Beside the locus the converter still gives the red away

A camera sees a white LED through its own three dyes, not the eye’s cones, and makes a colour error under it that depends on what the LED is made of. A run of essays has tried to read that error from an ambient sensor’s narrow channels before the photograph is taken, and a channel at each pump cannot see the recipe found the error simple in the recipe itself — a line in red-phosphor strength, colour temperature in mireds and their product reads every LED within ±0.07 ΔE₀₀ — and the sensor’s departures unable to read the red phosphor directly.

The converter gives the red phosphor away found the route round. To keep a white on the Planckian locus, each step of red phosphor moves the garnet converter twenty-nine to fifty-five nanometres shorter, and the 600 nm departure, where the converter’s long side falls, reads the converter within a quarter of its spread. Pooled over all thirty LEDs, the 600 nm departure, the mireds and their product read the red phosphor within ±0.12 of a spread of ±0.36, and the camera’s error read through that estimate came within ±0.074.

The tie holds because every one of those whites sits exactly on the locus, and a manufactured LED does not. Makers bin their LEDs by chromaticity, and a bin a few thousandths of Duv either side of the locus is common; an LED 0.003 above the locus has, for the same red phosphor, a different converter from one on it. The prediction was that a Duv of 0.003 moves the converter by roughly what a fifth of the red phosphor does, so that the line learned on the locus misreads the off-locus LEDs’ red by about a fifth of its range, and that their error, read through the misestimate, falls back to ±0.12. The alternative was that the locus line reads the off-locus LEDs as well, which would mean the departure carries the white’s Duv as well as its converter and the tie is not what the reading relies on.

The tie survives, and the error moves #

Rebuilt at a Duv of −0.003 and +0.003, the grid’s LEDs keep their converters within 1.8 nanometres of their locus twins’, where the first step of red phosphor alone moves the converter by more than six. The 600 nm line learned on the thirty locus LEDs reads the off-locus LEDs’ red phosphor within ±0.110 below the locus and ±0.117 above — against its own held-out ±0.120 on the locus — reading about a twentieth too little below and a twentieth too much above at every pump. Read through the estimated recipe, the off-locus LEDs’ camera error comes within ±0.096 and ±0.107, against ±0.074 on the locus; read through the true recipe it is ±0.082 and ±0.092, so most of the loss is not the misreading.

• The prediction’s mechanism fails. A Duv of 0.003 barely moves the converter; it moves the conversion.
• The alternative holds. The locus line reads the off-locus red as well as it reads the locus.
• The error does degrade, by a third to a half — less than the predicted ±0.12, and for a reason the prediction did not name.

Drawn as misreadings, the three sets of LEDs overlap. The locus LEDs, each read by the line fitted without it, scatter by ±0.12 about nought; the off-locus LEDs scatter by about as much, shifted a little down below the locus and up above it. At no red-phosphor strength do the off-locus misreadings leave the band the locus ones occupy. A camera maker who had learned the line on locus-built LEDs could apply it to the binned ones and lose nothing in the red phosphor’s reading but a small, predictable bias.

The bias has the Duv’s sign, and its size is about a twentieth of the full red-phosphor strength: −0.048 on average below the locus, +0.042 above. A twentieth of the range is an eighth of a step of the grid’s red phosphor, and it sits under the line’s own scatter of ±0.12, which is why the overall misreading barely changes.

The converter does not move #

The prediction reasoned from the locus solve: two numbers, the share of the blue converted and the converter’s peak, are set to put the white at one chromaticity, and moving the target should move both. It moves one of them. At every pump, temperature and red phosphor the off-locus converter’s peak sits within 1.8 nanometres of its twin’s, and on average 0.4 nanometres longer below the locus and 0.4 shorter above it — where the first step of red phosphor, from none to 0.3, moves it by six to twelve.

What absorbs the Duv is the conversion. An LED below the locus converts on average 2.6 hundredths less of its blue than its twin, and one above it 2.8 hundredths more, out of conversions between 0.36 and 0.81. A white above the locus is greener, and more converted light — the garnet’s broad yellow-green — makes it so without the converter’s shape changing at all. The locus solve has two handles, and a Duv pulls the one the red phosphor does not.

Small shifts, everywhere #

Across the whole grid the converter’s shift is a band two nanometres wide about nought, with no pump standing out. The cool whites shift most — about a nanometre and a half on average at 6,500 K, against half a nanometre at 2,700 — and the shift’s sign follows the Duv’s at two LEDs in three. Nowhere does a Duv of 0.003 move the converter as far as a tenth of the red phosphor’s full range does.

