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

Rank correlation — where it appears

A measure of how far two orderings of the same objects agree, computed from the ranks rather than from the values. Kendall's τ counts the pairs put in the other order and Spearman's ρ correlates the ranks themselves; the first is the right one wherever the question is how many comparisons would be decided differently.

Named by 17 essays across 4 fields — each of them below, with the objects they name alongside it.

What six of this collection's published numbers do when the unit changes. Six quantities, from six calculations that share nothing: a change of light after an observer has adapted, a camera profile's error, the gap between the two standard observers, a metameric pair under the lamp that breaks it, the same image on two papers, and an observer two seconds into a new room. Each is recomputed under all six units and every unit is calibrated onto ΔE2000's scale first, so the bar is not a change of units in the ordinary sense. The bar is the ratio of the largest reading to the smallest, and it runs from 1.71 to 2.30. Five of the six are printed in ΔE2000 by the essays that report them; the sixth is printed in CAM16-UCS, because the model it comes out of defines that unit.

A choice with no magnitude

An audit can multiply a width by 1.25 and report an elasticity. It cannot multiply CIEDE2000 by anything. Auditing a structural choice needs a different instrument, and building one shows that six published numbers in this collection each carry a factor of about two of unit-choice — after the change of scale has been taken out.

limits · Limits How far each unit is from being a rescaling of the one this collection publishes in. One row per unit on the menu. The bar is the root-mean-square scatter about that unit's own best rescaling of ΔE2000, over 374 pairs of surfaces differing by a fraction of a unit to about ten. A bar of zero would mean the unit is ΔE2000 in different money — every printed number would change and no conclusion would. ΔE2000's own row is zero by construction and is the check that the table is computed the right way round. The two units that divide a chroma difference by the chroma it was measured at, ΔE94 at 15 per cent and CAM16-UCS at 24, are closer to it than the three that do not, which run from 28 to 35. The split is by weighting and not by whether the unit is a matching difference or an appearance one.

The weighting is the disagreement

Five colour-difference formulae, three decades and two committees, and the single property that predicts which of them agree is whether a chroma difference gets divided by the chroma it was measured at. It sorts the menu exactly, it cuts across the distinction between a matching difference and an appearance one, and it halves the census's largest sensitivity.

difference · Metric The census along the line from ΔE76 to ΔE94, and past it. ΔE94 is ΔE76 with two weighting constants in it, and at zero those constants make every weight exactly one, so the two formulae are joined by a line rather than separated by a choice. The horizontal axis is how much of the published weighting is applied: 0 is exactly ΔE76, 1 is exactly ΔE94, and 3 is three times more weighting than anybody has proposed. The falling curve is the census's mean elasticity to how saturated its test set is, which drops from 1.11 to 0.74 — most of the fall happening before the published value is reached. The other curve is Kendall's τ against ΔE2000's ranking, and it peaks at w = 0.5, not at 1: the weighting that best reproduces the published ordering is about half the published weighting. There is no value of this dial that reaches ΔE2000, whose rotation term is not on this line at all.

A dial through a discrete menu

ΔE*94 is ΔE*ab with two weighting constants in it, and at zero those constants make every weight exactly one — so the two ends of the oldest disagreement in colour difference are joined by a line rather than separated by a choice. Walking it gives a derivative where a menu gives only a spread, and the derivative says the published weighting is on the far side of the interesting part.

difference · Metric Which steps of the census ranking a change of unit reverses. Every adjacent pair in the published census ranking that at least one unit puts the other way round. The bar counts how many of the five other units reverse it. The marker on the left says whether the test set had already declared the pair unresolved — a gap smaller than twice its own paired standard error, which is a statement about sampling over 125 surfaces and shares no arithmetic with a change of ruler. The two pairs every unit reverses are both flagged, which is the agreement. The pair at the bottom is the disagreement: the test set resolves it at 9.1 standard errors and four of the five units reverse it anyway, because a sampling error cannot see a change of ruler and a change of ruler cannot see a sampling error.

Two instruments and one ranking

A sampling error over a hundred and twenty-five surfaces and a change of colour-difference formula share no arithmetic at all, and they were asked the same question of the same table. Every adjacency the whole menu reverses had already been flagged as unresolved. And one the test set settles at nine standard errors is reversed by four of the five formulae, which is what makes them two instruments rather than one.

limits · Limits Thirty-six rooms as a viewer adapted to each would see them. Each cell is one room at a satin finish: wall colour down, lamp across. The large number is the share of the faces' chroma a viewer adapted to the room's own light loses, read through CIECAM16; the small number under it is what the room's light loses against the lamp's white. The viewer loses between 12.8 and 30.9 per cent, always more than the light, and the rows differ far more than the columns: a deep red room loses about twice what a green one does under every lamp.

