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

CIELAB — where it appears

A space built to make equal distances mean equal differences, in which they nearly do — and a space with no room in it. Its cube-root lightness and opponent axes were a compromise fixed in 1976, and the ellipses it was meant to make circular are still ellipses.

Named by 51 essays across 8 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 The sRGB transfer function, and the gamma 2.2 curve it is not. Code value against relative luminance. The sRGB function is piecewise — a short linear segment near black, then a 2.4 power law with an offset — and it is close to but not the same as a plain 2.2 power law. Half-way along the axis of stored values sits at 21 per cent luminance, and half the luminance of white is at code 188.

The midpoint is not half

Code 128 sits halfway along the sRGB scale and carries about a fifth of white's luminance. Half the luminance is code 188. Almost every gradient, blur and resize on the web gets this wrong, and the errors are visible once known.

difference · Metric One stimulus, three rooms, three appearances. The same XYZ in a dark, a dim and an average surround. The stimulus does not change and is drawn identically in all three panels; what changes is what CIECAM16 says it looks like. Predicted lightness runs from 65.6 to 57.5 — a spread of 8.1 — with chroma and colourfulness moving too. Colorimetry returns one answer here because it has nowhere to put the room.

A viewing condition is an argument

An appearance model takes a stimulus and a situation. The second argument is not a refinement of the first — it is the content of the claim, and a model that returns an appearance from a colour alone has assumed a room without saying which.

brain · Appearance A box tolerance and a ΔE tolerance around the same colour. A slice through CIELAB at L* 50, with the ΔE2000 = 1 contour traced point by point and a ±1 component box drawn over it. The contour is 2.1 times longer in one direction than the other, and the box is square. Of every sample either rule accepts, the two disagree about 74% — accepted by one specification and rejected by the other, on the same measurement.

A tolerance is a shape

A total colour difference below one, or every component within one — the two sound like the same requirement stated twice. They are different shapes, they disagree about most of what either accepts, and which one a supplier is held to is worth money.

difference · Metric The 1976 uniform chromaticity diagram, with the unreachable region marked. The u′v′ diagram: the same spectral locus and the same sRGB primaries as the 1931 picture, projectively transformed. Straight lines stay straight, so mixtures and the gamut triangle survive; what changes is the distribution of area, and the green region that dominates the 1931 diagram is much reduced. 8% of the cells sampled inside the locus are reachable at Y = 0.55; the rest are hatched.

The diagram was replaced in 1976

The CIE knew the 1931 diagram was badly distorted and published a better one. Half a century later almost every chromaticity plot in print is still the old one, and both are still printed filled edge to edge with colours no display can show.

matching · Gamut One palette under normal vision and two dichromacies. The same 7 colours simulated by the Brettel–Viénot–Mollon construction at severity 1.0. protanopia and deuteranopia collapse the red-green distinctions. This shows which discriminations survive, not what anybody sees.

Three cones, two axes

The retina does not send three receptor signals down the optic nerve. It sends a sum and two differences, and the reason falls out of the statistics of natural light rather than out of anything the eye intended.

eye · Cones Three lightness scales, seventy years apart. Munsell value, CIELAB's L* and CIECAM16's J against luminance, all rescaled to run 0 to 100. Each is somebody's answer to how evenly spaced lightness steps map onto light. They put the midpoint of the scale at 19.8%, 18.4% and 28.0% of the white's luminance respectively — close enough to be three measurements of one thing, far enough apart to be three measurements rather than one restated twice.

Colour by catalogue

A colour order system arranges physical samples on a regular lattice so a person can find one and name it. The space it samples is not regular, and everything interesting about such systems is what happens at that mismatch.

matching · Gamut The four unique hues, and the axes they are said to define. A constant-lightness, constant-chroma ring in CIECAM16, with the four unique hue anchors marked and CIELAB's a and b axes drawn through the same circle. If a* really were the red-green axis the anchors would fall on the crosshairs. Unique red sits 25° off, and the four are not 90° apart in any case. Hatched sectors are hues this display cannot reach at this chroma.

