A narrow channel has to be read on its own

Two sensors disagree about deep red, not lines asked whether a phone’s ambient-light sensor, disagreeing with its camera, could tell a lamp with lines in its spectrum from a smooth one — the classification a camera needs to choose a colour matrix and cannot make from its own white balance, because three numbers cannot see a line. With a clear channel the answer was yes, on all fourteen lamps, and the reason was deep red. The clear channel collects light past the camera’s infrared cut, the structured lamps among the fourteen happen to have little of it, and so a daylight with its far red trimmed was called structured and a white LED with a far-red emitter was called smooth. Without the clear channel, a red, green and blue sensor read something nearer to structure and broke at one per cent of calibration error.

That essay named a candidate that might do both. A narrow channel placed inside the camera’s range but between its peaks — near 480 nm, in the dip between an LED’s blue pump and its phosphor, or near 585 nm between a tube’s green and red lines — sees structure directly, and deep red cannot move it. It asked for three numbers: the margin on the fourteen lamps, the calibration error at which the classifier first fails, and whether the constructed lamps still cross the line.

All three are below, and the answer depends on something the proposal did not specify: how the fourth channel is read.

The same channel, read two ways #

Pooled into the sensor’s disagreement with the camera, a narrow fourth channel reads structure and does not buy the margin: the best of sixty placements first fails at one and a half per cent of calibration error, against four with a clear channel. Read on its own — its reading over the other three, against what the camera’s white predicts for that ratio — a channel at 450 nm first fails at four per cent and one at 500 nm at two and a half, and no constructed lamp in the test carries either across its line.

• Pooled, the three designs with narrow channels first fail at ±1.25 per cent, against ±1 for red, green and blue alone and ±4 with a clear channel.
• Pooled, they read structure: a far-red band added to a white LED or a three-emitter source moves their scores by at most 12 per cent and crosses none of them. With a clear channel, both cross.
• The narrow channel’s own departure on structured lamps is 35 to 61 times what one per cent of its gain moves it; its red, green and blue neighbours’ departures are 3.5 to 12 times. The pooled residual adds all four.
• Read alone, the channel at 450 nm separates the lamps 5.5 to one, the channel at 500 nm 4.8 to one.
• The 480 nm channel the proposal named does not separate the lamps when read alone: the halophosphate tube barely dips there.

Where the fourth channel can go #

The fourteen lamps are the ones the ambient-sensor essay scored: six smooth — two radiators, two daylights and a radiator tinted each way off the daylight locus — and eight structured, among them three phosphor white LEDs, an LED with a red phosphor, three kinds of fluorescent tube and a source built from three narrow emitters.

The candidates are where structured lamps depart from smooth ones, which is the same move as choosing a camera’s dyes for the job the camera will do rather than for cost, the way a sensor designed for its inverse chose them for white balance. At 480 nm the white LED sits at 24 per cent of its peak, in the trough between its blue pump and the rising phosphor; daylight there is near its own maximum. The halophosphate tube is at 40 per cent — its broad phosphor fills the region the LED leaves empty — which will matter. At 585 nm the triphosphor tube has a trough between its green and red lines and the three-emitter source is nearly dark. At 500 nm the LED is climbing out of its trough and the triphosphor tube is almost empty. And at 450 nm there is no dip at all: it is the peak of the LED’s blue pump, the place a white LED departs most from anything smooth, upward rather than down.

The sensor design is the red, green and blue sensor of the earlier essay, bare silicon under gaussian filters — like any set of dyes on silicon, a fourth observer that agrees with no standard one — with one more gaussian channel of a stated centre and width added to it. Every design is scored the same way: a linear map from the camera’s white to the sensor’s reading, fitted over thirty smooth radiators and daylights and never over any of the fourteen lamps, and a threshold at the geometric middle of the gap between the classes — the most favourable place it can go.

Pooled, the channel reads structure and nothing more #

The ambient-sensor essay’s statistic was the relative residual: predict all of the sensor’s channel shares from the camera’s white and take the length of the error. Adding a channel adds a row to that prediction and a term to that length.

The three designs with narrow channels do what the proposal asked of them in one respect. Far red does not move them: adding a band at 720 nm to the white LED and the three-emitter source changes their scores by at most 12 per cent, and both stay structured, where the two designs with a clear channel call them smooth. The proxy has stopped reading deep red.

They do not do the other thing. The gap between the classes is 0.024 to 0.034, where the clear channel had 0.076; and the first calibration error to misclassify a lamp is ±1.25 per cent for all three, against ±1 for red, green and blue alone and ±4 with a clear channel. One narrow channel buys a quarter of a per cent. Two narrow channels, at 480 and 585 nm together, buy the same quarter.

On a finer grid of calibration errors than the earlier essay swept, the clear channel’s first failure is at ±4 per cent rather than ±5; the earlier sweep stepped from three per cent to five and reported the first step that failed. Nothing about the comparison depends on the difference.

