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

Prediction — where it appears

A statement about a number written down before the number exists. Recording one where it was made, rather than deleting it afterwards, is what turns an arithmetic result into evidence — and in this collection the predictions have mostly been refused.

Named by 21 essays across 5 fields — each of them below, with the objects they name alongside it.

Every census row under five constructions of the same test set. A slope chart with 5 columns — lattice, coarse, fine, uniform, natural — and one line per change of light in the census, each line joining that row's mean residual under each construction. Four of the five columns describe the same region of surfaces walked at different densities or against different measures; the last is the clamped, realistic family, which is not linear in its parameters and is therefore answering a slightly different question. The levels move: between the coarse and fine lattices every row shifts by seven to nine per cent, in the same direction, which is a common-mode factor no published residual here has ever carried. The order almost survives. Inside the region exactly one pair crosses, and it is the pair the standard error had already flagged; under the clamped set two more cross, including one the error separates by nearly nine standard errors. The crossing lines are drawn heavy.

The instrument named the pair that moved

A standard error over a test set flagged four steps of the adaptation census as unresolved. Rebuilding the set three different ways reversed exactly one pair, and it was one of the four. Rebuilding it to a different rule reversed a pair the error separated by nearly nine standard errors — which is not a failure of the instrument but a statement of what it is about.

limits · Limits Eight conditions under which the model equation is exact, and how exact each one is. Each of the four departures vanishes if either of its two factors is empty, which is eight conditions. The axis is logarithmic in the residual that is left when the condition is imposed. Three of the eight are identities: the fluorophore's loading is zero so the emitted term is an empty sum, and a Lambertian surface or a uniform field makes the pairing's second argument identically zero. The other five are limits — a Gaussian excitation band has no edge, an opaque sample still has a kernel a few microns wide, a four-metre aperture is still finite, and the observer is small rather than absent at 380 nanometres. Each limit is drawn with the sequence its residual falls along as the condition is pushed, because a small number is not evidence of a limit and a falling sequence is.

Either factor being zero

Four different departures from the colour integral turn out to have one algebraic form — each is an inner product of something the sample does that the model has no slot for with something the light does that the model assumed away. Either factor being zero makes the departure exactly zero, and the sizes of the two factors decide neither how large it is nor which way it goes.

limits · Limits An aperture and a gloss lobe, apart and together. Six materials, each measured through a four-millimetre radius and each given a gloss lobe, alone and at the same time. The pale bar is what the two cost added together as if they were independent; the dark one is what they cost when both are present. Every material comes out below the sum, by between 0.8 and 3.6 ΔE₀₀. The two departures partly cancel: the aperture removes light that went into the material and came back out too far away, and the interface returns light that never went in at all. Measuring either one alone therefore overstates what both together do, which is the opposite of the way interacting errors are usually assumed to behave.

Two departures that partly cancel

A glossy translucent sample has two of this round's four departures at once, and the expectation was that they would compound. They do the opposite. An aperture takes light away that went into the material and came back too far out; an interface returns light that never went in at all — so measuring either alone overstates what both together do, on every material tested.

difference · Metric The six arguments a surface's response has, and the one this model keeps. A surface's response to light is a function of six arguments: the wavelength, direction and place the light arrives with, and the wavelength, direction and place it leaves with. The model every colour here is computed from keeps one number per wavelength, which means it takes the diagonal of the first pair, integrates the second away, and assumes the third pair equal. Each departure drawn here restores one of them. The fourth departure is not on the diagram: the wavelength grid is the range of the index that was kept rather than an index that was dropped, which is why it is the cheapest of the four to fix and was still not fixed.

The audit that changed the object

Three rounds audited this collection's numbers, its sets and its conventions, and each one ended by naming the same thing it could not reach — that a surface is a reflectance. This round reached it, found that four departures share one algebraic form, had two of its predictions refused, and discovered that its own hemisphere quadrature had been passing a convergence check by luck.

limits · Limits How far CIECAM16 moves a monochromatic hue when the light is brightened thirtyfold. A monochromatic stimulus at a relative luminance of 2 and of 60, read through CIECAM16 in one room, and the difference between its two hue angles. The model was never fitted to this effect and has it anyway: the shift is positive at the short end, negative through the greens, positive again in the yellows and reds, and crosses zero at 459, 495, 502, 570 nanometres. The marks are the invariant wavelengths the literature reports — 474, 506, 571.