Why there are four unique hues

Observers agree that four hues are elementary and that no colour is reddish green. Nothing in the three receptors predicts either fact, and the axes of every standard colour space miss the four by tens of degrees.

brain · Appearance Chromaticity-triangle area against CIELAB volume, both relative to sRGB. Two ratios for each space, both against sRGB. The upper bar is the area of the primary triangle on the CIE 1931 diagram; the lower is the volume of the gamut solid in CIELAB, computed by tetrahedral decomposition of the RGB cube at 24 cells per axis. Display P3 is 1.36× sRGB by area and 1.50× by volume; Rec. 2020 is 1.89× sRGB by area and 2.26× by volume. The chromaticity diagram divides luminance out, so its triangle is a projection along the axis the eye is most sensitive to — and a coverage percentage quoted on it is a statement about the shadow.

The triangle is a shadow

A display's gamut is drawn as a triangle on the chromaticity diagram, and coverage is quoted as a percentage of that triangle's area. Chromaticity has luminance divided out, so the triangle is a projection along the axis the eye cares most about — and the ratios computed on the solid are not the ratios computed on its shadow.

limits · Limits 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 The same medium at six path lengths. Beer's law at 6 depths of one absorbing medium. The absorption coefficient is a single spectrum and the only thing changing is how far the light travelled, yet the patches differ in hue by 7.8° as well as in lightness — because absorption is exponential in depth and the observer is linear, so the bands that survive at d = 8 are not a scaled copy of the ones that survive at d = 0.25. Path length belongs to the geometry, not to the substance, which is why this sits in a field about scenes.

The colour is in the thickness

Transmittance is exponential in path length and the observer is linear, so doubling the depth of an absorbing medium squares the transmittance rather than halving the colour. A translucent object therefore has no one colour — its thin edge and its thick middle are different spectra of the same substance, and the hue moves between them.

scene · Scene A clipped channel turns the hue of what is left. CIELAB hue shift against exposure for one saturated stimulus, measured against the same stimulus rendered without clipping. Nothing moves until the first channel reaches the ceiling at 0.25 stops; after that the recorded hue rotates by as much as 67 degrees, with nothing in the scene having changed colour.

A blown highlight turns

An exposure that clips nothing changes no hue at all. Once one channel reaches the ceiling the recorded hue rotates by as much as sixty-seven degrees, with nothing in the scene having changed colour — and every response a converter can make to that is an invention.

imaging · Capture What a press and a display can reach at L* = 50. A slice through both solids at lightness 50, with the press dashed and the display solid. The boundaries cross: the press reaches past sRGB in 15 of 48 directions and falls inside it in 31. That is the shape of every conversion between them — colours are lost in one direction and gained in the other, and the picture cannot show the gained ones, because it is being displayed on the gamut that cannot reach them. Those are hatched.

Neither gamut contains the other

Converting a picture to CMYK is described everywhere as a reduction, as though the press were a smaller version of the screen. Measured as solids, the press reaches eleven per cent of its volume outside sRGB while sRGB reaches fifty-four per cent of its volume outside the press — and in the cyans the press wins by seventy per cent of chroma.

applied · Delivery One pair of colours, blended in four spaces. Every row starts and ends at the same two colours and visits different ones in between. The chroma of the middle falls furthest in linear light, by 83 units below the straight line between the endpoints' chromas — the grey dead zone that every blue-to-yellow gradient has and that no endpoint mentions. Any colour a route visits that this display cannot show is hatched rather than clipped.

A gradient is a path

Two colours fix the ends of a blend and nothing else. The middle is decided by the space the interpolation happens in, and four spaces in daily use put the halfway point of one ordinary gradient as much as thirty-seven units of colour difference apart.

matching · Gamut The worst triple found, under ΔE2000. Three colours, plotted on the a–b plane of CIELAB. Going from a to c directly is 123.645; going via b is 60.528, which is 51.0 per cent shorter. A distance cannot behave that way, and this one is the formula every colour tolerance in industry is written in. The colours are far apart, which is where the violation is largest; the same search confined to tolerance scale finds a smaller one that has not gone away.

