Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Three numbers — page 11

An infinite-dimensional spectrum is projected onto three cone responses, and everything colour science can do — and every way it fails — follows from that single collapse.
What each design buys each family of lamps. The camera's colour error under each lamp, estimated with the lamp left out, and each family's held-out error under four designs: the ratio of the 500 and 450 nm channels alone; one line on the ratio and a 640 nm channel for every lamp; two stages with each lamp's family given; and two stages with the family found by the nearest other lamp. The census's halophosphate tube is missed by 0.405 under the ratio and 0.011 under either two-stage design; the LEDs go from ±0.127 to ±0.111 with the family given and ±0.155 with it found. What a camera does

Two stages catch the tube and bill the lamps they misplace

A narrow-channel estimate of a camera's colour error missed the halophosphate tube because the census held nothing like it, and a line that learned the tube's family from four siblings cost every other lamp. The proposed cure was two stages: recognise a lamp's family, then apply that family's own line. Given each lamp's family, it works as predicted and better — the tube missed by 0.011 ΔE₀₀, the fifty-four lamps estimated to ±0.133 against the ratio's ±0.141. Found by the nearest other lamp in the sensor's departures, the family is right for the tube and wrong for five lamps, and those five, missed by ±0.41, take the whole census back to ±0.162. The gain is real; the classifier is where it is lost.

 Three weights of one letter, placed by the stem's width in pixels. The four capitals E, H, L and T at three weights — stems six, ten and fourteen cells thick on the same spacing — set in black at 12, 18, 24, 30 and 48 point on a 300-dot page, sharpened on stored values rather than on light and read through the eye at forty centimetres, pooled over four sub-pixel positions: how much of the seen error the junctions of bar and stem hold, over their share of the outline, against the stem's width in pixels. What it takes to deliver it

A junction gathers by the width of its stem

Black type sharpened on its stored values gathers its error at the junctions of bar and stem, and gathers more as the letter grows. The mechanism offered for that was the stem growing past the eye's pool, which predicts that size is only standing in for the stem's width in pixels. Drawn at three weights and five sizes, the junctions' share ranks with the stem's width at 0.97 and with the point size at 0.85, and where two weights have stems within six tenths of a pixel of each other their junctions agree to within a tenth. The width is nearly the whole story. The part it leaves over is small and one-sided: at the same stem width, the heavier letter gathers a little more in five pairs of six.

 Every corner of the press against its own tangent cone, at two samplings of black. For the nine end colours of the margin census, how far each moves per unit of a four-unit margin: as a perfect corner of its own tangent shape would, on the press walked at thirty-three black levels, and on the census's press of nine. At thirty-three every corner sits within seven per cent of its cone; at nine every edge moved a fifth or more further, and the half-black yellow 63 per cent further. Matching and measuring

Nine black levels sharpened the press's corners

Three essays measured how far a safety margin pushes the corners of a coated press inward, and found them sharper than anything a smooth solid would have: the solid yellow moving nearly four times the margin, every corner past a cube's. Read by their ink recipes, three of the nine corners are vertices and six are edges, and each has a shape the press model's own derivatives state exactly. Against that shape the edges moved a quarter to two thirds too far. The reason is not the press. It is how the press was built: as three-ink solids at nine levels of black with nothing between them, so that every face black helps to make was scalloped. Walked at thirty-three levels, every corner moves within seven per cent of its own tangent shape, three of the nine are rounder than a cube's corner, and the solid yellow moves 3.2 times the margin, not 3.8.

