Seven channels carry their noise as lightly as three

A phone or a television that adapts its picture to the room reads the room’s light with an ambient sensor, a handful of photodiodes behind filters, and turns their signals into a chromaticity with a matrix. How wrong that chromaticity can be before a viewer sees the picture’s tint go wrong has been measured on the dimmest case in the census: a moon picture at a tenth of a candela per square metre, where the viewer’s eye is adapted mostly to the room and a declared white that is off shows as a tint in ΔEITP. A unit of that tint is the line between a reading that serves and one that does not.

No calibration reads the lamps and the LEDs together found that a three-channel sensor cannot be calibrated to read the room’s incandescent lamps and its white LEDs at once. Seven channels end the trade and lose the tube gave the sensor more channels, spread evenly from 445 to 650 nm, and found seven reading every lamp, mixture and LED within a unit — but losing the triphosphor tube, which had no relative in the census to learn from. Relatives, not twins, teach a sensor a tube added four triphosphor siblings and found the seven-channel sensor reading all fifty-one lights within a unit, each held out of its own fit.

Every one of those readings is noiseless. A real photodiode’s signal carries shot noise at low light and drifts with temperature, and the matrix that turns signals into chromaticity multiplies that noise by its weights. The expectation was that a fine sensor’s weights are large and of both signs, because it reads chromaticity from small differences between neighbouring channels. The prediction was that at a quarter of a per cent of noise on every channel the seven-channel sensor still reads every light within a unit, and that at one per cent it is no better than five channels, because its weights amplify channel noise about three times as much. If seven channels stayed ahead at one per cent, the gain the census found would survive the instrument it would be built as.

Seven channels hold at every level #

Read a hundred times with independent noise on every channel, the seven-channel sensor reads every one of the fifty-one lights within a unit at every reading at a quarter, a half and one per cent of noise; its worst reading is 0.33, 0.43 and 0.69. Five channels read 7, 9 and 22 lights beyond a unit at least once; three channels 40, 42 and 46; twelve channels 2 at every level.

• The first half of the prediction holds. At a quarter of a per cent no seven-channel reading of any light passes a third of a unit.
• The second half fails, by the widest margin it could. At one per cent seven channels lose nothing, where five lose twenty-two lights.
• The alternative named arrives. The census’s gain survives the instrument.

The four sensors’ lines sit almost where their noiseless readings put them. Three channels already read thirty-seven lights beyond a unit with no noise; noise adds nine more by one per cent. Twelve channels read two beyond a unit with no noise — the halophosphate tube and the broadband lamp — and noise adds none at any level. Five channels read six beyond a unit noiseless and lose sixteen more by one per cent. Seven channels read none beyond a unit noiseless and lose none.

So the ranking of sensors the census found without noise is the ranking with it, and the seven-channel sensor’s margin over five grows rather than shrinks as the noise rises. That is the opposite of what an amplifying matrix would do.

No sensor amplifies its noise much more than another #

The direct measure of amplification is how far the reported chromaticity moves for a given noise on the channels. If the seven-channel matrix multiplied noise three times as much as the five-channel one, its chromaticity would scatter three times as far.

Per unit of fractional channel noise, the reported u′v′ chromaticity moves by 0.081 for three channels, 0.116 for five, 0.084 for seven and 0.081 for twelve, the same to the third decimal at every level. At one per cent of noise that is a scatter of about eight ten-thousandths in u′v′ for three, seven and twelve channels and twelve ten-thousandths for five. Seven channels amplify their noise by a twentieth more than three broad channels and by a third less than five.

The figure has no trend in the count at all. Three channels and twelve differ by a factor of four in how finely they sample the spectrum and amplify alike; five sits above the others. What a sensor’s noise does to its reading is not set by how many channels it has.

The matrix that was expected to cancel does not #

The reason is in the weights. A matrix row that cancels — large positive contributions from some channels and large negative ones from others, summing to a small result — multiplies each channel’s noise by its contribution, and the noise adds while the signal subtracts. The ratio of the contributions’ summed sizes to their sum measures that: one means every contribution has the same sign, five means the result is a fifth of what the channels put in.

The three-channel sensor’s rows cancel by 1.51 for X, 1.37 for Y and 2.14 for Z, averaged over the census. The seven-channel sensor’s by 1.04, 1.00 and 1.13. The twelve-channel sensor’s by 1.08, 1.00 and 4.07. The premise of the prediction — that a fine sensor reads chromaticity from small differences between neighbouring channels — is false for seven channels. Its rows are sums of channels of one sign, almost without subtraction.

