The tube a third channel misses has no family

One ratio reads a peak and a trough gave a phone’s ambient-light sensor two narrow channels, at 450 and 500 nm, and asked how well they estimate the colour error the phone’s camera makes under whatever lamp is lighting the scene. Read as one number — the logarithm of the 500 nm reading over the 450 nm one, against what the camera’s own three channels predict for that ratio — they estimate the error under fifty-four lamps to ±0.141 ΔE₀₀, each lamp left out of the fit that estimates it. Structured lamps put a peak at 450 and a trough at 500, and the ratio adds the two.

Its worst miss was the halophosphate tube: 0.445, a third of the tube’s whole error. That tube’s colour trouble lies in its long-wave phosphor, not in the blue-green gap both channels read, and the lamps of the census also differ at long wavelengths — a tube’s orange and red lines, an LED’s broad phosphor tail, a lamp with a deep-red emitter. The prediction was that a third narrow channel near 610 nm, where the tube’s phosphor has a shoulder the LEDs lack, would halve the tube’s miss and take the whole estimate to about ±0.12, while channels nearer 580 add nothing. And it named the alternative: if no long-wave channel helps, the tube’s error belongs to something no narrow channel sees, and ±0.14 is the floor two numbers reach.

The long-wave gap holds a tenth, and the tube’s miss is the census’s #

No third channel halves the tube’s miss. At every centre from 560 to 660 nm and every width from 10 to 40, a third channel beside the ratio leaves the halophosphate tube’s miss above three quarters of the ratio’s alone, and the best of them takes the whole estimate from ±0.141 to ±0.128. At 610 nm, where the prediction put the answer, the camera already predicts the tube’s share of the sensor’s signal to within 0.013 in the logarithm. The tube’s error follows where its main phosphor band sits — 2.55 ΔE₀₀ with the band at 558 nm, 1.48 at 598 — and a 620 or 640 nm channel reads that position; but the census holds one halophosphate tube, and with it left out nothing remains to learn the relation from. Given four of its siblings, a 640 nm channel halves its miss, at a cost to every other lamp.

• The prediction’s floor is the right answer for the census as it stands: two numbers reach ±0.14, three reach ±0.13.
• The tube is not invisible to a narrow channel. Its error lives in the position of a broad band, and a channel on that band’s long-wave flank reads it.
• What cannot be seen is a relation the census never contains. A lamp unlike every other is estimated from nothing.
• Teaching it costs the others, because one straight line cannot serve two families.

The figure holds the whole argument. The joined points are one tube with its main band moved in ten-nanometre steps, and inside that family the channel’s departure and the camera’s error move together as cleanly as anything in this census does. The dots are the other fifty-three lamps, and nothing about their scatter says the same departure means the same error for an LED. An estimate fitted on the dots, with the tube left out, cannot find the line.

A third channel in the gap #

The sensor and camera are the earlier essays’: a phone camera modelled with its own three spectral sensitivities — a fourth observer, in the sense of a camera is a fourth observer, agreeing with the standard one nowhere exactly — and an ambient sensor whose narrow channels each report their share of the sensor’s signal against the share the camera’s three channels predict from the same light. That departure is zero for a lamp the camera can describe and grows where a lamp has structure the camera’s broad channels smooth over. The camera’s error under a lamp is its mean colour difference over a test set after white balance, the quantity two matrices do not reach a white LED found doubling under tubes and LEDs against a matrix fitted under the lamp itself.

The new channel is swept from 560 to 660 nm in ten-nanometre steps, 10, 20 or 40 nm wide. The estimate is a straight line in two inputs: the size of the ratio’s departure, as before, and the third channel’s departure, kept signed so that a lamp above the camera’s prediction and one below it can pull the estimate different ways. Each of the fifty-four lamps is left out of its own fit and estimated from the other fifty-three, as every earlier estimate of this camera’s error has been.

The best third channel buys under a tenth. A 10 nm channel at 580, 620 or 630 nm takes the estimate from ±0.141 to ±0.128; a 20 nm channel at 620, to ±0.128 as well; a 40 nm channel never below ±0.134. At 660 nm every width makes the estimate worse than the ratio alone, and a 10 nm channel does from 650, because the deep red is where the census’s lamps differ most in ways unrelated to the camera’s error: a tungsten lamp rises there, a three-emitter source has its red line there, and a white LED falls away. The prediction’s ±0.12 is not reached anywhere.

