Where the grid starts

A tabulation has a spacing and everybody argues about the spacing. It also has a starting point, and nothing in colorimetry has ever treated that as a variable. It is the variable that separates two errors which are otherwise reported as one number.

The claim #

Sliding a wavelength grid’s origin through one cell, at fixed spacing, is a measurement — and it is the only one that tells a sampling error apart from a quadrature error.

• A quadrature error does not care where the grid begins. It is a property of the rule and the smoothness of the integrand, and shifting the whole grid moves it only through the endpoints.
• An aliasing error cares about nothing else. Whether a sample point lands on a mercury line is a coincidence of arithmetic, and the answer changes completely when it does.
• The two are sixty times apart on the same step: 0.05 ΔE₀₀ of spread for a smooth light, 3.18 for a fluorescent tube, 35.0 for a three-laser projector.
• And refining the step does not help. A two-nanometre grid on the projector is worse than a five-nanometre one, because the question of which lines are caught has no limit as the spacing shrinks — only as the spacing passes the linewidth, which is two orders of magnitude away.

The experiment, stated exactly #

Take a light, a sample and an observer. Compute the colour on the grid 380, 385, … 780. Compute it again on 381, 386, … 781, then 382, then 383, then 384. Five grids of the same spacing, differing only in where they begin.

Every one of the five is a legitimate five-nanometre tabulation. Nothing distinguishes them; a laboratory that reported its spectra on any of them would be reporting a spectrum at five-nanometre intervals, and nothing in a data file or a standard says which offset was used. The spread across the five is therefore not an error in anybody’s procedure. It is the size of the arbitrariness the procedure contains.

For a red pigment under a 6500 K radiator that spread is 0.050 ΔE₀₀. Under tungsten it is 0.010. Under a white LED it is 0.0001, and under a three-emitter LED it is smaller than the last digit printed. Under a fluorescent tube it is 3.18, and under a three-laser projector 35.0.

What the smooth lights are actually showing #

The two thermal sources do move a little, and it is worth saying what that movement is, because it is not aliasing.

Shifting the grid shifts its endpoints. A grid beginning at 380 ends at 780; one beginning at 384 ends at 784. The integrand is not zero at either end, so the truncated range changes slightly, and the end cells are most of what a smooth light’s five-nanometre error is in the first place. The spread that survives on daylight is the endpoint term being dragged four nanometres.

That is confirmed rather than assumed. Halving the two end weights — the trapezoid rule, one line of arithmetic — takes the daylight spread from 0.050 to under 0.004, and does nothing at all to the fluorescent tube’s 3.18. Two mechanisms, told apart by a rule change, and the rule change is available to anybody at no cost.

So the honest reading of the smooth rows is that they show no aliasing, not a little. Their whole visible movement belongs to the range, and the range is a different decision with a different repair.

An average across origins is the wrong summary. The obvious response to a quantity that depends on an arbitrary choice is to average over the choice. It is the wrong response here and the reason generalises.

Averaging the fluorescent tube’s five origins gives 2.10 ΔE₀₀, and the individual values are 0.98, 4.05, 0.87, 3.14 and 1.46. There is no sense in which 2.10 is what a laboratory would get. Each laboratory gets one of the five, and which one is decided by a convention nobody wrote down. A number quoted as 2.10 with no spread beside it claims a precision the procedure does not have, and it claims it in the way that is hardest to catch: by being the right answer to a question about an ensemble that has one member.

This is the same shape as a mean over a test set standing in for its members, and it arrives from an unexpected direction. Nobody has thought of a tabulation as having members. It has five, or a hundred if the offset is continuous, and the spread over them is the honest report.

A shifted grid computes a different object. The laser projector makes the mechanism unmistakable because its lines are a fifth of a nanometre wide and its answer is not an approximation to anything.

Its emitters sit at 465, 532 and 638 nanometres. On the grid beginning at 380, exactly one of those is a grid point, so the arithmetic sees a single line at 465 and computes the colour of a monochromatic blue. On the grid beginning at 382, none of them is, and the arithmetic sees nothing at all — a spectrum of zeros, from which no colour can be computed and the machinery returns whatever the degenerate case gives. On the grid beginning at 384 the picture changes again.

