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

Just-noticeable difference — where it appears

The smallest change an observer detects reliably, usually defined at a stated proportion of correct responses. It is a threshold rather than a unit: differences well above it do not add up in units of it, which is what separates a threshold from a scale.

Named by 19 essays across 6 fields — each of them below, with the objects they name alongside it.

Three colour-difference formulae, disagreeing. ΔE76, ΔE94 and ΔE2000 for the same 9 pairs of colours. The largest disagreement between ΔE76 and ΔE2000 here is 26.6 units — larger than the threshold usually quoted for a just-noticeable difference, so the choice of formula can decide whether two colours count as matching.

How far apart are two colours

ΔE is meant to be a distance with the property that the same number means the same perceived difference everywhere. Three successive formulae have tried, they disagree with each other by more than a just-noticeable difference, and the disagreement decides real matching questions.

difference · Metric MacAdam's discrimination ellipses, drawn 10 times actual size. Twenty-five ellipses of colours indistinguishable from their centres. They are drawn at 10× because at true scale most are thinner than a line. Their areas vary by a factor of 74, which is the whole result: a step of the same size in xy means very different things in different places.

MacAdam measured it

In a perceptually uniform space the just-noticeable-difference contours would be circles of equal size. MacAdam's ellipses are neither, by a factor of eighty — and transforming them into each candidate space settles which spaces improved matters and by how much.

difference · Metric Threshold and suprathreshold contours, normalised to the same size. At five of MacAdam's centres: the measured just-noticeable-difference ellipse in grey and the ΔE2000 = 1 contour in gold, each scaled to the same mean radius so that only shape and orientation are being compared. A scale change preserves orientation exactly, so any rotation between the pair settles the question. They differ by 24° on average and by 70° at worst, and the ratio between their sizes varies 4.8-fold across the diagram — so no single factor turns one into the other.

A threshold is not a unit

MacAdam measured the smallest difference anyone could detect. ΔE2000 was fitted to how far apart plainly different colours look. The two are quoted interchangeably, and the contours they produce are not even the same shape.

difference · Metric Distinguishable colours in sRGB, counted under two difference formulae. The gamut volume divided by the volume of a ΔE = 1 ellipsoid, integrated over the solid because that ellipsoid changes size and orientation from place to place. Under the 1976 formula the answer is 195,720; under ΔE2000 it is 41,819 — 4.68 times fewer, from the same solid and the same lattice. Both assume perfect packing, which nothing achieves, so each is an upper bound rather than a count of anything. The gap between them is the result: "how many colours are there" is a question about a metric before it is a question about vision.

How many colours are there

Sixteen point seven million counts code values in a file format. Ten million distinguishable colours is a volume divided by the size of a just-noticeable difference — and the two difference formulae this site implements disagree about that size by a factor of nearly five.

difference · Metric The ΔE = 1 contour at points across the L* = 55 plane, magnified 14×. Each closed curve is the set of colours exactly one unit of difference from the dot at its centre, traced by bisection along 30 directions and drawn 14 times life size. Under ΔE2000 the contours run from 0.68 to 5.07 CIELAB units across this slice — a ratio of 7.44 — and they are not circles and not aligned with each other. A formula whose contours were circles of one radius everywhere would be claiming that CIELAB is uniform, which is what the 1976 formula claims and what the measurements refuse.

Where the formula is not smooth

A colour difference formula is a distance, and a distance ought to vary gently. CIEDE2000's does not everywhere — it carries a hue-rotation term with a hard edge in it, and the discontinuity sits where a great many industrial samples live.

difference · Metric A gradient of one hue, mapped into a press's gamut 2 ways. Chroma asked for along the bottom, chroma delivered up the side, for a ramp at lightness 55 and hue angle 25°. A colorimetric intent follows the diagonal until the press runs out and is flat afterwards — the flat part is a gradient arriving as a single colour. The perceptual intent is under the diagonal from the start, which is the price of never going flat.

No mapping preserves everything

More than half the sRGB solid is outside a press's gamut, so something has to be done with it, and there are two things that can be done. One leaves every reproducible colour exactly where it was and delivers a gradient as a flat area. The other keeps the gradient and moves every colour that needed no help by about three.

applied · Delivery Photons caught by one cone in one integration time, under D65. Each curve is one cone class, counting isomerisations during a 0.1 s integration through a pupil that closes as the light rises. At 1000 cd/m² a long-wavelength cone catches about 11522 and at 0.001 it catches 0.12 — which is where the square root of the count stops being a small correction and starts being the signal. The rate works out at 30 isomerisations per second per troland, inside the published band of 5–50.

Colour goes first in the dark

A cone reports a count, and a count carries the square root of itself as noise. Counting the photons says where colour vision stops — a chromatic difference runs out four hundred times sooner than a lightness difference of the same size — and says just as clearly that in daylight the eye is nowhere near that limit.

eye · Cones How often two colour-difference formulae disagree about which pair is worse. Pairs of colours sampled in CIELAB, compared two at a time. A rank inversion is a case where one formula calls pair A worse and the other calls pair B worse; no monotone rescaling of either can remove one. The left bar of each group is the rate over the whole space and the right bar is the rate among pairs sitting near a tolerance of ΔE 1, where the decision is actually made — and it is between 35 and 44 per cent, against a coin flip at fifty.

