A channel at each pump cannot see the recipe

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, from narrow channels read as departures: how far each channel’s share of the light sits from what a smooth lamp of the camera’s measured colour would give — the reading one ratio reads a peak and a trough built and radiators and LEDs are one family to the sensor extended to a merged family of smooth lamps and LEDs.

The LEDs in that census varied their blue pump and little else, and the line through them did not survive LEDs that varied more. LEDs varied in their phosphors fall off every line found the error a V about the Planckian locus, and LEDs built on the locus still fall off the line built thirty LEDs as a maker would — three pumps, four strengths of red phosphor, whites at 2,700, 4,000 and 6,500 K, each converter tuned until the white sat on the locus. The V went and the line did not come back: one line in the three departures missed the thirty by ±0.25. Within one pump it came back for one pump only — the 452 nm LEDs read to ±0.047, the 440 and 465 nm LEDs no better than ±0.15.

The obvious reading was that the 450 nm channel sits on the 452 nm pump’s line and nowhere near the others’. The proposal was a channel at each pump: ten-nanometre dips at 440 and 465 nm beside the 450, each pump’s LEDs read from the channel nearest its line. The prediction was that the 440 and 465 nm LEDs come within ±0.08 that way, as the 452 nm ones do from theirs, and that the pooled LEDs return within ±0.15. The alternative was that the 465 nm LEDs stay above ±0.2 even from their own channel, in which case the pump changes the camera’s error through the converter as well as through its own light.

The own channel helps one pump, and the recipe reads all three #

Read with the departure in the channel nearest their pump, beside the 500 and 640 nm departures and each LED left out, the 465 nm LEDs’ miss falls from ±0.264 with the 450 nm channel to ±0.135 with their own, and the 440 nm LEDs’ from ±0.171 to ±0.149; the 452 nm LEDs stay at ±0.047. Searched over every line in one, two or three of the five departures, the best reads the 440 nm LEDs to ±0.106 and the 465 nm ones to ±0.134. A line in the LED’s own recipe — red-phosphor strength, colour temperature in mireds and their product — reads every pump within ±0.07: the 440 nm LEDs to ±0.043, the 452 to ±0.055, the 465 to ±0.067. Pooled, with each LED’s pump known and its own channel used, the thirty are read to ±0.119, against ±0.248 for one line in the original three departures and ±0.186 for one line in all five. Along the difference between the 440 and 465 nm departures, every 465 nm LED lies apart from the rest, and the 440 and 452 nm LEDs overlap.

• The prediction fails for both pumps. Neither comes within ±0.08 from its own channel, nor from any three departures.
• The alternative fails too. The 465 nm LEDs fall to ±0.135 from their own channel, well under ±0.2.
• The reason is in neither reading. The camera’s error is simple in what the LEDs are made of, and the departures see only part of that.

The three rows say the same thing in different proportions. Each pump’s own channel is the best of the three blue channels for it, but the margin is small except at 465 nm. For the 440 nm LEDs the three channels are nearly interchangeable — ±0.149, ±0.171 and ±0.150 — which is not what a channel reading the pump’s light would give; a channel on the pump’s line should beat channels ten and twenty-five nanometres off it by more than a tenth. For the 465 nm LEDs their own channel is twice as good as the 450 nm one, and the 440 nm channel, twenty-five nanometres off, is between. And for the 452 nm LEDs, every blue channel reads them better than any channel reads either of the other pumps.

What the channels see #

Drawn against its own channel, each pump’s LEDs rise together — a larger departure, a larger camera error — but none of the three bands is narrow. A departure measures how far the light in a narrow channel sits from what a smooth lamp of the same colour would put there, and a pump line inside the channel adds light there by an amount set by how much of the pump survives the converter. That is one number, and the LEDs differ in two. At one pump, the white’s colour temperature sets the converter share and so how much pump survives; the red phosphor, at fixed temperature, changes the converter share again and adds its own long-wavelength band. Two LEDs with the same blue departure can differ in which of the two put it there, and the camera’s error, it turns out, cares.

The 440 nm LEDs’ departures spread furthest, from about 0.4 to 2.5, because a violet pump surviving the converter is a large departure from any smooth lamp’s blue; the spread is large and so is the scatter about any line through it. A channel can be on the pump and still read a quantity the error does not follow.