No channel halves the tube #

The prediction’s sharper test was the tube itself, and it fails more plainly. The ratio alone misses the tube by 0.445. The smallest miss any third channel leaves is 0.377, a 20 nm channel at 640 nm read by the size of its departure — seven hundredths better, where half was predicted. Around 610 nm, where the shoulder was expected to show, the miss is 0.442. The dotted line in the figure is the halving, and no channel of any width comes near it.

The tube stays the worst lamp or close to it throughout. With the best channels, the worst miss passes to the pink-tinted radiator, at 0.41 to 0.44: a smooth lamp whose error, like the tube’s, is larger than anything its spectrum’s narrow features predict. The lamps two channels miss are smooth to the camera met the same pair from the other side: the lamps a narrow channel cannot flag are the ones whose trouble is in a smooth shape, and a smooth shape is what three broad channels are good at describing and bad at correcting. A tinted radiator and a halophosphate tube are both a camera’s problem for a reason that lives in how their broad spectrum is weighted, not in a line or a gap.

At 610 nm the camera already predicts the tube #

The halophosphate tube departs from the camera’s prediction at 610 nm by 0.013 in the logarithm — a little over one per cent of its share of the signal, less than more than half the census’s structured lamps depart. Whatever makes the camera err under the tube, the camera’s own three channels predict how much of its light falls near 610 nm almost exactly. The shoulder the prediction pictured is not a narrow feature. The halophosphate coating is two broad bands, one near 480 and one near 578, each about a hundred nanometres wide — the tube a lamp is not a blackbody set beside a white LED of the same white and found rendering differently — and at 610 nm the tube’s spectrum is the smooth long-wave flank of the second band, which three broad channels describe well.

The triphosphor tube, by contrast, departs at 610 nm by 0.6: its europium red is a narrow cluster of lines, and a narrow channel on it sees something the camera’s channels smooth away. That is why channels near 610 nm do help the estimate a little, through the lamps they can see. They cannot help with a lamp whose trouble is not narrow — the same reason a straightened channel repeats the one beside it found a 570 nm channel adding nothing beside 450: a channel earns its place by seeing what the others cannot, and at a broad band’s flank the camera’s own channels already see nearly everything.

The tube’s error is where its band sits #

If the tube’s error is not a narrow feature, the question is what it is. The halophosphate model gives a direct handle: its main band’s position. Moved from 558 to 598 nm in ten-nanometre steps, the band takes the camera’s error under the tube from 2.55 to 2.22, 1.90, 1.64 and 1.48 ΔE₀₀ — falling at every step, by more than a whole unit in all. A band further into the red puts less of the tube’s light where the camera’s green and red channels disagree most with the observer, and the camera errs less.

The ratio of 500 to 450 nm does not see this. Across the family the ratio’s departure moves by small amounts and not in step with the error, because the family differs in the long-wave band and the ratio reads the blue-green gap. A 620 nm channel does see it: its departure rises at every step, from −0.47 with the band at 558 nm to +0.09 at 598, because the channel sits on the band’s long-wave flank and the flank moves past it. A 640 nm channel does the same more strongly. So the tube’s error is readable by a long-wave channel after all — as the band’s position, not as a shoulder.

A lamp with no family cannot be fitted #

That leaves the question why, if a 620 nm channel reads the tube’s error, the estimate does not use it. The answer is in the figure at the top of the essay. Across the census, a 620 nm departure means different things for different lamps. For the halophosphate family it means where the main band sits, and the error falls as the departure rises. For a white LED it means where the phosphor’s tail falls; for the triphosphor tube, how strong its europium lines are; for a radiator, very little. The census holds one halophosphate tube among fifty-four lamps, and when that tube is left out, the fifty-three that remain contain no lamp whose 620 nm departure is related to its error the way the tube’s is. The straight line fitted to them points the wrong way for the tube, or nowhere.

The test is to give the census the tube’s family. With four siblings — the main band at 558, 568, 588 and 598 nm — added to the lamps the estimate is fitted on, a 640 nm channel beside the ratio takes the tube’s miss from 0.445 to 0.215, a little under half. The prediction’s halving happens, with a channel thirty nanometres from where it was predicted and for a different reason. But the whole census pays for it: the fifty-four lamps’ held-out error rises from ±0.141 to ±0.162, because the line that now fits the halophosphate family is the wrong line for the LEDs. One straight line in three numbers cannot serve two families of lamp whose long-wave departures mean opposite things.