A difference is not a distance

Two earlier essays here said that CIEDE2000 violates the triangle inequality and left it at that. Searching for the violation finds a detour half the length of the direct route, a smaller one inside a five-unit ball, and a second defect nobody mentions — ΔE94 is not even symmetric.

difference · Metric 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 The eleven basic colour terms, at their quoted centroids. Each patch is the CIELAB centroid quoted for that term, converted to a stimulus and drawn — except blue, whose focal colour is outside the sRGB gamut and is therefore hatched rather than clipped, which is the rule for an unreachable colour everywhere else and applies here too. The centroids are rounded to 5 units in each coordinate, and moving them by that much either way leaves the same term outside.

A colour has a name

Every quantity here is a number, and the question a reader arrives with is what colour something is called. The eleven basic terms of English divide the space into territories that differ by a factor of five, the metric accounts for fifteen per cent of that, and one of the eleven focal colours cannot be shown on this page at all.

brain · Appearance A colour-order system's chips, sorted by what they would be called. A regular lattice in lightness, chroma and hue — the idealisation of a swatch book, and regular by construction because a person has to be able to find a page. Named, it comes apart: 1320 chips inside the gamut divide into 198 for grey and 66 for blue. And 22 per cent of one-step moves in the lattice change the name, so a page of a swatch book is not a page of a vocabulary.

A catalogue is not a vocabulary

A colour order system is a regular lattice, because a person has to be able to find a page. Sorted by what its chips would be called, that lattice comes apart into piles differing by a factor of three — and a fifth of all one-step moves in it change the name.

matching · Gamut The hue circle cut into names, at L 60 and C 40. Left, the arcs each name claims, drawn at the colour of their midpoints; right, the same arcs measured in ΔE00 by integrating the difference along the ring rather than in degrees. The widest is 5.3 times the narrowest in degrees and 4.5 times in colour difference, so the metric accounts for 15 per cent of the inequality and no more. Only the eight chromatic terms compete on this ring: at this chroma the achromatic three would otherwise take the region where no basic English term sits, which is a defect of the model and is named in the essay.

There is no word for that colour

The naming model built this phase gives a saturated cyan the name green, calls part of a chromatic ring grey, and puts one of the eleven focal colours outside what a display can show. Each failure is a measurement rather than a disclaimer, and together they say what a vocabulary is that eleven points and a distance are not.

limits · Limits How much of the gamut changes name, and what changed it. The eleven basic terms are quoted as centroids in CIELAB, and a colour is named by which one it is nearest. Two things nobody records decide the answer. Changing the distance function renames 20.5 per cent of the displayable gamut. Changing the room — the same colours, the same words, a different surround, through the appearance model — renames up to 26.8 per cent. The centroids were measured in one viewing condition and are applied here in every essay as though a name were a region of a space with no room in it.

A name moves with the room

Eleven basic colour terms are quoted as centroids in CIELAB and a colour is named by which one it is nearest. Two things nobody records decide the answer — which distance function is used, which renames a fifth of the displayable gamut, and which room the colour is in, which renames more than a quarter of it.

brain · Appearance How much of what a display can show each name owns. Every point on a 5-unit CIELAB lattice inside the sRGB gamut is given to its nearest centroid under ΔE00, and the shares counted. They run from 21.1 per cent for purple to 4.6 for blue, a factor of 4.6. The three terms that carry no chroma at all — black, grey and white — hold 20 per cent between them. A share here is a statement about the names and about the gamut they are counted over, and the gamut is sRGB.

A name in the model's own words

The eleven basic colour terms move when the room does, and the obvious objection is that they were being measured in a space with no room in it. Quoting them in an appearance model's own coordinates instead does not shrink the renaming — it grows it by four per cent — and the space alone renames an eighth of the gamut with the room held still.

brain · Appearance A radiance factor, split into the part that was reflected and the part that was not. The two components of what leaves a heavily brightened sheet under M₁ — D50 including its ultraviolet. The lower band is the reflected component, which is what a reflectance curve means and is everything a reflectance-based model can hold. The band above it is light emitted at wavelengths it did not arrive at, and its total is decided by how much ultraviolet the source had rather than by anything about the sheet's colour. The line at one is the boundary a reflecting surface cannot cross; the sum reaches 1.21 at 430 nanometres.

