A tilt moves every light's reading alike

A display that adapts its picture to the room reads the room’s light with an ambient sensor — a few photodiodes behind coloured filters — and turns their signals into a chromaticity with a matrix fitted to known lights. The census this essay continues judges that chromaticity by what it does to a dim picture: a moon at a tenth of a candela per square metre, where a declared white that is off shows as a tint in ΔEITP, and a unit of tint is the line between a reading that serves and one that does not. Relatives, not twins, teach a sensor a tube found a seven-channel sensor reading all fifty-one of the room’s lights and triphosphor siblings within that unit.

Seven channels carry their noise as lightly as three then gave every channel independent noise and found the seven-channel sensor reading every light’s every one of a hundred readings within a unit at one per cent of noise, because its matrix’s rows are sums of channels of one sign — contributions summing to 1.04, 1.00 and 1.13 times their total for X, Y and Z — where twelve channels cancel in Z by four times. Independent noise averages out across a sum of channels of one sign.

A drift does not. A real photodiode’s spectral response changes with temperature, and silicon’s tilts: as it warms its long-wavelength response rises and its short-wavelength response falls, so every channel’s gain moves together in a pattern set by its wavelength. A matrix fitted at one temperature reads the drifted signals as if they came from a different light. The prediction was that a tilt reaching ±1 per cent at the ends of the range leaves the seven-channel sensor within a unit for every light, and that the twelve-channel sensor, whose Z row cancels, loses more lights than seven at every tilt. If seven channels lost lights at ±0.5 per cent, a tilt would be to a fine sensor what independent noise had been expected to be, and its gains would need holding to a tenth of a per cent across temperature.

Seven channels hold to twice the tilt asked #

Tilted by ±0.5, ±1 and ±2 per cent at 650 nm — the 445 nm channel moving by about the same the other way — the seven-channel sensor reads every one of the fifty-one lights within a unit at every tilt; its worst reading goes from 0.26 untilted to 0.39 and 0.31 at −1 and +1 per cent and 0.56 and 0.49 at −2 and +2. The twelve-channel sensor reads two lights beyond a unit at every tilt, the five-channel sensor six to eight and the three-channel sensor thirty-two to forty.

• The seven-channel half of the prediction holds, at twice the tilt it was asked to survive.
• The twelve-channel half holds in its count and fails in its sense. Twelve channels read more lights beyond a unit than seven at every tilt — but the same two at every tilt, the halophosphate tube and the broadband lamp, which they misread with no tilt at all.
• The alternative does not arise. A half per cent of tilt costs no fine sensor a light.

The four sensors’ lines are flat or nearly so. Seven channels sit on nought; twelve on two. Five channels add one or two lights at the largest tilts, three channels a handful, but both were already reading many lights beyond the line — six and thirty-seven untilted — and the tilt moves lights that were already near it across. Nowhere does a tilt of a few per cent turn a sensor that served into one that does not.

Every sensor moves by the same amount #

At a tilt of one per cent the reported chromaticity moves by a mean of 8.4 ten-thousandths in u′v′ with three channels, 7.3 with five, 7.1 with seven and 7.0 with twelve; at half a per cent, half that, and at two per cent, twice. The shift is linear in the tilt to a twentieth, and it is within a fifth of the same for every sensor.

That is the opposite of what the noise found. Independent noise moved five channels’ chromaticity half again as far as seven’s and twelve’s, because it depended on how each matrix weighted its channels’ independent errors. A tilt is not independent across channels: it is a single smooth change to the sensor’s spectral response, and every sensor whose channels span the visible translates a smooth change of response into nearly the same change of reported colour. How finely the spectrum is sampled hardly matters to a change that is itself smooth across the spectrum.

Every light moves much the same way #

Tilted by +1 per cent, every light’s reading moves by about +6.0 ten-thousandths in u′ and +3.4 in v′ — towards the long wavelengths, as a sensor that sees more red and less blue would report — and no light’s shift lies more than 3.3 ten-thousandths from that mean. Tilted by −1 per cent, every shift reverses. The cloud of fifty-one shifts is tight beside its own distance from the origin, and it is the same cloud mirrored for the opposite tilt.