So the five answers are not five samples from a distribution around a true value. They are five different objects, and the true value is not among them. That is the precise sense in which aliasing is not an error: an error is a distance from an answer, and there is no answer in the arithmetic to be distant from.

The fluorescent tube is the same phenomenon at a survivable scale. Its lines are a nanometre wide against a five-nanometre spacing, so a grid point sits on a line about one time in five, and the strong 546-nanometre line contributes either a great deal or almost nothing depending on the offset. The 4.05 and the 0.87 in the sequence above are those two cases.

Why refining does not converge #

The natural instinct is that a finer grid must eventually be right, and it is true — eventually. The word is doing a great deal of work.

For the projector, the sequence of errors at steps of 1, 2, 5, 10 and 20 nanometres is 0.0001, 42.6, 37.2, 25.5 and 46.1. It is not monotone, it is not decreasing, and the one small entry is small because at one nanometre several lines happen to be caught rather than because one nanometre resolves a fifth-of-a-nanometre line. Halving the step from two to one changed the answer by a factor of four hundred thousand; halving it again would change it by an unrelated amount.

Convergence begins when the step falls below about half the linewidth — a tenth of a nanometre for these emitters, which is fifty times finer than any spectral data anybody publishes. Between five nanometres and a tenth of one there is no useful sense in which the calculation is approaching anything, and every intermediate answer is as arbitrary as the offset.

This is why a convergence check on the step alone can pass while telling nothing. Two nearby steps agreeing is evidence of convergence for a smooth integrand and is evidence of nothing for a spiky one, and the way to tell which case is in hand is to vary the offset rather than the step. The same lesson was learned in this collection about a hemisphere quadrature, where refining one knob tested one knob and rotating the light found a two and a half per cent error the refinement could not see.

Those two figures are the same statement in two coordinates. A light whose answer depends on where the grid starts is a light whose answer depends on whether a slit is present, because both questions ask whether the tabulation can hold what the light has. Neither question is about how many points there are.

The offset a real instrument has #

A physical spectrometer does have an origin, and it is not chosen for any reason connected to the spectra it will measure. It falls out of the grating’s angle, the detector array’s position and the wavelength calibration, and it is stable for a given instrument and different between instruments.

That has an uncomfortable consequence for inter-instrument agreement, which the field measures and reports as a matter of routine. Two instruments of identical bandpass, identical interval and perfect wavelength calibration will still report different spectra for a line source if their reporting grids are offset from one another — and the difference is not a calibration fault that could be trimmed out, because both are right.

What rescues real instruments is the same thing that rescues everything else in this round: the slit. An instrument with a five-nanometre triangular bandpass has already integrated away the structure that the offset would have made arbitrary, so its reported table is nearly offset-independent. The blur nobody wants is doing the work again, and the offset problem is severe exactly where the slit is absent — which is to say in calculations from formulae, which is to say in this collection.

What this collection does about its own offset #

Every spectrum here is evaluated at 380, 385, … 780, because that is what spectrumFrom does and it has never taken an argument. So the collection has one offset, chosen in the first week and never varied, and every figure it draws of a narrowband source inherits it.

The consequence is worth stating in its most awkward form. Any absolute colour this collection quotes for a discharge lamp or a laser primary would be a different number on a grid beginning one nanometre later, and by more than the tolerances the same collection writes its delivery essays in. That is not hidden in an approximation; it is a property of the arithmetic.

Two things limit the damage and neither removes it. Almost every figure here compares two computations done on the same grid, and the shared offset cancels most of what it would otherwise contribute — the pairing structure again, with the grid as one factor. And the sources with narrow lines are a small minority of what the collection draws, because most of its arguments are about daylight, blackbodies and reflectances.

Where the same shape appears #

Aliasing is a general phenomenon and colour has met it before, in a place that looks nothing like this one.

