The line had already undone the squeeze

A camera’s colour error under a white LED can be read from an ambient sensor before the photograph is taken, in two stages: the LED’s red phosphor from the sensor’s 600 nm departure, then the camera’s error from that estimate by a line each pump’s LEDs draw. The second stage has been built term by term across a manufacturing bin — a slope and a curve in Duv for each colour temperature, tested between its points and beyond them, then a level for each temperature, eleven terms a pump — and the red phosphor tilts the warm whites and bends nothing found where it still fails. At 2,700 K and twice the bin’s edge it reads the LEDs within only ±0.069 ΔE₀₀ below the locus and ±0.087 above; a red slope, the Duv times the estimated red, takes them to ±0.046 and ±0.067; and given the red phosphor each LED was built with, to ±0.032 and ±0.050.

That essay also read the first stage directly and found it squeezing the warm whites’ red. At 2,700 K an LED built with no red phosphor is read on average at 0.13 and one with a full red at 0.80, while at the cooler temperatures the reading follows the diagonal; and outside the bin the eleven-term line’s warm misses follow the red’s misreading times the Duv at r = 0.79. The correlation had two readings. Either the squeezed red was costing the second stage, or the squeeze, being nearly a straight-line function of the red, happened to trace a red-dependent slope the line was missing. Substituting the true red could not decide between them, because it also removed whatever the estimate carries about the spectrum beyond the red — and at 4,000 K that was worth more than the red itself.

What decides between them is an estimate that reads the red correctly and is still an estimate. The first stage’s form suggested how to build one. It carries a single Duv term, moving every LED’s estimate alike whatever its temperature, while the 600 nm departure at 2,700 K may move with the Duv by a different amount. The prediction was that the first stage given the Duv times the departure and the Duv times the mireds — two terms more, shared by every pump — reads a 2,700 K LED with no red phosphor outside the bin within 0.05 of nought and one with a full red within 0.05 of one, and that the red-slope line then reads the warm whites at ±0.006 within ±0.055 on both sides, 4,000 K staying within ±0.06. If the warm red stayed squeezed by a fifth or more, the departure at 2,700 K had stopped carrying the red off the locus at all.

The Duv’s terms leave the squeeze where it was #

With the Duv times the departure and the Duv times the mireds in the first stage, a 2,700 K LED outside the bin with no red phosphor is read at 0.14, one at 0.3 at 0.32, at 0.6 at 0.52 and with a full red at 0.80. With one Duv term those were 0.13, 0.32, 0.52 and 0.80. The first stage’s root-mean-square miss on the warm red is 0.129 both ways.

The prediction’s first half fails as completely as a prediction can: the two terms leave the squeeze unchanged at every built red. And the essay before had already given the reason without noticing it. The squeeze was there on the locus too, 0.132 of the red’s range at a Duv of nought; a term multiplied by the Duv is nought on the locus and so cannot repair anything there. What makes the warm reading different is not where the LED sits across the bin but which temperature it is.

The alternative the prediction named was that the departure at 2,700 K had stopped carrying the red. That is wrong too, and the figure’s third line shows it. With the square of the mireds over 250, alone and times the departure — a level and a departure slope for each temperature, two terms more — the same LEDs are read at 0.02, 0.29, 0.59 and 1.01. The departure carries the warm red perfectly well. What it does not do is carry it on the same scale as at 6,500 K, and a first stage with one departure slope across all three temperatures fits the slope the cooler LEDs need and applies it to the warm ones, where it is too shallow by about a third.

A level per temperature reads the red four times better #

Over every Duv the temperature-aware first stage reads the warm red within ±0.035 of its range, against ±0.129 with one Duv term; at 4,000 K within ±0.040, against ±0.085; at 6,500 K within ±0.020, against ±0.103. The figure’s two upper lines rise together from 0.10 or 0.11 below the locus to 0.14 or 0.15 above it. The third lies flat at 0.03 to 0.04 across the whole bin.

That is a first stage three to five times better at every temperature, for two terms shared by every pump. The converter gives the red phosphor away built the first stage’s reading on the observation that, on the locus, a red phosphor pushes the converter to a shorter peak, and the 600 nm departure reads the converter; beside the locus the converter still gives the red away carried it across the bin. The observation holds at each temperature with a different constant, and the first stage had been fitted as if the constant were shared. By the measure the first stage is judged on, the repair is complete.

The eleven-term line does not move #

The question the essay before left was whether the second stage was paying for the squeeze. With the squeeze gone and the estimate still an estimate, the answer can be read LED by LED.