The eye weights where the light is not

A brightened sheet returns a quarter more light than arrives at 430 nanometres, and it is three tenths of one per cent brighter for it. The luminous efficiency function is 0.017 there against 1.0 in the middle of the band, so the whole effect lands in the blue-yellow axis — brighter than white is a colour claim wearing a brightness word.

matching · Cones How much of a whiteness figure is the sheet, and how much is the lamp. Each bar is how many points of CIE whiteness a stock has above an unbrightened sheet of the same base, measured with an ultraviolet-included instrument. The dark part is what survives when the ultraviolet is removed — the part that is a property of the paper. On average 82 per cent of the scale is the pale part, which is a property of the instrument's lamp. A whiteness figure without a measurement condition beside it is therefore not a measurement of a sheet; it is a measurement of a sheet and a lamp, quoted as though it were the first.

Whiteness is mostly the lamp

The CIE whiteness formula ranks white samples the way people do, which is what it was built for and is not in question. What is in question is what it is a measurement of — take the ultraviolet out of the instrument and 82 per cent of the scale collapses, because the part that separates a premium sheet from an ordinary one was contributed by the lamp.

difference · Metric 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 The same formula, applied after six different changes of basis. CIELAB's arithmetic — divide by a white, take a cube root, difference the results — run on six of the bases the matching data leave free. A linear change of basis leaves every match alone; a cube root does not commute with one, so the space, and therefore every colour difference computed in it, depends on which basis was in place before the nonlinearity. CIELAB's own choice gives an axis ratio of 3.44 and the best row here is LMS (confusion points) at 2.60.

A difference needs a basis too

A linear change of coordinates leaves every colour match exactly where it was. A cube root does not commute with one — so a lightness–chroma space, and every colour difference computed in it, is a property of the basis that happened to be in place before the nonlinearity. CIELAB's basis was chosen in 1931 for reasons that had nothing to do with difference.

difference · Metric Which of this collection's own claims survive a change of basis, and which are about the paper. Nine sentences this site says, sorted by whether they mean the same thing after the observer's three curves are replaced by a nonsingular combination of themselves. 5 of the nine do. The four that do not are not thereby wrong — they are statements about a chosen set of coordinates, and they are true of those coordinates. What they cannot be is statements about the eye, which is how every one of them is usually read.

Which of these is a convention

Nine ordinary sentences from this collection, put through one test — do they mean the same thing after the observer's three curves are replaced by a nonsingular combination of themselves? Five survive and four do not, and none of the four is wrong, because each is a statement about a set of coordinates being read as a statement about an eye.

limits · Limits 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 Five answers to how far the ellipses are from circles. Five mean axis ratios on the same twenty-five measured ellipses, measured the same way in every row: the boundary points carried through, the longest radius over the shortest, averaged. What differs is which class of map is allowed. The first two rows are chromaticity diagrams, which divide by a sum; CIE xy as printed leaves 2.95 and the best diagram there is leaves 2.02. The last three are lightness–chroma spaces, which divide by a white point; CIELAB as specified leaves 3.44, the best space with no compression leaves 2.33, and the best space with a cube root in it leaves 1.61. Neither family contains the other, and only the last one gets below two.

A compression goes below the floor

Elsewhere this collection minimised the anisotropy of MacAdam's ellipses over every chromaticity diagram there is, found 2.02, and called the residual a property of the eye. It is a property of the eye seen through a projective picture. A cube root after the right basis reaches 1.61 on the same twenty-five ellipses.

difference · Metric 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 The minimum sits in a notch the width of the answer's reciprocal. The distance from the centre to the boundary, all the way round one MacAdam ellipse mapped into a lightness–chroma space built on the CIE RGB primaries. The curve has two broad maxima and two very narrow minima: the dip is about 2.5 degrees wide at a third above its floor, because the width of the minimum of an ellipse's radius is the reciprocal of its axis ratio, and this ratio is 41. Forty-eight sample points, marked, are spaced 7.5 degrees apart, so none of them lands in either notch and the smallest one found is 2.2 times the true minimum. The ratio comes out 18.59 where it is 40.76.

