Two channels estimate what one can only sort

The lamps two channels miss are smooth to the camera tested two narrow ambient-sensor channels, at 450 and 500 nm, against two families of white LED built to blind each of them. The families fell partly into the blind spots, and it did not matter: every lamp both channels missed left the camera’s colours as accurate as daylight did. The finding was about the target rather than the channels. The smooth-or-structured label that every channel had been fitted to was a stand-in for the camera’s colour error under a lamp, and an imperfect one — a pink-tinted smooth radiator gave the camera a larger error than a fluorescent tube.

That essay’s closing section asked for the obvious correction: score a channel against the error itself, and read it as an estimate rather than a verdict. It made a prediction about where such a channel would sit — “not at 450, where lamps differ most visibly, but in the 560 to 600 nanometre region, where the camera and observer disagree most.”

The same channel, used differently #

Over fifty-four lamps, a channel at 450 nm ranks the camera’s colour error better than one at any other centre, at 0.87; channels from 560 to 600 nm rank it at 0.79 to 0.83. As an estimate, the 450 nm channel alone predicts the error to within ±0.16 ΔE00 with each lamp left out of the fit, against ±0.30 for guessing the mean. Adding the 500 nm channel — which added nothing as a second vote — brings it to ±0.14 and cuts the worst miss from 0.74 to 0.46. The lamp’s white’s distance from the locus adds nothing. And deciding when to repair from the estimate raises four false alarms where the label raises fifteen, at the cost of missing one lamp the label catches.

• The prediction about placement is wrong. The band where the camera’s and the observer’s sensitivities differ most ranks the error well and no better than the channel already chosen.
• A channel that votes uselessly can estimate usefully. The 500 nm channel’s value was never in deciding smooth against structured; it is in saying how badly structured.
• Rank is not estimate. The 570 nm channel orders lamps by their error almost as well as the 450, and a straight-line estimate from it is worse than guessing the mean.
• An estimate trades false alarms for misses, and the trade is overwhelmingly in its favour at every tolerance tried — but one of its misses is a lamp the camera renders badly.

The fifty-four lamps and their errors #

The lamps are the ambient census’s fourteen — six smooth, eight structured — and the two families of the last essay: twenty violet-pumped LEDs and twenty cyan-filled ones. Under each, the camera’s error is what two matrices do not reach a white LED measured: the mean CIEDE2000 over the collection’s test colours, when the camera blends its two matrices at the lamp’s colour temperature and holds the lamp’s white neutral. The errors run from 1.09 under a broad violet LED to 2.37 under the triphosphor tube. The smooth census lamps sit between 1.14 and 1.62.

A channel is read exactly as the channel essays read it: its reading over the sum of the red, green and blue, as a logarithm against what the camera’s white predicts that ratio to be, with the prediction fitted over smooth training lights. What changes is only what the reading is compared with.

Where a channel knows most about the error #

The channel at 450 nm ranks the camera’s error at 0.87, and no centre does better. The band from 560 to 600 nm, between the lines, sits at 0.79 to 0.83 — high, and consistently so, as the prediction said it would be, but not highest. In between are three dead zones: at 470, 530 to 540 and 610 nm, channels whose departures have no relation to the error at all, or a negative one. At those wavelengths the lamps’ spectra differ from a smooth lamp’s in ways that do not affect the camera, or affect it in both directions across the families.

Why the 450 nm channel does so well is visible in what it sees. Every white LED’s pump and every tube’s mercury line puts a feature there, and those are the lamps whose errors are high. The violet LEDs that lack a feature at 450 are the ones the camera renders well. The channel was chosen for separating labels and it turns out to rank errors, because the labels were not a bad proxy — only an imperfect one.

A vote and an estimate are different uses #

The estimate is a straight line from one or more channels’ departures to the camera’s error, fitted over fifty-four lamps and scored by fitting it fifty-four times, each time with one lamp left out, and asking how wrong it is about that lamp. Guessing the mean is off by ±0.30 ΔE00, worst 0.92 at the triphosphor tube. The 450 nm channel alone is off by ±0.16, worst 0.74, also the triphosphor tube. The 450 and 500 nm channels together are off by ±0.14, worst 0.46 at the halophosphate tube. The white’s distance from the Planckian locus is worth nothing on its own — ±0.31, worse than the mean — and adds nothing to the channels.

