Across a bin, the camera's error bends with the temperature

A camera’s colour error under a white LED depends on what the LED is made of, and a run of essays has estimated it from an ambient sensor’s narrow channels before the photograph is taken. The converter gives the red phosphor away did it in two stages: read the LED’s red phosphor from the sensor’s 600 nm departure, then read the camera’s error from the estimated recipe — red phosphor, colour temperature in mireds and their product — by a line each pump’s LEDs draw. On thirty LEDs built exactly on the Planckian locus it came within ±0.074 ΔE₀₀.

Beside the locus the converter still gives the red away rebuilt every LED a Duv of 0.003 below and above the locus, where real LEDs are binned, and found the first stage robust and the second not. The 600 nm line read the off-locus LEDs’ red phosphor as well as the locus’s own, but their camera error came within only ±0.096 below the locus and ±0.107 above, and most of that loss remained when the true recipe replaced the estimated one. The camera’s error itself moves with the Duv: of twenty-nine recipes built on both sides, twenty-seven had their error rise on one side of the locus and fall on the other.

A sensor that reads a white’s chromaticity reads its Duv anyway, so the natural repair is to put the Duv into the error’s line and learn the line over all eighty-nine LEDs. The prediction was that one Duv term, beside the red phosphor, the mireds and their product, brings the LEDs beside the locus back to ±0.074 and costs the locus LEDs nothing, because the error’s shift is nearly linear in the Duv at each temperature. The alternative named was that the off-locus error stays near ±0.10, which would mean the shift depends on the recipe as well as the Duv and the line needs more than one term for it.

A curve for each temperature #

Learned over all eighty-nine LEDs with each left out, the two-stage reading comes within ±0.062 on the locus, ±0.120 below and ±0.106 above with no Duv in it; one Duv term makes those ±0.059, ±0.109 and ±0.085; a Duv slope that runs in mireds ±0.058, ±0.103 and ±0.052; a slope for each temperature ±0.058, ±0.079 and ±0.069; and a slope and a curve for each temperature ±0.073, ±0.056 and ±0.048. The camera’s error, averaged over each temperature’s recipes, rises across the locus at 2,700 K and falls at 4,000 and 6,500 K, and bends upward at all three — at 2,700 K more than three times as much as at 4,000, and at 4,000 more than three times as much as at 6,500.

• The prediction fails. One Duv term leaves the LEDs below the locus beyond ±0.10.
• The alternative is right in kind and wrong in its cause. The shift depends on the temperature more than on the red phosphor, and it is bent, not straight.
• The repair has a price. The ten-number line that reads both sides of the bin reads the locus itself worse than a line with no Duv in it.

Read model by model, the bars tell two stories at once. The locus LEDs’ bar shrinks as soon as the line is learned over the whole bin, from ±0.074 to ±0.062 with no Duv term at all, because each pump’s line now has nearly three times as many LEDs to fit; it then barely moves until the last model, where it rises to ±0.073. The off-locus bars shrink unevenly: above the locus a slope in mireds already halves the residual, below it nothing short of a curve per temperature does.

So the bin’s LEDs are not one population with an extra offset. They are three populations, one for each temperature, each with its own course across the locus, and the line has to be told which it is reading.

Warm and cool turn opposite ways #

The reason one term fails is in the error’s own shape. At 2,700 K the camera’s error averages 1.876 ΔE₀₀ below the locus, 1.931 on it and 2.057 above — rising across the locus and bending upward. At 4,000 K it runs 1.709, 1.645 and 1.600; at 6,500 K, 1.661, 1.616 and 1.574 — falling across the locus, and nearly straight at the coolest. A single Duv term can carry one slope for all three temperatures; the three have slopes of opposite sign.

A slope in mireds — the Duv multiplied by the mireds — can give the warm whites one sign and the cool another, and above the locus that is most of what is needed. Below it the curvature matters, because at 2,700 K the error bends strongly and the point below the locus sits well under the straight line through the other two. LEDs built on the locus still fall off the line found the camera’s error a V about the locus across LEDs varied freely in their recipe; at the scale of a bin the V shows as a bend whose strength and whose tilt both depend on the white’s temperature.

