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

Trade-off — where it appears

Two objectives over the same parameters whose optima are in different places, so improving one costs the other. Whether a decision is one is a measurement rather than an intuition: score each objective's own winner on the other and read the penalty.

Named by 20 essays across 7 fields — each of them below, with the objects they name alongside it.

Every basis against both objectives at once. A scatter with the mean adaptation residual across the illumination census on the horizontal axis and the mean axis ratio of MacAdam's ellipses in a lightness–chroma space on the vertical. Lower is better on both. The two winners sit at the two ends of an empty diagonal: the basis that adapts best leaves 7.70 on the vertical and the basis that discriminates best leaves 1.79 on the horizontal, each worse on the other objective than every published transform. The basis built from the dichromat confusion points is at (1.65, 2.60) — best at neither and within a factor of two of both floors, which no other entry in the picture manages.

No basis is good at both

The same nine numbers decide how well a von Kries gain reproduces a change of light and how nearly a lightness–chroma space makes the discrimination ellipses circles. Minimise either one and the other collapses. The basis built from the receptors is best at neither and is the only entry in the table respectable at both.

brain · Appearance How far from circles every basis leaves the ellipses. Eight bases ranked on the mean ratio of the long to the short axis of MacAdam's twenty-five discrimination ellipses, measured in a lightness–chroma space built on that basis. The range runs from 1.61 for best for discrimination to 7.70 for best for adaptation. The ordering is not the ordering on the other objective and is nearly its reverse.

One matrix doing two jobs

CIECAM16 adapts in CAT16 and then applies its response compression in the same axes, so a single matrix decides both how well the model handles a change of light and how uniform the space it produces is. The two jobs have different best answers, and the matrix was chosen against only one of them.

brain · Appearance The floor as a function of the exponent, and the fixed basis beside it. Two curves against the compression exponent on a logarithmic axis from 1 to 10. The lower curve is the best mean ellipse axis ratio any basis can reach with that exponent applied after it, and it falls from 2.33 at no compression to 1.66 at a square root and 1.61 at a cube root, then hardly moves — 1.57 at a tenth root. The upper curve is CIELAB's own basis at the same exponents and gets steadily worse, from 3.57 to 3.77. Almost everything a compression buys arrives with the first step away from linearity, and after that the exponent is choosing between 1.66 and 1.61 while the basis is choosing between 1.61 and 3.44.

The exponent was never the argument

A century of colour science has argued about whether the eye's response is a cube root, a square root or a logarithm. Minimise the anisotropy of MacAdam's ellipses over every basis, at each of eight exponents, and the floor moves by under three per cent between a cube root and a tenth root — while the basis moves it by a factor of two.

difference · Metric Every change of light in the census, under three bases. Fourteen changes of illumination, each drawn three times: the residual left by a gain in the basis built from the dichromat confusion points, in CAT16, and in the basis that minimises the average. The ordering between the three is the same on nearly every row, and — the part an average hides — the worst row is the same row for every basis nobody fitted, which is two bounces off the same wall. The difficulty belongs to the change rather than to the choice of axes — except for the fitted winner, whose worst row is D65 to a triphosphor tube instead. What a fit buys is not a uniform improvement; it is the abandonment of the one change everybody else is beaten by.

Which changes of light pay for it

A fitted adaptation basis beats the receptors by 0.68 units on average, and an average is a poor description of what it does. On six of fourteen changes of light it is worse, and the whole of its advantage comes from four — a tungsten lamp and three coloured walls.

light · Light Three published primary sets, and a fourth chosen for how it adapts. The spectral locus with four triangles inside it. sRGB covers 33.5% of the diagram and leaves an adapted observer 2.36 ΔE00; Display P3 covers 45.4% at 1.22; Rec. 2020 covers 63.3% at 1.09. The fourth triangle is the best adaptation basis available to a display asked to cover 63.5% of the diagram, at 1.02 — and it is a different triangle from Rec. 2020's rather than a smaller one. The largest triangle that fits at all covers 73.9%, which is where the axis of this argument ends.

Primaries chosen for their inverse

Moving a display's white point is a gain on its R, G and B, so a display adapts in the inverse of its own primary matrix — a basis chosen by committees for gamut coverage and phosphor availability. Pose the design problem properly and the answer costs one per cent of the gamut argument and reaches within two per cent of the best basis there is.

applied · Delivery The same optimum, along its narrowest direction and its widest. The adaptation objective along two straight lines through its own minimum, both of unit length in the nine coefficients. Along one of them the cost rises steeply; along the other the same step costs 8.0 times less, and a design constrained to move that way gives up almost nothing. That is why restricting the nine numbers to be the inverse of three realisable primaries — three degrees of freedom gone — costs about one per cent, while requiring them to hit the three dichromat confusion points costs seventy. Counting what a constraint removes predicts neither number; what matters is which way it points.

