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

Quadratic form — where it appears

A function of several variables in which every term is a product of exactly two of them, written as a symmetric matrix acting on a vector. It is what an ellipse, an ellipsoid and the curvature of a surface at a minimum all are, so its eigenvalues are the axes of whichever of those it is describing.

Named by 13 essays across 4 fields — each of them below, with the objects they name alongside it.

The points a ratio needs are proportional to the ratio. A scatter of 133 points on logarithmic axes, one per MacAdam ellipse under each of six coordinate systems. The horizontal position is that ellipse's true axis ratio; the vertical is the smallest sample size, from a sequence of doublings, at which the sampled ratio comes within one per cent and stays there. A line of slope 0.94 runs through them, against a predicted 1 — the minimum's notch is 0.88 σ₂/σ₁ radians wide, so resolving it takes a number of points proportional to σ₁/σ₂, and nothing about the basis or the ellipse enters beyond that. An ellipse with a ratio of two needs seventeen points and one with a ratio of twenty-six needs a hundred and ninety-two.

An extremum is not a sample

Three separate measurements in this collection took a maximum or a minimum over a sample of a set — forty-eight points round an ellipse, twenty-four directions out of an optimum, fourteen changes of light off a list. All three are wrong, all three are wrong in the same direction, and the error in each grows with the very quantity being measured.

limits · Limits 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 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 A quadratic is believed least far at the one place anybody takes one. One bar per basis: the radius, in the nine coefficients, within which the second-order model predicts the objective to within ten per cent in every one of eighteen directions. The shortest bar is the objective's own optimum, at 2.3×10⁻², and the longest is XYZ scaling at 1.1×10⁻¹ — several times further. The reason is not that the model is worse at a minimum but that it has less to do there: away from one the linear term is exact and carries most of the change, so a ten per cent error in the prediction takes longer to accumulate. It does not make a Hessian at a minimum wrong; it says the picture drawn from it describes the smallest neighbourhood in the table.

How far a quadratic can be believed

A second-order model has a radius inside which it describes a surface and outside which it does not, and that radius can be measured. Measured at eight places on one objective, it is smallest at the optimum — the one place anybody ever takes a Hessian.

limits · Limits Exactly flat everywhere, and eigenvectors at one point only. Two columns over the same nine places. On the left, how far from zero the objective's second derivative is along a row-scaling direction, on a logarithmic axis — it is between 10⁻⁹ and 10⁻⁶ of the largest eigenvalue at every one of them, which is a numerical zero. Scaling a row of the basis is a straight line along which the cost does not change, and that is true at every point, not only at the optimum. On the right, the angle between those three directions and the Hessian's own three smallest eigenvectors: 0.025 degrees at the optimum and up to 88 away from it. An invariance is a property of the function; being an eigenvector is a property of the function at a minimum, and the two coincide only where everybody computes.

Only the flat directions keep their names

Three of the nine numbers a colour match leaves free do nothing, and they do nothing everywhere — exactly, at every basis in this collection's table. They are the objective's own principal directions at one point only, and everywhere else the directions carrying the curvature have turned by tens of degrees.

matching · Gamut At a published matrix the slope arrives long before the bowl. One row per basis in this collection's table. Each row is a logarithmic axis of distance in the nine coefficients, with two markers: the radius at which the objective's curvature becomes as large as its slope, and the distance from that basis to the optimum. The first is between 3.2 and 108 per cent of the second. So over almost the whole journey from a published matrix to the best one, the surface is a slope and not a bowl — and a table of eigenvalues taken there describes a neighbourhood the optimum is nowhere near. XYZ scaling is the exception, at 1.08 of the distance, because its slope is the steepest in the table.

The slope arrives before the bowl

The adaptation transforms colour management actually uses are not optima of anything. At every one of them the objective has a slope, and the slope is the larger term over almost the whole distance to the best matrix — so a table of curvatures taken there describes a bowl nobody meets on the way anywhere.

applied · Delivery Downhill from every published matrix, one step at a time. Each curve is a steepest-descent walk from one of this collection's published bases, plotted as the objective against the distance walked in the nine coefficients. The horizontal line is the optimum. The first step of each walk is the long one — XYZ scaling closes 41 per cent of its whole gap in one — and every walk then flattens without reaching the line, because the valley floor is nearly flat and the steepest direction is nearly across it. Bradford starts closest and closes least: it is already in the flat part.

Downhill from a published matrix

Walking steepest descent from each adaptation transform in use closes between a quarter and nine tenths of its distance to the best one, and most of that in the first step. The direction it sets off in is eighty to eighty-seven degrees away from the answer, and that turns out not to be an artefact of the three directions nothing can see.

applied · Delivery The angle between two departures, which nothing in the audit records. Every pair of the six departures on every surface — 2520 pairs — binned by the angle between their two deviations in the local metric. The distribution reaches both ends: 440 pairs sit under thirty degrees and point almost the same way, and 615 sit above a hundred and fifty and point almost opposite. On 1095 of the 2520 the two together cost less than the larger of them alone. A table of magnitudes cannot say which case it is in.

A size is not a direction

An audit that reports magnitudes cannot say what two of them cost together. Over two and a half thousand pairs of observer departures the angle between them in the local metric runs from one degree to a hundred and seventy-nine, and on forty-three per cent of them the two together cost less than the larger of the two alone. Adding the angle predicts the composition to one and a quarter per cent; Pythagoras is out by twenty-eight.

matching · Gamut What one tolerance accepts, around four colours. The surface in tristimulus values that ΔE₀₀ 1.0 draws around four colours, each outline scaled to its own size so the shapes can be compared. The volumes they enclose differ by a factor of 1.0e+5 across the sRGB cube, and the longest axis of one shell is between 3.3 and 29.9 times its shortest. A tolerance is written as one number and is a different set of samples at every colour it is applied to.

What one number accepts

A delivery tolerance is written as a single colour difference and it acts on three tristimulus values, so what it actually accepts is a closed surface. Measured over sixty-four colours in the sRGB cube, the volume inside that surface varies by a factor of a hundred thousand and its longest axis is between three and thirty times its shortest. The same contract, applied to a dark colour and a light one, is two different requirements.

matching · Gamut Where a press's variation lies, and where the tolerance charges for it. Each printed patch's sheet-to-sheet variation under four stated mixes of press variation, split along the three axes of a one-unit ΔE₀₀ tolerance at that patch, tightest first. The upper bar of each pair is the share of the variation along each axis and the lower the share of the price, averaged over 29 patches. For an even mix the tightest axis holds 1.5% of the variation and pays 29% of the price, and the loosest holds 83% and pays 36%. For inking alone the tightest axis holds 2.0% of the variation and pays 34% of the price, and the loosest holds 79% and pays 28%. For a gain-led press the tightest axis holds 1.8% of the variation and pays 29% of the price, and the loosest holds 83% and pays 39%. For a trap-led press the tightest axis holds 1.6% of the variation and pays 29% of the price, and the loosest holds 82% and pays 35%.

A press is charged for the direction it barely moves

A press run varies almost entirely along the direction a colour tolerance forgives. Split along the tolerance's own axes under four stated mixes of press variation, its tightest axis holds about one per cent of the sheet-to-sheet variation and pays a quarter to a third of the price — and on a blue overprint four fifths, for the balance between two inking units rather than the level of either.

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

Named alongside it

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

AnisotropyColour differenceChromatic adaptationCIEDE2000CIELABCondition numberDegrees of freedomEigenvalueMacAdam's ellipsesConvergenceDeclared inputSampling

All concepts