What a tabulation step costs, by light, on vermilion
Drawn at its defaults, in the grid the integral is summed over. It takes 18 options, and the essays below show what each placement asks of it.
Called by 42 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.
What light isThe 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.
What light isA 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 uniformityA 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.
What light isWhere 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.
What light isTwo 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.
What light isWhich 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 uniformityThe 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.
What light isThe 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.
What light isA 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 doesThe 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.
What light isA 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 breaksThe 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.
What light isThe 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.
What light isThe 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.
What light isTwo 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.
What light isA 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.
What light isOne 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.
What light isA 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.
What light isA 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.
What light isThe 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isA 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.
What light isTwo 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.
What light isThe notch fit declares the true width, and the true width runs out
A lamp's table declares the narrowest line width its peaks admit, and builds on it a ceiling for how high any line can rise between the entries. A reversed tube breaks both: its lines are notched by cooler gas, a plain fit reads each notched line as a wider one, and on a one-nanometre table the ceiling falls below the true peak from a column depth of a half, until at two the table refuses every line and declares nothing. A fit with the notch in it reads the yellow's notch on that table, and the proposal was to declare the line it finds underneath — notch taken back out — with the prediction that the ceiling then holds to a depth of three. It holds to two. The notch fit returns the tube's own 1.2 nm core within two hundredths of a nanometre at every depth, clean and in noisy readings, and the ceiling built on it is safe to two and fails from two and a half, in the window beside the reversed blue line. So does the ceiling built on the true width itself, at the same depths to within a hundredth. The repair is perfect, and what fails past two is the ceiling's rule, which assumes the lines it bounds are not notched.
What light isA ceiling that knows the notch holds to the deepest column
A lamp's table promises a ceiling for how high any line can rise between its entries, and for a reversed tube — every line notched by a cooler column — that ceiling fails on a one-nanometre table from a column depth of two and a half, because its rule assumes the line it bounds has no notch. The notch fit already reads the yellow line's notch, and the proposal was to build the ceiling for a notched line instead: in each window, the factor for a slit-blurred line carrying that window's notch. The prediction was that it holds to a depth of four. It holds to five, the deepest column the census runs, at every depth and in every one of 440 noisy readings. The fit's depth must be read as an optical depth, −ln(1 − d); taken as the first-order depth it fits as, the ceiling fails from four and a half. Part of the safety is the fit's own error: read that way, a deep notch comes out too deep — 4.25 for a true 2.29 at depth five — and an overstated notch only raises the ceiling.
What light isA coarse table can borrow a notch, but not a phantom
A one-nanometre table can bound a reversed lamp by building its ceiling for the notch its own fit reads. A two-nanometre table cannot read the notch, and its ceiling, built for an unnotched line, fails from a column depth of three and a half. The proposal was to borrow the notch from the finer table, and the prediction that the coarse ceiling then holds to five with a wider margin. It does: 8 to 42 per cent above the reversed lamp's true peak at every depth, against the fine table's 1 to 29. But borrowed from the fine table's noisy readings, one notch in 440 sinks the coarse ceiling below the true peak — a notch of 0.13 the fit read in noise where there was no column at all. Under a wide slit a shallow notch flattens a line's top, two flanking entries see more of it, and a ceiling that assumes the notch is set lower; the phantom set it too low. Borrowing only notches the depth rule calls — deeper than 0.2 — keeps all 440 safe; borrowing those more than twice their error does not.
What light isA wide table needs no notch to borrow
A two-nanometre table of a reversed tube cannot read the notch a cooler column cuts in its lines, and its ceiling, built for an unnotched line, fails from a column of three and a half; borrowing the notch a one-nanometre table read rescues it at every depth. The prediction for coarser tables was that a shallow notch lifts their flanking-pair factor further — to a fifth at five nanometres — and that the five-nanometre ceiling on a borrowed notch still holds to five. Both hold: the rise is 15, 18 and 20 per cent at three, four and five nanometres, and every borrowed ceiling stays above the true peak. But a table three nanometres wide or wider never needed the notch. Its unnotched ceiling stands at least 4.7 per cent above the true peak at every column depth, because each of its entries carries so little of any line's peak that the ceiling built on them sits high above every line, reversed or not — and borrowing the notch takes up to twenty points of that margin away.
What light isA broad line brings the notch back to a wide table
A table of a reversed tube read through a slit three nanometres wide or wider never needs the notch a cooler column cuts in its lines: its ceiling, built for an unnotched line 1.2 nanometres wide, stays above the true peak at every column depth. The reason offered was that a wide entry carries too small a share of a narrow line's peak to be fooled, and the prediction was that broader lines undo it — that at a core of 3.6 nanometres the five-nanometre table fails from a column of about three, and that a table needs the notch wherever its slit is less than about twice the line's core. The five-nanometre table fails from a column of three, exactly. Across twelve tables of three cores and four slits, the column at which the unnotched ceiling first fails rises with the slit-to-core ratio — from a half at 0.56 to three and a half or four near 1.7 — and every table at a ratio of 2.5 or more holds; but a ratio of 2.08 still fails, so the threshold sits a little above two. With the true notch, every table holds.
What light isWings make a broad line easier to bound from its core
A spectral table read through a wide slit can still bound a reversed mercury tube's true peaks with a ceiling built for an unnotched line, so long as the slit is more than about two and a half times the lines' Gaussian core; at a ratio of 2.08 a five-nanometre table fails from a column depth of 3.5. A real discharge line has Lorentzian wings, which widen the width a table declares, and the prediction was that wings move that threshold out in proportion to the declared width, while a ceiling built on the declared width keeps it where it was. Both halves fail, the first in direction. With wings a tenth of a 2.4 nm core's width the five-nanometre table holds to 4.5, and with a quarter it holds at every depth, by eight per cent: wings take power from a line's peak and give it to the table's entry. A ceiling built on the wider declared width fails three of six winged tables that the core-built ceiling serves.
What light isWings rescue a three-nanometre table only to its deepest column
Lorentzian wings on a reversed tube's lines make every four-, five- and six-nanometre table's ceiling, built for the lines' 2.4 nm core, safer by eight to twelve points of margin a step, so that wings a quarter of the core's width carry all three to every column depth. The three-nanometre table, at a slit-to-core ratio of 1.25, is the one a bare broad line failed earliest, from a depth of two. The prediction was that it still fails with a quarter's wings, from about three, and holds at every depth only with a half's, because a narrow slit averages less of a line's skirt. The first half holds, half a step early: a quarter's wings fail it from 2.5. The second fails by a hair: with wings half the core's width the three-nanometre table holds to a depth of 4.5 and falls 1.1 per cent short at five. Its margin climbs by 6.6, 9.0 and 13.2 points a step — the smallest lifts of any table, from the lowest start.
What light isA ceiling built for the notch holds the three-nanometre table
A table of a reversed discharge lamp read through a three-nanometre slit cannot see the notch a cool column of vapour cuts into each line's core, and a ceiling built for an unnotched core — the bound on how high the lamp can peak inside each window — fails from a column depth of two, and even with broad wings at five. The prediction was that a ceiling built for each window's notched line holds at every depth with any wing, by more than two per cent. It holds at every depth with any wing. With bare lines it holds by three tenths of a per cent at its tightest, at a moderate column, beside mercury's violet lines. Knowing the notch costs the shallow columns and pays the deep ones, because a shallow notch flattens a line's top and a deep one splits it.