The model has a hue shift it was never given

A monochromatic light changes hue as it is brightened, except at three wavelengths that do not move — an effect measured since the nineteenth century and not among the things CIECAM16 was fitted to. The model has it anyway: brightening a stimulus thirtyfold moves its hue angle, and the places where the movement crosses zero land at 459, 495, 502 and 570 nanometres against the reported 474, 506 and 571. The sizes are another matter.

brain · Appearance 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 share of its colour each display hue loses to a quarter of its luminance in white. The twenty-four most saturated colours of an sRGB display, one every fifteen degrees of HSV hue, each scaled to a luminance of 20 and given white of a quarter that luminance: the share of its colourfulness lost, in CIECAM16, CIELAB and Oklab. All three put the fastest loss at an orange and the slowest at a blue. At the orange end they are close — CIECAM16 27 per cent, CIELAB 28 per cent, Oklab 21 per cent — and at the blue end CIECAM16's 1.7 per cent is a quarter of the others'.

White drains the blue last in every model

CIECAM16 says a dab of white costs a deep red wall four times the colour it costs a dark blue one, and the experiment proposed to test it asked whether observers lose colour in that hue order. Held at equal luminance and given equal doses of white, twenty-four display hues are ordered alike by CIECAM16, CIELAB and Oklab — an orange loses fastest and a blue slowest in all three. So the order cannot tell the models apart. What does is how much slower the blue is: CIECAM16 has it losing a sixteenth of what the orange loses, CIELAB and Oklab about a quarter. That ratio is the number an observer study has to measure.

brain · Appearance 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 Divide the faces and the tall room gains colour from a grey finish. How many of the census's 72 paints a satin finish on the grey faces makes more colourful, with every face divided into one to six patches a side. One patch a face is the census, where the tall room with its side walls painted gains for 0; from one to six patches a side it gains for 0, 4, 8, 11, 15, 15. The tall narrow slot, painted the same way, goes 22, 7, 11, 45, 72, 72, and the tall room with all four walls painted barely moves.

The tall room gains once its walls are divided

The room census solves each face as one patch, and it says a satin finish on the grey faces of a tall room with painted side walls makes the room more colourful for none of seventy-two paints. The true box disagreed: rays from points on the grey end walls said the finish sends the lamp's light towards the paint. Dividing every face into patches settles it. The end wall's upper patches, which see the lamp at a grazing slant, send its light across to the painted walls; its lower patches send it to the floor. The room gains for 0, 4, 8, 11, 15 and 15 paints as the faces are divided one to six times a side — the light, broad paints — and the narrow slot beside it goes from 22 to all 72.

scene · Scene Inside its family a 620 nm channel reads the tube's error; across the census it does not. The halophosphate tube with its main band at 558, 568, 578, 588 and 598 nm, joined in order: the camera's error under each falls from 2.55 to 1.48 as the band moves up, and the 620 nm channel's departure rises from -0.47 to 0.09 with it. The census's other fifty-three lamps, dots, lie nowhere near that line: the same departure means a different error for an LED.

The tube a third channel misses has no family

Two narrow sensor channels read as one ratio estimate a camera's colour error under fifty-four lamps to ±0.141 ΔE₀₀, and their worst miss is the halophosphate tube, whose trouble lies in its long-wave phosphor. A third narrow channel in the long-wave gap was predicted to halve that miss near 610 nm. No channel does, at any centre or width: the best buys under a tenth of the whole error and leaves the tube's miss above three quarters of what it was. At 610 nm the camera already predicts the tube's share almost exactly. The tube's error follows where its main phosphor band sits, and a 620 or 640 nm channel does read that — inside the tube's own family. The census has one halophosphate tube, and a lamp left out of a census that holds nothing like it is a lamp no channel can be fitted to.

imaging · Capture What each design buys each family of lamps. The camera's colour error under each lamp, estimated with the lamp left out, and each family's held-out error under four designs: the ratio of the 500 and 450 nm channels alone; one line on the ratio and a 640 nm channel for every lamp; two stages with each lamp's family given; and two stages with the family found by the nearest other lamp. The census's halophosphate tube is missed by 0.405 under the ratio and 0.011 under either two-stage design; the LEDs go from ±0.127 to ±0.111 with the family given and ±0.155 with it found.

Two stages catch the tube and bill the lamps they misplace

A narrow-channel estimate of a camera's colour error missed the halophosphate tube because the census held nothing like it, and a line that learned the tube's family from four siblings cost every other lamp. The proposed cure was two stages: recognise a lamp's family, then apply that family's own line. Given each lamp's family, it works as predicted and better — the tube missed by 0.011 ΔE₀₀, the fifty-four lamps estimated to ±0.133 against the ratio's ±0.141. Found by the nearest other lamp in the sensor's departures, the family is right for the tube and wrong for five lamps, and those five, missed by ±0.41, take the whole census back to ±0.162. The gain is real; the classifier is where it is lost.

imaging · Capture How many neighbours a lamp's family is read from, and what the estimate costs. The two-stage estimate's held-out error over the census's fifty-four lamps when a lamp's family is read from its one to six nearest neighbours, and used only if all of them agree (upper bars) or most of them do (lower), with the halophosphate tube's miss on each bar; otherwise the census-wide ratio line. The ratio alone is ±0.141 and the families given ±0.133. Blue bars catch the tube; orange ones lose it.

