Colour Science & Perception
what the eye settles, and what it cannot
Concept

Signal-to-noise — where it appears

The ratio of a measurement's true value to the scatter of its readings, quoted per channel of a sensor. A white balance and a colour matrix change it channel by channel, and a deep shadow's ratio decides how much of its noise a clip at zero discards.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

Correcting colour costs noise, and the two cannot both be least. Sweeping the colour matrix from its own diagonal — a white balance with no cross terms — to the full least-squares fit. Mean ΔE00 falls from 18.61 to 1.29; photon noise rises by a factor of 1.03. At 10000 photons per pixel.

Correcting colour costs noise

The matrix that turns a sensor's raw into XYZ has large off-diagonal terms of both signs, because that is what correcting a sensor which fails Luther's condition requires. Differences of large numbers are where noise grows, and the full correction amplifies photon noise by 1.66 times.

imaging · Capture Noise clipped at zero, averaged over a shadow, under tungsten. A grey ramp from black to ten per cent reflectance under tungsten, captured at three illustrative noise levels, with every negative raw reading set to zero before the readings are averaged over an area. Each line is the colour difference between that average and the noiseless grey. At high gain a half per cent grey is 1.04 off and a black frame 0.79; at very high gain the worst is 2.95, at 1.0 per cent. The same readings averaged before any clip come back exactly, at every level. The tint is gone once every channel sits several deviations above zero.

Clipped noise does not average away

Noise on a raw reading is as likely to fall below the true value as above it, which is why averaging an area removes it. A converter that sets negative readings to zero keeps the upper half and throws the lower away, and the mean of what is left is the signal plus a pedestal. With no black level error anywhere, a half per cent grey under a tungsten lamp comes out 1.04 colour differences off at high gain, 8.98 after a four-stop push — and a blur that removes every trace of the noise leaves the tint where it was.

imaging · Capture The statistic a converter would read, under each model. The log ratio of the two channels that are still open, against how bright the surface is, under the two models of what a highlight is. A matt surface over-exposed keeps its own ratio exactly — the line is flat, and it must be, because scaling every channel by the same amount leaves a ratio alone. A glossy surface carries a reflection of the lamp on top of its body colour, so its ratio slides towards the lamp's as the reflection strengthens: -0.086 of a log unit between a quarter of full scale and nine tenths. That slide is the whole of the evidence a converter has for choosing between them.

The converter can choose except where it matters

A converter rebuilding a clipped highlight has to say whether the surface was matt and over-exposed or glossy and carrying a reflection, and the evidence is whether its raw chromaticity slides towards the lamp's on the way up. Read through the sensor's own noise, the slide is clear on most of the chart from the few tens of pixels a specular highlight holds — and on the surfaces where it takes half a frame, confusing the two models costs more than the median. A ratio cannot see a common scale, and an exposure supplies one.

imaging · Capture The four places a clamp could sit are two pipelines. Every pair of clamp positions, with the largest difference their delivered values reach over a grid of raw inputs that includes negative ones. Two of the six are exactly zero: a clamp at zero commutes with the white balance, which is a positive scale applied channel by channel, and with the tone curve, which is monotone and fixes zero. It does not commute with the colour matrix, which is the only step that mixes the channels — so the four positions collapse to two, before the matrix and after it, and no measurement of any scene can say more than which side a converter is on.

Four places to clamp are two pipelines

The essay on clipped noise ended on a procedure: a black frame and a dim grey card at a high amplification would place each raw converter's clamp. A procedure is a claim that a measurement identifies something, and this one identifies less than it looks. A clamp at zero commutes with the white balance and with the tone curve and not with the colour matrix, so the four positions are two pipelines — and the measurement separates them, on a card at half a per cent of white rather than on the black frame.

imaging · Capture The same chart in two studios and in the room that holds both lamps. Each of the chart's twenty-four surfaces, with the pixels its slide needs at two standard deviations at high gain, on a logarithmic scale. Lit by a 3000 K radiator alone the worst surface needs 16,194; lit by daylight at 6500 K alone, 139,574. Lit by both, 50 and 50 per cent of the light, with a highlight of each lamp read together, the worst needs 10 and the median 6. The studios' hard surfaces sit at different places on the chart, and no surface is hard in both.

Two lamps decide what one lamp could not

Under one lamp a handful of surfaces cannot tell a glossy highlight from a matt over-exposure without most of a frame, because there the two differ by a scale a ratio cannot see. In a room lit evenly by a 3000 K lamp and daylight, every surface on the chart is decided from ten pixels. A body can sit at one lamp's white, not at two, and the rescue holds only while the second lamp carries a fifth of the light and its white sits far enough from the first.

imaging · Capture Seven channels keep every light within a unit through the noise. Every light of the enlarged census read a hundred times by sensors of three, five, seven and twelve channels whose signals each carry independent noise of a quarter, a half and one per cent, each light by the matrix fitted without it: how many lights are read beyond a ΔEITP unit at least once. 3 channels: 40, 42, 46; 5 channels: 7, 9, 22; 7 channels: 0, 0, 0; 12 channels: 2, 2, 2.

