Two estimates of one red each read their own end

A camera’s colour error under a white LED can be read from a small ambient sensor in two stages: first the LED’s red phosphor, from the sensor’s 600 nm departure, then the camera’s error, from that estimate and the LED’s colour temperature and distance from the Planckian locus. The line had already undone the squeeze built two versions of the first stage. One, with a single Duv term, reads a 2,700 K LED with no red phosphor at 0.13 and one with a full red at 0.80 — a third too close to the middle. The other, with a level and a departure slope for each temperature, reads them at 0.02 and 1.01, and the red at every temperature three to five times better.

The faithful estimate turned out to matter only to one line. The eleven-term error line — a level, a slope and a curve in Duv for each temperature, built across the bin’s points and beyond them — had absorbed the squeeze through its red-times-mireds term and did not move by more than 0.006 on any warm LED. The red-slope line, which adds the Duv times the estimated red with one coefficient a pump across all three temperatures, read the warm whites at ±0.006 within ±0.046 and ±0.067 ΔE₀₀ on the squeezed red and ±0.034 and ±0.057 on the faithful one; but the cold whites at a Duv of −0.003 within ±0.021 on the squeezed red and ±0.040 on the faithful. Each estimate was the better input at one end.

A second stage can be handed both estimates at once, and the obvious split follows the finding. The prediction was that the eleven terms on the squeezed estimate and the red slope on the faithful one — no more terms than now — keep the warm whites at ±0.034 and ±0.057 and the cold whites at ±0.021 or better at −0.003. If either end lost more than 0.005 of its gain, the two estimates would be carrying the same information in different proportions, and the choice between them would be a choice of which temperature to serve.

Split between two estimates, the line keeps both ends #

The split line reads the 2,700 K LEDs at −0.006 within ±0.034 and at +0.006 within ±0.057 — exactly the faithful estimate’s — and the 6,500 K LEDs at −0.003 within ±0.025, 0.004 worse than the squeezed estimate’s ±0.021 and 0.015 better than the faithful estimate’s ±0.040.

The prediction holds at the warm end exactly and at the cold end within its stated margin. The split costs the cold whites four thousandths where it was allowed five, and everything else in the figure lands where the two parent lines put it. By the alternative’s own test, the two estimates carry different information: the squeezed estimate’s departure from the red is worth something to the eleven terms at 6,500 K, the faithful estimate’s agreement with the red is worth something to the red slope at 2,700 K, and a line can take each from where it is.

Over all 208 LEDs the split line reads ±0.0346, against ±0.0371 on the squeezed red and ±0.0366 on the faithful one. Across the whole bin the split is the best of the three, or tied with the better of its parents, at every one of the seven Duv values. The improvement over either parent is small — a twentieth — and that is the expected size, since each parent was already near the split at one end and far from it only at the other.

One estimate, its slope scaled by the mireds, does as well #

A split line is the answer to a question about inputs. The finding that prompted it was about form: the faithful estimate helped the red-slope line only because that line’s one red-slope coefficient spans three temperatures, and so cannot undo a squeeze that changes with the temperature, as the eleven-term line’s red and red-times-mireds terms could. That suggests a second answer, with one estimate and no more terms. The squeezed red’s slope can be given the mireds as well — the Duv times the squeezed red times the mireds over 250, in place of the Duv times the squeezed red — so that the red slope steepens toward the warm end at the rate the squeeze does.

With the squeezed red alone and its slope times the mireds, the warm whites at ±0.006 are read within ±0.033 and ±0.056, against the split line’s ±0.034 and ±0.057; over all 208 LEDs within ±0.0346 against ±0.0346 — the same to a ten-thousandth. The dashed line lies on the split line’s at every Duv.

So the second estimate was never needed. The faithful first stage’s whole contribution to the red-slope line was a rescaling of the squeezed red at each temperature, and a rescaling that grows smoothly with the mireds is exactly what one product with the mireds supplies. The essay before found this property for the eleven-term line, which had absorbed the squeeze without being asked; the same property, applied to the one term that lacked it, makes the faithful estimate redundant in the term that wanted it.