Why the camera’s error turns this way the census does not trace. The camera’s dyes and the eye’s cones differ most in the long wavelengths, and a warm white has most of its light there; a white above the locus is greener, a white below it pinker, and at 2,700 K those shifts move light across the region where camera and eye disagree most. That is a plausible account, not a computed one.

Terms against misses #

Drawn against how many numbers each pump’s line fits, the residuals show what each added term buys. Below the locus the residual falls at every step, from ±0.120 with four numbers to ±0.056 with ten. Above it, it falls to ±0.052 with six — the slope in mireds — rises to ±0.069 with seven, a slope per temperature, and falls again to ±0.048 with ten. The locus’s own residual is flat from four numbers to seven and rises at ten.

The rise above the locus at seven numbers is the leave-one-out reading doing its job. A slope for each temperature is learned within one pump from the LEDs that temperature contributes on each side of the locus — four at most, two at 6,500 K — and with one of them left out it is noisier than a slope that runs smoothly in mireds and borrows from every temperature. Ten numbers bring in the curvature, which the data below the locus needs and which pays for its extra noise; seven numbers buy the noise without the curvature.

The cool whites below the locus #

Temperature by temperature, the misses below the locus sit where the slope in mireds fails to follow the error’s course. With a slope in mireds the LEDs below the locus are read within ±0.104 at 4,000 K and ±0.128 at 6,500 K; with a slope for each temperature, ±0.076 and ±0.079; with a slope and curve for each, ±0.046 and ±0.005. The 6,500 K figure is small because only six recipes can be built there on both sides and the curve passes nearly through each of their three points; it is a measure of how few cool whites the grid holds as much as of how well the curve fits.

The warm LEDs below the locus are the hardest at every model: ±0.082 with a slope per temperature, ±0.078 with the curves. At 2,700 K the error’s bend is strongest, and a curve fitted through three Duv values reads the middle of the bin better than its edges.

What the locus pays #

The ten-term line reads the locus LEDs at ±0.073 over all three pumps, against ±0.062 for the four-term line with no Duv and ±0.058 for the seven-term line. It is still as good as the locus-learned reading the earlier essay began from, ±0.074, so a camera maker who adopts it loses nothing on the locus against where the census started. Against the best a line learned across the bin can do on the locus, it loses a sixth.

That is the ordinary price of flexibility. A line with ten numbers per pump, fitted on twenty-nine or thirty LEDs with each left out, has fewer LEDs per number than the four-number line, and the extra numbers are spent on the off-locus LEDs’ curves. Every curve term adds to the line’s freedom near the locus as well, where it is not needed and is fitted to noise.

Every miss #

Drawn LED by LED, the two lines’ misses show where the curves earn their place. Under the slope in mireds, two LEDs miss by more than ±0.15, both below the locus at 6,500 K; under the curves, every LED is within ±0.14, and the largest misses have moved to the locus itself and to 2,700 K below it. The curves take the off-locus misses in and spread a little of their cost onto the LEDs that did not need them.

The pattern also says where a census for this line should spend its effort. The curve at each temperature is fixed by three Duv values; a fourth, at a Duv of ±0.0015 or ±0.006, would say whether the curve is real or only the line through three points, and the next essay’s question is exactly that.

How many numbers a line can afford #

This run of essays has asked the same question of every line it fitted: how many numbers can the LEDs it is learned on actually decide? A channel at each pump cannot see the recipe found the error simple in four numbers per pump — the recipe’s red phosphor, mireds and their product, and a constant — when the LEDs were all on the locus. LEDs varied in their phosphors fall off every line found a line learning a channel’s weight only once the LEDs varied in the way that channel reads. And a fit can be exact and empty is the standing warning beneath both: a line with as many numbers as the points that decide them will pass through every point and say nothing about the next.

The ten-term line sits close to that edge. At 6,500 K it has, within each pump, two recipes below the locus, two above and two on it, and a slope and a curve for that temperature that are decided almost entirely by those six LEDs. Its ±0.005 below the locus at 6,500 K is that closeness showing, and the leave-one-out reading is the only thing that keeps it honest: with each LED left out, the curve is fitted through five points and asked about the sixth, and it still reads it — which is evidence, but thin evidence, that the curve is the error’s and not the points’.

