A correction travels with its lines

A finer grid keeps the price of separating priced a choice every colour engine makes. It can store a sample’s reflectance and each lamp’s spectrum as separate tables and multiply them when a colour is needed, which lets one measurement of a sample serve under any lamp; or it can measure the sample’s reflected light under each lamp — the product — which is right only for that lamp. Under a fluorescent tube, on the site’s five-nanometre grid, the best pair of separate slits missed sixty-eight notch filters by 0.26 ΔE₀₀ on average and the product by 0.016: separating cost fifteen times, on every grid tried.

That essay closed on a middle course. Measure the product once, under a representative of a class of lamps; store the ratio of the product’s colour to the separable colour with the sample’s reflectance; apply the class’s ratio whenever the sample is used under a lamp of that class. One extra measurement per class, three extra numbers per sample. The prediction was that this recovers most of the factor of fifteen within a class — another tube with the same phosphors and lines of slightly different widths, or a different mix of the same mercury lines — and fails across classes, under a laser, because the error being corrected comes from where the lamp’s lines sit against the sample’s structure.

Most of the way for some tubes, and not for others #

Under tubes whose mercury lines are 0.8 or 2 nanometres wide instead of 1.2, or whose phosphor bed sits ten nanometres longer, the census tube’s correction removes 89 to 93 per cent of the separable error, leaving 0.019 to 0.028 against the product’s 0.016. Under a tube with the same four lines in a different mix it removes 63 per cent, and its worst notch is off by 0.49; along the way from one mix to the other the share falls at every step, to 35 per cent half as far again. Under a white LED, a three-emitter LED and a laser projector it leaves every lamp worse than no correction — the white LED five times worse.

• The prediction holds for the lines’ width and the phosphors’ position: the correction travels.
• It fails for the lines’ mix, which the proposal counted as within the class.
• Across classes it does not merely fail; it harms, because it carries a correction to tables that needed none.
• A class of lamps, for this purpose, is a set of line strengths, and a tube is in the census tube’s class to the extent its lines carry the same shares of power.

Eight lamps, one correction #

The census is the earlier essay’s: notch filters five, eight, twelve and twenty nanometres wide, centred every two nanometres from 530 to 562 — across the tube’s green mercury line at 546.1 — and each one’s colour computed three ways under each lamp. Separable is the best pair of the earlier census, the lamp tabulated through a five-nanometre slit and the sample through a one-nanometre slit, multiplied on the five-nanometre grid. Product is the reflected light tabulated through the five-nanometre slit and divided by the lamp’s table. Truth is the colour integrated on a tenth-of-a-nanometre grid with no slit at all.

The correction is three numbers per notch: the product’s X, Y and Z over the separable table’s, measured under the census tube. Under the census tube itself it turns the separable colour into the product exactly, and the error falls from 0.256 to 0.016. That is the point of it — it is the product measurement, stored in a form a separable table can carry.

The seven other lamps test how far that form travels. Four are tubes: the same three phosphor bands and four mercury lines with the lines 0.8 nanometres wide, or 2, at the same power; with the lines in a different mix; with the phosphor bed ten nanometres longer. Three are other kinds of lamp: a three-laser projector, a white LED with one phosphor, and a three-emitter LED.

What the lamps look like #

The census tube is three broad phosphor bands under four narrow lines: violet at 404.7, blue at 435.8, green at 546.1, yellow at 578.0. The notches all sit near the green line, and the separable table’s error comes almost entirely from it. The line can be in the sample and three numbers cannot see a line are the collection’s account of why: the lamp tabulated through a five-nanometre slit smears the line’s power across a slit’s width, the sample tabulated through a one-nanometre slit keeps its notch sharp, and their product on the grid mis-weights the notch where it meets the line.

The re-mixed tube carries the same four lines with the green one weaker, 1.7 against 2.4, and the violet, blue and yellow stronger. The moved bed shifts the phosphors ten nanometres longer and leaves the lines where they were. The LEDs have no lines at all: a blue pump and one broad phosphor, or three broad emitters.