That is a stronger statement than the split line’s success, and it changes what an engineer building the reading should do. The split line requires two first stages to be fitted, stored and run on every reading; the scaled slope requires one, and the same number of second-stage terms. Radiators and LEDs are one family to the sensor found a classifier’s whole cost repaid by treating two families as one; here a second first stage’s whole benefit is repaid by one product in the second.

The two estimates’ difference reads the cold whites #

The split line kept both ends but did not improve either beyond its better parent. If the two estimates carry different information, adding the faithful estimate to the split line as a term of its own, one coefficient a pump, puts the difference between the two estimates into the line — the part of the 600 nm departure’s variation with the temperature that the red phosphor does not explain.

With the difference in the line, the 6,500 K LEDs are read within ±0.016 at every Duv and on the locus within ±0.001; the split line’s worst was ±0.025 and the squeezed line’s ±0.021, and both read the cold whites’ locus within ±0.015. At −0.006 the cold whites go to ±0.009, at +0.003 to ±0.016, at the inner Duv values to ±0.003.

A reading within a thousandth on the cold locus, with every LED left out, is the tightest any line in this run of essays has reached on any temperature’s locus LEDs. It is the confirmation the essay before could only argue for. There the squeezed estimate read the cold whites’ error better than the faithful one while reading their red five times worse, and the explanation offered was that its departure from the red was carrying something the camera’s error depends on — the converter’s own peak and width, which shape the long-wave spectrum where the camera and the eye disagree. Given that departure as a term, the line uses it, and at 6,500 K it is nearly the whole of what was left. LEDs varied in their phosphors fall off every line found the recipe carrying more numbers than the departures can separate; at the cold end, one more of those numbers turns out to be readable after all, from the difference between two ways of reading the same channel.

The difference helps one temperature and leaves two #

At 2,700 K the term moves the worst Duv from ±0.057 to ±0.058, and at 4,000 K from ±0.056 to ±0.056; at 6,500 K from ±0.025 to ±0.016. The difference’s information is entirely at the cold end.

The size of the difference does not explain where it helps. The two estimates differ by ±0.122 of the red’s range at 2,700 K, ±0.074 at 4,000 K and ±0.098 at 6,500 K — largest at the warm end, where the term does nothing. Both first stages read the red from the same departure, the mireds and the Duv, and differ only in two terms in the square of the mireds; at 2,700 K their difference is the squeeze, which the eleven-term line had already undone, so the term repeats what the line knew. At 6,500 K the difference is something else: the squeezed estimate fitted the cold whites’ red worst and their error best, and the part of the departure it kept there is information the line had not yet been given. At 4,000 K the estimates differ least, and the difference carries least.

The middle temperature is where the red slope has cost since it was introduced, and nothing here touches it. At 4,000 K the worst Duv under every line in this essay lies between ±0.056 and ±0.060, against ±0.039 for the eleven-term line with no red slope at all. The red slope’s one coefficient a pump, however its input is chosen, still charges the middle for the ends.

Every LED with both estimates #

Five LEDs are missed by more than 0.1, three at 2,700 K and two at 4,000 K, the largest by 0.126. The 6,500 K column has become the narrowest band in the figure, a few hundredths wide at every Duv. The warm column still opens toward its outer Duv values, with the same red-free warm whites above the locus at its top that have led every list in this run; the 4,000 K column carries the two full-red LEDs below the locus that the red slope has missed throughout.

Those two groups are now the whole of the reading’s large errors, and they are different problems. The warm whites are LEDs the first stage reads correctly and the line still misses; the 4,000 K LEDs are the price of a term the warm whites asked for. Neither is a problem of inputs.

Where the reading now stands #

It is worth setting the run’s numbers side by side, since each essay in it changed one thing. The converter gives the red phosphor away read thirty LEDs built exactly on the locus, with a line learned on them, within ±0.074. Beside the locus the converter still gives the red away and across a bin, the camera’s error bends with the temperature carried that reading a few thousandths of Duv either side, and a level for each temperature brought the same thirty locus LEDs to ±0.025 by learning them alongside the bin’s. With both estimates in the line, all 208 LEDs — the eighty-nine fitted each left out and the 119 at Duv values never fitted — are read within ±0.035, and the thirty on the locus within ±0.025 again.

