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

Census — where it appears

A fixed table of cases a quantity is reported over, so that two computations of it are comparable. This collection's adaptation census is fourteen changes of light judged over a hundred and twenty-five constructed surfaces, and its rows are what every claim about adaptation here is measured on.

Named by 16 essays across 3 fields — each of them below, with the objects they name alongside it.

The worst case is wherever the box stops. Four horizontal tracks, one per parameter of a painted wall. Each track spans the range an ordinary paint is allowed to occupy, with a second, wider range drawn behind it, and two markers show where the search for the worst change of light came to rest under each. Under the narrower box the answer sits on the wall in centre and width; under the wider one, in centre, width, base. The residual rises monotonically towards a narrower notch at a shorter wavelength on a darker wall, so there is no interior maximum to find. The worst change of light is 21.3 ΔE00 under one box and 28.4 under the other, and the census's own worst row is 3.37.

The worst case is where the box stops

The worst change of light this collection quotes is two bounces off a green wall, and it is the worst of fourteen changes somebody wrote down. Searching the family those fourteen were drawn from reaches six times further — and does not stop, because the residual rises monotonically towards a narrower notch on a darker wall. There is no worst case in this family, and the number anybody quotes for one is a number about their own constraint.

scene · Scene Which of the collection's published quantities a departure can be pushed through. The six quantities the previous round recomputed under six different colour-difference units, and whether the same treatment works for a departure. Two do: the adaptation census and the metameric pair both take reflectances and a light, which is what a departure acts on. Four do not, and the reasons are different in each case rather than a single obstacle. A unit is a function applied to the answers, so it can be swapped at the end of any computation; a departure changes the object at the start, so it has to be accepted by every stage in between. That is the practical difference between auditing a convention and auditing a structure.

A departure is not a unit

The previous round audited six published quantities by swapping the unit they were quoted in — a function applied at the end of each computation. Nothing of that shape works here. A departure changes the object at the start, so every stage in between has to accept it, and only two of the same six quantities can take one at all. The fourth cannot even be expressed in the interface.

limits · Limits What a fourth emitter costs, by where it is put. A three-emitter LED with one more emitter added at each position from 470 to 610 nanometres, and for each lamp the number of the object-colour solid's directions that lose more than a per cent of their reach to a 40-nanometre transition limit. The three-emitter lamp itself costs 62 of 312. A fourth emitter in the middle of the blue-to-green gap, at 490 nanometres, costs 172; one at 540, beside the green emitter, costs 65. The two dips sit on the existing lines and the two peaks sit between them.

A fourth emitter spends the gap it fills

A three-emitter LED lets a pigment that takes forty nanometres to switch reach most of its ideal solid, because the lamp is dark where the pigment is slow. Adding a fourth emitter to render better takes that darkness back — but only if it is put in the middle of a gap. At 490 nanometres it costs 172 of the solid's 312 directions against the three-emitter lamp's 62, and renders two points worse. At 540 it costs three directions and renders two and a half points better.

limits · Limits The rescue is spent on light between the lines, not on width. The three emitters of a narrow-band LED broadened together, from their nominal widths up to six times them, plotted against the light left in the darkest of the lamp's two gaps as a share of its peak. Up is the share of the object-colour solid's directions that lose more than a per cent of their reach, at three transition limits, with each limit's cost under daylight marked at the right. At forty nanometres half of the rescue is gone by a floor of 7.4 per cent — emitters only 1.30 times their nominal width — and all of it by about a fifth. At twenty nanometres and at eighty there is little to lose either way.

The gap has to be dark, not the line narrow

A lamp whose light sits in three narrow emitters lets a blunt pigment reach most of its ideal solid, and the reason was given as the spacing of the lines. Broadening those emitters without moving them says otherwise. At 1.3 times their nominal width the lamp still looks like a line spectrum, its closest spacing has not changed at all, and half the rescue is gone — because the darkest point of the narrow gap has risen from one per cent of the lamp's peak to seven.

limits · Limits 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 The same gradients, priced by a penalty and by a projection. Each row is one gradient held inside a coated press from six starts. The pale dots are the penalised relaxation — a free step, with a price for leaving the press — and the dark ones are the projected relaxation, which takes the free step and then moves each point to the nearest printable colour. Across is how much longer than the free path each result is, logarithmic. On the 10 gradients whose straight line leaves the press, the penalty leaves 9 starts at more than twice the free length and the projection leaves none. The projection's best route costs a median 0.02% against the penalty's 0.13%, and its spread across starts is 0.95% against 132.5%.

A projection has no reason to detour

Holding a gradient inside a press by penalising the excursion turned the gamut's price from a number into a search: on five of ten crossing gradients some starting point leaves the relaxation trapped at more than twice the free length, and the spread across six starts runs to 271 per cent of the free path. Replacing the penalty with a projection — take the free step, then move each point to the nearest printable colour — leaves no trap on any gradient and a spread of 0.06 to 2.4 per cent.

matching · Gamut 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 How much of a coated press is left at each margin inside its boundary. The share of a coated press's printable volume in CIELAB that lies at least a given distance inside its boundary, for margins from half a unit to 32. A margin of one unit keeps 88%, two keep 80%, four 66% and eight 45%. At every margin up to 30 what is left is a single connected piece; at 31 units, with 0.14% of the volume left, it first splits, into a core of 623 cells and 2 fragments of one or two cells — the deepest point is 32.7 units inside, so what splits is the last crumb of the core, not a waist.

A margin costs a press its corners

A gradient held inside a coated press by projection was predicted to tear if the press were first shrunk by a safety margin, at any pinch narrow enough for the shrinking to cut. The press has no such pinch: shrunk by any margin up to thirty CIELAB units it stays in one piece, and projected routes on it are neither trapped nor torn. What a margin costs is concentrated at the corners — the solid yellow moves nearly four times the margin to get inside, like the tip of a 31-degree spike.

matching · Gamut Each end colour's move into the eroded press, over the margin, in CIELAB and in ΔE₀₀. The 9 end colours of the census gradients, each moved to the nearest colour of the press eroded by a margin, for margins of one, two and four. Squares: the margin in CIELAB units and the move in CIELAB. Circles: the margin in ΔE₀₀ and the move in ΔE₀₀. The dashed line at one is a flat face and the one at 1.73 a cube's corner. In CIELAB every end colour moves further than a cube's corner at two units, the solid yellow ×4.6, ×3.9, ×3.8. In ΔE₀₀ six of them move between 1.06 and 1.34 times the margin, and only the solid yellow, at ×1.91, ×2.04, ×2.10, is still sharper than a cube's corner.

A tolerance's own unit rounds the corners

Shrinking a coated press by a margin in CIELAB units made its corners pay several times the margin — the solid yellow nearly four. A print buyer states a margin in ΔE₀₀, and in that unit the corners pay about half as much: six of nine saturated end colours move within a third of the margin, and only the solid yellow is still sharper than a cube's corner. The prediction that the dark corners would stay sharp was wrong; they round too. What the tolerance's unit costs instead is volume — two ΔE₀₀ keep two thirds of the press where two CIELAB units kept four fifths.

matching · Gamut 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 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 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.

InterreflectionRadiositySpecularRank correlationPredictionBidirectional reflectanceChromaModelling assumptionReflectanceConvergenceForm factorChromaticity

All concepts