Under the eleven-term line, no warm LED outside the bin moves by more than 0.006 when the red is un-squeezed. The warm whites at ±0.006 are read within ±0.069 and ±0.087 before and after; the LED with no red phosphor at +0.006 is missed by 0.180 before and 0.181 after. Every grey point in the figure lies on the diagonal.

So the squeeze cost the eleven-term line nothing, and the correlation of 0.79 was the second reading after all: a misreading nearly proportional to the red, standing in for a slope the line lacked. The mechanism is in the line’s own terms. At a single temperature the squeeze is close to a straight-line map of the red — the estimate is about a constant plus two thirds of the red — and the second stage carries the estimated red and the estimated red times the mireds. Between them those two terms give the red a coefficient that changes in a straight line with the mireds, and the line’s level terms take up the constant. A squeeze that grows steadily toward the warm end is, to such a line, a different coefficient at each temperature, and it can learn that coefficient as easily as the true one. The line had learned the squeezed red’s scale at 2,700 K and paid it back.

That is worth stating as a property of two-stage readings generally. A first stage’s error is harmless to the second stage exactly to the extent that the second stage can represent its inverse. Two stages catch the tube and bill the lamps they misplace found the opposite case, where the first stage sorted lamps into families and a misplaced lamp received another family’s line: no term in the second stage could undo a wrong category. A wrong scale is a different kind of error. A line with the right products in it absorbs it without trace, and an engineer who repairs it buys nothing. Radiators and LEDs are one family to the sensor made the same point from the other side, where a classifier separating two families the second stage could have treated as one paid for every lamp it sorted.

The red-slope line gains #

The red-slope line is different in one respect that matters here. Its extra term, the Duv times the estimated red, has one coefficient a pump, shared by all three temperatures, and nothing multiplies it by the mireds. A squeeze at one temperature and not the others is exactly what one shared coefficient cannot undo.

On the un-squeezed red, the red-slope line reads the warm whites at −0.006 within ±0.034 and at +0.006 within ±0.057, against ±0.046 and ±0.067 on the squeezed red. Given the red each LED was built with, it read them at ±0.032 and ±0.050. Below the locus the estimate now reads the warm whites within 0.002 of the truth; above it, within 0.007. Thirty-eight of the 46 warm LEDs outside the bin move toward nought, and the largest moves are the ones at the ends of the red’s range, where the squeeze was largest.

So the prediction’s second half holds below the locus and not above it. The ±0.055 asked for on both sides is met at −0.006 by a wide margin and missed at +0.006 by 0.002 — close enough that the honest statement is the comparison, not the threshold: the estimated red now does almost exactly what the known red does, and what is left above the locus is the line’s, not the first stage’s.

The cold whites lose what the warm whites gain #

The repair has a cost, and it falls where the essay before found the known red falling: away from the warm end.

At 6,500 K the red-slope line reads the LEDs at −0.003 within ±0.021 on the squeezed red, ±0.040 on the un-squeezed red and ±0.046 on the known; at +0.006 within ±0.004, ±0.012 and ±0.011. At 4,000 K its worst Duv is ±0.057 on the squeezed red and ±0.059 on the un-squeezed. The cold whites are read twice as badly once their red is read right, and the un-squeezed estimate behaves there almost exactly like the truth.

That makes the earlier finding at 4,000 K a general one rather than an accident of one temperature. The squeezed first stage was a worse reading of the red at 6,500 K — ±0.103 against ±0.020 — and a better reading of the camera’s error. Its departure from the red was not noise. It is the part of the 600 nm departure that the red phosphor does not explain, which is the part that depends on the converter’s own peak and width, and the camera’s error at 6,500 K depends on that too. LEDs varied in their phosphors fall off every line found that the recipe has more numbers in it than the sensor’s departures can separate; here the departure, read as if it measured one of them, was quietly carrying another, and a first stage built to read the red faithfully threw it away.

The trade is exact enough to state. Of the two first stages, one is the better estimate of the red at every temperature, and the other is the better input to the error line at 6,500 K; the error line at 2,700 K prefers the first only through the one term that cannot absorb a rescaling.

Every LED on the un-squeezed red #

Four LEDs are missed by more than 0.1, two at 2,700 K and two at 4,000 K; the largest miss is 0.125, the warm white with no red phosphor pumped at 440 nm at +0.006. On the squeezed red six were, the largest 0.143. The two warm LEDs beyond 0.1 are the same two that led the list in every line of this run, the no-red whites above the locus at the 440 and 452 nm pumps; the two at 4,000 K are full-red whites at −0.003 that the red slope has been missing since it was introduced.

