The red phosphor tilts the warm whites and bends nothing

A camera’s colour error under a white LED can be read before the photograph is taken, from an ambient sensor’s narrow channels, in two stages. The first reads how much red phosphor the LED carries from the sensor’s departure at 600 nm, where a garnet converter’s emission falls away and a red phosphor’s rises. The second reads the camera’s error from that estimate, the colour temperature and the LED’s distance from the Planckian locus, by a line each pump wavelength’s LEDs draw for themselves. The converter gives the red phosphor away found the first stage on the locus; beside the locus the converter still gives the red away carried it a few thousandths of Duv either side, where real LEDs are binned.

The second stage then needed three repairs, each found by reading LEDs the line had not seen. Across a bin, the camera’s error bends with the temperature gave it a slope and a curve in Duv for each temperature. The bin’s curve holds between its points and breaks beyond them found that curve borrowing its bend from a level it could not place, and a level for each temperature lets the curve be the error’s gave each temperature its level with one term a pump, the square of the mireds. That eleven-term line reads 4,000 and 6,500 K within ±0.04 at every Duv from −0.006 to +0.006, and 2,700 K within ±0.03 inside the bin; but the warm whites at twice the bin’s edge only within ±0.069 below the locus and ±0.087 above, and the eight LEDs it misses by more than 0.1 are all warm.

The mean miss at 2,700 K was small, so what was left was scatter — recipe against recipe — and the one part of the recipe the line already estimates is the red phosphor. The prediction was that a warm white’s bend across the locus is set by its red phosphor, so that one more term a pump, the square of the Duv multiplied by the estimated red, brings the 2,700 K LEDs at ±0.006 within ±0.05 and every LED within 0.1, the cooler temperatures staying within ±0.04. The alternative was stated with it: if the warm whites’ scatter stayed near ±0.08, their bend depended on something the first stage does not estimate, and the sensor would have to read it directly.

The bend term moves the warm whites by a thousandth #

With the square of the Duv times the estimated red added to every pump’s line, the 2,700 K LEDs at −0.006 are read within ±0.068 and those at +0.006 within ±0.088, against ±0.069 and ±0.087 without it. Six LEDs are missed by more than 0.1, against eight, and the largest miss grows from 0.180 to 0.185.

The prediction fails, and fails cleanly: not a near miss but no effect. The bend term costs three coefficients across the three pumps and buys a thousandth at one end and loses one at the other. The figure’s dashed line lies on the eleven-term line’s at every Duv, inside the bin and beyond it.

That is evidence about the error’s shape rather than about the term. A term in the Duv’s square times the red describes LEDs whose curvature across the bin grows with their red phosphor: a strong-red warm white would bend more steeply about the locus than a weak-red one, symmetric on both sides. If the warm whites’ scatter had that form, the misses at −0.006 and +0.006 would have had the same sign for each recipe, and one coefficient a pump would have caught them. They do not. Of the eleven warm recipes built at both −0.006 and +0.006, eight are missed with opposite signs at the two ends — an LED with no red phosphor by −0.086 below the locus and +0.180 above it, at a 440 nm pump, and one with a full red by +0.106 and −0.093 — and the three that are not are the recipes at 0.6 red, whose misses are the smallest in the column. Opposite signs at opposite ends is the signature of a slope rather than a curve.

The red enters the slope #

Read that way, the term to try is the Duv itself times the red: a slope across the bin that steepens or flattens with the red phosphor, rather than a curve that deepens with it. It costs the same one coefficient a pump.

With the Duv times the estimated red in every pump’s line instead, the 2,700 K LEDs at −0.006 are read within ±0.046 and those at +0.006 within ±0.067; at the bin’s edges within ±0.056 and ±0.050, against ±0.068 and ±0.062. Below the locus the warm whites come within the prediction’s ±0.05. Above it they come a quarter of the way there and stop.

The asymmetry has a plain reading. Below the locus a white LED is pinker than a blackbody of its temperature, above it greener, and the red phosphor’s share of the light changes how far the camera’s long-wave channel and the eye’s long-wave cone disagree on that tint. LEDs built on the locus still fall off the line found the camera’s error a V about the locus whose depth depends on the phosphors; the slope term says the V is also tilted, and tilted by an amount the red phosphor sets. A term linear in the Duv can describe that tilt on both sides at once, and a term in the Duv’s square cannot describe it at all, which is why the second was useless and the first is not.

A slope that pays at the ends and charges the middle #

The red slope helps 6,500 K more than it helps the warm whites: at ±0.006 the cold whites go from ±0.029 and ±0.021 to ±0.010 and ±0.004, and at the bin’s edges from ±0.040 and ±0.020 to ±0.021 and ±0.009. At 4,000 K it costs: the LEDs at −0.003 rise from ±0.039 to ±0.057 and those at −0.006 from ±0.034 to ±0.045.