What an adapted viewer loses is set by the wall

A satin finish takes about a tenth of a room's colour, measured on the room's light against its lamp's white. Read through an appearance model by a viewer adapted to the room, the same thirty-six rooms lose between 12.8 and 30.9 per cent — 1.4 to 2.4 times as much — and the wall colour decides the multiplier: a deep red room loses twice what a green one does, under every lamp. A test on the bare wall predicts it. Add a little of the lamp's white to the wall's own colour and ask the model what that costs: the answer orders the thirty-six multipliers at 0.95.

scene · Scene The share a finish takes falls with how much light the wall returns. Seventy-two rooms under daylight, each with a different paint on two opposite walls — six hues, four band widths, three peak reflectances — at a satin finish. Across is the wall's luminance factor, the share of the lamp's light it returns; up is the share of the room's chroma the finish takes. The losses fall with the luminance factor at a rank correlation of −0.89, from 15.5 per cent on the darkest walls to 2.5 on the palest. The two ringed paints make rooms of the same matt chroma and lose 14.7 and 2.5 per cent.

A dark wall pays for a finish

A satin finish takes about a tenth of a room's colour, and the tenth varies. The natural explanation is that a weak colour loses a larger share of itself to the white light a glossy surface adds. Over seventy-two paints under one lamp that explanation orders nothing: the loss runs from 2.5 to 15.5 per cent, it follows how much light the wall returns at a rank correlation of −0.89, and it follows how colourful the wall is at −0.04. Two rooms equally colourful matt lose shares nearly six times apart, and the one that loses more is the darker.

scene · Scene What survives holding the lightness still. For each of five groups of rooms sorted by how much light the wall returns, the rank correlation of the finish's share with three properties of the paint, taken with the wall's luminance factor held. The wall's chroma runs from -0.90 among the darkest walls to 0.81 among the palest, crossing zero in the middle — the reversal. The band's width, which was the predicted mechanism, never leaves the range -0.15 to 0.22, and the census already contains four families whose bands are all the same width.

The finish adds the room's own colour

Among dark walls a more saturated paint loses a smaller share of its colour to a gloss finish, and among pale walls a larger one. The mechanism proposed for that reversal was spectral concentration — a narrow tall band — and the census that found it already contained four families of paints whose bands are all the same width. Holding lightness still, the band's width orders the losses at a rank correlation of 0.01. What does order them is that the light a finish adds has already bounced off the walls.

scene · Scene Where chroma stops protecting a wall, in five rooms. The seventy-two paints in each of five rooms, sorted by how much light the wall returns and read in overlapping windows of eighteen: the rank correlation of the share of colour a satin finish takes with the wall's chroma, lightness held. Below zero a more saturated paint loses less; above, more. The dots are where each room's curve crosses: cube at 0.36, corridor at 0.37, low room at 0.38, one wall open at 0.29, two walls open at 0.24. Solid lines are closed rooms of three shapes; dashed are the cube with one and two walls opened.

An open room hands over sooner

A gloss finish takes the least colour from a saturated dark wall and the most from a saturated pale one, and in a closed cube the sign changes at a wall returning about a third of the light. The prediction was that a less enclosed room would move that point up the lightness scale and that a room with a window would have no pale end. Opening one wall moves it down, from a luminance factor of 0.36 to 0.29, and opening a second to 0.24; stretching the room into a corridor or flattening it nudges it slightly up. The ambient does whiten, as predicted. A whiter ambient does not delay the colour term — it weakens it, so the other term takes over sooner.

scene · Scene The lamp's white, six walls and the light arriving at each. On the 1976 chromaticity diagram, in the closed cube under daylight: the lamp's white (centre), six saturated walls with bands centred from 450 to 650 nm (open circles), and the light arriving at each wall from the rest of the room, lamp included (filled). Each ambient lies on the line from the white to its wall, a fraction of the way along it: 450 nm 10 per cent, 490 nm 12 per cent, 530 nm 19 per cent, 570 nm 19 per cent, 610 nm 13 per cent, 650 nm 5 per cent.