So the locus has not improved since the level was placed, and the gains since have all been at the bin’s edges and beyond. That is the right place for them: a real bin’s LEDs are spread across it, and a reading that is excellent on the locus and poor at ±0.006 serves a lamp nobody manufactures. It also says where the run’s remaining error lives. Of the five LEDs still missed by more than 0.1, none is on the locus and none is at 6,500 K. Two stages catch the tube and bill the lamps they misplace named the habit of finding a reading’s errors on the lamps it misplaced; the warm whites here are not misplaced, by either first stage, and that is what makes them the next question rather than the last one’s.

What a calibration engineer should take from this #

Before adding an input, try the product the input stands in for. The faithful first stage helped one term because that term lacked a product with the temperature, and supplying the product with the existing input reproduced the second input’s entire benefit to a ten-thousandth. One first stage is cheaper to fit, store and maintain than two.

Two readings of one channel can carry two quantities. The two estimates read the same 600 nm departure with different terms in the temperature, and their difference read the cold whites’ error to a thousandth on the locus. A quantity a first stage discards as error against its target can be exactly the quantity the second stage needs — and the way to keep it is to give the second stage the difference, not to choose between the stages.

A term’s cost belongs to its form, not its input. The red slope’s charge to the 4,000 K LEDs survived every change of input in this essay. Where a cost survives a change of input, the fix is in the term itself.

How the lines were read #

Two first stages read the red phosphor by least squares, one from the sensor’s 600 nm departure, the mireds, their product and the Duv, the other with the square of the mireds over 250 and the departure times it added. The error line within each pump carries 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 — eleven terms — and a twelfth: the Duv times the squeezed red, times the faithful red, or times the squeezed red and the mireds over 250. The split line takes the eleven terms on the squeezed estimate and the red slope on the faithful; the difference line adds the faithful estimate as a thirteenth term. At nought and ±0.003 every LED is left out of both first stages and the error line; at ±0.0015 and ±0.006 every LED is read by stages fitted on all eighty-nine.

What this does not reach #

The difference is used, not decomposed. That the two estimates’ difference carries the converter’s peak and width at 6,500 K is the natural reading of why it reads the cold whites’ error; the converter’s quantities themselves are not read here.

Three temperatures, two of which the difference leaves alone. The term’s value is concentrated at one temperature, and with three temperatures fitted a term concentrated at one is close to a level for that temperature in disguise; whether it helps a cold white at 5,000 or 8,000 K is not tested.

A modelled camera and sensor. The finding that a second first stage’s benefit reduces to a product with the temperature is general wherever the second stage is linear; the sizes are this sensor’s.

Still open: whether a red slope per temperature frees the middle #

The red slope’s single coefficient a pump has charged the 4,000 K LEDs ±0.056 to ±0.060 at their worst Duv in every line that carries it, against ±0.039 without it, whatever red it was handed. The scaled slope gave it the mireds once; the middle temperature needs the slope to take any value at each temperature, which the square of the mireds supplies.

The calculation is the difference line with its red slope given the mireds over 250 and their square — two terms more a pump — with every LED left out of both stages as here. The prediction is that 4,000 K returns to within ±0.04 at every Duv, the cold whites stay within ±0.016, and the warm whites at ±0.006 stay within ±0.035 and ±0.058, because the middle temperature’s cost was a shared coefficient and not the red slope itself. If 4,000 K stays beyond ±0.05, the camera’s error at the middle temperature does not tilt with the red phosphor at all, and the red slope is a term for the two ends that the middle should be spared.

An input that was a product in disguise #

The habit here is about asking what a successful input actually contributed.

The split line did what it was built to do, and a reading that stopped there would have recorded two first stages as the answer. Asking why the faithful estimate helped the one term it helped — because that term lacked the temperature — led to a single product with the squeezed estimate that did the same, to the ten-thousandth. Asking whether the two estimates still carried different information, now that one was redundant as an input, led to their difference, which read the cold whites better than anything else in the run.

The failure mode is to credit an input with what its interaction with the model’s form supplied, and so to keep a whole stage in the pipeline for the sake of one missing product. The test is to replace the new input with the product it might be standing in for, and keep it only if the product cannot match it — and then to ask what is left between the two that neither replaces.