Seven channels end the trade and lose the tube

A display that adapts its black floor to the room has to know what light the room is lit by, and a room sensor must read the lamp to a hundred kelvin put a number on how well: a dark picture’s shadows, shown with a floor declared in the wrong chromaticity, take a visible tint once the sensor’s error in the CIE 1960 plane passes about 0.0045 at a tenth of a candela on the screen — the tint a warm room tints the black, not the halo found in every dark picture’s shadows when the screen’s black is declared in grey under a warm lamp. A sensor calibrated on lamps cannot read an LED then modelled a three-channel ambient sensor, pinned on two Planckian lamps, and found it reading every lamp and lamp-and-daylight mixture well and every white LED badly.

No calibration reads the lamps and the LEDs together tried every way of fitting that sensor’s nine-number matrix to the room’s lights — exact on two lamps, exact on three lights, least squares on eight, least squares on all forty-seven with each light held out of its own fit — and found the error moving from one family to the other without leaving. The census fit cut the LEDs’ worst error by more than half and quadrupled the lamps’. A fourth channel at 580 nm moved the trade inward, and still left lights beyond a unit.

Three channels give a matrix nine numbers to work with, and the lights differ in more ways than nine numbers can follow: a lamp’s smooth slope, an LED’s blue spike and the trough beside it, its yellow hump, a tube’s narrow bands. The prediction was that six channels — narrower, and spread from 445 to 650 nm — bring every lamp, mixture and LED under a unit, the LEDs within 0.003 and the lamps within 0.0025, and that no count below eight reads the halophosphate tube, whose broad bands and lines differ from every other light at once. The alternative named was a count beyond a unit falling by about the same number with every added channel, which would say the sensor’s problem is resolution in general and not any one feature.

Seven channels, and one light left #

With channels spread evenly from 445 to 650 nm, each one and a quarter times its spacing wide at half height, and the matrix fitted in least squares to the other forty-six lights, the lights beyond a unit number thirty-three with three channels, eighteen with four, six with five, two with six, one with seven and with eight, two with ten and three with twelve. Six channels read every LED within 0.0023 and every mixture, but leave the 2,200 K lamp beyond a unit and the lamps’ worst error at 0.0040. Seven read every lamp, mixture and LED within 0.001. The one light no sensor from five channels up can read is the triphosphor tube — read within a unit by three and four broad channels, and beyond it by every finer sensor.

• The prediction fails at six by one lamp. The LEDs come within 0.003 as predicted; the lamps do not come within 0.0025, and the warmest lamp is still beyond a unit.
• It fails the other way on the halophosphate. Six channels read it within a unit; eight do too.
• The alternative fails as well. The count falls by fifteen, twelve, four and one, not by a fixed number, and then rises.

The curve is a fall and then a floor with a lip. From three channels to five it falls by twenty-seven lights; from five to seven by five; from seven to twelve it rises by two. Two different things are happening along it. On the falling side, each added channel lets the matrix follow another feature that separates one kind of light from another, and the lights that differ from the rest only in that feature come in. On the rising side, nothing new is being learned; the matrix has more freedom than the census needs, and the lights that have no close relative in the census are read by a fit that bends further away from them.

The evenly spread three-channel sensor is worse than the hand-placed one the earlier essays used. Fitted the same way, it leaves thirty-three lights beyond a unit where the earlier sensor left thirteen, because its channels at 445, 547.5 and 650 nm sit at the ends of the visible rather than where the colour-matching functions peak. With three channels, where they sit decides nearly everything; by five, spacing evenly is good enough that placement stops being the question.

The trade dissolves #

The earlier essay drew its calibrations as points on a trade: wherever a fit put the LEDs, the lamps paid. Adding channels moves the point towards the corner rather than along the curve. Three channels err by 0.0105 on an LED and 0.0346 on a lamp; five by 0.0056 and 0.0073; six by 0.0023 and 0.0040; seven by 0.0009 and 0.0009. Each step improves both families at once.

That is what the trade was: not a conflict between the two families’ spectra, but nine numbers stretched over two families that needed more. With three channels, the matrix that reads a lamp’s slope and the matrix that reads an LED’s spike are different matrices, and least squares has to choose a compromise between them. With seven, one matrix can do both, because the signals now carry enough of each spectrum’s shape for a single linear map to recover its chromaticity whichever family it came from.

The point also explains why the fourth channel at 580 nm did little. It added a handle in the yellow, where the LEDs’ phosphor hump sits, and left the blue spike and the deep red with the same broad channels as before. Evenly spread channels add handles everywhere at once, and the earlier fourth channel, placed for one feature, bought one feature’s worth.

