The lamps two channels miss are smooth to the camera

A narrow channel has to be read on its own gave a phone’s ambient-light sensor one extra channel, a few nanometres wide, and asked whether it could tell a structured lamp — an LED, a fluorescent tube — from a smooth one without leaning on deep red, as the clear channel did. Pooled into the sensor’s disagreement with the camera, no placement worked. Read on its own, as a ratio against what the camera’s white predicts, a channel at 450 nm separated the census’s fourteen lamps and held until four per cent of calibration error, and one at 500 nm until two and a half.

That essay was explicit about why each worked, and so about where each would fail. The 450 nm channel works because every structured lamp in the set has a feature there — the blue pump of a white LED, a mercury line — and an LED pumped in the violet would not. The 500 nm channel works because every structured lamp is short of light there, between a pump and its phosphor, and an LED with a cyan emitter added to fill that dip for better colour rendering would not be. It proposed reading both, each alone, and calling a lamp structured if either departs: “a disjunction of two robust tests is nearly as robust as the weaker,” and the pair should cover both constructed families.

Each half of that prediction was half right.

Covered by one channel, missed by both, and harmless #

The 450 nm channel catches every cyan-filled LED, at more than 2.6 times its threshold; the 500 nm channel is blinded by a cyan emitter at 485 to 495 nm, as predicted. A violet-pumped LED needs a blue phosphor to make white, and that phosphor gives the 450 nm channel something to see — in half the violet designs. The other ten of twenty escape both channels. Every one of those ten leaves the camera’s colour error between 1.09 and 1.31 ΔE00, no worse than the smooth census lamps and below every structured one. The disjunction is right about all sixteen lamps with a margin and tolerance no better than the 450 nm channel’s alone.

• The second channel buys nothing. On the census plus one lamp of each family, the 450 nm channel alone is right with a margin of ×3.1; the disjunction is right with ×2.9.
• The violet family’s blind spot is real and harmless. A violet LED with broad phosphors has its structure at 410 nm, where the camera barely looks, and is smooth where it does.
• The classification was always a proxy. Even among the census lamps, a pink-tinted radiator gives the camera a larger error than the broadband tube.
• The right target is the camera’s error, and a channel that tracks it is worth more than one that tracks spectral shape.

Two families, built to blind one channel each #

The census is the ambient essay’s: six smooth lamps — radiators and daylights, two tinted off the locus — and eight structured — white LEDs, three kinds of fluorescent tube, a three-emitter source. Two families are added.

A violet-pumped LED has a die at 410 nm and three phosphors: a blue near 455, a green near 535 and a red near 630. Violet-pumped designs are sold for their colour rendering, because the violet die is far enough into the short wavelengths to be nearly invisible and the white is made from phosphors alone. A cyan-filled LED is the census’s neutral white LED — a 452 nm pump and a broad yellow phosphor — with a narrow emitter at 490 nm added, filling the gap between pump and phosphor. Each family is swept: the violet LED’s blue phosphor over centres from 435 to 475 nm and widths from 30 to 80; the cyan emitter over centres from 475 to 505 nm and powers from a twentieth to a third of the pump’s.

The violet LED that escapes has nothing distinctive at 450 or at 500. Its blue phosphor, centred at 465 and 45 nanometres wide, fills the region from 430 to 500 smoothly, and the violet die’s line at 410 sits below both channels. The violet LED that is caught has a narrower phosphor at 445, which makes a peak the 450 nm channel sees. The cyan-filled LED has its gap filled at 490, so the 500 nm channel sees light where every census LED was short of it — but it still has its pump at 452, and the 450 nm channel sees that as clearly as it sees any white LED’s.

What each channel reads #

The lower left square is what both channels call smooth. It holds the six smooth census lamps, as it should, and no structured census lamp. It holds none of the twenty cyan-filled LEDs: they sit well to the right, their 450 nm readings between 2.6 and 6.4 times threshold, while the 500 nm channel reads anything from a quarter of its threshold to four times it. And it holds ten of the twenty violet-pumped LEDs — every design whose blue phosphor is 60 nanometres wide or wider, and every design centred at 465, between the two channels.

So the prediction was right about the cyan family’s effect on the 500 nm channel and wrong about the violet family’s effect on the 450 nm channel. A violet die cannot make white alone; the phosphor that supplies the blue sits somewhere between 435 and 475 nanometres, and where it is narrow and near 450 it makes a feature the channel sees. The prediction treated a violet-pumped LED as a blue-pumped one with its pump moved. It is a different construction.

