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What a tabulation step costs, by light, on vermilion

The horizontal axis is the tabulation step in nanometres, from one to twenty; the vertical is how far the resulting colour is from the same integral taken at a tenth of a nanometre over the same range, in ΔE₀₀, on a logarithmic scale. Each line is one light. The three with no feature narrower than the step fall smoothly and stay below a tenth of a unit at five nanometres, which is the grid used throughout. The fluorescent tube and the laser projector do not fall at all: their lines are narrower than any step drawn here, so the answer depends on where the samples land rather than on how many there are. The sample is held at vermilion throughout.

What a tabulation step costs, by light, on vermilion. The horizontal axis is the tabulation step in nanometres, from one to twenty; the vertical is how far the resulting colour is from the same integral taken at a tenth of a nanometre over the same range, in ΔE₀₀, on a logarithmic scale. Each line is one light. The three with no feature narrower than the step fall smoothly and stay below a tenth of a unit at five nanometres, which is the grid used throughout. The fluorescent tube and the laser projector do not fall at all: their lines are narrower than any step drawn here, so the answer depends on where the samples land rather than on how many there are. The sample is held at vermilion throughout.

Drawn at its defaults, in the grid the integral is summed over. It takes no options at all, so every essay calling it gets this exact drawing.

Called by 22 essays

the blast radius of changing it

The index is a choice too

Every colour in this collection is a sum over eighty-one numbers running from 380 to 780 nanometres in steps of five. That index is not a property of the eye, the light or the sample — it is a tabulation, and it holds three separable decisions that behave completely differently from one another.

The slit is what makes it legal

Point-sampling a mercury line at five nanometres costs a colour difference of one unit, and a laser projector thirty-seven. Integrating the same spectrum through the five-nanometre slit every spectrometer already has costs 0.014 and 0.19. The blur anybody would remove if they could is what makes a coarse table honest.

A finer reading of a coarser table

Interpolating a five-nanometre spectrum to one nanometre helps a daylight calculation by a factor of five and harms a three-emitter LED by a factor of a hundred and twenty thousand. Both are the same operation on the same table, and which one happens is decided by a property of the light nobody records.

Difference and uniformity

A neutral has no grid

A perfectly flat reflectance computes to exactly the same colour on every wavelength grid, through every slit, at every origin, and for every observer — not nearly, but to the last bit of a floating-point number. The condition is an identity rather than a limit, and what makes it one is the white point.

Where the grid starts

Holding the step at five nanometres and sliding the grid's origin through one cell moves a fluorescent tube's computed colour by 3.18 ΔE₀₀ and a laser projector's by 35.0. Refining the step does not fix it and averaging over origins hides it. It is the one tabulation fault with no smooth error to cancel against.

Two ends and one is empty

Extending this collection's wavelength range down to 300 nanometres moves a red pigment under daylight by 0.502 ΔE₀₀. Extending it up to 830 moves the same colour by 0.00015. A fifth of a thermal source's power lies outside the range and almost none of its colour does, and confusing those two shares is how a range gets argued about instead of measured.

Which end to buy

Refining a five-nanometre grid to one buys a daylight calculation 0.05 ΔE₀₀ and widening its range buys 0.54. Under a fluorescent tube the same two purchases are worth 0.83 and 0.0001. The ranking reverses completely, and what decides it is one length compared against one other length.

Difference and uniformity

The normaliser carries the error too

A five-nanometre sum gets a red pigment's tristimulus value wrong by two hundredths of a per cent and its colour wrong by six hundredths of a unit. Those two numbers are not the same size because the grid appears twice in a colour — once in the sample and once in the white — and the two errors are largely the same error.

The endpoint term has a name

The five-nanometre error on a smooth light falls linearly with the step, which is not what a sampling error does. It is the half-cell at each end of a truncated range, it is first order where the sampling is second, and halving two weights removes fourteen fifteenths of it for nothing.

A grid is not a resolution

Ten essays into this collection there is one sentence about wavelength sampling, and it is that five nanometres is enough. Ten measurements later there are three decisions, three mechanisms, three repairs and two rankings, and the word resolution names none of them.

What the brain does

The grid under the census

The adaptation census is computed on eighty-one wavelengths from 380 to 780 nanometres. Under the daylight and blackbody sources it uses, the range is worth about half a colour difference on ordinary surfaces and the step about a twentieth — so the census carries a tabulation term as well as an observer one, and they are not the same size.

A lamp is two audits at once

A lamp's ultraviolet content decides what its tabulation costs and its blue content decides what its observer costs, and the two run in opposite directions with colour temperature. So no lamp is good for both audits and no lamp is bad for both, and a single figure of merit for either is a figure of merit for one property of a spectrum.

Where the model breaks

The conditions are the result

The round measured about twenty departures and established ten conditions. The departures are numbers that depend on a sample, a light and a construction; the conditions are exact, they hold under every substitution tried, and they are what a reader can act on. A size is a measurement and a condition is a mechanism.

The line can be in the sample

The rule for which tabulation defect to fix compares the light's narrowest feature with the grid's step, and it named its own failure case — a sample with structure narrower than the step. Given one, a two-nanometre notch under a thermal source with no feature at all costs 2.1 colour differences at five nanometres and moves 2.0 when the grid slides, which is what a fluorescent tube costs on a smooth pigment. Under that tube a notch twelve nanometres wide, more than twice the step, moves 8.1 when it sits on the mercury line.

The cost of a steep notch repeats every step

A Gaussian notch is safe on a five-nanometre grid once it is a couple of steps wide. A flat-bottomed notch with steep sides never is. Its cost on the grid rises and falls with its width, with the step as its period — 0.02 colour differences at ten nanometres, 1.99 at twelve and a half, 0.03 at twenty — and it does not die away as the notch widens. What sets its size is how fast the edges rise, and an interference filter's edges rise in under a nanometre.

Two slits are not one slit

A spectrometer's slit is what makes a coarse table honest, for a lamp and for a notched sample alike. But a colour is a sum over the product of the two, and a notch measured through one slit and a lamp measured through another are not the product measured through a slit. Under a smooth light the difference is nothing. Under a fluorescent tube a notch on the mercury line comes out 0.73 colour differences off from two slits — worse than no slit at all for some notches — and 0.02 off from one slit on the reflected light.

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.

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.

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.

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.

The tables cannot bound what they discarded

A colour computed from a lamp's blurred table and a sample's blurred table is wrong by the covariance the two blurs threw away, and Cauchy–Schwarz bounds a covariance by two variances. With the true variances the bound always holds and sits fifteen times above the error. With variances read from the tables it fails on six of sixty-eight notches under a fluorescent tube and twenty-two under a laser projector — on the line, where the error is largest. A blurred table does not carry the width of a line, and the covariance depends on it.

A declared width buys a factor of two

A colour engine given two separately blurred spectral tables cannot bound its own error from them, and the bound that always holds — the peak declared and nothing else — sits a median 196 times above the error under a fluorescent tube. Adding one number, the width of the lamp's narrowest feature, brings that to 86. It never fails on any declaration the lamp truly meets, it fails on 47 of 68 notches on one it does not, and its rank correlation with the error it bounds is 0.27.

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