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

Spectrophotometry — where it appears

Measuring reflectance or transmittance band by band rather than as three numbers, which is the only way to predict a sample under a light it was not measured under. The instrument's own bandwidth, sampling interval, geometry and lamp are all part of the answer.

Named by 35 essays across 6 fields — each of them below, with the objects they name alongside it.

A 20 nm bandpass, and what it does to the sample. The true reflectance, the same reflectance as a 20 nm instrument reporting every 20 nm returns it, and the slit function responsible, drawn at its own scale around 550 nm. The reconstruction is what every calculation downstream will use, and it differs from the truth by ΔE 2.97 under D65. Nothing in the file the instrument writes marks which values were measured and which were interpolated.

What the instrument reports

A spectrophotometer sees a sample through a slit of finite width, at a finite number of wavelengths. What it writes down is the truth convolved with its own bandpass, and nothing in the file says which values were measured and which were invented.

matching · Gamut One reflectance, four different infrared tails, and what the camera makes of each. The same visible reflectance continued past 780 nm to four different near-infrared values. A spectrophotometer reports only the left-hand part; an unfiltered sensor integrates all of it. The channel spread falls from 0.14 at a tail of 0.05 to 0.02 at 0.85, with nothing about the visible half changed.

Most things are pale in the infrared

A spectrophotometer stops at 780 nanometres and a black cotton shirt reflecting five per cent of visible light reflects more than half the near infrared. The measurement everybody has and the quantity a camera integrates are different quantities, and nothing in the first says so.

imaging · Capture What a coarse wavelength grid costs, by source. Colour error against grid size, for four sources through one reflectance. Daylight survives every grid tested: 0.67 ΔE00 even at 40 nm. A source with lines in it does not — the narrowband source reaches 16.2. The grid is not a property of the arithmetic; it is a claim about what the light has in it.

Five nanometres is a choice

Every integral here is taken in five-nanometre steps, and the interval has never had to be defended. Coarsening it to twenty costs daylight two hundredths of a colour difference and a fluorescent tube six and a half — and which way of coarsening is used decides a further factor of six.

matching · Gamut One tolerance decision, about pairs that agree less and less about the spectrum. Every point is a pair of samples that the reference observer reports as exactly ΔE00 1.0 apart — the same number, the same decision, the same line in the same specification. Along the axis is how far apart their two reflectances are. Up the side is the 95th percentile of what two hundred other eyes report. It runs from 1.13 to 2.32. The document records the horizontal line and not the axis it is plotted against.

A tolerance needs a second number

Two samples one colour difference apart can be read almost identically by everybody or two units apart by the worst-off twentieth, and which of those it is depends on how far apart their spectra are — a quantity every spectrophotometer has already measured and none of them prints. Adding it as a second field predicts the population three times better, and at the tolerances where it matters it is right about a third of the decisions the difference alone gets wrong.

difference · Metric What a halftone measures, against what it looks like. A 50 per cent screen, measured by an aperture that averages the patch and seen by an eye that blurs it first and compresses afterwards. At a coarse ruling the two disagree by ΔE00 11.1, and the patch looks 14.3 lightness units darker than it measures — the average of a concave function is below the function of the average, which is Jensen's inequality and not an illusion. The gap falls under a unit above 30 cycles per degree. That is a viewing distance, and the correction a press applies for dot gain has no distance in it anywhere.

Measured with an aperture, seen with an eye

A spectrophotometer averages a halftone patch and reports one reflectance. An eye blurs it first and compresses afterwards, and the average of a concave function is below the function of the average — so a resolvable screen looks darker than it measures, by fourteen lightness units at a coarse ruling and by nothing at a fine one. The correction a press applies is the same size and has no viewing distance in it.

applied · Delivery The same twenty-four samples, measured two standard ways. How far apart a 45°/0° instrument and a sphere with its gloss port closed are, on samples running from three per cent reflectance to seventy. The whole of the difference is the interface reflection — four per cent of the light, returned without ever meeting a pigment, thrown away by one geometry and collected by the other. It is the same four points in every row, which is why the disagreement is a property of how dark the sample is rather than of what colour it is: ΔE00 8.7 on the darkest samples against 1.93 on the lightest.