 From which starts a fit finds a lens band, two sessions. For a band of a tenth of a density at 415, 430 and 445 nm in a sixty-year-old's lens, the number of twenty simulated experiments of two sessions each in which a fit of all five numbers, started with the band's centre at each of 400 to 470 nm, finds the centre within twice its standard error. The ticks at the left edge are the same count for a fit started at the true observer: 18, 18, 18 for the three bands. What the eye does

A lost lens band goes to age or the macula

Two colour-matching sessions determine the centre of a band of extra absorption in an older lens to about five nanometres, if a fit can find it. Fitted in full from eight starts across 400 to 470 nm on twenty simulated experiments, two sessions find a band at 415, 430 or 445 nm from every start in fifteen to eighteen experiments of twenty, against eighteen for a fit started at the truth. The prediction was that a low band would be lost from high starts to the macula's flank. The start barely matters with two sessions, and a lost band does go somewhere — but a band at 415 or 430 goes to lens age, three to five years of it, and only a band at 445 goes to the macula. Every loss is a worse fit than the session's best, so eight starts find the band as often as the truth does. With one session the start decides.

 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. What light 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.

 Every corner's own shape, carried into ΔE₀₀, against the ΔE₀₀ census. For the nine end colours, the move per unit of margin a perfect corner of each one's tangent shape makes in CIELAB; the same shape carried into ΔE₀₀'s local metric at the end colour; and the move a margin of two ΔE₀₀ makes on the press walked at thirty-three black levels. The second and third agree at every corner, the edges to within a tenth of the margin and the vertices within a seventh; only the solid yellow is still sharper than a cube's corner. Matching and measuring

The tolerance unit opens every edge

On a finely sampled press every corner moves under a CIELAB margin as a perfect corner of its own shape would, the shape read off the press model's derivatives. A print buyer states the margin in ΔE₀₀, and the census in that unit had been explained one corner at a time. Carried into ΔE₀₀'s local metric, the same shapes explain all nine at once: each of the six edges' predicted move lies within a tenth of the margin of the census's, and each vertex's within a seventh. ΔE₀₀ opens the edges from 67–141 degrees to 123–157, nearly faces, because a unit along chroma is three to five CIELAB units at these colours. The solid yellow's cone opens only to 59 degrees and stays the one corner sharper than a cube's. Without its chroma term, ΔE₀₀ would leave every corner as sharp as CIELAB does, or sharper.

 How much of the halo the eye's own glare hides. For three pictures behind three diffusers, the share of a local-dimming panel's zones in which declaring each cluster's own floor misreads a five-per-cent grey step by more than ten per cent — as the panel delivers the steps, and with the veil the eye's own scattered light adds from every bright zone. The eye hides the whole misreading in six cases of nine, all three behind the window; what it leaves is behind the widest diffuser for the small highlight, and behind the two wider ones for the subtitles. Difference and uniformity

The eye's own glare hides most of the halo

A local-dimming panel's black can be declared as two floors, and the pair misreads only the diffuser's halo round the bright content — by up to a factor of 2.7. The halo sits next to the light, and light scattered inside the viewer's eye puts a floor of its own there. Added under every grey step, the eye's veil hides the whole misreading in six of nine cases: every window, and the highlight and subtitles behind the narrowest diffuser. It leaves a sixth of the misreading's share for a small highlight behind the widest diffuser, and a quarter for subtitles, all of it in the halo's inner half — not its outer edge, as predicted. The panel dims by a zone's brightest pixel and the eye veils by its average, so a line of text lights the backlight fully and the eye a quarter. The viewing distance cancels.

 How many directions each lamp costs a slow pigment, counted at three depths. For nine lamps, the share of the object-colour solid's directions in which a slope-limited pigment falls short of the ideal pigment's reach by more than one, three and five per cent. The broadband lamps' counts fall three- to eightfold from one per cent to five, and the emitters' and the triphosphor tubes' as much or more; the clear mercury lamp's falls least, from 14 per cent to 7, and at five per cent it costs more than the emitters' 3 and the tube's 2. Where the model breaks

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.