A constraint costs what it points at

Three primary chromaticities remove three of the nine numbers in an adaptation basis and cost one per cent. Three dichromat confusion points remove six and cost seventy. Counting what a constraint removes predicts neither, because an optimum is a long bowl and what matters is which way the constraint points.

limits · Limits The cheapest direction to give ground in is the flattest one. Six bars, one per direction the adaptation objective can see, showing how much of the other objective a fixed budget of adaptation buys if it is spent along that direction. The rate is the slope of the second objective divided by the square root of the first's curvature, so it rewards a direction the second objective wants and punishes one the first is stiff in. The flattest direction wins at 10.68 against 3.29 for the next best and 0.54 for the stiffest — a factor of 20. Spending 1 per cent of the adaptation optimum there moves the anisotropy from 7.70 to 5.02.

The trade only runs one way

Standing at the basis that adapts best, one per cent of adaptation buys forty-four per cent of the way to the discrimination floor. Standing at the basis that discriminates best, the same one per cent buys under two. The scatter that shows two objectives pulling apart looks symmetric and is not, and the asymmetry is what a committee choosing between them would most want to know.

brain · Appearance A population of receptor bases, in the plane the published ones live in. The two axes this collection scores an adaptation basis on — the residual across the illumination census on the horizontal, ellipse anisotropy on the vertical, lower better on both — with 200 extra points on it. Each is the basis a member of the population's own confusion points determine. The cloud is not a point: it runs from 1.22 to 2.24 ΔE00 horizontally, which is wider than the whole spread of the published transforms marked on it. The observer this site quotes sits inside the cloud and near one edge of it, and the sentence "the receptor basis costs seventy per cent" is a sentence about that one point rather than about the construction.

A trade between matrices, not people

Across the space of possible bases, adapting well and discriminating well pull in opposite directions — the two optima sit at the ends of an empty diagonal. Across a population of actual observers the same two costs move weakly together, at a correlation of +0.29. The trade-off is a property of the set of matrices somebody could choose, not of the eyes anybody has. One measurement inside the population does trade, and it is the macular pigment.

brain · Appearance Four departures from the model equation, each at an ordinary strength. What each of the four assumptions inside a colour integral costs, in ΔE₀₀, on a stated sample under a stated light. The wavelength index is a coated printing paper measured with and without the ultraviolet of D50; the range is the same paper integrated from 300 nanometres and from 380; the place index is a pigmented plastic through a four-millimetre radius; the direction index is an eggshell paint beside a window. The spread is a factor of 7.0. This is a ranking of four examples rather than of four departures — each of them can be made larger by choosing a more extreme sample, and the marble in the same collection of materials reaches 12.7 on the index that comes third here.

Which index to buy an instrument for

Four departures from the colour integral, ranked by what they cost and by what it would take to remove each one. The ranking by cost and the ranking by price are almost exactly reversed — the two largest are removed by specifying a lamp and by widening a table, and the two that need new hardware are the two smallest.

applied · Delivery 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 two tabulation choices over forty-two surfaces, under a 6500 K thermal radiator. Each column is one choice, measured over a family of forty-two analytic reflectances rather than on a single example: an absorption band of stated centre, width and depth. The four marks are the smallest, the median, the ninety-fifth percentile and the largest cost in ΔE₀₀, logarithmically. Under a smooth light the range is worth 9.1 times the step at the median, so a collection wanting one repair should widen its range rather than refine its step — and under a fluorescent tube the ranking reverses outright.

Which end to buy

Refining a five-nanometre grid to one buys a daylight calculation 0.05 ΔE₀₀ and widening its range buys 0.54. Under a fluorescent tube the same two purchases are worth 0.83 and 0.0001. The ranking reverses completely, and what decides it is one length compared against one other length.

light · Light The three cone absorptances at two settings of the pigment peaks. Solid and dashed are the same construction at the two ends of two standard deviations, and the L/M polymorphism on top. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 8.9 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

The peaks move the flanks

Shifting a cone's peak wavelength by three nanometres changes its sensitivity at its own maximum by almost nothing and on its flanks by several per cent, because a maximum is flat and a flank is steep. Every display primary sits on a flank, which is why a pigment polymorphism is worth 3.60 ΔE₀₀ under a laser projector.

eye · Cones The same white, matched at six primary widths. At every width the three primaries are solved to match D65 exactly for the reference member; the bands are what the population sees. A broad primary integrates the observer differences over a band and averages them away; a narrow one samples them at a point and passes them straight through. From 40 nm to 2 the ninety-fifth percentile rises from 11.3 to 17.9 ΔE00, monotonically, and the technology has been moving from left to right for thirty years.