Width and position travel #

Changing the lines’ width barely changes what needs correcting. From 0.6 to 2.4 nanometres wide, at the power the census tube’s lines carry, the separable table’s mean error moves only from 0.266 to 0.233: a line narrower than the five-nanometre slit is smeared to the slit’s shape whatever its own width. The census tube’s correction, measured at 1.2, keeps the corrected mean under 0.04 across the whole range, 0.016 at 1.2 and 0.036 at 2.4, and removes 85 to 94 per cent of the separable error.

The worst notch grows either side of 1.2 — to 0.14 at 0.6 nanometres and 0.31 at 2.4 — because the correction was measured on a line of one particular width and a notch sitting exactly on a line sees its width. For most notches, which sit off the line’s centre, the width does not matter.

Moving the phosphor bed barely matters either: 93 per cent recovered. The error being corrected is the line’s, and the phosphors under it are smooth enough that a separable table handles them without help.

The mix does not travel #

Re-mixing the lines is another matter. Moved a quarter of the way from the census tube’s mix to the re-mixed tube’s, the correction removes 89 per cent; half-way, 82; three quarters, 74; at the re-mixed tube, 63; half as far again, 35. It falls at every step, and faster the further the mix goes.

The reason is what the stored ratio is. For each notch it records how far the separable table’s colour sat from the product’s under the census tube, and that distance is set by how much of the notch’s colour came through the green line and how badly the separable table weighted it. Under a tube whose green line carries seventy per cent of the census tube’s power, the separable table’s error at the line is smaller, and the stored correction — sized for the stronger line — overcorrects it. Under a tube whose yellow line is stronger, notches near 578 carry an error the census tube barely had, and the stored correction is too small.

The proposal counted a different mix of the same mercury lines as the same class. For the correction’s purposes it is not. A class is the set of line strengths the correction was measured with.

Two members of a class #

The failure is not symmetric, and two measurements go most of the way to fixing it. The re-mixed tube’s own correction, carried back to the census tube, removes 74 per cent of the census tube’s separable error — more than the census tube’s correction removes under the re-mixed one, because the census tube’s stronger green line leaves a larger error for any roughly right correction to take out. Neither end is a good representative of the other.

A correction averaged from the two — the census tube’s three ratios and the re-mixed tube’s, notch by notch — removes 85 per cent of the error under the census tube, 79 under the re-mixed one, and 93 under the tube half-way between them, where it leaves 0.016, the product’s own error. It gives up a little at each end, where each end’s own correction would have been exact, and gains a great deal in the middle, where neither end’s would.

So a class represented by two members that bracket it serves every lamp between them about as well as a class of identical tubes serves itself. The cost is a second product measurement per class. What it cannot do is serve a tube outside the bracket: the average is still a correction for a particular range of line strengths.

Where the re-mixed tube goes wrong #

Away from the line the correction still works; on the line it leaves the most. For eight-nanometre notches under the re-mixed tube, the separable table is off by 0.35 at 540 nanometres and 0.63 to 0.65 at 546 and 548, where the notch sits on the green line. Corrected with the census tube’s ratios, those become 0.15 and 0.24. At 530 and at 560, where the notch misses the line, the separable error is nearly nothing and the correction leaves it there.

So the correction’s failure is concentrated exactly where the separable table’s error was largest, and where a correction was most wanted. The slit is what makes it legal found the notch sitting on a line the case every tabulation rule is written for; a correction carried between lamps inherits the same case.

The ratios an LED would need #

Under an LED the separable table hardly needs correcting. Its spectrum has no line — a blue spike and a broad phosphor, the shape a lamp is not a blackbody set beside the tube’s — and the product and separable colours of every notch agree to within four hundredths of a per cent in each tristimulus value; its separable error over the census is 0.051, already within three times the product’s. The census tube’s ratios depart from one by up to six tenths of a per cent where a notch meets the green line. Carried to the LED, they apply a correction about fifteen times larger than the LED’s table needed, in a pattern tied to a line the LED does not have, and the error rises from 0.051 to 0.256.

The three-emitter LED goes the same way, from 0.12 to 0.30. The laser projector, whose separable table is already badly wrong at 2.34 because its lines are a fifth of a nanometre wide — the narrowest of the primaries a screen is a poor lamp priced as a light — is made slightly worse, 2.60: the correction is tuned to a line at 546 and the laser’s green line is at 532.