A probe at the wall prices the finish

The light arriving at a painted wall from the rest of its room is what a gloss finish hands back, and a small probe held against the wall reads it. It lies on the line from the lamp's white to the wall's own colour, a fraction of the way along, and the fraction is set by how much light the wall returns, not by how colourful it is — as predicted. It is not the fixed fraction the prediction said: it runs from 2 to 46 per cent across seventy-two paints in one room. That variation is what makes it useful. Read in the matt room, it orders what a satin finish would cost more tightly than the paint's own lightness does, in every room tried.

scene · Scene The camera's error, and the two-channel estimate of it. Each of the fifty-four lamps as the camera's true mean colour error against the estimate from the 450 and 500 nm channels, fitted with that lamp left out. The held-out error is ±0.14 ΔE00 against ±0.16 from the 450 nm channel alone and ±0.30 from the mean. The dashed lines at 1.5 mark the repair decision: points right of the vertical and below the horizontal are lamps that need a repair and are not given one.

Two channels estimate what one can only sort

A narrow sensor channel was fitted to call lamps smooth or structured, and the label turned out to be a proxy for the thing a camera cares about: its own colour error under the lamp. Scored against that error directly, the best channel is still the one at 450 nm — the band where camera and observer differ most comes close and no closer. Read as an estimate rather than a verdict, a second channel at 500 nm, useless as a vote, earns its place: together they predict the camera's error to within 0.14 ΔE00 over fifty-four lamps, and deciding when to repair from that estimate raises a quarter of the false alarms the label raises.

imaging · Capture How many of 72 paints make a more colourful room, by where the finish goes. Six room shapes, each with five sets of painted faces and the rest mid grey, and 72 paints. Each cell is the number of paints for which a satin finish makes the room more colourful than matt, with the finish on every face, on the painted faces only, and on the grey faces only. With the finish on the painted faces, 29 of the 30 cells are zero; with it on the grey faces, 28 are not. The low room with its side walls painted — the room where a finish was found to add colour — gains for 21 paints with every face finished, none with only the walls finished, and 51 with only the grey faces finished.

A finish adds colour only where it covers grey

One low wide room was found in which a satin finish makes the room more colourful rather than less, and the explanation offered was dilution: the more of a room is grey, the more a finish's glancing return stands out. Painted area turns out not to decide it. Across six room shapes and five ways of painting them, a finish on the painted faces alone never adds colour, and a finish on the grey faces alone almost always does. The low room gains because its grey faces are most of it, and the whole finish is close to the sum of its two halves.

scene · Scene What straightening the 570 nm channel buys an estimate of the camera's error. The root-mean-square and worst held-out errors of linear estimates of the camera's error, over fifty-four lamps with the three-emitter source kept in: from the 570 nm channel as read and through three transforms, from 450 nm alone, and from 450 nm with a log-transformed 570 nm channel or with 500 nm. Every transform rescues 570 nm from worse than the mean; none reaches 450 nm; and beside 450 nm it adds nothing, where 500 nm does.

A straightened channel repeats the one beside it

A phone's narrow sensor channel at 570 nm ranks a camera's colour error under a lamp almost as well as the one at 450, and estimates it worse than guessing the mean, because one lamp reads ten times higher than any other. Read through a logarithm, the channel is rescued: its estimate improves from ±0.57 to ±0.23. It still does not reach the 450 nm channel's ±0.16, and put beside that channel it adds nothing at all, where a 500 nm channel cuts the error to ±0.14. Straightened, 570 nm sorts the lamps as 450 nm already does; and the two lamps it reads highest have the most different errors, which no transform of one reading can tell apart.

imaging · Capture Which way each narrow channel departs from what the camera predicts. Each lamp's two narrow channels, their departures from the camera's prediction kept signed. 37 of the fifty-four lamps, every structured one among them, lie in the lower right: more light than predicted at 450 nm, less at 500 — a peak and a trough. 8 lie in the upper left, the opposite pattern, most of them violet-pumped LEDs whose pump misses 450 nm. In both the departure of the two readings' ratio is the sum of the two distances. The 7 in the upper right are the most heavily cyan-filled LEDs, whose fill puts light at 500 nm, and two daylights near the origin.