That follows from what the channels are. Seven Gaussian channels spread evenly from 445 to 650 nm, each a quarter wider than its spacing, overlap enough that a positive weighting of them can follow the colour-matching functions’ main lobes closely: x̄’s long-wave lobe, ȳ and z̄ are each smooth humps over a stretch that two or three adjacent channels cover. Three broad channels cannot follow those shapes, so their matrix makes up the difference by subtraction — most of all for z̄, whose narrow short-wave peak three broad channels can approach only by taking one channel from another. Twelve channels can follow the shapes too closely: with more channels than the shapes need, the least-squares fit spends some of them cancelling, again in Z, and that cancelling would amplify noise if it did not happen in channels whose signals are small for these lights.

Noise lifts the worst reading by under a unit and a half #

At one per cent of channel noise, the worst noisy reading of any light lies 0.95 above its own noiseless reading with three channels, 0.69 with five, 0.59 with seven and 1.50 with twelve. Seven channels’ worst reading rises from 0.26 noiseless to 0.69, still a third of a unit inside the line; twelve channels’ rises from 1.56 to 3.06 for the broadband lamp it already misread.

Twelve channels lift their worst reading most, and the lift is on the light the twelve-channel sensor already reads worst. A light the matrix fits poorly is one whose channel signals the matrix weights in an unusual combination, and that combination, more than the matrix’s average cancellation, decides how far its noise carries. The amplification a single light suffers is the light’s own, and it is largest exactly where the noiseless reading is already worst.

Five channels lose the lights near the edge #

At one per cent of noise, twenty-two of the five-channel sensor’s fifty-one lights pass a unit at least once. Six of them already did with no noise; the other sixteen read between 0.39 and 0.95 noiseless, and thirteen of those pass in fewer than ten readings of a hundred. The 6,500 K LEDs pumped at 452 nm and the warmest triphosphor sibling, at 3,000 K, noiseless at 0.88 to 1.00, pass in a quarter to a half of their readings.

The noise does not make the five-channel sensor fail new lights so much as decide on which side of the line its near-misses fall. A sensor whose noiseless readings crowd up against a unit loses many lights to a little noise; a sensor whose readings sit well inside it loses none. Seven channels end the trade and lose the tube found the seven-channel sensor’s noiseless worst at a quarter of a unit, and a quarter of a unit is the margin that carries it through one per cent of noise.

The error grows in step with the noise #

For every sensor the chromaticity’s spread at one per cent is 3.93 to 4.03 times its spread at a quarter of a per cent: four times, within a fortieth. The reading’s error is linear in the noise across the range, so any level between these can be read off by proportion, and a noise of two or four per cent would scatter the seven-channel sensor’s chromaticity by about 0.0017 and 0.0034 in u′v′.

At what noise the seven-channel sensor would start losing lights is then a matter of its margin. Its worst light at one per cent, a 4,000 K LED pumped at 465 nm, reads 0.10 noiseless and 0.69 at its worst; if that lift grows in proportion to the noise, as the chromaticity’s spread does, the light would reach a unit near one and a half per cent. That is an extrapolation of a worst case from a hundred readings, not a measurement.

What a unit of tint asks of the instrument #

The line every reading here is held to has its own history, and it sets how much noise matters. A room sensor must read the lamp to a hundred kelvin derived it: a local-dimming panel in a warm room leaves a dark picture’s shadows tinted towards the lamp unless its black floor is declared in the room’s colour, and the declaration has to be right to about fourteen mireds — a hundred kelvin near 2,700 K — before the tint falls under a unit. Lamp and daylight together sit below the locus added that the declaration has to be a chromaticity, not a colour temperature, because an evening room’s mixture of lamp and daylight sits off the locus by more than the tolerance allows.

Measured against that line, the noise this census applies is modest. A scatter of eight ten-thousandths in u′v′ at one per cent of channel noise is about a fifth of the Duv of 0.003 a declaration may be wrong by, and about a seventh of the chromaticity change fourteen mireds makes near 2,700 K. The noise would matter if it added to a reading already near the line, and for the five-channel sensor it does; for the seven-channel sensor the noiseless readings sit far enough inside the line that a fifth of the tolerance cannot reach it.