 What closing the counters adds, stem width by stem width. At each stem width, how much larger a share of the seen error the closed spacing's junctions and stroke ends hold than the open spacing's. The junctions gain on average 0.043 and the stroke ends 0.011; width by width the junctions' gain runs from -0.046 to 0.115. What it takes to deliver it

A closed counter adds a little to a junction

Sharpened on its stored values, black type's junctions — the inside angles where a bar meets a stem — gather error by the stem's width in pixels, and at equal widths a heavier letter gathers slightly more. A heavier letter differs in two ways: it is smaller around its stem, and its counters are narrower. Holding the stem and moving the spacing separates them. At equal stem widths, closing the counters adds about four hundredths of their share to the junctions and one hundredth to the stroke ends, and twice as much to E and H, whose bars face each other across a counter, as to L and T. The prediction was the counter, and the counter it is — by an amount no larger than a sub-pixel position moves it. Drawing the letters apart found the stem census's light H with its crossbar floating.

 Counting the paint's light instead of the lamp's orders the rooms not at all. The thirty rooms of the painted census on divided faces, each placed twice: by the send number weighted by the paint's own light arriving at the grey faces, and by the send number weighted by the lamp's light, against how many of 72 paints a grey finish colours. The lamp-lit number ranks the wall and floor rooms at 0.63; the paint-lit one at -0.07, and over all thirty at -0.02. What a scene does

The paint's own light does not order the rooms

A satin finish on a room's grey faces colours some paint in 28 of the painted census's 30 rooms, and the number meant to say which rooms gain most — how much of the lamp's light the finish sends to the paint — orders the rooms solved on divided faces at only 0.63. The rooms it misplaced return mostly the paint's own light, so the proposal was to count that instead. Counted, the paint's light orders the rooms at −0.07. In every room with its floor painted the finish sends the floor's light away from the floor, and every paint gains; with the ceiling painted the paint is the lamp and the two numbers are the same number. Even the room's own response, read off the solve with the paint in place, orders the counts only at 0.71. The census has to be solved room by room.

 The blue corner and its edges, in both units. For six colours between the press's cyan and magenta — the blue corner, four edges beside it and the blue corner with half black — the move per unit of margin a perfect corner of each one's tangent shape makes in CIELAB, the move a four-unit margin makes on the finely sampled press, the same shape carried into ΔE₀₀, and the move a two-ΔE₀₀ margin makes. In CIELAB every one is already blunt, under 1.3; in ΔE₀₀ every one is within a tenth of a flat face. Matching and measuring

The blue corner has nothing to twist

A safety margin's cost at each corner of a coated press is predicted by the corner's own shape, read off its ink recipe and carried into whatever unit the margin is stated in — for nine corners, none of them blue. At a hue near 275 degrees ΔE₀₀ has a rotation term that couples chroma and hue, and the prediction was that leaving it out would miss the press's blue corner by more than a tenth of the margin. At the blue corner the term is strong: ΔE₀₀'s local matrix correlates a chroma step with a hue step by three quarters. And it changes the prediction by three hundredths. The blue corner and its edges are already blunt in CIELAB — none moves more than 1.3 times a margin — and a corner that is nearly a face has nothing for a shear to twist.

 The census read on the whole room and on the painted faces. For the thirty rooms on divided faces, how many of 72 paints a satin finish on the grey faces makes more colourful, read as the census reads it — every face's light together — and read on the painted faces alone. Read on the room, 11 rooms sit part-way; read on the paint, 28 rooms gain for all 72 and 2 for none, and no room sits between. What a scene does

The paint itself gains all or nothing

The painted census counts how many of seventy-two paints a satin finish on a room's grey faces makes more colourful, reading the room's colour as all of its light, and eleven of its thirty rooms sit part-way. Read on the painted faces alone — the light a viewer looking at the paint receives — no room sits part-way: twenty-eight gain for every paint and two for none. The change on the paint is never more than a per cent and a half, where the room's reading swings from −18 to +57 per cent, so the census's middle ground was the grey faces all along. A box with its floor painted and the same box with its ceiling painted gain the same colour to a hundredth of a point while the light they gain differs by up to eighty-two points; neither send number orders the size of the gain, and the paint's own brightness does.

 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. What light is

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.