An ellipse is not a ring of points

For eleven rounds of argument the distortion a colour space does to MacAdam's ellipses by mapping forty-eight points round each one and dividing the longest radius by the shortest. The minimum sits in a notch whose width is the reciprocal of the answer, so the method was accurate wherever the answer was small and short by a factor of two where it was large.

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 How wrong the ellipses would have to be for a pair to change places. One bar per adjacent pair in the uniformity table: the relative error on each ellipse's own axes at which that pair changes places in one draw in twenty. No error on the data is quoted anywhere — the question is inverted, so what is reported is how large an error would have to be, and a reader with an opinion about MacAdam's experiment can compare it with their own number. The nearest pair goes at 0.171; 2 of the 7 pairs do not reverse under any error this search covers.

How wrong would the data have to be

Twenty-five ellipses measured on one observer in 1942 are the ruler every colour space here is judged against, and they have never been given an error. Rather than invent one, the question is turned round, and asks how large an error would have to be before the ranking changed.

difference · Metric Twenty-five ellipses is a sample, and the score has an error bar. One row per colour space this collection ranks: the mean axis ratio its ellipses come out at, with the standard error of that mean over the twenty-five ellipses it was computed from. No literature is quoted — a mean of twenty-five numbers has a standard error those twenty-five numbers determine. The bars are far from equal: the best space carries ± 0.07 and the worst ± 1.56, because a space that makes the ellipses nearly circular makes all of them nearly circular and one that does not is dominated by whichever ellipse it handles worst.

Twenty-five is a sample of the diagram

A colour space's uniformity score is the mean of twenty-five numbers, and a mean of twenty-five numbers has a standard error those twenty-five numbers determine. Nothing has to be quoted to compute it, and three of the seven adjacent pairs in this collection's ranking survive it.

difference · Metric One surface dimmed sixteen times, and the two things that happen to it. A single surface, dimmed by successive halvings, with the lens age departure measured on it at every level. The tristimulus deviation falls by exactly the dimming factor — 16 times over the sweep, to the last bit, because the colour integral is linear in the stimulus. What a unit of that deviation is worth rises by 8.2 times over the same sweep. The colour difference the audit reports is the product of the two, and it falls by only 1.95.

A deviation is not a difference

The previous round priced twenty departures in colour differences and treated each number as a property of the thing that departed. Every one of them is a product of two factors — how far the reading moved, which is linear and belongs to the departure, and what a unit of that movement is worth where it landed, which is not linear and belongs to the colour. Dimming one surface sixteen times scales the first by exactly sixteen and the second by eight.

matching · Gamut The straight piece under the cube root, and where it stops. CIELAB's lightness against relative luminance, over the bottom 5.0 per cent of the range. Below Y = 0.008856 it is a straight line of slope 7.787; above it, a cube root. The two meet at L* 8 in value and in slope, exactly — the CIE's two constants are chosen to make that true. The dashed curve is the pure cube root, which reaches negative lightness before it reaches zero luminance and has an infinite slope there. The break is marked, and the axis runs to Y = 0.050.

The straight piece under the cube root

CIELAB's lightness is described everywhere as a cube root and below a stated luminance it is a straight line, spliced on with two constants chosen so the join is exact in value and in slope. Every black a delivery chain reaches is inside that straight piece — a press black at L* 2.4, a projected black at 1.1 — where the compression does not compress, the price of a deviation is flat to two parts in a thousand, and the second derivative the composition of two departures needs does not exist.

matching · Gamut Two primaries mixed, and the line a reader assumes they take. The additive mixture of two sRGB primaries, walked in twenty steps, plotted in the a and b of CIELAB. The filled points are where the light actually goes, which is exactly straight in tristimulus values because that is Grassmann's second law. The open points are the straight line between the two readings. They part company by 26.7 ΔE₀₀ at their furthest, at 60 per cent of the way along, and the half-and-half mixture misses the midpoint by 21.1.