This is what “fails across classes” turns out to mean. Not that the correction is useless outside its class but that it is harmful, in proportion to how much less correction the other lamp needed. A second slit buys a quarter found separate slits a partial repair; a correction carried to the wrong lamp is a repair applied to something that was not broken.

What a colour engine should store #

A correction per class, where a class is a mix of lines. Tubes of one make and phosphor blend share their line strengths closely, because the mercury lines’ ratios are set by the discharge and the blue and red phosphors’ by the blend; within such a family the correction travels, and it survives the tubes’ lines being wider or narrower as they age or run hotter. Tubes of different blends — a halophosphate against a triphosphor — are different classes even though both are “fluorescent”.

And a flag for whether a lamp has lines at all. The correction’s harm under the LEDs comes from applying it where the separable table was already close to right. A colour engine that knows the lamp is smooth should apply no correction; one that cannot tell should not apply a tube’s. Flicker sorts lamps the wrong way is the collection’s reminder that telling a lamp’s class from the lamp is itself a measurement with an error.

How the correction was computed #

Lamps are the collection’s analytic tube — three Gaussian phosphor bands and four mercury lines 1.2 nanometres wide — with the lines’ width changed at constant height-times-width, their four heights moved linearly between (0.55, 1.85, 2.40, 1.10) and (0.8, 2.4, 1.7, 1.4), or the bands’ centres moved by ten nanometres; and the collection’s laser projector, white LED and three-emitter LED. Notches are the collection’s notch filter at widths 5, 8, 12 and 20 nanometres and centres every 2 from 530.1 to 562.1. The separable colour multiplies the lamp through a triangular slit of half-width 5 and the notch through one of half-width 1 on the 5-nanometre grid; the product divides the reflected light through the 5-nanometre slit by the lamp’s table; truth is on a 0.1-nanometre grid with no slit. Colours are CIELAB against each lamp’s own white, differences CIEDE2000. The correction for a notch is the product’s X, Y and Z over the separable ones under the census tube, multiplied into the separable X, Y and Z under the other lamp.

What this leaves out #

The correction is three ratios per sample per class, applied to tristimulus values. A correction stored as a spectrum — the product’s table over the separable one, grid point by grid point — would carry more, and would be exact under any lamp whose lines sit where the census tube’s do; it would also cost a table per sample per class rather than three numbers.

The notches all sit near one line. A sample whose structure sits near the blue or violet line would see those lines’ strengths, and the re-mixed tube’s stronger blue line would move its correction in a different direction.

Real tubes differ in more than one thing at once. Their lines’ widths, their mix and their phosphors change together from one make to another, and the sensitivities here are one at a time.

Still open: whether a correction per line survives a re-mix #

The correction fails under a re-mixed tube because it is one number per tristimulus value for the whole lamp, and the error it corrects is several lines’ errors added. If the lamp’s lines are known — and a tube’s mercury lines are always at the same wavelengths — the error could be stored per line.

The calculation is a correction split by line: for each notch, the product-minus-separable difference computed separately with each of the census tube’s four lines alone added to its phosphor bed, stored as four vectors, and recombined under another tube by that tube’s own line strengths. The prediction is that this recovers more than ninety per cent of the separable error under the re-mixed tube, as the single correction does under width changes, because the per-line error depends on the line’s power linearly and on the notch’s position not at all. If it does, the class a correction needs is not a mix of lines but the lines themselves, and a colour engine could carry a tube correction to any tube it knows the line strengths of.

A correction is a measurement of a particular error #

The habit is about what a stored correction knows.

The correction here is exact under the lamp it was measured under, because there it is simply the product measurement in another form. Everywhere else it is a claim that the error it recorded is the error that would occur. Under tubes whose lines are wider or narrower that claim is nearly true, because the error depends on the lines’ power and not their shape. Under a tube whose lines have different powers it is partly false, and under a lamp with no lines it is entirely false.

The failure mode is to treat a correction as a property of the sample, when it is a property of the sample under a light. Stored with the sample it looks like a fact about the sample, and it travels only as far as the light it was measured under travels.