LEDs built on the locus still fall off the line

A camera turns its raw readings into colour with matrices blended by the lamp’s colour temperature, and under many lamps the result is off by a colour difference or two. A small sensor beside the lens can estimate how far off. The design this thread arrived at reads three of the sensor’s narrow channels as departures — at 450, 500 and 640 nm, from what a smooth lamp of the same colour would give, the first two read together as one ratio reads a peak and a trough proposed — places a lamp in a family by its nearest neighbour in those three numbers, and applies that family’s straight line, the two stages two stages catch the tube and bill the lamps they misplace introduced. Radiators and LEDs are one family to the sensor brought the estimate to ±0.127 ΔE₀₀ over the census’s fifty-four lamps, each left out in turn.

The census’s forty-four white LEDs varied their blue pump and little else. New LEDs miss the line their cousins drew found new LEDs that varied their phosphors missing the line by half again the census’s error, and LEDs varied in their phosphors fall off every line rebuilt the census on forty-eight LEDs varied independently in pump, converter share and red phosphor. The rebuilt line learned the red channel and lost everything else: the census’s own lamps were missed by ±0.294, the new LEDs by ±0.229, and no line in any of the sensor’s departures read the grid’s LEDs better than ±0.29. The camera’s error across that grid was a V about the Planckian locus, rising with the white’s distance on either side, and every one of the census’s own LEDs had sat on the V’s green arm.

The closing section’s diagnosis was the grid’s construction. It had varied its settings independently and let the white land where it landed, which put most of its LEDs off the locus in tints a maker would never sell as white. A maker does the opposite: chooses the converter for each pump and red phosphor so that the white sits on the locus at the colour temperature wanted. Built that way, the V should vanish by construction, and the prediction was that on the locus the camera’s error would be nearly linear in the departures again, the census’s held-out error would return within ±0.15, the new red-phosphor LEDs would land within ±0.13, and the heavier-converter LED, whose white sits 0.023 above the locus, would keep its miss.

On the locus the V is gone and the line does not come back #

Thirty of the thirty-six settings — three pumps, four red-phosphor strengths, three colour temperatures — can be built: the converter’s share and peak are solved until the white sits on the locus within a kelvin and a Duv of a hundred-thousandth, with the converter peaking between 520 and 600 nm as a garnet converter can. Each left out and estimated from the others, the thirty are missed by ±0.282 by the merged design’s line and ±0.248 by a line in all three departures — against ±0.291 and ±0.298 for the free grid. In the census in place of its own LEDs, they leave the census’s lamps missed by ±0.268, against ±0.294 for the free grid and ±0.127 for the census. Taken one pump at a time, the three departures read the LEDs pumped at 452 nm to ±0.047, at 440 nm to ±0.171 and at 465 nm to ±0.264. The five new LEDs are missed by ±0.152, better than from the census or the free grid, with the heavier-converter LED missing least and the strong-red LED most.

• The V was not the obstacle. Removing it by construction leaves the LEDs almost as unreadable as before.
• One pump is readable. The departures read LEDs that share the census’s own pump and fail on the others.
• Every part of the prediction about the new LEDs reverses or splits. The light-red LED lands within ±0.13; the strong-red one does not; the heavier-converter LED, predicted to keep its miss, loses it.

The first four bars are the prediction’s test. The merged design’s line misses the free grid by ±0.291 and the locus-built LEDs by ±0.282; a line in all three departures misses them by ±0.298 and ±0.248. A gain of about a twentieth is what removing the V buys. The LEDs’ own spread of camera error is ±0.426 on the locus, so a line in the departures explains a little over half of it — enough to see that the departures carry information and not enough to use them as an estimator.

The last three bars are where the information is. Fitted on the ten LEDs of one pump, a line in the three departures reads the 452 nm LEDs to ±0.047 — the census’s own accuracy and better — and the 440 and 465 nm LEDs to ±0.171 and ±0.264, against spreads of ±0.37 and ±0.31. Within one pump the converter and the red phosphor move the departures in ways a line can follow; across pumps the same departures mean different things.

Building each LED onto the locus #

The census’s white LED is a blue pump line twenty-two nanometres wide, a broad converter band, and optionally a red phosphor band at 632 nm. Two of its settings move the white: how much of the pump’s light the converter takes, and where the converter’s band peaks. With the pump and the red phosphor fixed, those two are solved together, by Newton’s method in the CIE 1960 plane, until the white lands on the locus at the temperature asked for — two numbers for two coordinates, as a maker tunes a garnet’s composition and loading. A warm white needs a heavy converter peaking far into the yellow: at 452 nm with no red phosphor, 78 per cent conversion peaking at 598 nm for 2,700 K, 44 per cent at 565 nm for 6,500 K.

