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

Convention — where it appears

A choice fixed once by agreement rather than derived, which everything downstream then depends on. The characteristic hazard is that a convention nobody wrote down is indistinguishable from a fact, and it stays that way until two parties who chose differently compare results.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

The same claim in nanometres of pigment, where no declared width can reach it. Five horizontal bars on a scale of nanometres, one per published chromatic-adaptation transform, each showing how far the medium-wave cone pigment's absorption peak would have to move for the receptors' own protan confusion point to land where that transform puts it. Zero is the measured peak. The bars run from -10.5 to 18.2 nanometres — in both directions, so two of the transforms want the pigment shorter and two want it longer. Drawn across them is the 25 nm separation between the L and M pigment peaks, which is the whole basis of red-green vision and is not a number this collection declared. The nearest transform asks for a displacement of 30 per cent of that separation, and the span across the table is 28.7 nanometres — larger than the separation itself. No population, cloud or standard deviation appears anywhere in the statement.

Five transforms and the space between them

Every appearance prediction here chooses one of five published adaptation transforms, and the five disagree about where a protanope's confusion lines meet by more than the distance between the two pigments the disagreement is about. That spread is itself a scale, and using it needs no population model at all.

brain · Appearance What the interface does to a reflectance, and the straight line it is taken for. The Saunderson relation between the reflectance inside a pigment layer and the reflectance an instrument reads off it, for a boundary of refractive index 1.50. The curve is the real map; the dashed line joins its two endpoints, which is the straight relation an additive pedestal assumes. They are 0.216 of a reflectance unit apart at their widest, which is 5 times the pedestal itself. The curvature comes from the k₂ term — light reflected back down into the layer from underneath the boundary — which is 0.60 where the outward reflection is 0.04.

A mixture in the variable nobody named

Kubelka–Munk works because absorption and scattering add over a mixture and reflectance does not. What adds is the absorption of the pigment layer, and what an instrument reports is that layer seen through an interface — related by a Möbius function rather than by a constant. Mixing in the reported variable instead of the internal one costs between three and eight ΔE₀₀, and no source this collection quotes says which variable its curves are in.

scene · Scene Six functions of wavelength, and the six different places they stop. Every table this collection integrates against, drawn over the range the body that published it defined it on. The scale is logarithmic so that the ultraviolet and the near infrared both fit. The bottom row is the range used here before the infrared band was added, and it is the intersection of the two rows that matter for an eye looking at a reflector — which is the right answer only while everything in the integral is being multiplied together. The daylight basis runs 80 nanometres further down than that intersection, and it was published that way because the ultraviolet in daylight is what makes a brightened sheet of paper glow. The analytic row is drawn to the edge of the plot because it has no edge: Planck's law is a formula and is exact at every wavelength, which is why illuminant A needs no table at all.

The grid is a range, not an index

Three of the four departures in this round restore an argument the model dropped. The fourth does not — the 380-to-780-nanometre grid is the range of the one argument the model kept, chosen in the first weeks and never revisited. It costs 6.70 ΔE₀₀ on a coated printing paper, it is the cheapest of the four to fix, and it is the one still unfixed.

light · Light Six ways of tabulating one notch on one mercury line. A 12-nanometre notch centred on a fluorescent tube's 546.1 nm line, its colour computed on a five-nanometre grid six ways, on a logarithmic scale. Point sampling costs 2.54 colour differences and two separate slits 0.727. Sharpening both blurred tables with the published three-term correction takes it to 0.188 — a real improvement, four times better — and one slit on the light the sample actually reflects gives 0.019. The correction recovers the part of the damage that is a blur, and the part that is left is not a blur.

A linear repair for a bilinear loss

The Stearns correction sharpens a table blurred by a triangular slit, and the obvious question was whether applying it to a lamp's table and a sample's restores the product the two of them are wrong about. It restores four fifths of the damage and cannot touch the rest: a three-term filter is linear, the covariance two separately blurred tables discard is bilinear in the two factors, and no linear operator applied to each factor separately produces a bilinear term. What is left is ten times the one-slit answer, at every position of the notch.

light · Light An interference notch filter, and where a five-nanometre grid lands on it. The transmittance of a Fabry-Pérot etalon of order 24 and finesse 20, drawn at a fifth of a nanometre, with the standard grid's points marked. Its features are 2.29 nanometres wide and spaced 22.9 apart, so the grid steps over them: between two adjacent grid points the transmittance rises and falls completely, and neither point records it. That is what a real coating looks like, and a Gaussian notch — which is what this collection's earlier work used — is a much gentler object.

A finer table is a worse table

A real interference filter is not a Gaussian notch. It is an etalon, with pass bands a nanometre or two wide spaced twenty-three apart, and a five-nanometre grid steps over them. Resampled from the maker's one-nanometre table its colour is out by five colour differences under every fill-in rule — the three rules agree to three decimals, because none of them is ever handed a sample inside a feature. The same filter measured through a five-nanometre slit is out by 0.03.

light · Light 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 How often the choice of correction changes a pair's grade. Eighteen metameric pairs walked to each of eleven reference mismatches, with the share whose index falls in a different band under the two corrections the standard allows. At an exact match the share is zero and must be: there is nothing for either correction to correct. It rises to 44 per cent at a reference mismatch of 2, which is the quality a dyehouse reaches rather than the quality a laboratory constructs. The banding is a five-step convention at 0.5, 1, 2 and 3, stated here rather than quoted, and how much the count depends on it is drawn separately.

The ambiguity is largest where the index is used

The metamerism index is defined for a pair that matches exactly under the reference light, no real pair does, and the two corrections the standard allows for the residual give different answers. Over eighteen pairs at eleven match qualities the gap is nearly a function of the mismatch alone — its middle half spans a factor of 1.4 at the mismatches a dyehouse reaches — so it could be tabulated. And it is largest exactly there: zero at a laboratory's match, and re-grading eight pairs in eighteen at a trade's.

matching · Gamut 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

Named alongside it

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

Colour differenceInterpolationWavelength gridBandpassCIEDE2000CIELABColour appearanceDeclared inputGradientInstrumentMeasurement conditionPerceptual uniformity

All concepts