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

Linear model — where it appears

A description of a spectrum or a reflectance as a weighted sum of a small fixed set of basis functions. Its dimension is what every counting argument about recovery turns on, and it is a measured property of a population of surfaces rather than a modelling convenience.

Named by 14 essays across 6 fields — each of them below, with the objects they name alongside it.

How short of determining the light a photograph is, as the scene grows. Each cell is the number of unknowns left over after every equation the image supplies: three sensors, a three-dimensional illuminant, and reflectances confined to a linear model of the dimension on the left. At one and two dimensions more surfaces close the gap. At three the gap never closes, because each further surface adds three equations and three unknowns; at four it widens. The count is arithmetic and has no algorithm in it.

An image does not determine the light

A photograph of a scene under one illuminant gives three numbers per surface and asks for the illuminant plus three numbers per surface. The count closes only if reflectances lie in a two-dimensional model, and no number of surfaces helps — at three dimensions the alternative scenes can be written down, and they reproduce every sensor response exactly.

scene · Scene An instrument, as the only thing it really is. The 3 filters a bank of that size puts across the visible range, each drawn against wavelength. Everything the instrument can report about a spectrum is 3 numbers — the integral of the light against each of these — so the set of spectra it cannot tell apart is everything orthogonal to all 3 of them, which is 78 dimensions of the 81 this site works in. Three of these is a colorimeter in spirit; the eye is three of them too.

Three numbers cannot see a line

An instrument that returns three filtered readings of a spectrum determines a three-dimensional projection of it and is exactly blind to the other seventy-eight. On daylight that costs almost nothing; on a fluorescent tube, three quarters of the lamp lies in the part no reading reaches, and adding filters recovers it slowly.

light · Light The colour is right long before the spectrum is. The colour error of the projection, against the number of readings. At twelve readings the fluorescent tube's colour is right to 0.48 ΔE00 while 66% of its spectrum is still unmeasured. That is the trap in one line: a reconstruction good enough to pass any colorimetric check will predict a match under a second illuminant that does not happen, because the part it got wrong is exactly the part a different lamp weights differently.

The colour is right first

A reconstruction of a lamp from twelve filtered readings gets its colour right to half a unit while two thirds of its spectrum is still unmeasured. That combination is not a partial success — it is the exact condition under which a spectral prediction made from the reconstruction will be confidently wrong.

light · Light What each fitted thing in these essays carries, what its data fix, and what is left. Three columns per row: how many numbers the model has, how many the stated data determine, and the difference — the dimension of the family that fits equally well. The third column is the one nobody publishes. A zero there does not mean the model is right; it means it is determined, which is a much weaker property and is compatible with being determined badly, as the camera row is.

A fit can be exact and empty

Every fitted object here reports one number, the residual on the data it was fitted to, and every one of them has two more that nobody publishes — how many of its parameters the data actually determine, and how large the family of equally good answers is. The third column is where the failures live.

limits · Limits The arguments a standard observer does not have. Seven choices inside a set of colour-matching functions, each with the shape it takes and what it is worth in ΔE₀₀ on a red pigment under a 6500 K radiator. Six are measurements: a field size, an age, a macular density, a cone optical density, three peak wavelengths and a rod contribution. The seventh is not — a change of basis is a change of curves and not a change of observer, and its entry is exactly zero because the space an experiment measures is what an observer is. Printing that zero beside the others is the clearest statement of what the other six are measurements of.

Three curves for one space

Rotate a set of colour-matching functions by an arbitrary invertible matrix, undo the rotation at the end, and the computed colour is identical to eight parts in a thousand million million. An observer is a three-dimensional subspace, not a set of curves, and the literature keeps reopening a question that is a theorem.

eye · Cones A departure against how far the sample sits from the light. The sample is mixed with a flat reflectance, from the flat one at the left to its own at the right, and two observers differing in the macular pigment look at each mixture. The straight line is the distance between their relative cone excitations, and it is straight to 0.0 per cent: the departure is a pairing, and scaling one factor scales the product. The curved line is the same sequence in ΔE₀₀, which is not a linear function of the excitations and cannot be — it has cube roots in it and a chroma weighting underneath. The identity is about the eye; the curvature belongs to the unit.

A departure is straight in the excitations

Walk a sample a quarter of the way from the light towards its own reflectance and exactly a quarter of the observer disagreement remains — in cone excitations, to two parts in a hundred. In ΔE₀₀ the same quarter leaves 0.347 where proportionality wants 0.428, and the discrepancy belongs entirely to the unit.

difference · Metric With the level placed, the line bends as the error does. At each temperature, the curvature in Duv of the camera error's mean course through nought and ±0.003, beside the curvature each slope-and-curve line implies, averaged over the three pumps. At 2,700 K the course bends by +7,916, the line without a level by +2,803 and the line with one by +8,198; at 4,000 K the course bends by +2,196, the line without a level by +14,591 and the line with one by +2,195; at 6,500 K the course bends by +363, the line without a level by −13,349 and the line with one by +445.