A red phosphor pulls the white towards red, and the converter must peak greener to pull it back. At 6,500 K, a red phosphor of 0.6 needs a converter peaking near 513 nm and full red one near 490, or none reaches the locus at all; no garnet converter emits there, so those six — the cool whites with strong red phosphor, at every pump — are not LEDs a maker would build, and are left out. That is the first difference from the free grid, which included combinations no maker would choose: here every LED is one that could be sold as a white.

Where the information lives #

Of the three departures, the one at 450 nm is the LEDs’ best single reader of the camera’s error, correlated with it at 0.82 across the thirty. Plotted against it, the three pumps do not share one band: at a departure near 0.7 the 452 nm LEDs carry errors near 1.55 and the 440 nm ones 2.1 to 2.3. What the census measures is the outcome: a line fitted within the 452 nm pump reads its LEDs as well as the census read its own, and within either other pump it misses by three to five times as much.

Why is not settled by this calculation. One reading is geometric: the sensor’s 450 nm channel sits on the 452 nm pump’s line, so for those LEDs its departure measures how much pump light the converter leaves, the setting that most changes the camera’s error; for a pump at 440 or 465 nm the channel sits on the line’s flank, where its reading moves with the pump’s position as much as its strength. Against that reading, the census’s own forty-four LEDs — designs with an added violet or cyan line placed anywhere from 435 to 505 nm — were read to ±0.127, pump positions and all. What every locus-built LED has that those did not is a converter retuned with its pump and its red phosphor to hold the white, so that pump, converter and red move together; the census’s LEDs moved one thing at a time. Which of the two the departures cannot follow is the question the closing section sets.

An offset between pumps is the whole of the pooled miss #

The size of the pooled line’s miss follows from that picture without any further fitting. Where the 452 nm and 440 nm LEDs share a 450 nm departure near 0.7, their camera errors differ by about 0.6 to 0.7 ΔE₀₀. A single line through both has to pass between them, and the best it can do is split the difference: each group is missed by about a third of a unit in opposite directions, before any scatter within a pump is counted. A third of a unit is close to the ±0.28 the pooled line actually reports, so most of the locus-built LEDs’ unreadability is that one offset between pumps, and the scatter along a pump’s own line is the smaller part.

That is a stronger statement than “the departures read one pump”. It says the departures carry much of what each pump’s own LEDs need, and fail above all to tell the pumps apart in level. A line that knew which pump it was looking at would need only a different intercept for each, three numbers where the merged design has one, and the within-pump misses — ±0.047 at 452 nm, larger at the others because their LEDs spread more widely across the flank of the channel — would be what remained. The information missing from the sensor is a label, not a measurement, and a label is the cheapest thing a second channel can supply.

The census and the new LEDs #

In the census in place of its own LEDs, the locus-built thirty do slightly better than the free grid’s forty-eight. The census’s lamps, each left out in turn, are missed by ±0.268 — the locus LEDs by ±0.283 and the radiators, daylights, tubes and emitters together by ±0.217 — against ±0.294 with the free grid and ±0.127 with the census’s own LEDs. The new radiators and daylights stay within ±0.13 throughout, the worst at 0.106. The smooth family’s line weights the 640 nm departure — the long-wave channel the tube a third channel misses has no family added for the halophosphate tube — at −0.37, between the census’s −0.13 and the free grid’s −0.74.

The five new LEDs do better than from either earlier pool, ±0.152 against ±0.196 from the census and ±0.229 from the free grid. Four of the five are pumped at 452 nm, and the locus-built pool’s 452 nm LEDs are the ones it reads well.

One by one, they trade places. The heavier-converter LED, the new LED furthest from the locus with a Duv of +0.023, missed by +0.18 from the census and −0.17 from the free grid, and misses by +0.07 from the locus: the prediction had it keeping its miss, and it is the best estimated of the five. The strong-red LED, nearest the locus at +0.004, missed by −0.24 from the census and −0.33 from the free grid, and still misses by −0.24 — the worst of the five. The light-red LED lands within ±0.13 at −0.09, as predicted; the deep-pump LED, pumped at 445 nm, misses by +0.19, more than from the free grid’s −0.06.

