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

Every essay — page 7

Page 7 of 40, 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 best possible 3×3, and the patches it makes worse. Each row is one patch printed on a brightened sheet, measured under both conditions. The pale bar is how far apart the two measurements are; the dark bar is what is left after the best least-squares 3×3 over the whole set has been applied. It leaves 23 per cent of the mean, and — the part a mean hides — it makes 4 patches worse than doing nothing. The solids are the ones it damages: the ink blocks the ultraviolet, so a solid barely disagrees between the two conditions and the correction has no business touching it. A matrix has no way to apply itself only where the paper is showing.

A tolerance cannot cross a condition

If two measurement conditions disagree by seven units, the obvious repair is a correction matrix fitted between them. The best least-squares 3×3 over seventeen printed patches leaves 23 per cent of the disagreement and makes five patches worse than doing nothing — because the term it is trying to remove is proportional to how much paper is showing, and no linear map on three numbers can express that.

7 figures 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.

6 figures 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.

8 figures 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.

8 figures 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.

8 figures 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