Which of two is worse

Two colour-difference formulae disagree about which of two pairs is the larger difference in thirteen per cent of comparisons overall — and in forty-three per cent of comparisons among pairs sitting near a tolerance of one unit, which is where every acceptance decision is actually made.

difference · Metric One colour difference, at four places in the visual field. The same pair of colours — ΔE00 22.0 at the fovea — with each of its three components divided by that channel's own threshold scaling at the stated eccentricity. What is left at 20° is 4.4, and its hue has turned by 26 degrees, because the red–green part is divided by more than the blue–yellow part. The swatches are the predicted colours, drawn where a reader will look straight at them; the figure states a prediction it cannot stage.

A difference has no place

A colour difference formula answers a question about two patches somebody is looking straight at. Move the same pair ten degrees into the periphery and a third of it is left — and it has turned twenty-four degrees of hue, because the three channels give out at three different rates.

difference · Metric What a dither mask is worth, read as components, in two dimensions. Five luminance ramps, each quantised to 8 bits with and without a high-passed mask of the same power. The bars are the most visible single sinusoidal component of the error, as a multiple of the contrast that component needs to be seen: above the line at one it is visible. The mask lowers it by 20–22×, on every ramp — which the one-dimensional model on this site says it does not, and that disagreement is the finding.

Every threshold was measured with a grating

An earlier essay here claimed that the model cannot explain why dither works, and named two missing pieces. One of them was real and worth thirteen times the guess; the other was not needed. The piece nobody named was the detector — and reading the same model two ways changes the answer by a factor of fifty.

limits · Limits How large a step changes the name, across the ab plane at L* 60. At each point, the smallest ΔE00 step in any direction after which the probability of two people using the same word has halved. It runs from 4.8 to 33.8 units across this one plane, in eight quantised levels: the palest cells are where a name is finest — a short step changes it — and the strongest are the middles of large territories, where a colour can move twenty units and keep its word. The ragged edge is the sRGB boundary at this lightness rather than a property of the vocabulary. The boundary softness is a stated parameter of the model, and the map barely moves when it is changed fourfold, because what sets this quantity is how far apart the centroids are.

A name is not a threshold

Two colours have to move about ten times a just-noticeable difference apart before people stop calling them the same thing, and how far varies threefold across one plane of the space. A tolerance and a word are answering different questions, and nothing in colorimetry converts between them.

brain · Appearance MacAdam's ellipses, drawn on one diagram. The twenty-five measured discrimination ellipses at 10× actual size, on CIE xy (1931). Mean axis ratio 2.95 — one would mean every contour is a circle — and a size spread of 10.42 between the largest and the smallest. Both numbers depend on the plane, which is why the 1976 revision existed; neither can be taken to one, which is why the revision did not finish the job.

No diagram makes them circles

Every chromaticity diagram is a projective picture of the same measurement, so how badly MacAdam's ellipses fail to be circles can be minimised over the whole family of them. The best plane there is still leaves the average ellipse twice as long as it is wide — which makes the residual a fact about the eye rather than about anybody's choice of primaries.

difference · Metric Three numbers for one set of ellipses, and which of them is which. Two curves and a horizontal line, against the size the ellipses are drawn at. The line is the analytic axis ratio — the ratio of the singular values of the map's own derivative, which is what "does this space make discrimination contours circles" means. The upper curve is a very finely sampled ring, which sits 0.6 per cent above the line at full size and converges onto it as the ellipse shrinks, because the gap between them is the second-order distortion of the map across a real ellipse rather than an error. The lower curve is the forty-eight-point sample used for this until now: it does not converge onto anything, because its error is set by the sample and not by the size.

Three numbers for one ellipse

How far a colour space is from making a discrimination contour circular has three different answers — what a coarse sample of the boundary reports, what a converged sample of a contour of stated size reports, and what the map's own derivative says. They differ by up to a factor of two, they mean different things, and only one of them is what the question is asking.

difference · Metric Where on the scale the units disagree. The reference pairs split into bands by how far apart they are in ΔE2000, with each unit's root-mean-square relative departure from the published one plotted per band. Every unit is calibrated once, over the whole sample, so a band is not refitted and the shape is the effect rather than an artefact of fitting. Every one of the five falls: the disagreement is proportionally largest on the pairs that are closest together, which is the opposite of what being fitted to threshold data would suggest. The appearance unit is the extreme case, at 91 per cent on the narrowest band and 17 on the widest, because CAM16-UCS raises its distance to the power 0.63 and a power below one inflates small differences against large ones. In absolute terms every curve here runs the other way — the widest band disagrees by 1.16 to 2.37 ΔE₀₀-equivalent against 0.14 to 0.68 on the narrowest — so which reading is right depends on whether the published quantity is a level or a ratio. This is the mechanism behind the census's own behaviour, where the mildest rows spread furthest across the menu.