A narrow primary buys a disagreement

The observer audit decomposes what a display costs a population. Narrowing the primaries raises the pigment-peak departure monotonically, moving one raises or lowers the macular departure, and the two respond to different design variables — so a wide gamut and an observer-robust display are bought with the same money.

matching · Gamut The share of each light outside 380–780 nanometres, as power and as visual product. Two measurements of the same truncation. The upper bar is the fraction of the light's radiant power that lies outside the range; the lower is the fraction of the product of light, sample and observer — which is what a colour is made of. A thermal radiator puts a fifth of its power outside and about a thousandth of its colour, because the observer is zero where most of that power is. The gap between the two bars is the observer's own tails doing their job, and reading the upper number as though it were the lower is how a range gets argued about without being measured.

A lamp is two audits at once

A lamp's ultraviolet content decides what its tabulation costs and its blue content decides what its observer costs, and the two run in opposite directions with colour temperature. So no lamp is good for both audits and no lamp is bad for both, and a single figure of merit for either is a figure of merit for one property of a spectrum.

light · Light One instrument, one slit, two requirements. The slit's width swept, with three measurements on a logarithmic scale: a smooth sample under the line lamp, where only the lamp's structure is at stake; the notched sample under a smooth lamp, where only the notch is; and the real case, both at once. The lamp wants a slit of 5 nanometres and the sample wants 1, and each wants what it wants for the same reason: a slit should spread a feature the grid cannot resolve and leave one it can. The real case is best at 3 nanometres — which is neither requirement's answer — and costs 0.247 there, an order of magnitude more than either requirement alone.

One slit, two requirements

A line lamp wants a wide slit, because a wide slit spreads a line where a coarse grid can see it. A notched sample wants a narrow one, because a wide slit fills the notch the grid could have resolved. An instrument has one slit. Measured on the two requirements separately the best widths are five nanometres and one; measured on the two together the best is three, which is neither — and it costs twelve times what the lamp alone would cost and thirty-four times what the sample alone would.

light · Light What a gamut charges a gradient, against what the relaxation's noise is. For each gradient, how much longer the best path that stays inside sRGB is than the free shortest path, as a percentage of the free one. The shaded band is the relaxation's own noise, measured by relaxing the same free path from two different starting points: 0.42 per cent at its worst. Every excess is inside it. So holding a gradient inside a display's gamut costs nothing measurable in the metric's own units, on any of these pairs — including the ones whose free path is outside at most of its points.

A gamut charges a gradient nothing

On red to green the shortest path happened to stay inside sRGB where the straight line did not, and that was called a coincidence of mechanism. Made into a measurement it is not a coincidence: constraining a path to stay inside the gamut costs less of the metric's own length than the relaxation's own noise, on every gradient tested — including the ones whose free shortest path is outside at fifteen of its twenty-one steps. The gamut's boundary and the metric's cheap region point the same way, and the price of the constraint is nothing.

matching · Gamut Every pair of slits, over 68 notches. The colour error, in ΔE₀₀ from the truth, for every pair of slit widths — the lamp's table blurred through the width down the side, the sample's through the width across — on 68 notches, the mean over all of them under a fluorescent tube. Circle area follows the error. The best pair is 5 nm on the lamp and 1 nm on the sample, at 0.26; the best single slit, on the diagonal, is 5 nm at 0.36. One slit on the reflected light, at 5 nm, averages 0.016.

A second slit buys a quarter

A line lamp wants a five-nanometre slit and a notched sample a one-nanometre slit, so an instrument with a slit for each should do much better than one with a single compromise. Over sixty-eight notches under a fluorescent tube, it does better by 28 per cent. One slit on the reflected light does fifteen times better than the best pair, and an oracle choosing the best pair for every notch is still six times worse. The error was never that the factors were flattened; it was that they were flattened separately.

light · Light 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 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 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

Named alongside it

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

BasisSpecificationChromatic adaptationCIELABOptimisationCAT16Display gamutMacAdam's ellipsesPerceptual uniformitySpectral structureThe von Kries transformWavelength grid

All concepts