An instrument has a geometry

Every reflectance here arrives through a model with a bandpass, a sampling interval and no position at all. Real instruments say where they were standing, and the two standard answers disagree by ΔE00 8.35 on a dark gloss sample — a difference that adds rather than multiplies, and that no adaptation removes.

matching · Gamut A sample with two reflectance curves, and neither below one. The apparent reflectance of an optically brightened sample, measured under D65 and A. It exceeds 1 — the shaded band — which no reflector can do: more light leaves at these wavelengths than arrives at them, because the sample absorbs in the violet and re-emits in the blue. And the two curves differ, so the sample has no single reflectance to store. The effect drawn here is a floor: most of the excitation band lies below 380 nm, outside the range computed here.

A surface that is not a multiplication

Every argument here about what light does to a surface begins by multiplying two spectra together. A surface with a brightener in it takes light at one wavelength and returns it at another, so it is a full operator rather than a diagonal one — and it does not have a reflectance at all.

scene · Scene A fluorescent sample is a matrix, and a reflectance is only its diagonal. The Donaldson matrix of a brightened sheet: how much light leaves at each wavelength for light arriving at each wavelength. A reflecting surface has entries on the diagonal and nowhere else, which is exactly the statement that light leaving at 440 nanometres arrived at 440. The block off the diagonal is the fluorophore — it takes light between about 305 and 420 nanometres and returns it between 400 and 500, wherever in that band it was absorbed, which is why the block is a rectangle rather than a smear along the diagonal. A spectrophotometer that reports a reflectance is reporting the diagonal and folding the block into it at whatever weight its own lamp happened to give.

A reflectance is a diagonal

A reflecting surface returns light at the wavelength it arrived at, so its whole description is one number per wavelength. A fluorescent one returns it somewhere else, so its description is a square matrix — and the curve every instrument reports is that matrix's diagonal with the rest of it folded in at whatever weight the lamp happened to give.

light · Light Six sheets, four standard measurement conditions, and every pair of them. How far apart two measurement conditions put the same sample. Darker is further. The top row has no brightener in it and every pair agrees to within a unit except the ones involving M₃, whose difference is cross-polarisation removing the interface reflection and is a change of geometry rather than of spectrum. Every row below it disagrees, and the M₁ against M₂ column reaches ΔE00 7.7 — several times any tolerance a printer would accept, on one sheet measured twice by two instruments that are both working correctly. The numbers under the columns are the means.

An instrument brings its own light

ISO 13655 names four measurement conditions and they are usually read as four degrees of care. They are not. They are four different quantities — on an unbrightened sheet all four agree to a rounding, and on a brightened one they are up to seven and a half units apart, with every instrument in calibration and every answer correct.

applied · Delivery An instrument, as the only thing it really is. The 3 filters a bank of that size puts across the visible range, each drawn against wavelength. Everything the instrument can report about a spectrum is 3 numbers — the integral of the light against each of these — so the set of spectra it cannot tell apart is everything orthogonal to all 3 of them, which is 78 dimensions of the 81 this site works in. Three of these is a colorimeter in spirit; the eye is three of them too.

Three numbers cannot see a line

An instrument that returns three filtered readings of a spectrum determines a three-dimensional projection of it and is exactly blind to the other seventy-eight. On daylight that costs almost nothing; on a fluorescent tube, three quarters of the lamp lies in the part no reading reaches, and adding filters recovers it slowly.

light · Light The colour is right long before the spectrum is. The colour error of the projection, against the number of readings. At twelve readings the fluorescent tube's colour is right to 0.48 ΔE00 while 66% of its spectrum is still unmeasured. That is the trap in one line: a reconstruction good enough to pass any colorimetric check will predict a match under a second illuminant that does not happen, because the part it got wrong is exactly the part a different lamp weights differently.

The colour is right first

A reconstruction of a lamp from twelve filtered readings gets its colour right to half a unit while two thirds of its spectrum is still unmeasured. That combination is not a partial success — it is the exact condition under which a spectral prediction made from the reconstruction will be confidently wrong.

light · Light The share of a sample's reflectance an aperture recovers, by how wide it is. Six materials, and the fraction of each one's true reflectance that a measurement recovers through an aperture of the stated radius. The horizontal axis is logarithmic in millimetres; the vertical is a share, so 1.0 is the whole of it. The dashed line is a 4-millimetre radius, which is about what a hand-held spectrophotometer has. At that aperture coated paper reads 98 per cent of its own reflectance and candle wax reads 48. Every curve approaches one from below and none of them reaches it: the kernel's tail is what is being cut, and it falls as one over the aperture rather than exponentially.

