The collection

Every essay — page 17

Page 17 of 40, continuing through the fields in the same order.

What light is What the eye does Matching and measuring Difference and uniformity What the brain does What a scene does What a camera does Where the model breaks What it takes to deliver it

Series Observers Named objects Refutations Search

What light is

A spectrum is a function, not a colour. Illuminants, reflectance, blackbody radiation computed from Planck's law, and what has already been discarded before the eye is reached.

What an instrument's slit width does to a tabulated colour. The horizontal axis is the full width of a triangular slit, from zero — perfect point sampling — to twenty nanometres; the vertical is the distance from the true colour, logarithmic. For a smooth light the lines are flat: a slit narrower than any feature changes nothing. For a line spectrum they fall off a cliff at the left. Point-sampling a mercury line at five nanometres costs 1.26 ΔE₀₀ and integrating the same spectrum through a five-nanometre slit costs 0.014. A spectrometer does not sample a spectrum; it integrates one, and the blur everybody would remove if they could is what makes a five-nanometre table safe.

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.

6 figures

What three fill-in rules cost when a five-nanometre table is read at one. A five-nanometre table read at one nanometre by three rules — hold the value, straight lines, a cubic through four points — each compared with the same tenth-nanometre reference. The dashed rule is the answer obtained by summing the table as it stands, at 0.00000 ΔE₀₀. Two of the three interpolations are worse than not interpolating. That is not a paradox: the summation's error is already small because the normaliser cancels most of it, and an interpolator introduces a shape the original curve did not have, which the cancellation cannot touch. A finer grid is not more resolution when the table is not finer.

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.

6 figures

How much the answer moves when the 5-nanometre grid is slid through one cell. Each bar is the spread of one light's colour across five grid origins, all at the same 5-nanometre step, in ΔE₀₀. A smooth light barely moves, and what movement it has is the end cells rather than the sampling. The fluorescent tube moves by 3.18 units and the laser projector by 35.0, because their emission lines are narrower than the step and whether a sample lands on one is a coincidence of arithmetic. This is the measurement that separates a quadrature error from an aliasing error, and no average over origins can substitute for it.

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.

5 figures

What each end of the 380–780 nanometre range costs, by light. Two bars per light, on a logarithmic axis: the upper is what extending the range down to 300 nanometres moves the answer, the lower what extending it up to 830 does. The asymmetry is the whole figure. A thermal source has about a fifth of its power outside this collection's range and almost all of it at the long end, where the observer is already zero; what costs money is the short end, where the observer is small but not zero and daylight is still strong. A light with no ultraviolet — an LED lamp, a laser — pays nothing at either end, which is the pairing again: a range only costs what the light puts in it.

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.

7 figures

The two tabulation choices over forty-two surfaces, under a 6500 K thermal radiator. Each column is one choice, measured over a family of forty-two analytic reflectances rather than on a single example: an absorption band of stated centre, width and depth. The four marks are the smallest, the median, the ninety-fifth percentile and the largest cost in ΔE₀₀, logarithmically. Under a smooth light the range is worth 9.1 times the step at the median, so a collection wanting one repair should widen its range rather than refine its step — and under a fluorescent tube the ranking reverses outright.

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.

6 figures

The rectangle sum against the trapezoid sum, under a 6500 K thermal radiator. Colorimetry's summation is the rectangle rule at the tabulated points. The trapezoid rule differs from it by exactly one thing — half a cell at each end of the range — and the gap between these two lines is therefore that term and nothing else. At five nanometres it is a factor of 14.3, which means the number everybody calls a sampling error is mostly a truncation error wearing the step's clothes. On a light whose lines are narrower than the step the two rules agree to three decimal places, because there the error really is the sampling.

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.

6 figures

What a tabulation step costs, by light, on paper. 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 paper throughout.

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.

6 figures $The share of each light outside 380–780 nanometres, as power and as visual product. Two measurements of the same truncation. The upper bar is the fraction of the light's radiant power that lies outside the range; the lower is the fraction of the product of light, sample and observer — which is what a colour is made of. A thermal radiator puts a fifth of its power outside and about a thousandth of its colour, because the observer is zero where most of that power is. The gap between the two bars is the observer's own tails doing their job, and reading the upper number as though it were the lower is how a range gets argued about without being measured.$

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.

6 figures

A 2-nanometre notch, and what two five-nanometre grids record of it. The reflectance of a sample with a 2-nanometre notch at 552.3 nm, drawn finely from 535 to 570 nm. The dark ticks are the samples of a five-nanometre grid starting at 380 nm and the pale ticks those of the same grid started half a step later. The true minimum is 0.06; the first grid's deepest sample reads 0.70 and the second's 0.08. The sample has put a line into a calculation whose light has none.

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.

7 figures

What a five-nanometre grid costs a steep-sided notch, against its width, under a 6500 K source. The cost of a five-nanometre grid starting at 380 nm, against a reference at two hundredths of a nanometre, for a sample with a flat-bottomed notch at 552.3 nm under a 6500 K thermal radiator, against the notch's width from 2 to 30 nm, for edges rising in 0.4, 2.2, 6.6 nm. With the steepest edges the cost is 0.02 at 10 nm, 1.99 at 12.5 and 0.03 at 20: it rises and falls with the step as its period and does not die away as the notch widens. With the softest edges it stays under 0.10 at every width.

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.

7 figures

A 12-nanometre notch at 546.1 nm under a fluorescent tube, tabulated five ways. The cost against a tenth-nanometre reference, on a five-nanometre grid, of a notched sample under a fluorescent tube, mercury lines on a phosphor bed, when the two factors of the colour are tabulated as points, when the lamp alone is measured through a five-nanometre slit, when the sample alone is, when each is measured through its own slit, and when the light the sample reflects is measured through one slit. The costs are 2.535 for both sampled at points, 0.379 for the lamp through a slit, 2.724 for the sample through a slit, 0.727 for both through their own slits, 0.019 for the product through one slit. The tick on the two-slit bar is the same two tables summed at a tenth of a nanometre, 0.749: what the separate slits leave is not the grid's.

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.

7 figures

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.

6 figures