The 500 nm channel’s contribution is the point. A narrow channel has to be read on its own found it a good classifier on the census; the last essay found it a useless second vote, adding a way for smooth lamps to be misread and no lamp the 450 nm channel missed. As an estimator it is the most useful addition there is.

The lamp the 450 nm estimate gets most wrong is the triphosphor tube, whose camera error of 2.37 is by far the census’s largest, and it is also the lamp with by far the largest departure at 500 nm: its three narrow phosphors leave almost nothing between blue and green. A vote does not need to know that the triphosphor tube is worse than the other structured lamps — it is structured either way. An estimate does, and the 500 nm channel is where the information is.

Rank is not an estimate #

The 570 nm channel ranks the error at 0.83, and a straight-line estimate from it is off by ±0.57 — worse than guessing the mean. The lamps are in the right order along its axis: small departure, small error; large departure, large error. But the relation is steep at small departures and flat at large ones, and the three-emitter source’s departure is many times anyone else’s, so a straight line fitted through the cloud is pulled far from most of it. Left out of the fit, the three-emitter source is predicted an error of more than five.

Leave the three-emitter source out of the census and the 570 nm estimate is off by ±0.18, worst 0.58, against ±0.17 for the 450 nm channel on the same fifty-three lamps. The whole of its failure as an estimate was one lamp. Its departure at 570 nm is 2.1 on the channel’s log scale, almost three times the next lamp’s 0.73, and a straight line has to choose between that lamp and the other fifty-three. That makes the 570 nm channel not a worse channel but a fragile one: as good as the 450 nm channel on ordinary lamps, and wrecked by the first lamp whose mid-spectrum structure is extreme — which is exactly the kind of lamp a camera most needs an estimate for.

That is the difference between the two ways of scoring a channel. A rank correlation asks whether a channel orders lamps correctly; an estimate asks whether it can say how much. The prediction about 560 to 600 nm was about ordering, and it was roughly right about ordering. The channel a camera can use is the one that says how much, and there the 450 and 500 nm pair is far ahead.

What the estimate does to the repair decision #

A camera does not want a number for its own sake. It wants to know whether to apply a repair — a third matrix, a correction fitted for structured lamps — and a repair helps the lamps that need it and costs the ones that do not.

At a tolerance of 1.5 ΔE00, twenty-two of the fifty-four lamps need a repair. Deciding from the label — structured means repair — raises fifteen false alarms, repairing cyan and violet LEDs the camera was already rendering well, and misses one lamp, the pink-tinted radiator at 1.62. Deciding from the held-out estimate raises four false alarms, all cyan-filled LEDs estimated just above the line, and misses four: the pink radiator, the broadband tube at 1.53, one violet LED at 1.59, and the halophosphate tube at 1.90.

At every tolerance tried the estimate makes fewer mistakes in all: at 1.4, two false alarms and one miss against ten and one; at 1.6, one and two against nineteen and one. The label’s single miss is always the pink radiator, which no spectral channel sees, because its error comes from its tint rather than from structure. The estimate’s cost is at 1.5, where it misses the halophosphate tube — a lamp the camera renders at 1.9, which the label would have repaired. Whether that trade is worth it depends on what a false alarm costs against a miss, and that cost has been measured: one row for every lamp costs the lamps that lose least found a repair applied where it was not needed costing those lamps several times what it gave the others.

An estimate fitted on known lamps, used on new ones #

The leave-one-out scores fit the estimate on fifty-three lamps drawn from all three groups and ask it about the fifty-fourth. A camera maker’s situation is harder: the estimate is fitted on the lamps available when the camera is designed, and it meets the lamps that reach the market afterwards. The two constructed families stand in for those well, since they were built to fall outside what the census’s channels were chosen for.

Fitted on the census’s fourteen lamps alone and used on the forty family lamps it has never seen, the two-channel estimate is off by ±0.14 ΔE00, with a worst miss of 0.31. The 450 nm channel alone is off by ±0.19 and guessing the census mean by ±0.32. The estimate carries a bias — it expects the new lamps to be about a tenth of a unit worse than they are, because the census’s structured lamps are worse on average than the families’ — and a bias in that direction is the safe one for a repair decision, since it errs towards repairing.