Families, worst first #

Taken family by family, three of the four fall nearly in parallel. The LEDs’ worst error falls from 0.0105 at three channels to 0.0007 at twelve, the Planckian lamps’ from 0.0346 to 0.0005, the mixtures’ from 0.0093 to 0.0006. All three pass under the one-unit line between five and six channels, and none rises at any count past seven.

The tubes do not fall with them. Their worst error is least at six channels, 0.0064, and from there it rises: 0.0111 at seven and eight, 0.045 at ten, 0.147 at twelve — fourteen times the worst LED’s error at three channels, from the sensor that reads the LEDs best. The three families that improve are well represented in the census, thirty LEDs, nine mixtures and five lamps, each with neighbours; the tubes are three, each unlike the others and unlike anything else, and when one is held out of its own fit there is nothing left in the census that resembles it.

The tube nobody else resembles #

The triphosphor tube is read within a unit by three channels, at 0.87, and by four, at 0.99, and beyond it by every sensor from five channels up — 1.68 at five, 2.4 at seven and eight, ten at ten, thirty-three at twelve. The halophosphate tube runs the other way at first, from more than six units with three channels to under one from six to eight, and then turns back up past a unit at ten. The broadband tube, whose phosphor is the nearest of the three to a smooth spectrum, is read from five channels to ten.

What a held-out fit asks is whether the sensor can read a light it has not been shown. For a light with relatives — an LED at a temperature between two others, a lamp between two lamps — the answer improves as the sensor grows finer, because the relatives teach the matrix what that kind of spectrum looks like. For a light with no relatives, the matrix fitted to everything else has to extrapolate, and a matrix with more freedom extrapolates further. The triphosphor tube’s reading at twelve channels is a large linear combination of twelve signals whose weights were chosen to fit lamps, LEDs and mixtures exactly, and nothing in that fit constrains what those weights do to a spectrum shaped like no other.

This is held-out validation doing its job, and the rise is not an artefact to be explained away. A sensor fitted at the factory to a set of lights will meet, in a room, lights that were not in its set, and the tube’s curve is what happens to them. The broad three-channel sensor reads the triphosphor tube well precisely because it can do so little: its nine numbers are too few to fit the census’s features closely, and a matrix that does not follow features cannot overreact to one it was never shown.

Where the channels fall on a tube #

Drawn behind the eight-channel sensor, the two spectra show the mechanism. The LED’s phosphor hump spreads across four channels at once, and its blue spike across two; a small shift in either moves several signals together, in proportions the census’s other LEDs have already shown the matrix. The tube’s narrow green and red bands each fall mostly within a single channel, on its slope rather than at its peak, so that a channel’s signal from the tube depends on exactly where a band sits relative to that channel’s centre.

With three channels a hundred and twenty-eight nanometres wide, every band sits deep inside a broad channel and its precise position barely matters. With eight channels thirty-seven nanometres wide, the green band at 545 sits on the falling side of one channel and the rising side of the next, and the ratio between them is a steep function of the band’s position — a function the matrix learned nothing about, because no other light in the census has a band there. The finer the channels, the more a tube’s reading is decided by details the census never contained.

The warmest lamp comes in last #

Among the lights that do come in, the last is the warmest. Every Planckian lamp is beyond a unit with three channels; the 2,200 K lamp stays beyond it until seven — 7.2 units at three, 4.9 at four, 2.7 at five, 1.27 at six, 0.24 at seven — while the 3,000 to 8,000 K lamps are all in by six.

The 2,200 K lamp is the census’s warmest light and the furthest from every LED, which are built from 2,700 K up, and held out of its own fit it has only the 2,700 K mixture and the 3,000 K lamp for neighbours. It is a milder case of the tube’s: a light at the edge of the census, read by extrapolation rather than interpolation. The mixtures, by contrast, sit between their own ends — lamp and daylight together sit below the locus placed them on a curve from the lamp to D65 — and every one of them is read from five channels on. Unlike the tube, its spectrum is smooth and the same shape as its neighbours’, so a finer sensor reads it better rather than worse; it simply needs more channels before the fit is close enough to carry to the edge.

How wide each channel is #

The census’s channels are one and a quarter times their spacing wide at half height, and the curve’s shape does not depend on that choice. With channels as wide as their spacing, or half as wide again, the counts beyond a unit at five, six and eight channels lie within two lights of the census’s: six, two and one at one and a quarter; five, two and two at one; seven, four and one at one and a half. The widths differ only in which of the tubes are lost past eight channels — all three at ten with the narrowest, the triphosphor alone with the widest — and all three bring the 2,200 K lamp in at seven. The fall, the floor and the tube’s lip appear at every width.