By family, the shift is 7.75 ten-thousandths for the lamp-and-daylight mixtures, 7.69 for the Planckian radiators, 6.90 for the white LEDs, 6.53 for the fluorescent tubes and 7.19 for the triphosphor siblings — a fifth between the least and the most. Lights with spiky spectra, the tubes and LEDs, move a little less than smooth ones, because a spike lies in one or two channels and the tilt’s effect on those few is less than its effect spread across all of them; but every family moves in the same direction by nearly the same amount.

A drift that moves every light the same way is a drift a single measurement can remove. If one known light — a reference LED in the device, or the display’s own white reflected back — were read at the sensor’s working temperature, the offset between its reported and true chromaticity would be the tilt’s offset for every other light within three ten-thousandths. A sensor calibrated on lamps cannot read an LED found a calibration’s errors depending on which lights it was fitted to; the tilt’s error depends on almost nothing about the light, which is what makes it the easier of the two to correct.

Twelve channels lose only what they always lost #

The twelve-channel sensor’s two failing lights, the halophosphate tube and the broadband lamp, read 1.29 and 1.56 untilted, and neither moves by more than 0.24 across the seven tilts; no other light passes a unit at any of them. The twelve-channel sensor’s cancelling Z row, which the prediction expected to make it fragile, costs it nothing under a tilt: the tilt changes every channel’s gain smoothly, so the large positive and negative weights in that row change together and their cancellation survives.

Seven channels end the trade and lose the tube found finer sensors reading the halophosphate tube worse past eight channels, the cost of a matrix fitted too closely to lights that do not include the tube’s kind. That cost is in the calibration and the tilt neither adds to it nor removes it. Cancellation makes a matrix sensitive to errors that differ between channels; it is indifferent to errors that change all its channels in step.

The worst reading rises, and stays under a unit #

Seven channels’ worst reading rises from 0.26 untilted to 0.56 at −2 per cent and 0.49 at +2; twelve channels’ worst, the broadband lamp’s, ranges from 1.45 to 1.80; five channels’ from 2.62 to 2.98; three channels’ from 8.15 to 9.47. The tilt lifts the seven-channel sensor’s worst by three tenths of a unit at two per cent, about half what one per cent of independent noise lifted it.

The worst reading is not always worse with a tilt. A tilt of +0.5 per cent lowers seven channels’ worst from 0.26 to 0.23, and −1 per cent lowers the twelve-channel sensor’s broadband lamp from 1.56 to 1.45 while +1 per cent raises it to 1.67: a small tilt can move a light’s reading towards its true colour as easily as away from it, because the calibration’s own error for that light had a direction. That is a reminder that the sensor’s untilted readings already carried errors of the same order as the tilt’s.

What seven ten-thousandths is worth #

The tilt’s shift can be set against the tolerances the census has derived, and it is small against all of them. A room sensor must read the lamp to a hundred kelvin found the declaration of a warm room’s light needing to be right to about fourteen mireds — a hundred kelvin near 2,700 K — before the moon picture’s tint fell under a unit, and fourteen mireds near 2,700 K is a move of about five thousandths in u′v′, six or seven times the shift a one-per-cent tilt makes. Lamp and daylight together sit below the locus added a Duv tolerance of about 0.003, which in u′v′ is about six times the same shift.

So a tilt of one per cent spends about a sixth of the room’s tolerance, and a tilt of two per cent a third. The figures are rough — a tolerance in mireds and one in Duv are not the same direction in the chromaticity plane, and the tilt’s shift has a direction of its own — but they agree on the scale. That is why seven channels survive it: their untilted readings sit a quarter of a unit inside the line, and a third of the tolerance moves them less than that. The five-channel sensor, whose readings crowd up against the line, loses one or two lights at the largest tilts for the same reason it lost lights to noise. A sensor’s margin is what any error is paid out of, and the tilt, being smooth and shared, costs every sensor about the same.