Seven channels carry their noise as lightly as three

An ambient-light sensor of seven narrow channels, fitted to a room's forty-seven lights and four triphosphor siblings, reads every one of the fifty-one within a ΔEITP unit of a dim moon picture's tint, each light held out of its own fit. A fine sensor was expected to read chromaticity from small differences between neighbouring channels, with large weights of both signs that multiply every channel's noise. The prediction was that seven channels stay within a unit at a quarter of a per cent of channel noise and are no better than five at one per cent. The first half holds. The second fails completely: at one per cent seven channels still read every light's every one of a hundred readings within a unit, while five lose twenty-two lights. The premise was wrong. Seven channels' matrix barely cancels — its contributions sum with one sign — and per unit of noise it moves the reported chromaticity no more than three broad channels do.

difference · Metric A tilt moves every light's reading much the same way. Each of the fifty-one lights' shift in reported u′v′ when the seven-channel sensor's gains are tilted by +1 and −1 per cent. At +1 per cent every light moves by about 6.0 in u′ and 3.4 in v′, ten-thousandths, towards the long wavelengths, and no light lies more than 3.3 from that mean; −1 per cent mirrors it.

A tilt moves every light's reading alike

An ambient-light sensor of seven narrow channels reads all fifty-one of a room's lights within a ΔEITP unit of a dim moon picture's tint and keeps them there through one per cent of independent noise on every channel, because its matrix barely cancels. A drift shared by the channels does not average out: as a photodiode warms its response tilts, rising at long wavelengths and falling at short. 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. Seven channels hold to ±2 per cent, the worst light at 0.56. Twelve loses two lights at every tilt — the two it misreads with no tilt at all — and none to the tilt. Every sensor's reported chromaticity moves by about seven ten-thousandths in u′v′ per per cent of tilt, every light nearly the same way.

difference · Metric An offset draws every light toward one colour. The seven-channel sensor's reading of each of the fifty-one lights in CIE 1960 uv with no offset and with a five-per-cent offset, beside the colour the sensor reports for equal signals in every channel, at 0.210, 0.316. Every light moves straight toward it, by 4.9 to 11.3 per cent of the way.

A dark offset pulls every light toward one colour

An ambient sensor of seven narrow channels reads all fifty-one of a room's lights within a ΔEITP unit of a dim picture's tint through independent noise and a gain tilt, because its matrix barely cancels. A dark current is different again: it adds the same signal to every channel whatever the light, and at the dim end of a room's lighting it is a larger share of a weak channel than of a strong one. The prediction was that an offset of a quarter of a per cent of the strongest channel costs the seven-channel sensor lights, while three broad channels move least. Seven channels hold every light to two per cent. Three move furthest. And the offset does something simpler than either sensor's design suggests: it draws every light's reading straight toward the one colour the sensor reports for equal signals in every channel.

difference · Metric The further from the equal-signal colour, the sooner a light goes. For each of the 19 lights the seven-channel sensor loses by an eight-per-cent offset, the offset at which it crosses a unit against its reading's distance from the colour the sensor reports for equal signals. The correlation is −0.93; the nearest lost light, triphosphor 3500 at 0.035, crosses at 7.9%, the furthest, planck 2200 at 0.091, at 2.1%.

A light's distance from the equal-signal colour sets when it goes

An uncorrected dark offset draws every light an ambient sensor reads straight toward the one colour its matrix reports for equal signals in every channel, and a seven-channel sensor holds the room's fifty-one lights within a ΔEITP unit to an offset of two per cent of each light's strongest channel. The prediction was that the first light goes near three per cent and is the warmest, that each light's crossing offset falls with its distance from the equal-signal colour, and that two readings a light predict its crossing within a fifth. The warmest goes first — the 2,200 K radiator — but at 2.1 per cent. The twelve 2,700 K LEDs go together, between 3.9 and 4.4. Crossing follows distance at r = −0.93, and a straight line through each light's tints at one and two per cent predicts its crossing within 8 per cent, always early. Below half a per cent nothing moves at all.

difference · Metric

Named alongside it

The objects these essays reach for when they reach for this one.

Shot noiseCamera rawColour-matching functionsΔEITPHeld-out validationLinear modelNoiseClippingIdentifiabilityWhite balanceCamera sensorColour matrix

All concepts