A dark offset pulls every light toward one colour

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. Seven channels end the trade and lose the tube and relatives, not twins, teach a sensor a tube built a seven-channel sensor that reads all fifty-one of the census’s lamps, LEDs, tubes and mixtures within that unit.

Two essays then asked what disturbs it. Seven channels carry their noise as lightly as three found it keeping every light within a unit through one per cent of independent noise on every channel, because its matrix’s rows are sums of channels of one sign and hardly cancel. A tilt moves every light’s reading alike tilted every channel’s gain across the spectrum, as a photodiode’s response tilts when it warms, and found it holding to ±2 per cent, every light’s chromaticity shifting by nearly the same small vector. Both disturbances multiply the signal or scatter it about its value; seven channels hold every light within a unit through both, the worst at 0.56.

A dark current does neither. It is the signal a photodiode gives with no light at all, set by temperature and junction rather than by the light, and an uncorrected reading carries it as the same addition to every channel. 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. The prediction was that an offset equal to a tenth, a quarter and a half of a per cent of the strongest channel’s signal costs the seven-channel sensor lights from a quarter of a per cent, its narrow end channels taking the offset as a large share of a weak signal, while three channels, whose broad channels all see much of every light, move least. If seven held at half a per cent, a matrix of one sign would be turning an offset, as it turned the tilt, into a shared shift.

Seven channels hold to two per cent #

With an offset of a tenth, a quarter, a half and one per cent of each light’s strongest channel signal, the seven-channel sensor reads every one of the fifty-one lights within a ΔEITP unit, its worst light at 0.27, 0.27, 0.29 and 0.42. At two per cent its worst is 0.94, and still no light is lost. At five per cent it loses sixteen.

The prediction fails by a factor of eight. A quarter of a per cent was expected to cost lights and costs none; the first light is lost somewhere between two and five per cent. The five- and twelve-channel sensors tell the same story from their own starting points: neither loses a light it read correctly until two per cent, where twelve channels lose one; at five, five channels lose thirteen and twelve channels sixteen.

The seven-channel sensor’s worst light barely moves to half a per cent — 0.26 to 0.29 — and then rises steeply, to 0.42 at one per cent and 0.94 at two. That is not the shape of a channel’s weak signal being swamped, which would hurt from the smallest offset up. It is the shape of an error growing steadily on lights that began well inside the unit, until the ones it grows fastest on cross it.

An offset draws every light toward one colour #

What an offset does to the reading turns out to be geometrically simple, simpler than anything the noise or the tilt did.

Under every offset, at every channel count, each light’s reported chromaticity moves straight toward a single colour — the colour the sensor’s matrix reports when every channel’s signal is equal — the cosine of the angle between the move and that direction above 0.998 for every light. At a five-per-cent offset the seven-channel sensor moves each light 4.9 to 11.3 per cent of the way there.

The reason is linear algebra, and it holds for any sensor read through a matrix. An offset adds the same number to every channel, which is a vector of equal signals; the matrix turns that vector into one fixed tristimulus value; and adding a fixed tristimulus value to every light’s reading mixes every light with one colour, in proportion to how large the offset is beside the light’s own signal. In chromaticity, mixing with a fixed colour moves every point along the straight line toward it. The share of the way each light moves depends on how much signal it gives against the offset — and since the offset here is set as a share of each light’s own strongest channel, it depends on how unevenly the light spreads its signal across the channels.

So the offset’s damage has a direction, and the direction is the same for every light: toward the sensor’s equal-signal colour. A light far from that colour moves far; a light near it barely moves.

The equal-signal colour lies at the blue end of the lights #

The seven-channel sensor’s equal-signal colour lies at u 0.210, v 0.316; the five- and twelve-channel sensors’ within 0.002 of it; the three-channel sensor’s at 0.187, 0.302. The fifty-one lights lie between u 0.195 and 0.290 and v 0.301 and 0.357. Every sensor’s equal-signal colour sits at the bluish edge of the census, near its coolest daylights, and the three-channel sensor’s beyond even those.