That is why the 600 nm line keeps working. The departure reads the converter, the converter carries the red phosphor, and the Duv slips past both through the conversion. It still reaches the departure a little: a white converted more strongly has a little more light at 600 nanometres relative to the rest of the spectrum, and the 600 nm departure shifts by 0.002 to 0.017 between an off-locus LED and its twin, where one step of red phosphor shifts it by about 0.06 at 4,000 K. That small shift is the bias in the red phosphor’s reading.

A bias with the Duv’s sign #

Pump by pump, the bias is the same in sign and nearly the same in size. Every pump reads too little red below the locus and too much above it, by 0.040 to 0.055. The 440, 452 and 465 nm pumps agree within a hundredth, which says the bias comes from the converter’s side of the spectrum — the one thing the three pumps share — and not from the blue.

A bias of known sign is the easiest kind of error to remove. A sensor that measures a white’s Duv, which any ambient sensor reading chromaticity already does, could subtract about fifteen times the Duv from its red-phosphor estimate and leave the reading’s scatter as it is on the locus. The census does not need to: the bias sits inside the reading’s own uncertainty, and the larger loss lies elsewhere.

Where the error is lost #

Read the camera’s error the way the earlier essay did — the red phosphor estimated from the 600 nm line, then the error from the estimated recipe by the line each pump’s locus LEDs draw — the off-locus LEDs come within ±0.096 below the locus and ±0.107 above, against ±0.074 on it. Pump by pump, the 440 and 452 nm LEDs are read to about ±0.10 at both Duvs, the 465 nm LEDs to ±0.076 below and ±0.108 above.

The comparison that locates the loss is the true recipe. Give each off-locus LED its real red phosphor and read its error by the same pump’s locus line, and it comes within ±0.082 below the locus and ±0.092 above — more than four fifths of what the estimated recipe leaves. The red phosphor’s misreading adds a little; most of the loss is that the line from recipe to error, learned on the locus, does not quite fit the LEDs beside it.

The camera’s error is not a function of the recipe alone #

The reason is the camera, not the sensor. An LED and its off-locus twin differ in the camera’s colour error by up to 0.18 ΔE₀₀, with the same pump, the same red phosphor and the same colour temperature — only the conversion, and so the white’s tint, changed. The camera’s error depends on where the white sits as well as on what the LED is made of, and a line from recipe to error learned on whites that all sit on the locus has no term for it.

Radiators and LEDs are one family to the sensor and new LEDs miss the line their cousins drew are the reminder that this is the census’s standing weakness. Every line in this run of essays has been learned on a population and tested on its own members held out, and an LED binned a few thousandths off the locus is a member of a slightly different population. Here the sensor’s half of the reading carries across that difference almost untouched, and the recipe’s half does not.

A tilt across the locus, not a valley on it #

The census has met the camera’s error off the locus once before. LEDs built on the locus still fall off the line varied LEDs freely in pump, converter and red phosphor and found the camera’s error forming a V about the Planckian locus — larger the further a white sat from it either way — which is why makers’ practice of building on the locus mattered to the estimate at all. LEDs varied in their phosphors fall off every line then rebuilt the estimate on such LEDs and found the sensor’s 640 nm channel learning what the census had never shown it.

At the scale of a bin the V is not what the off-locus LEDs show. Of the twenty-nine recipes built on both sides, twenty-seven have their camera error rise on one side of the locus and fall on the other; two rise on both, none falls on both. Within a Duv of 0.003 the error tilts across the locus rather than sitting at the bottom of a valley on it, and the tilt’s direction depends on the recipe. A V across a wide range of Duv and a tilt across a narrow one are the same surface seen at two scales, with the V’s floor lying beside the locus for most recipes rather than on it.

That is what makes the error line’s loss recipe-dependent and its repair simple. A tilt is linear in the Duv at each recipe, which is the kind of dependence a line can carry once it has a Duv term; a valley would need a squared term and LEDs on both sides to fit it. A matrix is fitted under one light is the general warning beneath all of this: a camera’s own colour error is a property of the light it is used under, and every census that estimates it is estimating a function of the light, sampled where the census happened to sample it.

