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

Every essay — page 9

Page 9 of 54, continuing through the fields in the same order.

What light is What the eye does Matching and measuring Difference and uniformity What the brain does What a scene does What a camera does Where the model breaks What it takes to deliver it

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Difference and uniformity

How far apart two colours are, whether the space you measured in was uniform, and how MacAdam's ellipses settle the question by measurement.

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.

7 figures 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.

7 figures 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.

7 figures 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.

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

A lattice is a quadrature rule

Walking a set of test surfaces more finely does not converge on a better answer, because refining a lattice under a constraint changes which corners of the region get sampled and not only how densely. The lattice used here turns out to be a two per cent biased estimate of the integral it stands for.

8 figures The error on a gap is not the two rows' errors added. Two bars for each of the 13 adjacent pairs in the census ranking. The upper, shorter bar is the standard error of the gap taken as a paired difference — the same 125 surfaces score both rows, so a surface that is awkward under one change of light is usually awkward under the other and the difference is quieter than either. The lower bar is the two rows' own errors added in quadrature, which is what comparing error bars by eye amounts to. Pairing is worth a factor of 1.78 on average and 3.36 on the pair it helps most, and it is the difference between 6 adjacencies unordered and 4. The gain is largest where the two rows are two daylights or two tungstens, because then the surfaces they find awkward are nearly the same surfaces.

The error on a gap is not the errors at its ends

Comparing two rows of a table by looking at whether their error bars overlap is the wrong comparison, and here it is wrong by a factor of up to 3.4. The same 125 surfaces score both rows, so the difference between them is quieter than either — and how much quieter is a measurement of how alike the two rows are.

8 figures How far each census row moves when the test set's own description does. A grid of bars, one row per change of light in the census and one bar in each row per number that describes the region the test surfaces are drawn from: how saturated they are, how bright, and how far the two modulations may go together. A bar's length is the elasticity — the proportional change in the published residual for a proportional change in that number. Saturation runs from 0.49 to 0.91 and brightness averages 0.104, so a test set's chroma range is nearly everything and its lightness range is nearly nothing. For scale, the largest elasticity found anywhere among this collection's five declared population widths is about a half — and those at least have declared ranges, while these three numbers have never been quoted with one.

Saturation is nearly everything

The set of test surfaces has three numbers describing it, and only one of them matters. How saturated the surfaces are carries an elasticity of about 0.7 on every result computed over them; how bright they are carries 0.10. A test chart's chroma range decides its answer and its lightness range does not.

9 figures How far each unit is from being a rescaling of the one this collection publishes in. One row per unit on the menu. The bar is the root-mean-square scatter about that unit's own best rescaling of ΔE2000, over 374 pairs of surfaces differing by a fraction of a unit to about ten. A bar of zero would mean the unit is ΔE2000 in different money — every printed number would change and no conclusion would. ΔE2000's own row is zero by construction and is the check that the table is computed the right way round. The two units that divide a chroma difference by the chroma it was measured at, ΔE94 at 15 per cent and CAM16-UCS at 24, are closer to it than the three that do not, which run from 28 to 35. The split is by weighting and not by whether the unit is a matching difference or an appearance one.

The weighting is the disagreement

Five colour-difference formulae, three decades and two committees, and the single property that predicts which of them agree is whether a chroma difference gets divided by the chroma it was measured at. It sorts the menu exactly, it cuts across the distinction between a matching difference and an appearance one, and it halves the census's largest sensitivity.

7 figures The census along the line from ΔE76 to ΔE94, and past it. ΔE94 is ΔE76 with two weighting constants in it, and at zero those constants make every weight exactly one, so the two formulae are joined by a line rather than separated by a choice. The horizontal axis is how much of the published weighting is applied: 0 is exactly ΔE76, 1 is exactly ΔE94, and 3 is three times more weighting than anybody has proposed. The falling curve is the census's mean elasticity to how saturated its test set is, which drops from 1.11 to 0.74 — most of the fall happening before the published value is reached. The other curve is Kendall's τ against ΔE2000's ranking, and it peaks at w = 0.5, not at 1: the weighting that best reproduces the published ordering is about half the published weighting. There is no value of this dial that reaches ΔE2000, whose rotation term is not on this line at all.

A dial through a discrete menu

ΔE*94 is ΔE*ab with two weighting constants in it, and at zero those constants make every weight exactly one — so the two ends of the oldest disagreement in colour difference are joined by a line rather than separated by a choice. Walking it gives a derivative where a menu gives only a spread, and the derivative says the published weighting is on the far side of the interesting part.

8 figures 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.

9 figures 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.

7 figures 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.

The departures are larger than the tolerance

A delivery tolerance is written around one ΔE₀₀ and every one of this round's four departures is above it on ordinary material. A specification that names an illuminant, an observer and a tolerance, and does not name a measurement condition, an aperture and a field, has written a number that two honest laboratories can miss each other on by more than the number itself.

7 figures