That transfer is the strongest evidence here that the channels are reading the right thing. An estimate that held only on the lamps it was fitted to would be a description of the census; one that holds on families built to defeat the census’s channels is a measurement. Flicker sorts lamps the wrong way found a classifier that sorted the census correctly for the wrong reason and would have failed on the first lamp driven differently. A camera is a fourth observer is the underlying reason the 450 and 500 nm readings can transfer: they measure how a lamp’s spectrum differs from a smooth one in the places where the camera’s three sensitivities and the observer’s collapse it differently, and that relation belongs to the camera and the observer, not to any list of lamps.

What a sensor designer should take from it #

Keep the 450 nm channel, and add the 500 nm one for the estimate, not for the vote. The pair that the last essay found no better than one channel at classifying is clearly better at estimating, and an estimate is what a repair decision needs.

Fit the channels to the camera’s error, not to a label. The fit here needs only the camera’s colour error under a set of training lamps, which a camera maker can measure; the label was a guess about which lamps those would be, and the guess was wrong often enough to raise fifteen false alarms in fifty-four.

And choose a channel by what it can estimate. Two sensors disagree about deep red, not lines found a classifier reading the wrong property; this finds one reading the right property in a form a straight line cannot use. The test for a channel is its held-out error as an input to the estimate a camera will actually make.

How the estimates were computed #

The fifty-four lamps and the channel readings are those of the lamps two channels miss are smooth to the camera. The rank correlation is Spearman’s, over the fifty-four lamps. Each estimate is an ordinary least-squares straight line with an intercept, from the stated inputs to the camera’s mean colour error; its held-out error is found by refitting it fifty-four times with one lamp left out each time and predicting that lamp, and the figures report the root mean square and the largest of those fifty-four errors. The white’s distance from the locus is the absolute Duv of the lamp’s correlated colour temperature. The repair decisions compare the held-out estimate, and the 450 nm channel’s structured label, with the camera’s true error at each tolerance.

What this leaves out #

Fifty-four lamps is a small training set for anything beyond a straight line, and the held-out scores show how much the three-emitter source alone can pull a fit. A camera maker would train on many more lamps, including real ones.

The camera’s error is a mean over test colours. An estimate of the worst colour, or of skin tones, might want different channels; the method would be the same.

And the lamps are partly constructed. Two of the three groups were built to test the channels’ blind spots, which is not how lamps are distributed in the world. An estimate fitted to a realistic mix would weight the families differently.

Still open: whether a channel read nonlinearly makes 570 nm useful #

The 570 nm channel ranks the camera’s error almost as well as the 450 nm one and estimates it worse than the mean, because its relation to the error is curved. A logarithm, a square root or a saturating function applied to its reading might straighten it.

The calculation is this estimate with each channel’s departure passed through a small family of monotone transforms before the fit — its logarithm, its square root and a saturating curve with one fitted constant — scored by the same leave-one-out error. Leaving the three-emitter source out has already shown what 570 nm is worth on ordinary lamps; the open question is whether a transform lets it keep that value with the extreme lamp in the census. The prediction is that a transformed 570 nm channel reaches the 450 nm channel’s ±0.16 over all fifty-four lamps, and that the pair of 450 nm and transformed 570 nm does better than 450 and 500 nm, because 570 nm is where the camera’s green and red channels see the lamps’ mid-spectrum structure and 500 nm only sees whether the blue-green gap is empty. If no transform helps, a sensor designer should treat 570 nm as a channel that needs a guard against its own extremes, and 450 and 500 nm as the pair that does not.

A label is an estimate someone rounded #

The habit is about asking what information a binary decision discarded before trusting it.

The smooth-or-structured label was a rounded estimate of the camera’s error: lamps expected to be bad and lamps expected to be fine. Rounding threw away how bad, and with it the use of any channel whose information was about degree rather than kind. The 500 nm channel looked worthless because it was scored against the rounded quantity; scored against the unrounded one, it was the most useful addition available.

The failure mode is to judge an input by whether it changes a rounded answer. An input that says how much is invisible to a test that only asks which, and a camera’s decisions are made on how much.