That one fact explains both failures of the prediction. The sixteen lights the seven-channel sensor loses at five per cent are every one of the twelve 2,700 K LEDs, the 2,200 K radiator, two warm lamp mixtures and the 3,000 K triphosphor sibling — their u averaging 0.263, against 0.218 for the lights it keeps — and every one of them read within 0.24 with no offset. The warm lights lie furthest from a bluish equal-signal colour, so the offset pulls them furthest, and it is distance from that colour, not the difficulty a light already posed, that decides which are lost. But the offset does not attack a channel; it attacks a direction, and the seven-channel matrix’s virtue — rows of one sign that do not amplify — does nothing against a disturbance that the matrix itself turns into a colour. The seven-channel sensor holds to two per cent not because it resists the offset but because its readings start well inside the unit, the worst at 0.26, and two per cent raises its worst reading to 0.94.

Three broad channels move furthest #

At one per cent the three-channel sensor’s readings move by 6.48 ten-thousandths on average, against 5.23 for five channels, 5.24 for seven and 5.51 for twelve; every sensor’s shift doubles with each doubling of the offset, within a twentieth. The prediction had three channels moving least.

The three-channel sensor’s equal-signal colour lies furthest from the lights, and every light is pulled toward it, so its readings move furthest. Its broad channels do see much of every light, as the prediction said, which makes the offset a smaller share of each channel’s signal; but the matrix that turns three broad, overlapping channels into tristimulus values has to separate them, and it reports an equal-signal input as a colour bluer than any of the census’s lights. The breadth that reduces the share is outweighed by the direction the matrix sends it.

For three channels a small offset helps #

The three-channel sensor reads thirty-seven of the fifty-one lights beyond a unit with no offset, thirty-five at one per cent, thirty-two at two and thirty at five, with one light newly lost at five; at ten per cent it reads thirty-eight beyond a unit, seven of them new. Up to five per cent, an offset makes it better.

The census found long ago that three channels cannot read the room’s lights, and their failures have directions too. The five lights the three-channel sensor recovers at two per cent were each misread away from its equal-signal colour, the cosines between their errors and the direction away from it 0.77 to 0.96. An offset pulls every reading toward that colour, which is back toward where those readings should have been. It is the same mechanism as before, acting on a sensor whose errors happen to point the other way; it is not a calibration anyone should rely on, and at ten per cent the pull overshoots.

Against the tolerances the census set #

The offset’s shifts can be set against the tolerances the run of essays derived for this picture, as the tilt’s were. 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 a hundred kelvin near 2,700 K before the moon picture’s tint fell under a unit, and lamp and daylight together sit below the locus added a tolerance of about 0.003 in Duv, which is measured in the same CIE 1960 uv plane as the shifts here.

For the seven-channel sensor, the mean shift is 5.2 ten-thousandths at a one-per-cent offset, 10.3 at two and 24.6 at five; the largest, on the warmest lights, 22.5, 43.9 and 102. Against a Duv tolerance of 0.003, the average light moves a sixth of it at one per cent, a third at two and five sixths at five; the warmest light three quarters of it at one per cent, one and a half times it at two and more than three times it at five. The tolerance is a guide here and not a boundary: at two per cent the warmest light has moved one and a half Duv tolerances and is still read within a unit, because the tolerance was derived for a white declared off the locus, across it, and the offset moves a warm white along a line toward the blue that runs partly along the locus instead.

The comparison with the tilt is instructive in the other direction. A one-per-cent tilt moved every light by nearly the same vector, about seven ten-thousandths, so the warmest light moved no further than the coolest. A one-per-cent offset moves the average light less than that and the warmest light three times as far, because the offset’s move grows with a light’s distance from the equal-signal colour and the tilt’s did not. Two disturbances of similar mean size, then, with opposite structures: the tilt a uniform translation of the whole census, which a single reference reading could remove; the offset a contraction of the census toward one point, which one dark reading removes.