An aperture is a filter

A measuring aperture throws away the light that came back outside it, and the light that comes back furthest is the light at the wavelengths the sample absorbs least. So an aperture does not attenuate a translucent sample evenly — it attenuates its peaks more than its troughs, which is a filter whose transmission curve the sample itself decides.

applied · Delivery The share of a sample's reflectance an aperture recovers, by how wide it is. Six materials, and the fraction of each one's true reflectance that a measurement recovers through an aperture of the stated radius. The horizontal axis is logarithmic in millimetres; the vertical is a share, so 1.0 is the whole of it. The dashed line is a 8-millimetre radius, which is about what a hand-held spectrophotometer has. At that aperture coated paper reads 99 per cent of its own reflectance and candle wax reads 66. Every curve approaches one from below and none of them reaches it: the kernel's tail is what is being cut, and it falls as one over the aperture rather than exponentially.

Either disc can be the wide one

Every standard on translucent samples says to illuminate a larger area than is measured, and explains it by saying that light leaks out of the lit spot. That is true and it is not the reason, because the instruction works equally well the other way round — measuring a larger area than is lit gives a reading just as exact. The error is a product of two apertures and either one being wide kills it.

applied · Delivery Four departures from the model equation, each at an ordinary strength. What each of the four assumptions inside a colour integral costs, in ΔE₀₀, on a stated sample under a stated light. The wavelength index is a coated printing paper measured with and without the ultraviolet of D50; the range is the same paper integrated from 300 nanometres and from 380; the place index is a pigmented plastic through a four-millimetre radius; the direction index is an eggshell paint beside a window. The spread is a factor of 7.0. This is a ranking of four examples rather than of four departures — each of them can be made larger by choosing a more extreme sample, and the marble in the same collection of materials reaches 12.7 on the index that comes third here.

Which index to buy an instrument for

Four departures from the colour integral, ranked by what they cost and by what it would take to remove each one. The ranking by cost and the ranking by price are almost exactly reversed — the two largest are removed by specifying a lamp and by widening a table, and the two that need new hardware are the two smallest.

applied · Delivery 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.

light · Light The conditions under which an observer's departure is exactly zero. A departure of the observer is the pairing of something belonging to the observer with something belonging to the stimulus, so emptying either factor empties the product. The axis is logarithmic in what is left when the condition is imposed. Six rows empty the stimulus's factor — a perfectly neutral sample is the same colour for every observer, at any age and any field size — and two empty the observer's, since a gain on each cone and a change of basis are both absorbed exactly. All eight are identities rather than small numbers. The last two are the same two conditions imposed in a published cone space rather than in the observer's own, and they are worth eight and thirteen units: the identity is about the eye, and the arithmetic everybody uses is in somebody else's coordinates.

The instrument is one observer exactly

A spectrophotometer has no observer uncertainty. It computes through the 1931 tables to the last bit, reproducibly, forever — which is its whole value and is also why it cannot report the three colour differences of observer spread between itself and whoever is looking at the sample.

applied · Delivery 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.

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 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 Three bounds against the error they bound, over 68 notches under a fluorescent tube. Each notch placed across by its actual colour error from blurring the lamp and the sample separately, and up by a bound on that error, both on logarithmic scales; the dashed diagonal is where a bound equals the error, and a valid bound sits above it. Cauchy–Schwarz with the true window variances is above the diagonal on every notch, a median 14.7 times the error. Estimated from the blurred tables it falls below on 6 of 68, as low as 0.45 of the error. The Bhatia–Davis bound from the tables and declared ranges is above on every notch and a median 196 times the error.

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.

light · Light What declaring a narrowest feature buys, and where it stops being true. The median looseness of a Cauchy–Schwarz bound whose lamp variance is bounded by a declared narrowest feature, against the width declared, for a fluorescent tube and a three-laser projector. Each lamp's own Bhatia–Davis bound — the peak declared and nothing else — is the upper dashed line, and the bound with the true variances is the lower one. The marks are the width each lamp's lines actually have. Declaring it truly takes the tube from ×196 to ×86 and the projector from ×30 to ×14. The open circles are declarations the lamp does not meet, where the bound falls below the error.