A level for each temperature lets the curve be the error's

An error line with a slope and a curve in Duv for each colour temperature read white LEDs inside a manufacturing bin within ±0.045 ΔE₀₀ and LEDs at twice the bin's edge only within ±0.16, though the camera's own error stays on its parabola there. Its curve was borrowed: the line's shared terms could not place each temperature's level, and the square of the Duv absorbed the gap. The prediction was that one more number a pump, the square of the mireds, brings the locus LEDs within ±0.05 and those at ±0.006 within ±0.08. It brings the locus to ±0.025 — three times better than the locus-only line that started this run of essays — and the outer LEDs to ±0.049 and ±0.058. The curvature the line now implies at each temperature is the error's own within a fifth; what it leaves is the warm whites' scatter at the bin's edges.

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 The red phosphor bends nothing and tilts the warm whites. The 2,700 K LEDs' residual at each Duv under the eleven-term line, with the square of the Duv times the estimated red phosphor added, and with the Duv times the estimated red added instead. The bend term leaves ±0.006 at ±0.068 and ±0.088, against ±0.069 and ±0.087; the slope term takes them to ±0.046 and ±0.067.

The red phosphor tilts the warm whites and bends nothing

An error line with a level, a slope and a curve in Duv for each colour temperature reads white LEDs at 4,000 and 6,500 K within ±0.04 ΔE₀₀ across and beyond a manufacturing bin, but warm whites at twice the bin's edge only within ±0.087. The prediction was that the red phosphor sets a warm white's bend, so that one term more a pump — the square of the Duv times the estimated red — brings them within ±0.05. It moves them by a thousandth. The red phosphor enters the slope instead: the Duv times the red takes them to ±0.046 below the locus and ±0.067 above. And the rest points upstream. The first stage reads a warm white's red phosphor squeezed toward the middle of its range, the warm misses follow that misreading times the Duv at r = 0.79, and given the red each LED was built with the warm whites come within ±0.05 — though the same substitution doubles the error at 4,000 K.

imaging · Capture The squeeze belongs to the temperature, not the Duv. For the 2,700 K LEDs at a Duv of ±0.003 and ±0.006, the first stage's mean estimate of the red phosphor against the red each was built with, by three first stages; the dashed line is a perfect reading. With one Duv term, no red and a full red are read at 0.13 and 0.80; with the Duv times the departure and the mireds at 0.14 and 0.80; with a level and a departure slope for each temperature at 0.02 and 1.01.

The line had already undone the squeeze

An ambient sensor's first stage reads a 2,700 K white LED's red phosphor squeezed about a third toward the middle of its range, and the camera-error line's warm misses outside the bin follow that misreading times the Duv. The prediction was that giving the first stage the Duv times its 600 nm departure and times the mireds would undo the squeeze. It changes nothing: the squeeze belongs to the temperature, and a level and a departure slope for each temperature read the warm red four times better, within ±0.035. But the eleven-term error line does not move — no warm LED by more than 0.006 — because at one temperature a squeeze is a rescaling its red and red-times-mireds terms had already undone. Only the red-slope line, whose one coefficient a pump spans three temperatures, gains: to ±0.034 below the locus, within 0.002 of the red known. And the cold whites pay for it.

imaging · Capture Split between two estimates, the line keeps both ends. The warm whites at ±0.006 and the cold whites at −0.003, read by the red-slope line on the squeezed red, on the un-squeezed red, and split — the eleven terms on the squeezed estimate, the red slope on the un-squeezed. The split line reads the warm whites at ±0.034 and ±0.057 and the cold at ±0.025; the squeezed line reads them at ±0.046, ±0.067 and ±0.021, the un-squeezed at ±0.034, ±0.057 and ±0.040.

Two estimates of one red each read their own end

Two first stages read a white LED's red phosphor from an ambient sensor: one squeezes the warm whites' red a third toward the middle, the other reads it four times better, and a camera-error line with a red slope prefers the first for cold whites and the second for warm. The prediction was that a line handed both — the eleven-term line on the squeezed estimate, the red slope on the faithful one — keeps each end's gain. It does, within 0.004 at the cold end. But the second estimate is not needed: the squeezed red alone, its red slope multiplied by the mireds, reads every LED within a ten-thousandth of the split line. And the two estimates' difference is worth a term of its own. Added to the line, it reads the cold whites within ±0.016 at every Duv and on the locus within a thousandth, and moves the warm whites and the middle by nothing.

imaging · Capture 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.

Held-out validationColour-matching functionsOverfittingCamera rawΔEITPIdentifiabilityNull spacePlanckian locusShot noiseSignal-to-noiseWhite LEDMetameric black

All concepts