to the offset. The pull does; the tint does not, because the tint is the length of the pull added to the light’s own error, and a length of a sum lies flat while one term is small and climbs once it dominates. Read that way, the first half per cent costs nothing, the crossing comes from the straight part, and two readings on the straight part predict it.

The failure mode is to predict the size of an error from the size of the one part that changes, when the error is the length of that part added to another. The test is to write the error as a sum of its parts, see which dominates over the range in question, and extrapolate only from the stretch where the changing part does.