The mixture line bows

Grassmann's second law says an additive mixture is exactly linear in tristimulus values, and tested here it is exact to floating point. Nothing downstream of the three numbers preserves it. The physical half-and-half mixture of two colours sits a median of 5.5 colour differences from the midpoint of their two readings and up to thirty; on a green and a blue display primary it is twenty-one, which is a quarter of the distance between them.

matching · Gamut A hue circle through a per-channel curve. 28 colours on a circle of constant lightness 55 and chroma 38, each put through the tone curve one channel at a time and read back. The curve is a function of a single number and has no idea what hue is, and it rotates the circle by up to 4.4 degrees — largest at hue 260 — while raising chroma by a factor of 1.27 and lightness by about 1 units.

A contrast control is three controls

A tone curve is a function of one number at a time and knows nothing about hue. Applied to each channel separately it rotates a hue circle by up to seventeen degrees, raises chroma by a factor of 1.27 at ordinary strength, and lifts lightness — so a photographer who moves a contrast slider has moved three things and the interface names one of them. All three scale with the curve's strength, monotonically, and the hue rotation depends on which hue it is.

imaging · Capture Which way each tolerance is tightest, colour by colour. Sixty-four colours on a lattice through the sRGB cube, with lightness across the bottom. For each, the tolerance ΔE₀₀ 1 is pulled back into tristimulus values and its shortest and longest axes are found. The filled points are the angle between each colour's shortest axis and the direction common to all of them: a median of 15 degrees, ninety per cent within 32. The open points are the same for the longest axis, at 23. The common short axis is 3.1 degrees from X up and Y down, which is the direction of a*.

A tolerance has a grain

A colour tolerance pulled back into tristimulus values is a long thin shape, and across the whole sRGB cube its shortest axis points the same way, fifteen degrees off at the median — the direction in which X rises as Y falls, which is the direction of a*. So what one colour difference allows depends on which way a delivery drifts. An instrument's X filter may age 1.4 per cent before the tolerance is used up, its Z filter 5.1, and the light on the sample 9.3.

matching · Gamut What a small neutral deviation costs as the grey darkens, in two lightness scales. The price of a fixed fraction of deviation on a neutral — how much lightness it is worth per unit of luminance — from L 40 down to L 0.25, both axes logarithmic. CIELAB's L is a straight line below L 8 and its price there is exactly flat. The appearance model's J′ has no straight piece: its price keeps rising, by a factor of 5.0 across the same range, as the luminance to the power -0.442 — the derived exponent is -0.441 in an average surround.

The appearance model has no straight piece

CIELAB's lightness is a straight line below L* 8, so a deviation near black has a fixed price and a black has a floor under it. CIECAM16 has no such piece. Its lightness near zero goes as the luminance to a power set by the room, the price of a deviation rises without limit as the black deepens — as Y to the power −0.44 in a lit room and −0.58 in a dark one — and on a projected black a lightness unit in the model allows under half the luminance change a unit of L* allows.

matching · Gamut The straight line between two colours, and the formula's own shortest path. Five gradients seen from above, in the a and b plane of CIELAB: the straight line between the two colours in grey, and the shortest path under ΔE₀₀'s own local metric in colour. The lightness coordinate bows too and is not drawn. red to green saves 3.6 per cent and leaves the straight line by 13; blue to yellow saves 7.5 per cent and leaves the straight line by 21; cyan to magenta saves 3.9 per cent and leaves the straight line by 10; black to white saves 0.0 per cent and leaves the straight line by 0; red to blue saves 0.3 per cent and leaves the straight line by 4 CIELAB units at the widest. Black to white is the flat case: its shortest path is the straight one.

The straight line is not the shortest gradient

A colour difference formula says what a small step costs at every colour, and that is enough to ask which path between two colours is shortest. It is not the straight line: between red and green the shortest path bows thirteen CIELAB units away, saves four per cent — and stays inside sRGB on every step where the straight line leaves it. The formula's own answer for the two endpoints, meanwhile, is neither length.

matching · Gamut Two uniform spaces, two answers: cyan to magenta. The gradient from above, in the a and b plane. The grey line is the straight path in CIELAB; the two coloured curves are the shortest paths under ΔE₀₀'s own local metric and under CAM16-UCS's. They are 15.7 CIELAB units apart at their furthest, against bows from the straight line of 9.9 and 12.2. Both spaces are published as uniform and both are used to decide what lies between two colours; they do not agree.