Those two warm LEDs mark what the first stage can no longer be blamed for. Their red is now read at 0.01 and 0.04, within the first stage’s noise, and they are still missed by more than a tenth. Whatever the camera’s error does for a red-free warm white greener than the locus is not in the error line’s terms and not in the estimate’s. LEDs built on the locus still fall off the line found the camera’s error a V about the locus whose depth the phosphors set; a red-free warm white is the recipe whose long wavelengths come from the converter alone, so whatever sets that depth for it is the converter’s and nothing in the red’s estimate can reach it.

What a calibration engineer should take from this #

Before repairing a first stage, ask whether the second can already undo its error. A rescaling of the estimate at each temperature is undone by any second stage that carries the estimate and its product with the temperature. The squeeze was real, large and four times reducible, and repairing it changed the eleven-term line by under a hundredth.

A more faithful estimate can be a worse input. The un-squeezed first stage reads the red three to five times better at every temperature and halves the error line’s accuracy on the cold whites. The quantity a first stage is fitted to and the quantity the second stage needs are different, and the first stage’s residual against the first can be information about the second.

Correlation with a misreading is not evidence of harm when the misreading tracks the quantity. The warm misses’ correlation with the misreading times the Duv was 0.79, and un-squeezing the red moved those misses by nothing, because at one temperature the misreading was nearly a straight-line function of the red.

How the stages were read #

Each 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; the Duv-aware stage adds the Duv times the departure and the Duv times the mireds, and the temperature-aware stage the square of the mireds over 250 and the departure times it. Each second stage reads the camera’s error within each pump from the estimated red, the mireds and their product, the square of the mireds over 250, and the Duv and its square each times one, the mireds over 250 and their square — the eleven-term line — with the Duv times the estimated red added for the red-slope line. At nought and ±0.003 every LED is left out of both stages; at ±0.0015 and ±0.006 every LED is read by stages fitted on all eighty-nine. The red-known lines skip the first stage and use the built red in fitting and reading alike.

What this does not reach #

The cold whites’ preference is shown, not decomposed. That the squeezed estimate’s departure from the red carries the converter’s peak and width is the natural reading of why it reads the camera’s error better at 6,500 K; it is not shown here by reading those quantities.

One error line’s form decides who gains. The red-slope line gains from the un-squeezed red because its one red-slope coefficient spans three temperatures; a red slope per temperature would absorb the squeeze as the eleven-term line does, and the comparison would change.

A modelled camera and sensor, three temperatures. The squeeze’s size is this sensor’s; the property that a second stage absorbs a rescaling it can represent is general.

Still open: whether the two first stages can each do their own job #

The squeezed first stage reads the camera’s error better at 6,500 K, the un-squeezed one at 2,700 K, and the only term that tells them apart is the red slope. A second stage can be handed both estimates at once.

The calculation is the red-slope line given both the squeezed and the un-squeezed red — the eleven-term line on the squeezed estimate and the red slope on the un-squeezed, one term a pump no more than now — with every LED left out of both stages as here. The prediction is that the warm whites at ±0.006 keep the un-squeezed red’s ±0.034 and ±0.057 and the cold whites the squeezed red’s ±0.021 or better at −0.003, so that every temperature is read at the better of the two lines. If either end loses more than 0.005 of its gain, the two estimates are carrying the same information in different proportions rather than different information, and the choice between them is a choice of which temperature to serve.

Two estimates of one red each read their own end made that calculation: the split keeps both ends, one estimate with its slope scaled by the mireds does as well, and the two estimates’ difference reads the cold whites to a thousandth on the locus.

A repair that bought nothing it was meant to #

The habit here is about checking whether a downstream stage needed the upstream fix.

The first stage’s squeeze was found by reading it directly, its cause was found by asking which variable it followed — the temperature, not the Duv — and its repair was clean: two terms, four times better. By the first stage’s own measure, it was the most successful calculation of the run. By the measure that matters, the camera’s error, it bought nothing in the line it was aimed at, a little in the line with one shared coefficient, and a loss at the cold end.

The failure mode is to judge a stage of a pipeline by its own error, when the next stage’s form decides which of its errors are visible. The test is to repair the stage and read the pipeline’s output, not the stage’s — and before that, to ask whether the next stage’s terms can already represent the repair.