A cyan emitter blinds the 500 nm channel when it sits at 485 to 495 nanometres with a fifth of the pump’s power or more. At a twentieth of the pump’s power it barely moves the reading; at 475 or 505 it sits beside the channel rather than in it. The channel’s blind spot is narrow and the family has to be built into it, but real cyan-filled LEDs are built into it, since 485 to 495 is exactly where the gap they are designed to fill is deepest.

What the disjunction costs #

On the census alone, the 450 nm channel has a margin of ×5.5 and fails at four per cent of calibration error; the 500 nm channel ×4.8 and two and a half per cent; the disjunction ×5.3 and two and a half per cent. The disjunction takes its tolerance from the weaker channel — any gain error on either channel can flip a smooth lamp over that channel’s threshold — and its margin from whichever smooth lamp reads highest on either. It is, as the proposal said, nearly as robust as the weaker test, and that is the problem: it is exactly as robust as the weaker test, and no more robust than the stronger one alone.

With the violet-pumped and cyan-filled lamps added, the 500 nm channel is wrong about both, the 450 nm channel is right with a margin of ×3.1, and the disjunction is right with ×2.9. Both fail at two and a half per cent, because the violet LED reads only 1.3 times the 450 nm threshold and a small gain error puts it under. The second channel adds a way for smooth lamps to be misread and no lamp the first channel missed.

What the camera would have got wrong #

This is the part the proposal did not ask about, and it changes the reading of everything above. The classification into smooth and structured was never the goal. Two matrices do not reach a white LED found that a camera blending two colour matrices by colour temperature renders colours worse under structured lamps, and the classifier was proposed so that a camera could know when to switch to something better. The quantity it stood for is the camera’s colour error under the lamp.

The camera’s errors under the six smooth census lamps run from 1.14 to 1.62 ΔE00, and under the eight structured ones from 1.53 to 2.37. Under the ten violet-pumped LEDs that both channels miss they run from 1.09 to 1.31 — no worse than the smooth lamps, and for two of them better than any. Under the violet LEDs the channels do catch, 1.15 to 1.59: mostly in the smooth range too, so within that family most detections are false alarms by the measure that matters. Under the cyan-filled LEDs, 1.29 to 1.83, and it is the most powerful cyan emitters, the ones that blind the 500 nm channel, that bring the error down to 1.3 or 1.4, because a filled gap makes the spectrum smoother where the camera looks.

The misses are the broad-phosphor designs, and broad phosphors are what make a lamp smooth for the camera. The only violet design with a camera error in the structured range, 1.59, has a narrow blue phosphor at 475 — and both channels catch it. The violet die’s line at 410 nanometres is structure, but it is structure where the camera’s blue channel has almost no sensitivity, so the camera’s colours do not suffer from it and the channels do not need to see it.

That makes the result a statement about the proxy rather than about the channels. The census’s classes were a stand-in for camera error, and an imperfect one even among its own lamps: the pink-tinted radiator, a smooth lamp, gives the camera an error of 1.62, more than the broadband tube’s 1.53. Two sensors disagree about deep red, not lines found one classifier reading deep red rather than structure; a classifier can also read structure that the camera does not care about. Neither is what a camera needs.

What the classifier is for #

A classifier that tells a camera which lamps are structured is only worth building if the camera does something different when it is told. Three things it might do have been priced. Flicker sorts lamps the wrong way was the first attempt at the telling, and it sorted lamps by how their power was delivered rather than by their spectra. A camera is a fourth observer is the reason any of this is needed: a camera’s three sensitivities are not the observer’s, so a lamp’s spectral detail reaches the two differently, and the camera’s error depends on which detail and where.

The repairs a camera can apply once it knows are several. A third matrix fitted under a structured lamp, used for every lamp the classifier flags, is the one the two-matrix essay proposed. For the frame’s corners, a corner is corrected by one row found a colour-dependent correction that is right under the lamp it was fitted for, and one row for every lamp costs the lamps that lose least found that pooling it across lamps charges the ones that needed it least. Every one of those repairs is an improvement for the lamps it targets and a cost for the others. A classifier’s false alarms apply a repair where none was needed, and its misses withhold one where it was.