Bins and lines #

The practical reading turns on how LEDs are sold. A sensor calibrated on lamps cannot read an LED worked with white LEDs that sat on the locus by construction, as every one in this census until now did. A real white LED carries a chromaticity bin that spans Duv values of a few thousandths either way. A camera’s lighting estimate that has to work under any LED a room might hold has to work across the bin.

The answer here is that it does, with a cost. The 600 nm departure’s reading of the red phosphor is robust to the bin, its bias small and of known sign; the camera’s error, which is what the estimate is for, is read a third to a half worse across the bin than on the locus, and the fix for that is not a better sensor channel but a line from recipe to error learned across the bin — which needs no more than LEDs built off the locus to learn it from.

What a camera maker should take from this #

The 600 nm reading of the red phosphor survives binning. An estimate learned on locus-built LEDs reads LEDs a Duv of 0.003 either side as well as the locus LEDs themselves, with a bias of a twentieth of the red phosphor’s range that a measured Duv would remove.

Learn the error line across the bin, not on the locus. The camera’s error moves by up to 0.18 ΔE₀₀ between an LED and its off-locus twin, and a line that knows only locus whites cannot see that. The sensor reads the Duv anyway; the error line should use it.

Expect ±0.10, not ±0.074, across a real bin — until the error line is learned there.

How the LEDs were built and read #

Each LED is the census’s: a blue pump at 440, 452 or 465 nanometres, a garnet converter with a peak between 520 and 600 nanometres and a share of the blue converted, and a red phosphor at strength 0, 0.3, 0.6 or 1. On the locus, the conversion and the converter’s peak are solved so that the white sits on the Planckian locus at 2,700, 4,000 or 6,500 K; here the same two numbers are solved, by the same Newton’s method, so that it sits a Duv of 0.003 below or above the locus at the same temperature, measured along the locus’s normal in the CIE 1960 plane. One LED pumped at 465 nanometres with no red phosphor at 2,700 K cannot be built below the locus with a converter in range, leaving twenty-nine there and thirty above. The camera’s error and the sensor’s departures are read for each as the census reads them. The red-phosphor line is the least-squares fit of the strength to the 600 nm departure, the mireds and their product over all thirty locus LEDs; the error line is the fit of the error to the estimated or true strength, the mireds and their product over each pump’s locus LEDs. The locus figures are the earlier essay’s, each LED left out.

What this does not reach #

One Duv each way. A Duv of 0.003 is a typical bin’s half-width; a wider bin would show whether the bias and the error’s shift grow in proportion.

A fitted line, not held out, for the off-locus LEDs. The off-locus LEDs are read by lines fitted on all thirty locus LEDs, which never contained them; the locus figures are each held out of their own fit. The comparison is fair in that neither set is read by a fit that saw it.

The camera’s error at one illuminant level. Every error is the census’s mean ΔE₀₀ over its test colours under the LED; how it shifts off the locus for the most saturated colours alone is not read.

Still open: whether the Duv belongs in the error’s line #

The camera’s error moves by up to 0.18 ΔE₀₀ when an LED moves 0.003 off the locus, and a line from recipe to error learned only on the locus cannot follow it. An ambient sensor that reads chromaticity reads the Duv directly.

The calculation is the error line refitted over all eighty-nine LEDs — on the locus and either side of it — with the Duv as a fourth term beside the red phosphor, the mireds and their product, the red phosphor estimated from the 600 nm line as here, each LED left out. The prediction is that the Duv term brings the off-locus LEDs back to the locus’s ±0.074 and costs the locus LEDs nothing, because the error’s shift off the locus is nearly linear in the Duv at each temperature. If the off-locus error stays near ±0.10, the shift depends on the recipe as well as the Duv, and the error line needs the off-locus LEDs’ product terms too. Across a bin, the camera’s error bends with the temperature makes that calculation.

A tie and what it was holding #

The habit here is about the conditions a reading was learned under.

The 600 nm departure reads the red phosphor because, on the locus, the red phosphor and the converter are tied. The worry was that the tie was the reading: break the locus and the reading breaks. It did not, because the locus constraint and the red phosphor pull on different handles of the LED, and moving off the locus pulled the one the reading does not use. What did move was a second relation the essays had been using without naming as locus-bound — the camera’s error as a function of the recipe.

The failure mode is to check a learned reading for the constraint it was suspected of depending on, and not for the other relations learned under the same conditions. The test is to move the condition and read every stage of the chain again — here the converter, the red phosphor’s reading, and the error from the true recipe — so that the stage that breaks is found rather than assumed.