Worst case — where it appears
Named by 29 essays across 8 fields — each of them below, with the objects they name alongside it.
A mean is not a worst case
Every adaptation number this collection publishes is an average over objects, and the reader asking whether adaptation will fail them is asking about the object it fails on. That object costs between 1.9 and 4.0 times the published figure, and how uneven a change of light is across objects turns out to be a property of the change rather than a constant.
An extremum is still not a sample
Two rounds ago three measurements turned up that took a maximum over a sample of a set and were short by up to a factor of two. The same error was live in a fourth place the whole time, on the set of surfaces every adaptation number is averaged over, and it is short by up to a third.
Every worst surface sits on a declaration
Bounding the wall in a painted room produced a real worst case — the residual turns over at a band six nanometres wide because a narrower band returns too little light. Bounding the surfaces the residual is averaged over produces nothing of the kind, because all fourteen answers sit exactly on two numbers somebody typed and the one constraint that comes from the world never binds at all.
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.
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.
The ranking is not stable
On a red pigment under daylight the six observer departures run from 2.38 down to 1.20 ΔE₀₀. Over forty-two surfaces two of them change places, the top two separate, and every one spans between a factor of ten and a factor of thirty-five. A chart of six bars is a chart of one example.
A size is not a direction
An audit that reports magnitudes cannot say what two of them cost together. Over two and a half thousand pairs of observer departures the angle between them in the local metric runs from one degree to a hundred and seventy-nine, and on forty-three per cent of them the two together cost less than the larger of the two alone. Adding the angle predicts the composition to one and a quarter per cent; Pythagoras is out by twenty-eight.
A lattice has no derivative
Every departure priced here was priced by perturbing something and reading the answer, which requires the thing being perturbed to have a derivative. A file written on a code lattice does not have one — its output is flat almost everywhere and jumps on a set of measure zero — so quantisation can be bounded and never propagated. The bound is 1.15 colour differences at eight bits per channel against a mean of 0.19, and it is worst where the encoding spends fewest codes.
Two converters and one highlight
The clip is the only step in a raw pipeline that destroys information rather than moving it, and it is the step whose position varies most between converters. Below the sensor's ceiling its position changes nothing at all, exactly. Above it, clipping at the sensor and clipping after the matrix land twenty-one colour differences and seventy-eight degrees of hue apart, and which hues are affected is a property of the camera's own dyes.
The budget adds two units
This collection publishes a three-stage error budget for a colour-management chain and prints its sum. Two of the three stages are colour differences between stimuli and the third is a distance between appearances, and the conversion between those is not a constant. Re-measured in one unit over the same colours the chain has four stages rather than three, its end-to-end error is 5.05 against a sum of 7.66, and the profile's contribution where a job actually lands is 3.2 times its published average.
Quadrature is exact in one room
An observer allowance is built by adding the departures in quadrature, which assumes they are mutually perpendicular. Over fifteen pairs their angles run from 18 degrees to 179 and hardly any are perpendicular. Measured against the real combination, quadrature is too small in a cinema by eight per cent and too large under the sky by fourteen, crossing at an adapting luminance of 22 candelas a square metre — and on individual surfaces it is out by a third in both directions in every room.
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.
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.
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.
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.
A serif moves the error and adds little
When a scanned page is sharpened on its stored values, the error the eye sees lies on the dark side of every edge, so black sans-serif type does not gather it at its corners. The worry was that a serif changes that: every stroke end gains a bar and every bar an inside angle, the configuration where large letters did gather error. It does not. Black serif type's brackets hold three quarters of their share of the error and its serif tips half. Reversed, the serif tips hold two and a half times theirs. Across both, serifs lengthen the outline by a tenth and the error by a few per cent — they move it, and in black type away from the corners.
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.
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.
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.
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.
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.
A mercury lamp's few losses are deep
Every census of what a slow pigment can reach has counted the directions of the object-colour solid it misses by more than one per cent, and by that count a clear mercury lamp — four visible lines and nothing else — spares a pigment as well as three narrow emitters do. Read for depth, it is the one lamp of nine whose losses do not fall away: half of what a slope-limited pigment loses under it, it loses by more than five per cent, and its mean shortfall is half as large again as under the emitters or a triphosphor tube. Every one of those losses is the same kind of colour: one that reflects the mercury's green line at 546 nm and absorbs its yellow line at 578, or the reverse, 32 nm apart. The dark-share rule that ordered the counts orders the depths less well.