That is why the camera’s error is the right yardstick for a miss and not the label. The ten violet LEDs the channels miss would, if flagged, have had a structured-lamp repair applied to colours the camera was already getting right — a false alarm by the only measure a camera has. Missing them is the correct answer. The violet LEDs the channels catch, which the camera also renders like daylight, are the classifier’s real errors in this family, and they are errors a second channel does not remove.

What a sensor designer should take from it #

One narrow channel at 450 nm, read alone, is the design. It separates the census, catches every cyan-filled LED, catches the violet designs whose narrow blue phosphor would trouble the camera, and misses only lamps the camera renders as well as daylight. A second channel at 500 nm adds a way to be wrong and no lamp to be right about.

Test a classifier against the camera’s error, not against a label. The census’s labels were chosen by construction — LEDs and tubes structured, radiators and daylights smooth — and the camera’s error is what they were chosen to predict. A new family should be scored on that directly, as here, and a classifier’s miss judged by whether the missed lamp costs the camera anything.

And the next question is a channel that reads the camera’s error itself. The camera’s error depends on how a lamp’s spectrum interacts with the camera’s three channels, not on where its features are. A narrow channel has to be read on its own placed channels where features were; a channel placed where the camera’s error is most sensitive would be a different design.

How the lamps and channels were modelled #

The ambient sensor is the red, green and blue channels of the ambient essays on bare silicon — Gaussians at 615, 535 and 465 nanometres — with a narrow Gaussian channel ten nanometres wide at 450 or 500. For each design, the ratio of the narrow channel’s reading to the sum of the three broad channels is predicted from the camera’s white by a linear fit over smooth training lights, radiators from 2500 to 6500 K and daylights from 4000 to 10000 K, and a lamp’s departure is the absolute logarithm of its measured ratio over the predicted one. Each channel’s threshold is the geometric middle between the census’s most departing smooth lamp and least departing structured lamp. The disjunction calls a lamp structured when either channel’s departure exceeds that channel’s threshold.

The calibration tolerance is the smallest error on a grid from a quarter of a per cent to ten per cent at which some combination of the five channel gains — each raised, lowered or left alone by that error, with the three broad channels shared by both narrow ones — misclassifies a lamp. The camera’s colour error is the mean CIEDE2000 over the collection’s test set when the camera’s two matrices, fitted under illuminant A and D65, are blended at the lamp’s correlated colour temperature with the lamp’s own white held neutral.

What this leaves out #

The violet and cyan designs are constructed. Real violet-pumped LEDs vary in their phosphors and real cyan-filled LEDs in their emitter’s position and width; the sweeps cover a plausible range of each, not a catalogue.

The camera’s error is its mean over a test set. A camera designer might care about the worst colour, or about skin, and the ranking of lamps by those could differ; the census’s smooth lamps set the standard in the same terms either way.

And the classifier’s threshold is fitted on the census. A threshold fitted on the camera’s error instead — flag a lamp when its predicted error exceeds a tolerance — would be a regression, not a classification, and would need a training set of lamps whose camera errors are known.

Still open: whether a channel can read the camera’s error directly #

The camera’s error under a lamp depends on how the lamp’s spectrum projects onto the camera’s three sensitivities against how it projects onto the observer’s, and a narrow channel placed where those two projections differ most would read that directly rather than reading features. The census suggests where to look: the errors are largest for lamps with narrow features between 520 and 620 nanometres, where the camera’s green and red channels overlap the observer’s differently.

The calculation is a sweep of one narrow channel’s centre across the visible range, scored not by how well it separates the census’s classes but by how well its departure predicts the camera’s error across the census and both families — the rank correlation of reading with error. The prediction is that the best placement is not at 450, where lamps differ most visibly, but in the 560 to 600 nanometre region, where the camera and observer disagree most; and that such a channel would flag the pink-tinted radiator, which the census calls smooth and the camera renders worse than a tube.

A proxy has to be checked against what it stands for #

The habit is about asking, when a classifier misses something, whether the thing it missed mattered.

The proposal set out a clean test: two blind spots, two families constructed to fall into them, and a disjunction to cover both. The census confirmed one blind spot, refuted the other and found the disjunction no better than one channel — and all of that was measured against labels. Measured against the camera’s error, which the labels were chosen to predict, the misses were lamps the camera handled well and the extra channel protected against nothing.

The failure mode is to optimise a classifier’s agreement with its labels after forgetting why the labels were drawn. A label is a hypothesis about the quantity that matters, and a new family is a chance to test the hypothesis as well as the classifier.