The disagreement is at the near end

Every colour-difference formula on the menu was fitted to threshold data, so the expectation is that they agree about pairs an observer can only just tell apart and diverge on large differences. They do the opposite. Proportionally the disagreement is largest at the near end, by a factor of six for the appearance unit, and the cause is an exponent of 0.63.

difference · Metric MacAdam's twenty-five ellipses, measured in each unit. The uniformity instrument used here, applied to units rather than to spaces. The upper bar is anisotropy — the mean over the twenty-five of the largest radius divided by the smallest, where 1 would be a circle. The lower is spread — the largest mean radius divided by the smallest across all twenty-five, which asks whether a step of the same size means the same thing in different parts of the diagram. Reading down the three CIELAB-based formulae in the order they were published, the anisotropy falls 3.42 → 2.89 → 2.74 and the spread rises 3.24 → 3.59 → 4.12: the weighting divides a difference by the chroma it was measured at, which equalises directions at a point and unequalises magnitudes between points. Neither number is scaled, so no calibration is applied here. CAM16-UCS is ahead on both.

A unit rests on a space that was ranked

This collection ranks three colour spaces by how nearly they make MacAdam's ellipses circles, and CIELAB comes last. It then publishes every difference it computes in a formula built on CIELAB. Turning the same instrument on the formulae rather than the spaces shows the repair works — and that it buys roundness by paying in evenness.

matching · Gamut Pairs built to sit exactly on a ΔE2000 tolerance, read in every other unit. Twenty-four pairs of surfaces, each constructed by walking one member along a fixed direction until the difference is exactly 1.0 ΔE2000 under D65. The bar spans what those same pairs read in each unit, after calibration, with the tick at the mean. ΔE2000's own row is a point at 1.0 by construction. Every other unit spreads them: ΔE*uv reads them from 0.69 to 1.75, so a contract written at "one unit" accepts and rejects a different set of deliveries depending on which unit it means. CAM16-UCS rejects all twenty-four: it reads the closest of them at 1.43.

A tolerance is a boundary through pairs

Twenty-four pairs of surfaces built to sit exactly on a ΔE2000 tolerance of one read from 0.69 to 1.93 in the other five units. A contract quoting "one unit" without naming the formula does not become slightly wrong in another; it accepts and rejects a different set of deliveries, and in one case rejects every single one.

applied · Delivery The length of one grey ramp, cut into more and more steps. A neutral ramp from L 1 to L 100 cut into between one and ten thousand equal steps, each step measured and the steps added, in four units, both axes logarithmic. ΔE*ab and the model's Euclidean J′a′b′ give the same length at every step count, 99 and 96. ΔE₀₀ settles at 74.6 once the steps are small. The power-corrected ΔE′ does not settle: 25 in one step, 137 in a hundred, 755 in ten thousand, growing as the number of steps to the power 0.37.

A distance raised to a power has no length

CAM16-UCS's colour difference is its Euclidean distance raised to the power 0.63 and multiplied by 1.41. That is still a metric — the triangle inequality holds on every one of four thousand random triples — and it has no length. A grey ramp from black to white measures 25 units in one step, 137 in a hundred and 755 in ten thousand, growing as the number of steps to the power 0.37, and halving the size of a step triples the number of steps that fit.

difference · Metric What is left of one colour difference when the two colours alternate. 23 pairs built at exactly ΔE₀₀ 1.000, alternated at a rate, with each part of the difference scaled by its own temporal channel and the formula then applied unchanged. At rest every pair is the flat line at one. By 7 hertz the median is 1.11 and the pairs run from 0.57 to 3.04 — a factor of 5.3 between pairs the formula calls identical. By sixty hertz the largest of them is 0.15.

A difference has no rate

A colour difference formula answers for two patches that are both there and stay. Alternate the same two colours and the difference is not scaled but taken apart: the colour half is gone by fifteen hertz and the lightness half is four times louder at eight, so twenty-three pairs the formula calls identical run over a factor of six at the rate the eye is best at, and are worth nothing at all above sixty.

difference · Metric Twenty-three pairs at one ΔE₀₀, read in two units that know the light level. Twenty-three pairs of surface colours, each exactly one ΔE₀₀ apart, on a display whose white runs from 1.5 to 10,000 cd/m² across, with a background at a fifth of the white. ΔE₀₀ has no argument for the light and stays at one. The median ΔEITP rises from 1.01 to 2.75 and flattens near the top, and the median CAM16-UCS distance from 0.76 to 1.20.

Two units with a light level disagree about lightness

ΔE₀₀ has no argument for how bright a display is. Two colour differences do: CAM16-UCS takes the room's adapting luminance, and ΔEITP — the difference defined for high-dynamic-range television — takes the stimulus's own absolute luminance. Twenty-three pairs at exactly one ΔE₀₀ grow in both as the display brightens, 2.1 times in ΔEITP and 1.5 in CAM16-UCS from a 5 to a 5,000 cd/m² white. But in ΔEITP the lightness part grows fastest, 2.7 times, and in CAM16-UCS it does not grow at all.

difference · Metric

Named alongside it

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

CIEDE2000ThresholdΔEToleranceMacAdam's ellipsesCIELABPerceptual uniformitySpecificationChromaColour differenceAppearance modelCalibration

All concepts