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.

light · Light Narrowing the slit makes the declared-width bound looser, not tighter. The median ratio of the declared-width bound to the error it bounds, over sixty-eight notches under each lamp, with the slit and the grid narrowed together from five nanometres to one and each lamp declaring its true line width; the dotted lines are the bound with both variances known. Under the tube the declared bound goes from ×86 at five nanometres to ×253 at one, while the true-variance bound stays between ×12 and ×15. Under the projector the declared bound lies on its true-variance bound at every slit.

A narrow slit shrinks the error, not the bound

A colour engine given two separately blurred spectral tables can bound its own error if the lamp declares its narrowest feature, and under a fluorescent tube that bound sits 86 times above the error at a five-nanometre slit. Narrowing the slit was supposed to close the gap, because a comb of narrow features stops fitting in a narrow window. It does the opposite. At one nanometre the error falls 27-fold and the bound only sevenfold, so the bound is 253 times loose and certifies no notch within a colour difference. The looseness was never the comb. It is the lamp's one declared peak, charged to every window.

light · Light The ceiling a table reads for each window, against the lamp it came from. The fluorescent tube between 525 and 590 nanometres, with the green and yellow mercury lines, and for each window of a one- and a five-nanometre slit the highest the lamp could be there according to the table's own entries. The dashed line is the tube's peak, which the global declaration charges to every window. At one nanometre the ceiling follows the lamp, a median 2.05 times each window's true peak; at five it sits on the global peak for twenty nanometres either side of each line.

A narrow table can vouch for its own peaks

The bound a colour engine can put on its bandpass error is loose because a lamp declares one peak for its whole spectrum and the bound charges it to every window. A table measured through a slit about as narrow as the lamp's lines can do better: no feature of the declared width can be blurred below a computable fraction of its height, so the table's own entries cap how tall the lamp can be in each window. At one nanometre that ceiling is never below the truth and sits twice above it, where the global peak sat twenty times above. The bound halves, and for the first time it certifies notches within a colour difference. At five nanometres it buys nothing at all.

light · Light A finer grid shrinks both arrangements, and the gap between them stays. The mean error over sixty-eight notches of the best fixed pair of separate slits and of one slit on the reflected light at the grid's own step, on grids of one, two and five nanometres, under the tube and the projector. The tube's best pair falls from 0.256 to 0.019 and reaches the product's five-nanometre error of 0.016 without passing it; the product on the same fine grid is 0.0007. Under the projector the pair at one nanometre is 0.135.

A finer grid keeps the price of separating

A sample's reflectance and a lamp's spectrum measured through separate slits can be reused under any other lamp; measured together through one slit on the reflected light, they are fifteen times more accurate and belong to that one lamp. A finer grid was expected to close the gap, because a one-nanometre table barely blurs anything. It closes most of the distance to the old answer and none of the distance to the new one. On a one-nanometre grid the best separate pair under a fluorescent tube is as good as one slit on the product used to be at five, and one slit on the product on the same fine grid is twenty-nine times better again.

light · Light A correction measured under one tube, carried to seven other lamps. For each lamp, the mean error over sixty-eight notch filters of the separable table (upper bar), of the separable table corrected by the ratio of product to separable colour measured under the census tube (middle), and of the product measured under the lamp itself (lower). Under the census tube the correction is the product exactly; under tubes with other line widths or a moved phosphor bed it removes nine tenths of the error; under a tube with its lines re-mixed, 63 per cent; under the laser and the LEDs it makes things worse.

A correction travels with its lines

Keeping a sample's reflectance and a lamp's spectrum as separate tables costs fifteen times the error of measuring their product under a fluorescent tube. The middle course proposed was to measure the product once under a representative tube, store the ratio of product to separable colour with the sample, and apply it under any tube. Under tubes whose lines are narrower or wider, or whose phosphors sit elsewhere, it removes nine tenths of the error. Under a tube with the same lines in a different mix it removes under two thirds, and less the further the mix moves. Under an LED or a laser it makes things worse — a white LED five times worse than no correction. A class of lamps, for this purpose, is a mix of lines.

light · Light How well the table's entries around the 546 nm line fit each width of line. For the tube's green mercury line, 1.2 nm wide, the seven table entries around it fitted by a line of each width from 0.2 to 12 nm through the same slit on a sloping background, centre, height and background free: the best fit's miss. Through a one-nanometre slit the miss has a sharp minimum at the true width and only widths near it pass half a per cent; through two, a broader dip; through three and five the curve is flat from the narrowest width tried up to about two, and any of them passes.