The recipe is enough #

The test of whether the error is hard or the channels are blind is to read the error from what the LEDs are made of. Every LED in the grid has a recipe: a pump, a red-phosphor strength and a colour temperature, with the converter solved to put the white on the locus. Within one pump, a line in the red phosphor, the mireds and their product reads the camera’s error within ±0.07 for all three pumps — and the 440 nm LEDs, the hardest pump for the departures, the best of all, ±0.043, under half what the best three departures manage.

So the camera’s error under these LEDs is not complicated. It is nearly a plane in red phosphor and mireds, with a twist where the two meet, and a sensor that could read an LED’s recipe would read its error. The departures read the 452 nm LEDs as well as the recipe does — ±0.047 against ±0.055 — because at that pump the combination of red phosphor and temperature that sets the camera’s error happens to be the combination the 450, 500 and 640 nm channels see. At the other two pumps it is not, and no rearrangement of the five channels makes it so.

Drawn by recipe the 440 nm LEDs make a fan. Each strength of red phosphor gives a nearly straight line in mireds — warmer whites, larger errors — and the lines fall and steepen as the red phosphor rises: with none the error climbs from 2.3 at 6,500 K to 2.6 at 2,700, with the full strength from 1.3 at 4,000 K to 1.7 at 2,700. The red phosphor lowers the camera’s error by up to a whole unit at fixed temperature, because it fills the long end of the spectrum where a phosphor-converted LED is weak and the camera’s matrices, built under smooth lamps, expect light. The departures see the red phosphor through the 640 nm channel and the temperature through the blue channels, but the two readings mix in each channel, and with a violet pump the mixing is the wrong way round.

The departures cannot read the recipe either #

If the camera’s error is a simple function of red phosphor and temperature, and the departures read the 452 nm LEDs’ error to ±0.047, the departures might be expected to read the 452 nm LEDs’ recipe as well. They do not, at any pump. A line in the own channel with the 500 and 640 nm departures, each LED left out, misses the red-phosphor strength by ±0.27 at 440 nm, ±0.39 at 452 and ±0.27 at 465, against a spread of ±0.36 — at the middle pump it does worse than guessing the average. It misses the colour temperature by 52, 76 and 75 mireds against a spread of 83.

That settles what the 452 nm result was. The departures at that pump do not read the recipe; they read the error directly, because the one combination of red phosphor and temperature the error follows happens to be a combination the three channels’ readings move along. Neither number alone is in the departures, and the error, which is a mixture of both, is. At the other two pumps the error is a different mixture — the fans in red phosphor and mireds have different slopes at each pump — and the departures’ own mixture no longer lines up with it. Why the two line up at 452 nm and nowhere else is a fact the census reports and does not explain.

This is the identifiability problem a band in the lens reads as age or as macula met in an observer’s colour matches: two causes that move a reading together cannot be told apart by that reading, and what can be recovered is only the combination the reading follows. Here the reading is three departures and the causes are two parts of a recipe, and whether the combination they recover is the one the camera’s error needs is a matter of where the pump sits.

Knowing the pump #

Pooled over all thirty, the question becomes how much is lost by not knowing which pump an LED has. One line in the original three departures misses by ±0.248; adding the 440 and 465 nm departures to that line brings it to ±0.186; knowing each LED’s pump and reading it from its own pump’s line brings it to ±0.119 — inside the ±0.15 the prediction set for the census, on the LEDs alone. Knowing the recipe brings it to ±0.056. The two extra channels buy less as a fifth and sixth term in one line than as a switch that picks which line to use.

That is the shape a family’s line runs through its neighbours found for fluorescent tubes: a line serves a family whose members differ in one way, and a census that holds several families needs to know which family a lamp is in before it applies a line. The pumps are three families, each with its own line, and the question is whether the sensor can tell them apart.

Two blue channels sort one pump #

The simplest sorting is a ratio of two blue channels, and the difference between the 440 and 465 nm departures is that ratio on the departures’ logarithmic scale. It separates the 465 nm LEDs from the rest with room to spare — every one below −0.7, every other LED above −0.02. It does not separate the 440 from the 452 nm LEDs: the violet pump’s LEDs run from 0.56 to 2.86 and the middle pump’s from −0.01 to 1.31, so a warm 452 nm LED with no red phosphor reads like a cool 440 nm LED with plenty. The same mixing of temperature and red phosphor that blinds each channel to the recipe blinds the pair to the pump.