No calibration reads the lamps and the LEDs together found a three-channel sensor’s errors of several thousandths coming from its calibration — a matrix that could not serve both kinds of light at once — and those errors dwarf the tilt’s. Three channels misread thirty-seven lights untilted and a tilt of two per cent moves the count by five either way. The order of priorities this sets for an instrument designer is plain: the calibration first, the channel count second, the noise and drift last, each an order of magnitude below the one before for a sensor of seven channels.

The shift’s direction is also a diagnostic, and a cheap one, because it needs no knowledge of the room’s light at all. A sensor that reports every light a few ten-thousandths towards red, by about the same amount, is showing the signature of a gain tilt; a sensor whose errors vary from light to light in direction is showing a calibration’s. Two sensors disagree about deep red, not lines found two real sensors’ disagreement concentrated in their deep-red response, which is the end of the spectrum a temperature tilt moves most; a device that compared its reading of a known light at two temperatures would see the tilt directly, as a shift of the same size and direction for whatever light it compared.

What a sensor designer should take from this #

A gain tilt of a few per cent does not undo a seven-channel sensor. At ±2 per cent across the range every light stays within a unit; whether that covers a real device’s temperature range depends on its diodes and filters, which this census does not model.

Correct a drift with one reference reading. A tilt moves every light’s reported chromaticity by nearly the same vector; one known light read at the working temperature measures that vector for all of them.

Cancellation is a risk for independent errors, not shared ones. The twelve-channel sensor’s cancelling row did not make it fragile to a tilt; check a matrix’s cancellation against the errors the instrument actually has.

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 to 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; its matrix for each light is the least-squares fit from channel signals to XYZ over the other fifty lights at unit luminance. The light’s own signals are multiplied channel by channel by one plus the tilt times the channel centre’s distance from 550 nm over a hundred nanometres, so that the 650 nm channel’s gain changes by the tilt and the 445 nm channel’s by 1.05 times it the other way; the tilted XYZ is read as the moon picture’s worst tint at a tenth of a candela against the light’s true chromaticity. The shift is the distance in u′v′ between the tilted and untilted readings.

What this does not reach #

Other drifts. A tilt is one shape; a channel whose filter drifts in wavelength, or a dark current that adds the same signal to every channel at low light, is a different error and changes the reading differently.

Correlated noise and tilt together. Real drift comes with noise; the two have been read apart.

The reference reading itself. A single-light correction is proposed here, not computed; how well one reading removes the tilt for every other light, after the reference’s own error, is not read.

Still open: whether a dark offset undoes the fine sensor #

A tilt multiplies every channel’s signal and changes each by a share; a dark current adds the same signal to every channel whatever the light. At the dim end of a room’s lighting — the evening the moon picture models — a fixed offset is a larger share of a weak channel’s signal than of a strong one’s, and a channel that sees little of a light gains the most.

The calculation is every light read by the four sensors with a dark offset added to every channel equal to a tenth, a quarter and a half of a per cent of the strongest channel’s signal for that light, each light by the matrix fitted without it. The prediction is that the seven-channel sensor loses lights from a quarter of a per cent, its narrow channels’ weak signals at the spectrum’s ends taking the offset as a large share, while three channels, whose broad channels all see much of every light, move least. If seven channels hold at half a per cent, their matrix of one sign turns an offset, like a tilt, into a shared shift, and a fine sensor’s only weakness is its calibration.

A dark offset pulls every light toward one colour made that calculation: seven channels hold to two per cent, three move furthest, and every light is drawn toward the sensor’s equal-signal colour.

An error judged by its sign across the channels #

The habit here is about matching an instrument’s weakness to the error it will meet.

The prediction carried over the previous calculation’s idea of fragility — a matrix that cancels amplifies error — and expected the cancelling twelve-channel sensor to suffer most from a drift. It suffered not at all, because a drift moves every channel in step and a cancelling sum is unmoved by a change its terms share. Independent noise and shared drift are different errors, and the same matrix is fragile to one and robust to the other.

The failure mode is to carry an instrument’s measured sensitivity to one kind of error over to another kind, when the two differ in how they are correlated across the instrument’s parts. The test is to ask, for each error the instrument will meet, whether it changes the parts independently or together, and to read the instrument’s response to each separately.