A narrow table declares its own lines

A bound on a colour table's bandpass error needs one number from the lamp's maker: its narrowest feature, 1.2 nm for a fluorescent tube. Through a slit about as narrow as that feature, the table's entries around each line carry its width. Measured to half a per cent, a one-nanometre table's entries admit only line widths within four per cent of the truth, and a bound built on the narrowest of them is within two per cent of the bound on the maker's number. Through two nanometres the table still declares a usable width. Through three it cannot rule out a line five times narrower, and the bound it supports is the global one. The failure is as sharp as predicted, and never unsafe.

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 Wings raise the width a table declares, and it follows the line's half maximum. The width the tube's table declares for itself at half a per cent — the narrowest width any of its peaks admits — as every mercury line is given Lorentzian wings from nothing to half its core's width, through one and two nanometre slits, beside the Voigt line's own full width at half maximum. With no wings both slits declare under 1.2 nm; with wings a quarter of the core's, 1.36 through 1 nm and 1.41 through 2 nm, against a half-maximum width of 1.37. Through one nanometre the table declares nothing at the widest wings: every line is refused.

A line's wings widen the width a table declares

A narrow table declares its lamp's narrowest line width by fitting each peak with a Gaussian, and the worry was a line whose profile is not Gaussian: a mercury line broadened by collisions has Lorentzian wings on its core, every line shares them, and a Gaussian fit can only take the wings in by widening. It does. With wings a quarter of the core's width, the table declares 1.36 nm for a 1.2 nm core — close to the winged line's own half-maximum width, 1.37. The prediction called that the unsafe direction. It is not: the ceiling and the variance the earlier bound builds on the declared width never fall below the winged lamp's own, because through a slit a winged line behaves like the wider line the table says it is.

light · Light Where the ceiling on a self-absorbed line holds, and where it does not. In the windows within four nanometres of the 546 nm line, the smallest ratio of the ceiling the tables put on a window's peak to the absorbed line's true peak, built on the narrowest width the table admits for the line and on the emitted 1.2 nm, through each slit. On the admitted width the ceiling holds to a depth of three and falls below the peak at 3.5 (×0.981) and 4 (×0.885) through two nanometres. On 1.2 nm it holds through two nanometres at every depth.

A reversed line is read as one line

A strong mercury line in a dense discharge absorbs its own centre on the way out: at an optical depth of one its top is flat, and beyond that it dips, until at four the line is two humps with a trough eighty per cent deep between them. The prediction was that a one-nanometre table refuses such a line from a depth of about one and a two-nanometre table reads it as a Gaussian about the width of its flat top, safely. Both hold, and the second only up to a point. A two-nanometre table admits the line at every depth to four, and from three and a half the width it admits is too wide for the humps: the ceiling built on it falls eleven per cent below the line's true peak. The only line that could make a declared width unsafe was one narrower at its peak than its entries suggest — and a reversed line is exactly that.

light · Light Where a two-nanometre table's ceiling on a notched line fails. In the windows within four nanometres of the 546 nm line, the ceiling built on the narrowest width a two-nanometre table admits for the line, as a multiple of the absorbed line's true peak, behind columns a whole, a half and a third as wide as the line. The uniform column's ceiling crosses below one at a depth of 3.42 and ends where the table refuses the line; the half and third columns' cross at 2.34 and 2.35 and are never refused, falling to ×0.73 and ×0.80 at a depth of five.