So the sensor can be told one useful thing: whether an LED’s pump is long. Sorting the 465 nm LEDs out and reading the remaining twenty as one family, the census would lose part of what knowing all three pumps buys, and the 440 nm LEDs would still be read no better than their own pump’s line allows.

What a sensor designer should take from this #

A channel on a pump’s line is not a reading of the pump. Two sensors disagree about deep red, not lines found a sensor’s verdicts on lamps decided by a band its designers had not looked at, and the same is true here of the pump channels. The departure at a pump’s wavelength tells how much pump survived, and that depends on two parts of the recipe at once. Adding channels at more pump wavelengths adds more readings of the same mixture; the 440 and 465 nm channels together buy a quarter off one line’s miss.

The camera’s error under locus-built LEDs is easy, and the difficulty is the sensor’s. Within a pump, the recipe reads the error to a twentieth of a unit. A sensor that could measure red-phosphor strength and colour temperature separately — the temperature it already has from its colour channels, the red phosphor from a channel placed where the red phosphor and the yellow converter differ most — would read the error from them directly.

How the LEDs were read #

The thirty LEDs are the locus grid’s: pumps at 440, 452 and 465 nm, red phosphor at nought, 0.3, 0.6 and one, whites at 2,700, 4,000 and 6,500 K, each converter’s share and peak solved to put the white on the Planckian locus; the six strong-red cool whites that need a converter peaking below 520 nm are not built. Their camera errors are the census’s: the mean colour difference over the test surfaces between the camera’s rendering under the LED, with its matrices blended by the LED’s colour temperature, and the surfaces’ true colours. The departures at 440 and 465 nm are computed exactly as the census’s at 450, 500 and 640 are — a ten-nanometre Gaussian channel’s share of the camera’s raw signal, against the share a smooth lamp of the same camera white would give, on a logarithmic scale; at 450 nm the computation here reproduces the census’s own departure to the last digit. Every line is fitted in least squares with each LED left out and read by the line fitted on the others.

What this does not reach #

Ten LEDs a pump. A line with four numbers fitted on nine LEDs is lightly constrained, and the best-of-many search over departures is flattered by its own choice; the ±0.106 and ±0.134 are the best of twenty-five lines, not a design chosen in advance.

A recipe the sensor cannot read. The recipe line is a diagnosis, not a design: no sensor here reads red-phosphor strength directly.

Other cameras. The camera is the census’s one modelled camera, the one the lamps two channels miss are smooth to the camera and its successors read; a camera whose matrices were built under LEDs would carry a different error and might be read by different channels.

Still open: whether a channel where the converters differ reads the red phosphor #

The red phosphor moves the camera’s error by up to a unit at fixed temperature, and the departures see it only through the 640 nm channel, mixed with the temperature. A garnet converter peaks between 530 and 570 nm and falls through 600; a red phosphor peaks near 630 nm. Their ratio is largest between 600 and 620 nm, where the converter is falling and the red phosphor rising.

The calculation is a departure in a ten-nanometre channel at 610 nm added to the census, the red-phosphor strength read from it and the 640 nm departure within each pump, each LED left out, and the camera’s error read from the estimated red phosphor and the colour temperature the sensor’s colour channels report. The prediction is that the 610 nm channel reads the red-phosphor strength within a tenth of its range at every pump, and that the error read through it comes within ±0.08 for the 440 nm LEDs — near the recipe’s own ±0.043 — because the temperature is already known. If the 610 nm channel reads the red phosphor no better than the 640 nm one, the converter’s own tail mixes into every red channel and the recipe cannot be read from narrow departures at all. The converter gives the red phosphor away makes that calculation.

Testing the channels against the answer #

The habit here is about separating a hard problem from a blind instrument.

The earlier essay found a line failing and the natural response was more channels, placed where the failing LEDs differ — at their pumps. The channels helped a little, and a little is ambiguous: it could mean the error is hard and the channels are doing what can be done, or that the error is easy and the channels are looking at the wrong thing.

The failure mode is to keep adding instruments to a problem without asking what the right instrument would see. The recipe line answers that question here, because the census knows what each LED is made of. An error that a line in three recipe numbers reads to ±0.043 is not a hard error; it is an error the sensor’s readings do not resolve. The test is to read the target from its true causes before reading it from the instrument, and to compare the two.