One ratio reads a peak and a trough

Two narrow sensor channels, at 450 and 500 nm, estimate a camera's colour error under a lamp to ±0.14 ΔE₀₀; the proposal was to read their ratio as one number instead. Taken as one channel's departure over the other's, the ratio is worthless — no better than the mean, because for smooth lamps both departures are nearly nothing. Taken as the departure of the two readings' ratio from what the camera predicts for it, one number estimates the error to ±0.141, as well as the two channels as two. The reason is a sign: under every structured lamp the channels depart in opposite directions, a peak at 450 and a trough at 500, and the ratio adds them; where a cyan fill lifts 500, it subtracts them, as the error falls. And no five per cent calibration error moves any of these estimates by a hundredth.

imaging · Capture Where a grey finish sends the lamp's light orders which rooms it colours. The thirty rooms of the census, each placed by one number computed from its box and lamp alone — how much more of the lamp's light a satin finish on the grey faces sends to the painted faces than their matt return did — and by how many of 72 paints a finish on the grey faces makes more colourful. Over the twenty-four rooms with a wall or the floor painted the rank correlation is 0.92, and the vertical line at zero sorts them: right of it every room gains for 29 paints or more, left of it for 22 or fewer. The six rooms with the ceiling painted, drawn hollow, all sit left of the line and five of them gain at every paint.

A grey finish mirrors the lamp onto the paint

A satin finish on a room's grey faces makes the room more colourful in almost every arrangement, and the explanation offered was that it re-weights each grey face towards the paint it sees at a slant. That number, computed from the geometry alone, orders the rooms at a rank correlation of 0.24. The number that orders them is the same slant pointed the other way and lit: how much of the lamp's light the finish sends to the paint. It orders the rooms with a wall or floor painted at 0.92, and its sign sorts them without an exception.

scene · Scene The painted census's counts with one patch a face and with the faces divided. For each of the thirty rooms — six boxes, each painted five ways — how many of 72 paints a satin finish on the grey faces makes more colourful with one patch a face, as the census solved it, and with every face divided five times a side (six for the two tall boxes). 12 counts do not move; 9 move by twenty paints or more, shaded — darker where the divided room gains more. Some paint gains in 28 rooms both ways, and every paint in 13 with one patch and 17 divided.

Divided faces keep the census's count and lose its rule

The painted-room census, solved with one patch a face, found that a satin finish on a room's grey faces adds colour in 28 of 30 rooms, and a later essay found one geometric number whose sign sorted which rooms gain. Re-solved with every face divided into twenty-five or thirty-six patches, the count survives exactly — 28 rooms, though two of them trade places — and the finish colours every paint in 17 rooms instead of 13. The rule does not: the send number, now matching the true box's own to a rank correlation of 0.96, orders the divided counts at 0.63, and rooms on the wrong side of its zero gain for every paint. Nine of the thirty counts move by twenty paints or more.

scene · Scene Counting the paint's light instead of the lamp's orders the rooms not at all. The thirty rooms of the painted census on divided faces, each placed twice: by the send number weighted by the paint's own light arriving at the grey faces, and by the send number weighted by the lamp's light, against how many of 72 paints a grey finish colours. The lamp-lit number ranks the wall and floor rooms at 0.63; the paint-lit one at -0.07, and over all thirty at -0.02.

The paint's own light does not order the rooms

A satin finish on a room's grey faces colours some paint in 28 of the painted census's 30 rooms, and the number meant to say which rooms gain most — how much of the lamp's light the finish sends to the paint — orders the rooms solved on divided faces at only 0.63. The rooms it misplaced return mostly the paint's own light, so the proposal was to count that instead. Counted, the paint's light orders the rooms at −0.07. In every room with its floor painted the finish sends the floor's light away from the floor, and every paint gains; with the ceiling painted the paint is the lamp and the two numbers are the same number. Even the room's own response, read off the solve with the paint in place, orders the counts only at 0.71. The census has to be solved room by room.

scene · Scene The census read on the whole room and on the painted faces. For the thirty rooms on divided faces, how many of 72 paints a satin finish on the grey faces makes more colourful, read as the census reads it — every face's light together — and read on the painted faces alone. Read on the room, 11 rooms sit part-way; read on the paint, 28 rooms gain for all 72 and 2 for none, and no room sits between.

The paint itself gains all or nothing

The painted census counts how many of seventy-two paints a satin finish on a room's grey faces makes more colourful, reading the room's colour as all of its light, and eleven of its thirty rooms sit part-way. Read on the painted faces alone — the light a viewer looking at the paint receives — no room sits part-way: twenty-eight gain for every paint and two for none. The change on the paint is never more than a per cent and a half, where the room's reading swings from −18 to +57 per cent, so the census's middle ground was the grey faces all along. A box with its floor painted and the same box with its ceiling painted gain the same colour to a hundredth of a point while the light they gain differs by up to eighty-two points; neither send number orders the size of the gain, and the paint's own brightness does.

scene · Scene

Named alongside it

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

ChromaInterreflectionSpecularCensusPredictionRadiosityCalibrationModelling assumptionReflectanceBidirectional reflectanceColour differenceTest set

All concepts