The dashed line in the figure above is every placement of a 10-nanometre channel from 450 to 640 nm, pooled. None rises above ±1.5 per cent, and most sit within a quarter of a per cent of the red, green and blue line, above or below it. At widths of five and twenty nanometres the census is the same: the best pooled placement anywhere is ±1.5 per cent. Where the channel goes barely matters. The pooled residual cannot use it.

Why pooling wastes it #

A residual is a sum of squares over channels, and a calibration error in any channel adds to it whether or not that channel carries a signal.

With a clear channel, every channel’s departure is 14 to 33 times its own one-per-cent drift. That is the part of the earlier result nobody had looked at: the clear channel’s deep red does not stay in the clear channel. The sensor’s reading is a set of shares, so when a lamp is short of deep red the clear share falls and every other share rises, and all four channels carry the signal. A calibration error of a few per cent in any one of them is small against it.

With a narrow channel at 480 nm, the narrow channel’s departure is 61 times its drift — the best-placed channel in any design here. Its red, green and blue neighbours’ departures are 3.5, 9.4 and 3.7 times theirs. A narrow channel collects a few per cent of the sensor’s light, so a lamp’s dip moves its own share hugely in relative terms and everyone else’s hardly at all. The residual then adds the narrow channel’s large clean signal to three small noisy ones, and a calibration error in red or blue — which the narrow channel’s signal cannot outweigh, because it is measured in a different channel — misclassifies a lamp almost as soon as it would have without the narrow channel: at ±1.25 per cent rather than ±1.

The mechanism is general. A statistic that pools channels is as robust as its worst signal-to-drift ratio, not its best, unless each term is weighted by what it carries. The clear channel was robust in a pooled residual because it spread its signal into every channel. A narrow channel concentrates its signal into one.

Read on its own #

The repair is to stop pooling. Read the narrow channel’s reading over the sum of red, green and blue, a ratio a calibration error in any one channel moves by about that error, and compare it with what the camera’s white predicts for that ratio from the same smooth lights. The size of the logarithm of the two is the score.

At 450 nm, the smooth lamps sit within 0.056 of zero and every structured lamp departs by more than 0.3. The four white LEDs read high by 0.87 to 0.96, their blue pump putting far more light in the channel than any smooth spectrum with their white could. The three-emitter source reads 0.81 high, the triphosphor tube 0.62, and the halophosphate and broadband tubes 0.37 and 0.31, on the mercury line at 436 nm and, in two of them, a blue phosphor band beside it. The ratio of the classes is 5.5 to one, and the first calibration error to misclassify a lamp is ±4 per cent — the clear channel’s figure.

At 500 nm every structured lamp reads low: the LEDs by 0.33 to 0.47, the three-emitter source by 0.87, the triphosphor tube by 1.90, the broadband and halophosphate tubes by 0.21 and 0.19. The smooth lamps sit within 0.039. The ratio is 4.8 to one and the first failure ±2.5 per cent.

Back in the sweep, the solid line is every placement read alone. It is far higher than the dashed line wherever it works — ±4 per cent at 450, ±2.5 at 500 — and at several centres it does not work at all, marked with crosses: 470, 480, 530, 550 and from 600 to 630 nm, the channel read alone cannot separate the lamps, because some structured lamp has no feature there.

The place the proposal named does not work alone #

One of those crosses is at 480 nm, the dip the proposal named. Read alone, a channel there does not separate the classes: the halophosphate tube departs from the smooth prediction by 0.06, and the most departing smooth lamp, the radiator tinted green, by 0.062.

The spectra say why. A white LED’s dip at 480 nm is a gap between two emitters, and a halophosphate tube has no such gap — its broad phosphor band covers the region the LED leaves dark, and it reads nearly as a smooth light would. A dip is a feature of one family of lamps, not of structured lamps in general, and the fourteen include a family without it. In the pooled residual the tube was still called structured, because the other three channels carried enough; alone, the narrow channel had nothing to say about it.

The places that work alone are the places every family departs in the same direction. At 500 nm every structured lamp here is short of light, whether it has a gap there, a slope there or a trough between lines. At 450 nm every one has a feature there — an LED’s pump, a tube’s mercury line and blue phosphor, an emitter. Neither is between the camera’s peaks in the sense the proposal meant. The best of them is on a peak.

Constructed lamps #

The ambient-sensor essay’s decisive test was lamps built so that deep red and structure come apart, and it is the test a narrow channel was proposed to pass.

The clear channel, pooled, is wrong on all five — both far-red lamps called smooth, and all three trimmed radiators and daylights called structured. The 480 nm channel, pooled, is wrong on one: a radiator with everything above 660 nm rolled off is called structured, because its red channel reaches that far. Both channels read alone are right on all five. The 500 nm channel is not crossed by any trimming down to 640 nm; the 450 nm channel is crossed only when a radiator loses everything above 640 nm, which removes visible red that a lamp’s own colour would show, and is not the construction the earlier essay was worried about.