A notch in a line is never refused

A dense discharge's green mercury line loses its centre to the atoms between the glow and the tube wall, and those atoms are cooler than the glow, so their absorption is narrower than the line it cuts. With the absorbing layer half or a third as wide as the line, a one-nanometre table still refuses the line from an optical depth of one, exactly where it refused the broad dip. A two-nanometre table never refuses it at all: to a depth of five, with ninety-eight per cent of the centre gone, one Gaussian fits its entries within a seventh of a per cent. And the ceiling built on the width it reads fails from a depth of about two and a third — a whole unit sooner than behind a column as wide as the line — and never recovers, falling to three quarters of the true peak.

light · Light Where a reversed tube's table declares a width its ceiling cannot stand on. With every mercury line behind its own column, in proportion to its strength, the ceiling built on the width the whole table declares — the narrowest any of its peaks admits — as a multiple of the reversed lamp's true peak in the worst window, through one and two nanometres behind columns as wide as the lines and half as wide. The one-nanometre lines stop where the table refuses every peak and declares nothing. Through two nanometres behind the cooler column the table never stops declaring, and its declaration is unsafe from a depth of two and a half, falling to ×0.69.

A reversed tube declares from its least reversed line

A spectral table declares its lamp's line width as the narrowest width any of its peaks admits, and that declaration survived every reversed line so far because the tube's other lines were left unreversed to set it. A real discharge reverses its strong lines together. With every mercury line behind its own column, in proportion to its strength, the prediction was that a one-nanometre table would refuse every strong line and declare from the weak violet line, and a two-nanometre table would declare an unsafe width from a green-line depth of two and a half. The violet line is never read as a peak at all. The yellow line, the least reversed of the three that are, sets the declaration at every depth and widens it as it reverses. Through one nanometre the table refuses the blue and green lines and goes on declaring from the yellow — unsafely, in the violet's window, the one line the table never checked. Through two nanometres behind a cooler layer it refuses nothing, and from two and a half its declaration fails in the blue line's window, down to seven tenths of the true peak.

light · Light A stricter fit makes the reversed tube's declaration fail sooner. Through two nanometres, with every mercury line behind its own column, the ceiling built on the width the table declares — as a multiple of the reversed lamp's true peak in the worst window — at fit thresholds of half, a fifth and a tenth of a per cent, behind columns half as wide as the lines and as wide. Every stricter threshold lowers the ceiling at every depth; behind the cool column it falls below the true peak from a depth of two instead of two and a half.

A stricter fit declares a wider line

A two-nanometre table of a mercury lamp whose every line is self-reversed admits every reversed line as a Gaussian within half a per cent, and declares a width its own ceiling cannot stand on from a depth of two and a half. The obvious repair is to fit harder. The prediction was that at a tenth of a per cent the table refuses the reversed blue and green lines, admits the least-reversed yellow, declares unsafely from it between — and that a threshold that strict would refuse even unreversed lines on a real instrument's noisy table. The refusals come where predicted: the blue from a depth of three, the green from four. But the stricter threshold makes the declaration less safe, not more. It narrows the range of widths each line admits from below, so the yellow, never refused, declares a wider width at every depth, and the ceiling built on it falls below the true peak from a depth of two instead of two and a half. And a tenth of a per cent survives a quiet instrument: no unreversed line is refused until the table's noise reaches half a per cent — and then the first line refused is the yellow, the one that declares.

light · Light One nanometre finds the yellow's notch; two find a spike. The depth of the notch a reversed-line fit finds at the reversed tube's 578 nm yellow peak, every line behind a column half its width, on the clean one- and two-nanometre tables at eleven column depths. On the one-nanometre table the notch is -0.02 of the line when nothing is reversed and grows at every depth, to 0.99. On the two-nanometre table it is negative at every depth — from -4.2 to -0.06 — a spike added to sharpen a wide Gaussian, not a notch cut from a reversed one.

Two nanometres hide the yellow's notch from any fit

A plain Gaussian fit admits the reversed tube's 578 nm yellow line at every threshold on a two-nanometre table, and the ceiling built on its width fails from a column depth of two and a half. A fit with the notch in it — an emitted Gaussian with an absorbed one taken out of its centre — might see what the plain fit cannot. The prediction was that two nanometres leave too few entries to determine the notch, and one nanometre finds it from a depth of two. Two nanometres hide it completely: the best notch fit carries a negative notch, a spike, at every depth, and the readings a two-standard-error rule calls reversed are the same noise draws at every depth, each fitting a notch that erases the whole line. One nanometre finds it sooner than predicted — in every reading from a depth of one and half of them at a half — and its fitted depth climbs with the column. But its standard error cannot be trusted with seven entries and four numbers: the rule calls a third or more of unreversed readings reversed. The notch's fitted depth itself separates them cleanly.

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