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

Metric axioms — where it appears

The four properties a distance must have, of which the colour-difference formulae in industrial use keep two or three. Symmetry and the triangle inequality are the ones the industrial formulae give up, which means the distance between two colours can depend on which is called the standard.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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 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.

CIEDE2000CIELABPerceptual uniformityColour differenceΔEGradientSpecificationInterpolationAssertionAuditChromaCIECAM16

All concepts