New LEDs miss the line their cousins drew is what happens when a line that passed its own held-out test meets LEDs from outside its census. The off-locus LEDs were exactly that kind of test for the locus-learned line, and it failed it by a third to a half. The ten-term line has not yet met its outside test, and the bin’s inner and outer edges are where it will. A matrix is fitted under one light made the same point about a camera’s colour matrix: the error quoted is the error where it was fitted.

What a camera maker should take from this #

Learn the error line across the bin. With no Duv in it at all, a line learned on LEDs built on and beside the locus reads the locus LEDs better than one learned on the locus alone, ±0.062 against ±0.074.

Give the Duv a slope and a curve for each temperature, or give it nothing. A single Duv term, the obvious choice, buys almost nothing below the locus, because warm and cool whites turn opposite ways across it. A slope in mireds is enough above the locus; below it, the bend at 2,700 K needs its own term.

Expect to pay a sixth on the locus for the bin. Ten numbers per pump read every LED within ±0.14 and both sides of the bin within ±0.056, and the locus LEDs at ±0.073 instead of ±0.062.

How the lines were fitted #

The eighty-nine LEDs are the thirty locus-built LEDs of the earlier census and the fifty-nine built at a Duv of −0.003 and +0.003, the camera’s error and the sensor’s 600 nm departure read for each as there. In every model each LED is left out of both stages. The first stage reads the red phosphor by least squares from the 600 nm departure, the mireds, their product and — except in the pooled model — the set Duv, over the other eighty-eight; the second reads the error within the LED’s pump from the estimated red phosphor, the mireds and their product, with the Duv as nothing, as one term, as a term and its product with the mireds, as a term and its products with the mireds over 250 and its square, and with those three again multiplied by the Duv. The mean errors at each temperature average the recipes built at all three Duv values — eleven at 2,700 K, twelve at 4,000 and six at 6,500 — and the slope and bend are the finite differences across ±0.003.

What this does not reach #

Three Duv values. Each temperature’s curve is fitted through −0.003, nought and +0.003. Whether it describes the error between them, or only passes through those three points, a fourth value would test.

Three temperatures. The slope and curve are given separately to each of the grid’s temperatures; a white at 3,000 or 5,000 K would need them interpolated in mireds, which the census does not try.

A fixed grid. The red phosphors and pumps are the census’s; a real bin holds LEDs of recipes the census never built.

Still open: whether the curve holds between its points #

Each temperature’s curve across the bin is fitted through three Duv values, and at 6,500 K it reads the LEDs below the locus within ±0.005 — close enough to the three points to suggest it is passing through them rather than describing the error.

The calculation is the grid rebuilt at a Duv of −0.0015 and +0.0015, inside the bin, and −0.006 and +0.006, outside it, read by the ten-term line learned on the eighty-nine LEDs as they are, with none of the new LEDs in its fit. The prediction is that the inner LEDs are read within ±0.06, as the outer ones of the fit were, because the error’s course across the bin is smooth, and that the outer LEDs at ±0.006 are read worse than ±0.10, because a curve fitted on ±0.003 extrapolates its bend. If the inner LEDs are read worse than ±0.08, the curves are the line through three points and not the error’s course, and the bin needs more Duv values before its line can be trusted.

A term for each thing that changes #

The habit here is about adding a variable to a model.

The off-locus LEDs differ from the locus ones by one number, the Duv, and the obvious repair was to add one term for it. That term assumes the Duv does the same thing to every LED. Here it did opposite things at warm and cool temperatures and bent differently at each, so one term averaged two opposite slopes into almost nothing.

The failure mode is to add a variable to a model as a single term when its effect depends on the other variables already there. The test is to look at the effect within each level of those variables before choosing its form — here, the error’s course across the locus at each temperature — and to enter it as it is found, a slope and a curve for each, then to measure on held-out cases what the extra terms cost where they were not needed.