A classifier that abstains gives up the tube

Found by the nearest other lamp, a lamp's family cost a two-stage estimate of a camera's colour error more than it bought, because five lamps were placed confidently in the wrong family. The proposed repair was a classifier that abstains — falls back on the census-wide line — unless a lamp's three nearest neighbours agree. It abstains on the wrong lamps. The misplaced ones have three agreeing neighbours, all white LEDs; the lamp it gives up is the halophosphate tube the two stages were built to catch, whose third neighbour is a cyan LED, and its miss returns from 0.011 to 0.445. Two agreeing neighbours look better than knowing every family. Chosen without looking at the lamp being estimated, the rule is worth exactly what the ratio alone is worth.

imaging · Capture Three family lines through the departures, and whom they claim. Every lamp of the census and the halophosphate tube's four siblings by its 450 and 640 nm departures, coloured by family, with the principal line of each family that has three or more members — the radiators, the white LEDs and the halophosphate tubes — solid over the stretch its members span and dashed beyond. Circled are the 29 lamps whose nearest line is another family's. The radiators' short line runs along the LEDs' long one and, extended, through the violet-pumped LEDs.

A family's line runs through its neighbours

The halophosphate tubes differ from one another by one number — where their main phosphor band sits — and so lie along a line in the three departures an ambient-light sensor reads. The proposal was to classify a lamp by the family whose line passes nearest, which would keep the tube a vote of neighbours gives up. It keeps the tube, under every version that does not rescale the distances. It also places eighteen white LEDs on the radiators' line, because that line — fitted to six radiators close together — runs along the LEDs' own and, extended, straight through the violet-pumped ones; and the radiators' estimate, extrapolated to them, misses the triphosphor tube by 7.6. Clamped to the stretch each family spans, the lines still cost ±0.285; given to the halophosphate family alone, its line claims a cyan LED and costs ±0.179. The one line classifier that comes in under the ratio line's ±0.141 does so by two thousandths, having placed the tube with the LEDs.

imaging · Capture What treating radiators and LEDs as one family does to the estimate. Held-out error over the census's fifty-four lamps for the ratio line alone, for the two-stage estimate with the separate families found by the nearest lamp and given, and with radiators and LEDs merged into one family of smooth sources, found by the nearest lamp, with the halophosphate tube's miss. Merged, the estimate comes to ±0.127 — below the ratio's ±0.141 and below the separate families even when every lamp's family is given — and keeps the tube.

Radiators and LEDs are one family to the sensor

An ambient-light sensor's three narrow departures see the collection's six smooth radiators and forty-four white LEDs along one line: the radiators' own short line lies on the LEDs' long one. Estimating a camera's colour error family by family, the census had treated them as two, and every classifier that tried to tell them apart paid for it. Merged into one family of smooth broadband sources, found by the nearest other lamp as before, the two-stage estimate comes to ±0.127 over fifty-four lamps held out — below the ratio line's ±0.141, and below the ±0.133 the separate families managed when every lamp's family was simply given. The halophosphate tube is still caught. The radiators, which the prediction said the merge would cost a little, are estimated better inside it than inside their own family, because a three-coefficient line fitted over five radiators was worse for them than the census-wide ratio line.

imaging · Capture How far the merged design misses each lamp it never saw. For fourteen lamps outside the census — radiators and daylights, tinted and filtered lamps, white LEDs varied in their phosphors, and a three-band LED — the camera's colour error less the estimate the merged design makes from the ambient-light sensor's departures, fitted on the whole census. The dotted lines are the census's own held-out error, ±0.13. The radiators, daylights and tinted lamps fall inside or on it; 4 lamps fall outside, every one of them an LED.

New LEDs miss the line their cousins drew

An ambient-light sensor's three departures can estimate a camera's colour error under a lamp, and merging radiators and white LEDs into one smooth family brought that estimate to ±0.127 over fifty-four lamps, each left out in turn. Every one of those lamps had relatives in the pool. Fourteen lamps the census never saw test what the relatives were worth. The prediction was that new smooth lamps would land on the line and tinted incandescents fall off it by four tenths or more. The radiators and daylights land on it, within ±0.13 each. The tinted lamps do too — the pink-glass incandescent, placed among the LEDs by its nearest neighbour, misses by 0.12. The lamps that fall off are the new white LEDs: ±0.20, half again the census's error, two of them by a quarter. Their misses follow the sensor's 640 nm channel at a slope of −0.72, explaining four fifths of their scatter, because the line learned what that channel means from LEDs varied only in their blue pump. Adding the fourteen to the pool does not help.

imaging · Capture

Named alongside it

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

Camera rawHeld-out validationLeast-squaresModelling assumptionRank correlationWhite LEDSpecularBidirectional reflectanceCensusRadiosityChromaFluorescent

All concepts