 Where a red window's veil leaves the halo tinted: σ 2. The 32 by 18 zones behind a window filling a fifth of a dark room drawn in red, with a diffuser of spread 2: the zones where the cluster declaration's misread floor, seen through the eye's red veil, tints a grey by more than one ΔEITP — 18.9 per cent of the screen, at worst 4.5 — and those still misread in lightness by more than ten per cent, 4.3 per cent. Difference and uniformity

A coloured highlight tints the halo it cannot hide

A local-dimming panel lights a zone for its brightest channel, so a pure red or blue highlight drives the backlight as hard as a white one and sends a fifth or a fourteenth of the light into the viewer's eye. The prediction was that a saturated highlight's veil would hide the halo's misread greys as a white one does, and add a colour. It hides far less. Behind the middle diffuser a white highlight hides the whole misreading; drawn red, blue, green or even yellow — nine tenths of white's luminance — it hides none of it. The yellow fails because a veil hides a grey step only in the cone signals it reaches, and yellow light reaches a fifth of the short-wave one. And where the veil does fall on a misread floor, it colours the error: the darkest greys are tinted by up to fifteen ΔEITP, and behind a red window, where the veil hides the lightness almost entirely, a fifth of the screen is tinted by more than a unit.

 How sharp each corner of two displays is, in CIELAB and in ΔE₀₀. For the three primaries and three secondaries of sRGB and BT.2020, the move per unit of margin that a perfect corner of each one's tangent shape makes in CIELAB and carried into ΔE₀₀ by its full local matrix. The blue is ×3.35 and ×3.17 in CIELAB, against the press's blue at ×1.25; in ΔE₀₀ it is ×1.42 and ×1.26. ΔE₀₀ brings every corner of both displays under ×1.9. Matching and measuring

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.

 Condensing and shortening each carry half of what closing the counters added. At equal stem widths, the mean gain in the junctions' share of the seen error from the extended letter to the condensed one at one height, from the tall letter to the short one at one width, and from the open spacing to the closed one, which changed both together. The first two are +0.023 and +0.023; together +0.047, against +0.043 for both at once. Each half is within one and a half standard errors of nought. What it takes to deliver it

Width and height each carry half the counter's gain

Closing a sharpened letter's spacing at a fixed stem width raised its junctions' share of the error the eye sees by about four hundredths, and the spacing moved the letter's width and height together. A type family's condensed and extended faces separate them: the counters across the letter narrow while its height and stem stay put. The prediction was that the condensed letters' junctions would carry the whole four hundredths. They carry half — 0.023 — with the stroke ends moving by four thousandths. Shortening the letter at one width carries the other half, also 0.023, and the two add to within four thousandths of what closing both did. Each half is found where its own counter lies — condensing gains most in the H, shortening in the E — and the T, which has no counter across it, gains nothing when condensed. Neither half, on its own, is larger than one and a half times the scatter a sub-pixel position produces.

 The best band pigment for a direction that splits the green from the yellow. Under the clear mercury lamp, the direction of the object-colour solid with the pattern −−+− over the 405, 436, 546 and 578 nm lines that the three-band pigment reaches least well, searched from 680 starts: its reflectance, 5.4 per cent short of the ideal, with bands at 607 nm, 427 nm, 607 nm; and with a fourth band, 4.3 per cent short. The fall between the green line and the yellow is made by stacking bands on one side of it, never by a narrow peak between two. Where the model breaks

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.

 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. What light is

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.

 What ΔE₀₀'s chroma length alone does to each corner of two displays. For the six corners of sRGB and BT.2020, the move per unit of margin their tangent shapes predict in CIELAB, with ΔE₀₀'s chroma length alone, and with its lengths along lightness, chroma and hue together. The chroma length alone brings every corner but the yellow under ×1.35; it leaves the yellow at ×1.78 and ×1.80, and the full metric at ×1.75 and ×1.78. Matching and measuring

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.