Far red moves the 450 nm reader by 1.3 per cent and the 500 nm reader by at most 1.8 per cent at the largest amount added. The proxy is the property.

What a sensor designer can take from it #

The choice between a robust classifier and a correct one was not a choice between sensors. It was a choice of statistic. The earlier essay’s red, green and blue design and every design with a narrow channel carried enough information to classify structure; what broke them was a residual that weighted a channel with nothing to say as heavily as one with everything.

A narrow channel is worth adding and worth reading separately. At 450 nm it matches the clear channel’s calibration tolerance without reading deep red; at 500 nm it gives up some tolerance and becomes immune to any trimming of the red. A sensor could carry both and require them to agree.

And test with lamp families, not lamps. The 480 nm channel would have passed every white LED in the set and failed on one tube, and the ambient-sensor essay’s clear channel passed every real lamp in the set and failed on every construction. In both cases the failures were a family that happened to be absent or present — which is the lesson flicker sorts lamps the wrong way drew about a cross-tabulation that passes on the cases that exist.

How the channels were modelled and read #

The camera is the modelled silicon sensor with colour dyes and an infrared-cut filter used throughout the imaging essays, and a matrix is fitted under one light is where its profile was fitted. The ambient sensor’s red, green and blue channels are gaussian filters at 615, 535 and 465 nm, widths 40, 45 and 35 nm, on bare silicon; the narrow channel is a gaussian of the stated centre and width on the same silicon. The lamp spectra are those of two matrices do not reach a white LED, from 380 to 780 nm.

The pooled score is the earlier essay’s: each instrument’s reading reduced to shares of its channels, a least-squares map from the camera’s three shares to the sensor’s, fitted over seventeen radiators and thirteen daylights, and the length of the prediction’s error over the length of the reading. The score read alone is the absolute natural logarithm of the narrow channel’s reading over the sum of red, green and blue, divided by a linear prediction of that ratio from the camera’s three shares fitted over the same thirty lights.

Calibration error multiplies each channel by one plus, minus or zero times a stated fraction, every combination tried, on a grid from a quarter of a per cent to ten per cent. A design’s tolerance is the first step on that grid at which any lamp is misclassified.

What this leaves out #

A narrow channel is a dim one. A 10-nanometre channel collects a few per cent of what a broad one does, and the scores here are noiseless. In a dim room, a phone’s ambient sensor integrating for a few milliseconds reads a narrow channel with photon noise that a statistic divided by that channel amplifies. The calibration tolerance is the right test for a bright room and an incomplete one for a dark one, and a narrow channel’s noise floor is a property of real parts not modelled here.

The sensitivities are modelled, gaussian filters on a generic silicon curve, and a real interference filter narrow enough for this has a centre that moves with the angle light arrives at — the corner of the frame has another filter priced that for a camera. A sensor under a diffuser sees light from every angle, which broadens a narrow channel by an amount set by the filter’s design.

The lamps are fourteen and the families six. Every result here, including the failure at 480 nm, is a statement about which families are in the set. A white LED pumped at 405 nm rather than 450 would have no pump in the 450 nm channel at all, and it is not in the set.

Still open: whether two narrow channels cover a missing family #

The 450 nm channel works because every structured lamp in the set has a feature near 450 nm, and a white LED pumped in the violet would not. The 500 nm channel works because every structured lamp in the set is short of light near 500 nm, and a lamp built with an emitter there — a cyan-filled LED of the kind sold for higher colour rendering — would not be.

The two channels fail on different lamps, which suggests reading both, each alone, and calling a lamp structured if either departs. The prediction is that the pair covers both constructed families without giving up much tolerance, because each reading’s calibration error is independent of the other’s and a disjunction of two robust tests is nearly as robust as the weaker. The computation is this one with a violet-pumped LED and a cyan-filled LED added to the structured set, the pair read as a disjunction, and the three numbers — margin, first failing calibration error and the constructions — reported for the pair and for each channel alone. If the pair holds, a sensor with two narrow channels read separately is a classifier of spectral structure. If it does not, the families a sensor has to be tested against are a longer list than any sensor designer can write down.

A channel is worth what the statistic lets it say #

The habit is about the statistic, not the instrument.

A fourth channel was proposed as a better instrument, and as an instrument it was: its own departure on structured lamps was the cleanest signal any design here carried, more than sixty times its drift. Put into the statistic the earlier classifier used, it improved nothing that mattered, because that statistic added every channel’s error to every other’s and a clean signal in one of four terms was outvoted by the noise in the other three. The same channel, read by a statistic that asked it alone, matched the best design there was.

The failure mode is to judge a measurement by the classifier it was first put into. An instrument that carries information can be wasted by a combination rule that weights it like everything else, and a design comparison that holds the rule fixed will report the waste as a property of the instrument. The move is to ask, for each channel, what it carries against what it drifts by, and to read the channels that carry the signal in a way that does not dilute them.