So the term the warm whites asked for is not free. It is fitted once a pump across all three temperatures, and its single coefficient has to serve a tilt that is large and of one sign at 2,700 K, small and of the same sign at 6,500 K, and something else at 4,000 K. The middle temperature pays for the ends. A version with a red slope per temperature — the Duv times the red times the mireds and their square — would free it, at three coefficients a pump rather than one, and with only eighty-nine LEDs to fit it on that is the kind of line a family’s line runs through its neighbours warned about: one that describes its sample well and is read where the sample does not reach.

There is a better reason not to take that step yet. The red slope reaches the prediction below the locus and falls short above it, and a line that falls short on one side only is usually reading a variable that is itself wrong on that side.

The first stage squeezes the warm whites’ red #

The red phosphor enters the second stage as an estimate, read by the first stage from the 600 nm departure, the mireds, their product and the Duv, by one line across all three temperatures. Its miss can be read directly, since each LED was built with a known red.

At 4,000 and 6,500 K outside the bin, an LED built with no red phosphor is read on average at −0.04, and one with a full red at 1.03: the reading follows the diagonal. At 2,700 K the same LEDs are read at 0.13 and 0.80, and those built at 0.3 and 0.6 at 0.32 and 0.52. The warm whites’ red is squeezed toward the middle of its range, by about a third — and not only outside the bin. The first stage’s root-mean-square miss at 2,700 K is 0.132 of the red’s range on the locus itself, 0.098 at −0.006 and 0.149 at +0.006, against 0.075 to 0.12 at the cooler temperatures. On the locus the squeeze costs the reading nothing visible, since the warm locus LEDs are read within ±0.026; whatever it does to the second stage, it does through the terms that grow with the distance from the locus.

The first stage’s form suggests where the squeeze might come from, though nothing here isolates it. Its single Duv term moves every LED’s estimate by the same amount per unit Duv whatever its temperature; but the 600 nm departure at 2,700 K, where most of the LED’s light is already long-wave, moves with the Duv by a different amount from the departure at 6,500 K, and its link to the red phosphor weakens as the converter’s own emission fills the band. A line that does not know the Duv changes the departure’s meaning differently at each temperature reads a warm white’s departure as if it were a cool white’s, and the strong and the weak red are read as nearer each other than they are.

The warm misses follow the misread red times the Duv #

If the squeeze is what remains, the second stage’s warm misses should follow it, and in a particular form: a misread red enters the line through every term the red multiplies, and the term whose effect grows with the distance from the locus is the slope. So the miss should follow the red’s misreading times the Duv.

It does: across the 46 warm LEDs outside the bin, the eleven-term line’s miss correlates with the red’s misreading times the Duv at r = 0.79. With the misreading alone the correlation is 0.19, and with the red each LED was built with −0.10. The eight LEDs beyond 0.1 sit at the figure’s two corners: LEDs with no red phosphor read as carrying a little, above the locus, and LEDs with a full red read as carrying less, below it.

The red the LED actually carries, alone, says nothing about its miss. What follows it is how wrongly the first stage read that red, and on which side of the locus. But the correlation has two readings, and it cannot choose between them by itself. At 2,700 K the misreading is very nearly a straight-line function of the red — the squeeze lifts the weak reds and lowers the strong ones in proportion — so the misreading times the Duv is also, very nearly, a constant times the Duv less a constant times the red times the Duv. The misses may follow the misreading because the misreading is harmful, or because it happens to trace a red-dependent slope the line is missing. The first reading puts the fault in the first stage, the second in the second.

The red known cures the warm whites and harms the middle #

The test of that reading is to hand the second stage the red phosphor each LED was built with, in place of the first stage’s estimate, refitting both the eleven-term line and the red-slope line on the true reds.

With the red known, the eleven-term line reads the warm whites at ±0.006 within ±0.050 and ±0.067, and the red-slope line within ±0.032 and ±0.050: the prediction’s ±0.05, reached on both sides. With the estimate, the same line reads them at ±0.046 and ±0.067, a third worse above the locus. But at 4,000 K the known red is worse than the estimate: the eleven-term line reads +0.003 and +0.006 within ±0.064 and ±0.056 given the true red, against ±0.024 and ±0.024 given the estimate.

The second half of that is the more instructive. At 4,000 K the estimate carries more about the camera’s error than the truth does, because it is not really an estimate of the red. It is a reading of the sensor’s 600 nm departure, and that departure sees the whole long-wave shape of the spectrum — the converter’s peak and its width as well as the red phosphor’s share — which is exactly the region where the camera and the eye disagree. The built red is one number of the recipe; the departure is the spectrum’s own answer. LEDs varied in their phosphors fall off every line found that no line in the sensor’s departures alone reads a grid of recipes better than ±0.29; here the departure, filtered through a red phosphor stage, is reading something about each LED that its recipe card would not have said.

So the substitution does not settle which reading of the correlation is right. Handing the line the true red removes the squeeze, and removes along with it whatever the estimate carries about the spectrum beyond the red; the warm whites gain, the middle temperature loses, and the two effects cannot be told apart from this comparison. What would tell them apart is an estimate that reads the red correctly and is still an estimate.

Every LED with the red in the slope #

Under the red-slope line, six LEDs are missed by more than 0.1, the largest by 0.143; under the eleven-term line eight were, the largest by 0.180. The cold whites’ spread is visibly the narrowest in the census, the 4,000 K spread a little wider than before, and the warm whites still open toward their outer columns, most of all above the locus.