Two uniform spaces disagree about between

ΔE₀₀'s own local behaviour says which colours lie between two colours, and so does CAM16-UCS's. They do not agree. On cyan to magenta the two shortest paths run fifteen CIELAB units apart — more than either runs from the straight line — and on that gradient ΔE₀₀ would rather have the straight line than CAM16-UCS's answer. Getting the comparison at all takes noticing that one of the two has no local metric: its published difference is a Euclidean distance raised to the power 0.63, and a power below one has an infinite derivative at zero.

matching · Gamut 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 How far each model turns a display colour's hue as white is added. The twenty-four most saturated colours an sRGB display makes, one every 15° of HSV hue across, each mixed with the display's white at the same luminance down to a fifth of its purity. Up, how far that mixture's hue angle turns in CIECAM16, CIELAB and Oklab. From red to a fifth of its purity CIECAM16 turns −9.4° and Oklab −10.2°; CIELAB −18.1°. From the display's blue, Oklab turns +16.3°, CIECAM16 +3.6° and CIELAB −8.9°.

White turns a hue, and the models part at blue

Mix a saturated light with white and its hue changes as well as its saturation — the Abney effect, and the reason lines of constant perceived hue curve in a chromaticity diagram. Asked what adding white does to the twenty-four most saturated colours a display makes, CIECAM16, CIELAB and Oklab all turn the hue. About reds they agree: CIECAM16 turns within a degree of Oklab, whose hue was fitted to observers' constant-hue judgements. About the display's blue they do not: Oklab turns sixteen degrees, CIECAM16 four, and CIELAB nine the other way.

brain · Appearance Three uniform spaces, three answers: red to blue. The gradient from above, in the a and b plane: CIELAB's straight line, and the shortest paths under ΔE₀₀'s local metric, under CAM16-UCS's and under Oklab's, whose shortest path is its own straight line carried back into CIELAB. They bow from CIELAB's line by 14.4, 42.5 and 42.6 units. ΔE₀₀'s and CAM16-UCS's paths run 28.3 apart at their furthest, Oklab's runs 29.0 from ΔE₀₀'s and 5.6 from CAM16-UCS's, and the space furthest from the other two here is ΔE₀₀.

A third space breaks the tie only once

ΔE₀₀ and CAM16-UCS disagree about which colours lie between two colours. Oklab was the obvious tie-breaker, and it breaks the tie on one gradient of five: on red to blue its path runs within six CIELAB units of CAM16-UCS's and twenty-nine from ΔE₀₀'s. On red to green and cyan to magenta it keeps to CIELAB's straight line while both others bow, and on blue to yellow it bows further than either. Each of the three spaces is the odd one out somewhere. Asking also exposed a CAM16-UCS path that had never converged.

matching · Gamut Six starts for every held gradient, and where each one lands. Each row is one gradient held inside a gamut, relaxed from six starts: the straight line, the free shortest path, and the straight line bent towards and away from the neutral axis and up and down in lightness. Each dot is how much longer than the free shortest path that start's result is, on a logarithmic scale from a hundredth of a per cent to a thousand; the ring marks the best. The first 10 rows are gradients between colours on a coated CMYK press's boundary whose straight line leaves the press: their best routes cost a median of 0.12% and at most 1.1%, and on 5 of them some start lands at more than twice the free length. The next 3 are the display gradients held inside sRGB, whose best cost at most 1.3% and whose starts spread by at most 4.2%. The last 6 are press gradients that never leave, where no start is trapped.

A press makes the cheap route a search

Holding a gradient inside a display's gamut cost nothing measurable, and the essay that found it credited the gamut's convexity. A press's gamut is not convex: 216 of 630 straight lines between colours on its own hue ring leave it. Holding gradients inside the press still costs little at best, a median of a tenth of a per cent. But the best route depends on where the relaxation starts, the free path is the best start on one gradient of ten, and on half of them some start is trapped at more than twice the free length. On the display no start is trapped.

matching · Gamut 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 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 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.

Perceptual uniformitySpecificationCIEDE2000ΔEMacAdam's ellipsesGamutLightnessColour differenceToleranceThresholdCIECAM16Gamut mapping

All concepts