A halftone screen is a sampling of an image, and two screens at slightly different angles produce a moiré that is a beat between two sampling grids. The mathematics is the same: a periodic sampler meeting structure near its own period, producing an answer that depends on relative phase. Printers solve it by choosing screen angles that put the beat where the eye cannot resolve it, which is an admission that the phase cannot be removed and can only be steered.

The spectral case has no equivalent steering, because there is nothing to steer towards. There is no offset that is right for both mercury and a laser primary, and no convention that could be adopted fleet-wide to make the arbitrariness go away. What can be done is what the instruments already do, which is to filter before sampling.

That collapse is the third face of the same fact and the one that explains why aliasing is so much more expensive than its share of the spectrum suggests. Every other tabulation fault is partly refunded by the division that turns a tristimulus value into a colour; this one is not, because the refund requires the two sums to be wrong in the same way and a shifted grid makes them wrong about different lines.

What was computed, and how #

The five offsets are 0, 1, 2, 3 and 4 nanometres against a five-nanometre step, which covers exactly one cell. Each grid runs from its own start to 780 or the last point below it, so the grids differ in their final point as well as their first — that is what a real re-tabulation looks like, and pinning both ends would have measured something else.

The spread is reported as a maximum minus a minimum rather than a standard deviation. Five points do not support a distributional claim, and the quantity anybody needs is how far apart two legitimate answers can be.

The gate this family carries is written as a ratio rather than as a ceiling: the least spiky of the spiky lights must move more than twenty times the most spiky of the smooth ones. The first version was an absolute ceiling on the smooth lights and it failed, because a smooth light does move — through its endpoints — and a gate written as a ceiling was measuring the range while claiming to measure aliasing.

Where the model stops #

The lines here are Gaussians of stated width and real emission lines are not. A pressure-broadened mercury line has Lorentzian wings that extend far beyond its nominal width, and a wing is exactly the part a coarse grid does catch, so a real tube is somewhat kinder to a shifted grid than this one is.

The continuous version of the experiment is not run. Five offsets sample one cell coarsely, and the true spread over all offsets is at least as large as the one reported and could be larger. A finer offset sweep is cheap and was not done, which makes every number here a lower bound.

And this measures a computed spectrum’s sensitivity to a computed grid. What a real laboratory’s reported spectra do under re-tabulation is a different experiment involving real instruments, and nothing here substitutes for it.

The generalisation #

The habit is about which knob to turn when checking a numerical result.

The standard convergence test refines the discretisation and watches the answer settle. It is necessary and it is not sufficient, because a discretisation has more parameters than its resolution: an origin, an alignment, an orientation, a phase. Refining tests one of them and leaves the others at whatever value they were first given. A quantity that must not depend on a parameter is a better test than a parameter that must not change the quantity, and the second is what refining gives.

The way to find the right knob is to look for a symmetry the problem has and the discretisation does not. A wavelength integral does not care where a tabulation begins; a tabulation does. A hemispherical integral over an isotropic surface does not care about azimuth; a quadrature grid does. Both of those found errors in this collection that refinement had passed.

Who found it, and when #

Nyquist’s condition is from 1928 and Shannon’s statement of it from 1949, and neither needed a colour to be interesting. The application to spectral data is old enough that the CIE’s recommended practice takes it for granted and discusses the consequences rather than the theorem.

What is genuinely absent from the literature is the offset as a reported quantity. Spectral data formats record the interval, the range and often the bandpass; none of them records where the grid began relative to anything, because there is nothing for it to be relative to. That absence is not an oversight — for a properly filtered measurement the offset carries no information — and it becomes an omission exactly when the data did not come from an instrument.

Where the ladder goes next #

The step, the origin and the slit are measured, and every one has come back the same way: what a tabulation costs is a product of what the index does and what the light has in it. The remaining decision is the range, and it is the one this collection has named as outstanding for two rounds without measuring what the alternative would buy.

The two ends of a range are not alike, and the asymmetry between them is a factor of three thousand.