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.
A display's blue is sharp where the rotation is not
ΔE₀₀'s rotation term shears chroma into hue near a CIELAB hue of 275 degrees, and at a press's blue corner it had nothing to act on because the corner was blunt. The prediction was that a display's blue — an additive primary — would be a sharp corner, and that there the rotation would change how far a safety margin moves it by more than a tenth. The first half holds: sRGB's blue moves 3.35 times a CIELAB margin and BT.2020's 3.17, against the press's 1.25, and insetting the curved gamut directly confirms it to the second decimal. The second half fails, twice. A display's blue sits at 306 degrees, thirty past the rotation's centre, where the term grips a quarter as hard; and ΔE₀₀ blunts the corner to 1.42 by compressing its chroma sevenfold. Along the edge from blue to cyan the hue does pass through 275 and the grip reaches −0.87 — where the edge is nearly a face. Nowhere near either display's blue does the rotation move a prediction by four hundredths.
A pair of lines asks for a cliff, not a gap
Under a clear mercury lamp, every direction of colour a slow pigment loses by more than five per cent asks it to reflect one of the lamp's green and yellow lines, 32 nanometres apart, and absorb the other. The prediction was that a pigment of two absorption bands with a narrow gap between them — a peak no single transition can make — would recover those deep losses. No direction asks for a peak. The lamp's four lines leave only one transition within a hundred nanometres of another, so what a splitting direction needs is a cliff: one flank falling from reflecting to absorbing in the twenty-five nanometres between the green line's last strong sample and the yellow's first. One band at the census's depth takes 27.8 to fall that far; two stacked at one centre take 24.9. Searched from 680 starts, the census's own band model does exactly that — stacks its bands on one side of the pair — and reaches further than the census found in seventeen directions of twenty-five. What stays deep spends a band on the blue.
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.
The yellow is sharp where the tolerance unit does not compress
A safety margin moves each corner of a display's gamut inward by a multiple of itself that depends on the corner's shape. In CIELAB, sRGB's blue is its sharpest corner and its yellow among its bluntest; carried into ΔE₀₀, the order turns over, and the yellow becomes the sharpest corner of both sRGB and BT.2020. The prediction was that the yellow keeps its sharpness because its faces run along lightness, where ΔE₀₀'s weighting is largest at the yellow's L* of 97. Taking ΔE₀₀ apart length by length says otherwise. Its lightness weighting changes the yellow's move by nothing. Its chroma compression — the term that opens every other corner — does, on its own, bring every other corner of both displays under 1.35 times a margin, and leaves the yellow at 1.78. Of the yellow's three directions only one runs along chroma; at the other corners two or three do, and a metric that compresses chroma pulls those together and flattens the corner between them.
A harder search pulls the band pigments away from the slope limit
Every count in the pigment-limit censuses — how many directions of the object-colour solid a slow pigment cannot reach within one per cent of the ideal — compares a slope limit solved exactly with two band models searched from a few structured starts. Under the mercury lamp a search from 680 starts had already found the band model reaching further in seventeen of twenty-five directions. Searched from 594 starts under all nine lamps, both band models reach further in between three fifths and nineteen twentieths of the directions they lost, and their counts fall by a sixth: 395 to 335 for bands in nanometres, 370 to 311 for bands in energy, while the slope limit stays at 463. The rankings the censuses reported hold where they were stated. What moves is everything measured against the slope limit: its lead over the band models grows from a sixth to more than a third, the coating's odd ranking — bands losing more than the slope limit — disappears, the triphosphor tube passes the three emitters for bands in nanometres, and under the mercury lamp the band models' deep losses halve.
Named alongside it
The objects these essays reach for when they reach for this one.
Spectral structureBandpassBoundDeclared inputSpectrophotometryModelling assumptionWavelength gridFluorescentInstrumentMeasurement uncertaintySamplingSpecification