A sensor calibrated on lamps cannot read an LED and no calibration reads the lamps and the LEDs together found the three-channel sensor’s errors on LEDs pointing one way, which is why one shared correction could not fix them. The offset is a disturbance whose errors point toward one place rather than one way, and that is the structure a subtraction before the matrix removes exactly.

What a sensor designer should take from this #

A dark offset is a colour, not a noise. Added to every channel, it is turned by the matrix into the sensor’s equal-signal colour and mixed into every reading. Its effect can be predicted for every light from one number per sensor — where that colour lies — and removed entirely by one dark reading subtracted before the matrix.

The matrix’s shape does not protect against it. The seven-channel sensor’s freedom from cancellation made it robust to noise and to a tilt; against an offset it is no better than five or twelve channels, because the offset does its damage after the matrix, as a colour.

Place the equal-signal colour among the lights. A sensor whose equal-signal colour sat at the centre of the room’s lights would lose least to an uncorrected offset; one whose equal-signal colour sits at the edge, as every sensor here does, pulls its warmest lights furthest.

How the sensors and lights were read #

The census holds fifty-one lights — lamp mixtures, Planckian radiators, white LEDs, fluorescent tubes and triphosphor siblings. Each sensor’s channels are Gaussian sensitivities between 445 and 650 nm, three, five, seven or twelve of them; each light’s signals are integrated on a one-nanometre grid, and each light is read by the matrix fitted by least squares to the other fifty, so that no light is read by a matrix that saw it. The offset adds to every channel the same amount, a stated share of that light’s largest channel signal. The reported tristimulus value is the matrix times the offset signals; its tint is the moon picture’s ΔEITP error when the display declares that white, and the shift is the distance in CIE 1960 uv between the readings with and without the offset. The equal-signal colour is the matrix, fitted on all fifty-one lights, applied to a vector of ones.

What this does not reach #

The offset is set as a share of each light’s own signal. A real dark current is fixed in absolute terms and becomes a larger share as the room dims; reading the census at a fixed absolute offset over a range of light levels would turn the share here into a light level.

Equal offsets in every channel. Real channels differ in area and junction, and their dark currents differ with them; an offset unequal across channels is a different vector, turned by the matrix into a different colour.

Uncorrected readings only. A dark reading subtracted before the matrix removes the whole effect; the essay measures what happens when that step is skipped.

Still open: how dim the room can get before the offset costs a light #

Seven channels hold every light to two per cent and lose sixteen at five. The offset’s damage to each light is its pull toward the equal-signal colour, in proportion to the offset over that light’s signal, so each light has its own offset at which it crosses a unit.

The calculation is the seven-channel sensor at offsets between two and five per cent, every light’s tint read as here, each light’s crossing offset located by bisection and set against its distance from the equal-signal colour and its tint with no offset. The prediction is that the first light is lost near three per cent and is the census’s warmest, that the crossing offset falls with the light’s distance from the equal-signal colour, and that a light’s tint grows nearly linearly in the offset from its own starting value, so that the crossing is predicted within a fifth from two numbers a light. If the crossings do not follow the distance, the tint does not grow along the pull’s direction alone, and the moon picture’s sensitivity to a white’s error differs by direction.

A light’s distance from the equal-signal colour sets when it goes made that calculation: the warmest light goes first, at 2.1 per cent, the twelve warm LEDs together near four, and two readings a light predict each crossing within a twelfth.

A disturbance that was a colour #

The habit here is about asking what a disturbance looks like after the operation that follows it.

The prediction reasoned about channels: a weak channel takes a fixed offset as a large share, so a sensor with narrow channels should suffer. That is true before the matrix and irrelevant after it. After the matrix, an equal addition to every channel is one fixed colour added to every reading, and every light’s error points toward it. Read that way, the offset’s whole effect follows from where one colour lies, and none of the channels’ individual shares matters.

The failure mode is to analyse a disturbance where it enters a system and not where it leaves. The test is to push the disturbance through the system’s linear part on its own — here, one vector through one matrix — and see what it has become by the time it reaches the output.