A sensor calibrated on lamps cannot read an LED found the first sensor’s errors in the calibration, not in the electronics: a three-channel sensor fitted to two lamps misread phosphor LEDs by several thousandths. That remains the larger source of error for every sensor here. Three channels’ noiseless worst reading is 8.6 units, which no plausible noise approaches; even twelve channels’ worst, 1.56 for the broadband lamp, is a calibration failure that noise only adds to. Only for seven channels, whose calibration is good enough to leave a margin, does the noise become the question at all — and the answer is that it uses about two thirds of the margin at one per cent.

What a sensor designer should take from this #

More channels need not mean more noise. A seven-channel sensor whose channels overlap enough to follow the colour-matching functions builds its matrix from sums, not differences, and carries its noise as lightly as a three-channel one.

Watch for cancellation, not channel count. The ratio of a matrix row’s contributions to their sum is a direct, cheap check on noise amplification; here it flagged the twelve-channel sensor’s Z row as the one place a fine sensor cancels heavily.

A margin is what survives noise. Five channels lose sixteen lights to one per cent of noise because their noiseless readings crowd up against the line; seven lose none because theirs sit a quarter of a unit inside it.

How the readings were made #

The census is the room’s forty-seven lights — nine daylight-and-lamp mixtures, five Planckian radiators, thirty white LEDs and three fluorescent tubes — and four triphosphor siblings at 3,000, 3,500, 4,000 and 5,000 K. A sensor of n channels has Gaussian channels centred evenly from 445 to 650 nm, each 1.25 times the spacing wide at half height. For each light, its matrix is the least-squares fit from channel signals to XYZ over the other fifty lights, each at unit luminance; the light’s own signals are then multiplied channel by channel by one plus a seeded normal deviate times the noise level, a hundred times, and each noisy XYZ is read as the moon picture’s worst tint at a tenth of a candela per square metre against the light’s true chromaticity, as the census reads every reading. The spread is the root-mean-square distance in u′v′ between the noisy readings and the noiseless one. A matrix row’s cancellation is the sum of the sizes of its channels’ contributions to one light’s reading over the size of their sum, averaged over the lights, for the matrix fitted on all fifty-one.

What this does not reach #

Correlated noise. Each channel’s noise here is independent. A temperature drift moves every channel’s gain together, and a shared drift is invisible to a ratio of channels but not to a matrix that subtracts; it would treat the twelve-channel sensor’s Z row worse than this does.

Noise that is not proportional. Shot noise grows as the square root of the signal and dark noise not at all; at the lowest light a dark channel’s noise is a larger share of its signal than a bright one’s.

Real filters. Gaussian channels are smooth and evenly spaced; real interference filters have side lobes and real dyes have tails, and either could make a matrix cancel more. Two sensors disagree about deep red, not lines found a phone sensor’s clear channel collecting deep red that a camera’s infrared cut discards — a tail no Gaussian channel has, and one a matrix would have to subtract.

Still open: whether a drift shared by every channel undoes the fine sensor #

Independent noise averages out across channels whose weights all have one sign; a drift that moves every channel’s gain together does not average at all, but it moves every channel in proportion, which a matrix of one sign turns into a change of luminance more than of chromaticity. A drift that tilts the channels — the long-wave channels’ gain rising with temperature as the short-wave channels’ falls, as silicon’s response does — is different.

The calculation is every light read by the three-, five-, seven- and twelve-channel sensors with every channel’s gain multiplied by one plus a tilt in proportion to its centre’s distance from 550 nm, reaching ±0.5, ±1 and ±2 per cent at the ends of the range. The prediction is that the seven-channel sensor stays within a unit for every light at ±1 per cent and that the twelve-channel sensor, whose Z row cancels, loses more lights than seven at every tilt. If seven channels lose lights at ±0.5 per cent, a tilt is to a fine sensor what independent noise was expected to be, and the instrument needs its gains held to a tenth of a per cent across temperature.

A premise that was a picture, not a calculation #

The habit here is about where a prediction’s mechanism came from.

The prediction rested on a picture of how a fine sensor reads colour: by small differences between neighbouring channels, with large weights of both signs. It was a reasonable picture, true of some sensors, and nobody had looked at this sensor’s weights. When they were looked at, the seven-channel matrix turned out to be nearly all positive sums, and the noise it carried was the three-channel sensor’s.

The failure mode is to predict a result from a mechanism’s usual form without checking that the instrument at hand has that form. The test is to compute the one quantity the mechanism depends on — here, how much the matrix cancels — before predicting what the mechanism will do.