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

Spectral resolution — where it appears

How fine a feature in a spectrum has to be before an instrument stops seeing it. For a surface viewed under a light it is set by the light rather than by the observer: a smooth source averages fine structure away, and a source with narrow lines samples it instead.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Which fourth dimensions are expensive, and how fine is too fine to matter. Two curves on axes of how many half-cycles a cosine fourth basis function makes across the visible band, against what it costs the matrix theorem in ΔE₀₀, at a fixed ten per cent amplitude. Both curves touch zero at exactly one and two half-cycles: those are the family's own second and third basis functions, so a fourth coefficient along them adds no dimension and a change of light stays exactly a matrix. Between them the cost climbs, reaches a maximum, and — for the smooth source — falls away again, because structure finer than the scale on which three broad cone sensitivities differ integrates to nearly nothing. The two curves part company at the fine end. Under daylight-to-tungsten the cost has fallen by a factor of 2.2 from its peak; under daylight-to-a-triphosphor-tube it has barely fallen at all, because a source with three narrow emission lines has structure of its own at that scale for the surface's structure to beat against. The observer is identical in both curves.

A fourth dimension has a shape

How much a fourth reflectance dimension costs spans a factor of nine across four equally plausible shapes at one amplitude, and the expensive ones are not the shapes a variance figure would identify. The band that hurts is set by the illuminant rather than by the eye, which is why a triphosphor tube and a tungsten lamp disagree about it.

light · Light The census as the surfaces stop being three-dimensional. A slope chart with three columns — a test set with no fourth reflectance dimension, one with a fourth dimension at ten per cent amplitude, and one at twenty — and a line per change of light. Almost every line rises: a surface with structure the observer's three channels cannot follow is a surface an adaptation gain handles worse. Two lines are drawn heavy. daylight to a three-primary display rises fastest, by 90 per cent, because a source made of three narrow lines is precisely the instrument that cannot see a fourth reflectance dimension. And daylight to a triphosphor tube falls — the only row that does — because a triphosphor tube already samples the spectrum at three places, so extra structure in the surface is partly averaged away rather than added. The order of the middle of the table is not the same at the two ends; the extremes do not move.

The row a fourth dimension improves

Giving the test surfaces one more degree of freedom makes almost every change of light harder for an adapted observer — but not all of them, and which one it helps depends entirely on what the extra dimension looks like. A triphosphor tube is improved by one shape and hurt more than anything else in the census by another.

light · Light One instrument, one slit, two requirements. The slit's width swept, with three measurements on a logarithmic scale: a smooth sample under the line lamp, where only the lamp's structure is at stake; the notched sample under a smooth lamp, where only the notch is; and the real case, both at once. The lamp wants a slit of 5 nanometres and the sample wants 1, and each wants what it wants for the same reason: a slit should spread a feature the grid cannot resolve and leave one it can. The real case is best at 3 nanometres — which is neither requirement's answer — and costs 0.247 there, an order of magnitude more than either requirement alone.

One slit, two requirements

A line lamp wants a wide slit, because a wide slit spreads a line where a coarse grid can see it. A notched sample wants a narrow one, because a wide slit fills the notch the grid could have resolved. An instrument has one slit. Measured on the two requirements separately the best widths are five nanometres and one; measured on the two together the best is three, which is neither — and it costs twelve times what the lamp alone would cost and thirty-four times what the sample alone would.

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 Every pair of slits, over 68 notches. The colour error, in ΔE₀₀ from the truth, for every pair of slit widths — the lamp's table blurred through the width down the side, the sample's through the width across — on 68 notches, the mean over all of them under a fluorescent tube. Circle area follows the error. The best pair is 5 nm on the lamp and 1 nm on the sample, at 0.26; the best single slit, on the diagonal, is 5 nm at 0.36. One slit on the reflected light, at 5 nm, averages 0.016.

A second slit buys a quarter

A line lamp wants a five-nanometre slit and a notched sample a one-nanometre slit, so an instrument with a slit for each should do much better than one with a single compromise. Over sixty-eight notches under a fluorescent tube, it does better by 28 per cent. One slit on the reflected light does fifteen times better than the best pair, and an oracle choosing the best pair for every notch is still six times worse. The error was never that the factors were flattened; it was that they were flattened separately.

light · Light A pair is refused once it is split by about a grid step. Two 1.2 nm lines centred on 578 nm and split by 0.3 to 4 nm, tabulated through slits of one, two and three nanometres on grids of the same step, and the least miss of any single line fitted to the entries round the peak. Each slit's curve crosses half a per cent between three quarters of its own step and one full step: a one-nanometre table refuses a pair split by 0.9 nm, a two-nanometre table one split by 1.8, a three-nanometre table one split by 3.

A table can refuse a pair it cannot resolve

A narrow spectral table reads each of its peaks as one line and declares the line's width from the entries round it, and the assumption under that is that every peak is a line. Split the mercury tube's yellow into the real pair at 577 and 579 nm and the table notices: through one and two nanometre slits no single line fits the entries within half a per cent, so the table refuses to read a width there. Swept apart, a pair is refused once it is split by about the grid step and admitted when it is closer — always as one wider line, up to 3.3 times too wide. Through three nanometres the pair reads as a 3.2 nm line, and a 3 nm europium band beside it reads the same.

light · Light

Named alongside it

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

BandpassSpectral structureSpectrophotometryWavelength gridFluorescentInstrumentAliasingBasisChromatic adaptationDimensionalityIlluminantMeasurement condition

All concepts