Nothing in that pattern follows the distance from the locus, which the free grid’s V had suggested would matter. The heavier-converter LED is estimated well because it is pumped at 452 nm and its converter is the census’s own; the strong-red LED is estimated badly although it sits on the locus, because no locus-built LED at 452 nm has its combination of a standard 565 nm converter and a heavy red phosphor — every locus LED with full red phosphor has its converter tuned greener to hold the white.

What this means for the sensor #

Among white LEDs built as a maker builds them, the camera-error sensor’s three channels read those pumped at 452 nm and not those pumped at 440 or 465, though the V about the locus is gone. A sensor meant to estimate a camera’s error under the white LEDs a building actually has — pumped anywhere from about 440 to 465 nm — needs either a channel that reads the pump’s position as well as its strength, or a design that places a lamp by its pump before applying a line — with the risk a classifier that abstains gives up the tube found in any design that places lamps, of placing some confidently in the wrong group. A family’s line runs through its neighbours found the merged design only as good as the neighbours a lamp’s family gives it; here the neighbours that matter are the ones with the same pump.

The census’s own ±0.127 was earned on its own forty-four LEDs, violet- and cyan-pumped designs whose departures its line had learned. That is a reason to treat that number as a property of the census rather than of the sensor, and to quote the sensor’s accuracy with the kinds of LED, and the pumps, it was demonstrated over.

How the LEDs were built and read #

Each LED is the census’s white LED model: a blue pump line twenty-two nanometres wide at 440, 452 or 465 nm, a converter band 118 nm wide taking a share of the pump’s light, and a red phosphor band at 632 nm at strength nought, 0.3, 0.6 or one. For each pump, red strength and colour temperature, the converter share and the converter band’s peak are found by Newton’s method on the white’s CIE 1960 coordinates, started at 0.6 and 565 nm, until the white lies within a millionth of the locus point at that temperature; a solution whose converter peaks outside 520 to 600 nm is not built. Thirty settings survive.

Each LED’s camera error and sensor departures are computed exactly as the census’s own. The locus pool is the census with its forty-four LEDs replaced by the thirty, its other lamps unchanged; the merged design — nearest neighbour in the three departures, then the family’s line in the departure ratio and the 640 nm departure — is run with every lamp left out in turn, and the new lamps are estimated from the whole pool. The lines on the LEDs alone are fitted with each left out, in the merged design’s two inputs and in all three departures.

What the census does not reach #

One converter shape. The converter is the census’s single broad band, its peak moved; a real garnet’s band changes width and shape as its composition moves its peak.

Pumps at three wavelengths. The pump effect is read from three pumps; a grid of pumps every few nanometres would show whether the departures’ reach falls off smoothly away from 450 nm or abruptly.

A modelled camera. The camera’s matrices and the sensor’s channels are the census’s models, and a real sensor’s channel widths decide how far its flank reading reaches.

Still open: whether a channel at each pump reads them all #

If the 450 nm channel reads only the LEDs pumped onto it, a sensor with channels at 440, 450 and 465 nm could read every pump’s LEDs from the channel nearest its line — or, more simply, read the pump’s position from the ratio of two blue channels and place each LED among its own pump’s neighbours.

The calculation is this census with the sensor’s departures read at 440 and 465 nm as well as 450, the locus-built LEDs placed by the nearest pump channel and each pump’s line fitted on its own ten, each LED left out. The prediction is that the 440 and 465 nm LEDs come within ±0.08 when read from their own channel, as the 452 nm LEDs are from theirs, and that the census’s held-out error with the locus LEDs returns to within ±0.15. If the 465 nm LEDs stay above ±0.2 even from their own channel, the pump’s position changes the camera’s error through the converter’s absorption as well as through the pump’s own light, and no blue channel alone will read it. A channel at each pump cannot see the recipe makes that calculation.

A clean construction tests a diagnosis, not a hope #

The habit here is about what a fix designed from a diagnosis can and cannot show.

The free grid’s failure came with a picture — a V about the locus — and the picture suggested a cause: whites off the locus. Building the grid on the locus removed that cause completely, which is what made it a good test. The camera’s error stayed unreadable, and the diagnosis was shown to be incomplete before any further fix was built on it.

The failure mode is to take the most visible pattern in a failure as its cause and design the next attempt around removing it. The test is to remove the pattern by construction and see whether the failure goes with it. Here it did not, and the residue pointed somewhere the picture had not: at the pump, which the picture of the V had never separated out.