The camera’s estimate met the same tube #

This is not the first instrument in this collection to lose a tube for want of relatives. A run of essays estimating a camera’s colour error from an ambient sensor’s narrow channels met the halophosphate tube in the same position. The tube a third channel misses has no family found that no added channel, at any centre or width, rescued the estimate under that tube, because its error followed where its main phosphor band sat and nothing else in the census had a band there. A family’s line runs through its neighbours then gave the tube siblings — halophosphate tubes differing in that one band’s position — and found them lying along a line the estimate could follow.

The cost of that cure is the warning for this one. Two stages catch the tube and bill the lamps they misplace recognised each lamp’s family before applying the family’s own line, and the tube was caught while five misplaced lamps paid for it. Adding relatives to a census changes what the fit does to every other light, and new LEDs miss the line their cousins drew is the reminder that a census’s relatives teach only as far as the next light resembles them. The ambient sensor’s matrix and the camera’s estimate are different instruments with the same weakness: each is a linear map learned from a census, and each is only as good off the census as the census is wide.

What a sensor maker should take from this #

Seven evenly spread channels end the lamp-and-LED trade. A sensor with seven channels from 445 to 650 nm, fitted to a census of lamps, mixtures and LEDs, reads every one of them within a ΔEITP unit at a tenth of a candela, each held out of its own fit. Six channels nearly do; five leave half a dozen lights out.

More channels than that buy nothing on the census and cost on whatever is not in it. Past seven, the lights the census represents read no better, and a light it does not represent reads worse at every added channel. A maker who adds channels must add training lights with them — the more freely the matrix can bend, the more lights it has to be shown to keep it straight.

A line-spectrum source needs relatives. No sensor here reads the triphosphor tube unless it is too coarse to read the LEDs. The remedy is not in the sensor; it is in the census the sensor is fitted to.

How the sensors were built and read #

Each sensor has n Gaussian channels with centres from 445 to 650 nm at equal steps, each channel’s full width at half height 1.25 times the step — 128 nm for three channels, 85 for four, 64 for five, 51 for six, 43 for seven, 37 for eight, 28 for ten, 23 for twelve — on the 5 nm grid. Its matrix from signals to XYZ is the least-squares fit, at unit luminance, to the census’s lights: nine mixtures of a 2,700 K Planckian lamp and D65, thirty white LEDs built onto the Planckian locus from three pumps, three red-phosphor loadings and 2,700 to 6,500 K, five Planckian lamps from 2,200 to 8,000 K, and halophosphate, triphosphor and broadband fluorescent tubes. Every light is read by the matrix fitted to the other forty-six, and the reading’s error is its distance from the light’s own chromaticity in the 1960 plane; the tint is the moon picture’s worst ΔEITP in its shadows with the black floor declared in the sensor’s chromaticity, at 0.1 cd/m² on the screen, as the earlier essays computed it.

What this does not reach #

Noiseless channels. Every signal is exact. A real sensor’s channels carry noise and drift, and a matrix with large weights — the kind that misreads the tube at twelve channels — amplifies them; a finer sensor’s practical floor sits above the one drawn here.

Even spacing. The channels are spread evenly. A sensor maker would place them, and a placed five-channel sensor might match an even seven.

Three tubes. The tubes are the census’s thinnest family, and a census with more of them is exactly the one that would decide whether the tube’s curve is a property of tubes or of being alone.

Still open: whether a tube with relatives is read #

The triphosphor tube is misread by fine sensors because, held out of its own fit, nothing in the census resembles it. A real census would hold several: tubes of the same phosphor family at different colour temperatures, whose bands sit in the same places in different proportions.

The calculation is this census with four triphosphor siblings added — the same three phosphors mixed for 3,000, 3,500, 4,000 and 5,000 K — the sensors refitted with each light held out, and the tubes’ tints read at every channel count. The prediction is that with siblings in the census the triphosphor tube is read within a unit at seven and eight channels, and that the rise past eight shrinks for it while the halophosphate, still alone, rises as before. If the triphosphor tube stays beyond a unit, the problem is not that it is alone but that its bands fall between channels whatever the census teaches, and a fine sensor cannot read narrow-band light at all. Relatives, not twins, teach a sensor a tube makes that calculation.

The light the fit was not shown #

The habit here is about where a model’s accuracy is measured.

Every one of the census’s three richest families improves with every channel up to seven and holds past it. Judged by its typical light, the twelve-channel sensor is the best of the eight. Judged by the light least like the others, it is the worst — thirty times worse than three channels — and the difference between the two judgements is the whole of what the census can say about lights it has not seen.

The failure mode is to measure a flexible model’s accuracy on the cases it resembles and report that as its accuracy. Held-out validation guards against reading a light with its own fit, but not against a census whose held-out cases all have twins inside it. The test is to look at the case with the fewest relatives — here, a tube held out of a census with no other tube like it — and to read the model’s accuracy there as well as at the median.