That last shape is the one the red-known line narrows and the estimated line does not. Every one of the remaining large misses is a warm white with its red at one extreme of the range and read toward the other — the pattern two stages catch the tube and bill the lamps they misplace found in a two-stage reading’s faults, landing on the lamps the first stage misplaced and billed to the second. Whether that pattern is cause or coincidence here is the question the squeeze leaves.

What a calibration engineer should take from this #

A term that does nothing is evidence about the error’s shape. The bend term was the natural reading of “the warm whites bend by recipe” and it moved nothing. The sign of each recipe’s misses at the two ends of the bin said, before any new term was fitted, that the dependence on the red was a tilt and not a curvature.

Read the first stage’s miss directly before blaming the second. In a two-stage reading the second stage’s residual can carry the first stage’s errors multiplied by whatever the second does with its input. Here the multiplier would be the Duv, so a squeeze at the warm end would appear as a scatter growing with distance from the locus — exactly what a missing curvature looks like. Measuring the first stage’s miss found the squeeze; it did not yet show that the squeeze was what the second stage was paying for.

A known input is not always a better input. Handing the line the true red cured the warm whites and doubled the error at 4,000 K, because the sensor’s reading of the red was carrying part of the camera’s error that the recipe does not carry. An estimate that has been validated as an estimate can be quietly working as a measurement of something else, and replacing it with the truth removes that.

How the lines were read #

The first stage reads the red phosphor’s strength by least squares from the sensor’s 600 nm departure, the mireds, their product and the Duv over all LEDs in its training set. The second stage reads the camera’s error within each pump by least squares from the estimated red, the mireds and their product, the square of the mireds over 250, and the Duv and its square each multiplied by one, the mireds over 250 and their square — eleven terms — with one more term for the bend line (the Duv’s square times the estimated red) or the red-slope line (the Duv times the estimated red). At nought and ±0.003 each LED is left out of both stages and read by the lines fitted on the other eighty-eight; at ±0.0015 and ±0.006 every LED is read by the lines fitted on all eighty-nine. For the red-known lines the first stage is skipped and the built red used in both training and reading. The squeeze is the first stage’s mean estimate for each built red over the LEDs at ±0.003 and ±0.006, at 2,700 K and at the cooler two together; the correlation is Pearson’s over the 46 warm LEDs at those Duv values.

What this does not reach #

One slope coefficient a pump, shared by three temperatures. The red slope’s cost at 4,000 K is a fact about that sharing, not about the term; a slope per temperature might keep the gain at the ends without the cost, and would also have three times the room to describe the sample rather than the error.

Neither the squeeze’s cause nor its cost is isolated. That the first stage’s single Duv term is what squeezes the warm red is the form’s natural reading, and that the second stage pays for the squeeze is one of two readings of a correlation; neither is shown here by fitting a first stage without the restriction, which is the next calculation.

A modelled camera and a modelled sensor. The red phosphor’s tilt across the bin is this camera’s; the habit of reading the first stage before the second is general.

Still open: whether the first stage can stop squeezing the warm red #

Outside the bin the warm whites’ red is read about a third too close to the middle, while at the cooler temperatures it follows the diagonal; the first stage’s one Duv term moves every LED’s estimate alike, and the 600 nm departure does not move alike.

The calculation is the first stage given the Duv times the 600 nm departure and the Duv times the mireds — two terms more, shared by every pump — with the red-slope line read on its estimate, every LED left out as here. The prediction is that an LED with no red phosphor at 2,700 K outside the bin is then read within 0.05 of nought and one with a full red within 0.05 of one, and that the red-slope line reads the warm whites at ±0.006 within ±0.055 on both sides — near the red-known line’s ±0.032 and ±0.050 — while 4,000 K stays within ±0.06. If the warm red is still squeezed by a fifth or more, the 600 nm departure at 2,700 K has stopped carrying the red off the locus at all, and the first stage needs a second channel, at 620 or 660 nm, to read it there.

The line had already undone the squeeze made that calculation: the Duv’s terms leave the squeeze where it was, a level per temperature removes it, and the eleven-term line, which had absorbed it, does not move.

A term that bought nothing #

The habit here is about reading a null result as a measurement.

The warm whites’ scatter was attributed to the one recipe variable the reading carried, the prediction named the form that attribution implied, and the form did nothing. Rather than reach for more terms, the misses were read at the two ends of the bin, where their signs said the dependence was a tilt; the tilt helped, but less on one side than the other; and the asymmetry pointed upstream, to a first stage that reads the warm whites’ red squeezed. With the red known, the warm whites came in within the prediction, and the cooler temperatures went out — which says the first stage matters, without yet saying how.

The failure mode is to read a two-stage residual as the second stage’s, and to add terms to the stage where the error shows rather than the stage where it starts. The test is to measure the first stage’s miss on the quantity it estimates, at the places where the second fails — and then, because a miss that tracks the quantity can stand in for it, to repair the first stage and see whether the second’s failure moves.