What a sensor designer should take from this #

Two narrow channels, read as one ratio, are close to what a linear estimate can do with this census, and a third channel is worth a tenth at most. The place to spend a third number is not another wavelength.

The tube’s miss is a statement about the census, not about the sensor. A camera’s colour error under a lamp is a function of the lamp’s whole spectrum, and a handful of narrow channels can estimate it only where the lamps the estimate was fitted on include ones like the lamp in front of it. Two channels estimate what one can only sort found two channels predicting the camera’s error to within 0.14 over the census, each lamp left out in turn; that number is an average over lamps most of which have relatives in the census, and it says little about one that has none. A lamp family with one member is a lamp the estimate has never seen, whatever channels it has.

If halophosphate tubes matter, the remedy is a second stage, not a better channel. The sensor already separates structured lamps from smooth ones, as a narrow channel has to be read on its own established; a sensor that also recognised the halophosphate family — two broad bands and mercury lines is a distinctive shape — could apply that family’s own line, fitted on that family, and leave the LEDs’ line alone. The census here says such a line exists and that a 640 nm channel is what it would read.

How the census was computed #

The fifty-four lamps, the camera’s error under each and the sensor’s channels are those of the earlier essays on this sensor: fourteen named lamps and forty generated LEDs, the error a mean CIEDE2000 difference over a test set after the camera’s white balance, and each narrow channel a 10 nm triangle whose share of the sensor’s signal is compared, in the logarithm, with a linear prediction from the camera’s own three channels fitted on a training set of smooth lamps. A third channel of width 10, 20 or 40 nm is centred at every ten nanometres from 560 to 660. Each estimate is ordinary least squares with an intercept, the ratio’s departure in magnitude and the third channel’s signed departure as inputs, and each lamp’s residual is taken from a fit on the other fifty-three. The tube’s family is the halophosphate model — bands at 480 and 578 nm with powers 0.45 and 1 and widths 90 and 100, mercury lines at 0.55 of their relative strength — with the second band moved; the family’s departures are computed through the same channels and the same predictions.

What this leaves out #

The estimate is linear. A nonlinear estimate — a small tree, or a line per recognised family — might use a long-wave channel for the halophosphate family without costing the LEDs, and this census does not try one; the two-stage remedy above is a prediction, not a result.

The family is one parameter. Real halophosphate tubes differ in their antimony and manganese content, which moves both bands and their proportion; the census moves only the main band, and a real family would be less tidy.

The census is one camera’s. A camera whose red channel reached further into the long wave would already see the band’s position in its own three numbers, and its error under the tube, like the tube’s departure from its prediction, would be smaller.

Still open: whether a second stage keeps the LEDs’ estimate #

The census suggests an estimate in two stages: recognise which family a lamp belongs to, then apply that family’s own line. The calculation is that estimate on this census, with the tube’s four siblings added: the lamps classified into families by their 450, 500 and 640 nm departures, each family’s line fitted on its own members, and every lamp left out of both its classification and its line. The prediction is that the halophosphate tube’s miss falls to about 0.2, as it did with one line, while the fifty-four lamps’ error stays near the ratio’s ±0.14 rather than rising to ±0.16 — because the LEDs keep their own line. If the classification itself misplaces lamps, the gain goes to the misplaced ones as extra error, and the census will say how many a family needs before its line is worth having.

A floor measured on a sample is the sample’s floor #

The habit is about what a held-out error can promise.

The prediction offered two readings of a failure: either a long-wave channel catches the tube, or the tube’s error lies where no narrow channel sees and ±0.14 is the floor. Both treated the floor as a property of the sensor. It is not quite either. The tube’s error is visible to a long-wave channel, and ±0.14 is still the floor — because the floor was measured on a census with one tube in it. Leaving a lamp out and estimating it from the rest asks how well the estimate transfers to a lamp like the others; for a lamp like none of them, it asks nothing the channels could answer.

The failure mode is to read a held-out floor as a limit of the instrument when it is a limit of the sample. A census that contains one member of a family can say how badly that family is estimated, but not whether it could be estimated better.