 How sharp each corner of two displays is, in three units. For the six corners of sRGB and BT.2020, the move per unit of margin that a perfect corner of each one's tangent shape makes in CIELAB, in ΔE₀₀ and in CAM16-UCS at the display's white, 64 cd/m² and an average surround. CAM16-UCS keeps the yellow the sharpest corner of both displays, at ×1.66 and ×1.66, and blunts the blue — the sharpest corner of sRGB in CIELAB — to ×1.17 and ×1.10, the bluntest of all twelve. Matching and measuring

An appearance model flattens the blue by its hue

A safety margin moves a corner of a display's gamut inward by a multiple of itself set by the corner's shape, and the unit the margin is written in decides the shape. In ΔE₀₀ the yellow is the sharpest corner of an sRGB and a BT.2020 display, because its sharpness lies across hue where ΔE₀₀'s chroma compression cannot reach it. CAM16-UCS compresses colourfulness less steeply, and the prediction was that it would leave the blue sharper than the yellow. It leaves the yellow sharpest, at 1.66 times a margin on both displays, and flattens the blue to 1.17 and 1.10 — the bluntest corner of all twelve. Its colourfulness does what the prediction said: on its own it would sharpen the blue past the yellow. Its hue scale undoes that. At a display's blue one degree of CIELAB hue is three and a half degrees of the appearance model's, and a unit that counts hue that finely there spreads the blue's directions apart until the corner is nearly a face.

 How sharp each corner of two displays is, in four units. For the six corners of sRGB and BT.2020, the move per unit of margin that each corner's tangent shape predicts in CIELAB, ΔE₀₀, CAM16-UCS and ΔEITP, the last with the white at 320 cd/m². ΔEITP keeps the blue the sharpest corner of both displays, at ×3.12 and ×3.24, where CAM16-UCS makes it the bluntest at ×1.17 and ×1.10. At the green, cyan, magenta and yellow the three tolerance units agree within a quarter. Matching and measuring

A quantiser keeps the blue a spike

A gamut margin moves a display's corners inward by multiples set by the unit it is written in. ΔE₀₀ and CAM16-UCS both blunt an sRGB or BT.2020 display's blue, CAM16-UCS to nearly a face, and the prediction was that ΔEITP — the difference built on ICtCp for high-dynamic-range television — would blunt it too. It does the opposite. With the white at 320 cd/m² the blue is ΔEITP's sharpest corner on both displays, moving 3.12 and 3.24 times a margin, about as far as in CIELAB; at the green, cyan, magenta and yellow the three tolerance units agree within a quarter, and at the blue they span a factor of three. ΔEITP's lengths alone would blunt the blue as ΔE₀₀ does. What keeps it sharp is the perceptual quantiser: ICtCp's chroma rows sum to nought, so under a response that compresses like a logarithm, dimming the blue primary is a step in intensity and nothing else. Replace the quantiser by a linear response and the blue falls to 2.15; by a logarithm, it rises to 3.39. A brighter display sharpens it further.

 How far the merged design misses each lamp it never saw. For fourteen lamps outside the census — radiators and daylights, tinted and filtered lamps, white LEDs varied in their phosphors, and a three-band LED — the camera's colour error less the estimate the merged design makes from the ambient-light sensor's departures, fitted on the whole census. The dotted lines are the census's own held-out error, ±0.13. The radiators, daylights and tinted lamps fall inside or on it; 4 lamps fall outside, every one of them an LED. What a camera does

New LEDs miss the line their cousins drew

An ambient-light sensor's three departures can estimate a camera's colour error under a lamp, and merging radiators and white LEDs into one smooth family brought that estimate to ±0.127 over fifty-four lamps, each left out in turn. Every one of those lamps had relatives in the pool. Fourteen lamps the census never saw test what the relatives were worth. The prediction was that new smooth lamps would land on the line and tinted incandescents fall off it by four tenths or more. The radiators and daylights land on it, within ±0.13 each. The tinted lamps do too — the pink-glass incandescent, placed among the LEDs by its nearest neighbour, misses by 0.12. The lamps that fall off are the new white LEDs: ±0.20, half again the census's error, two of them by a quarter. Their misses follow the sensor's 640 nm channel at a slope of −0.72, explaining four fifths of their scatter, because the line learned what that channel means from LEDs varied only in their blue pump. Adding the fourteen to the pool does not help.

 The blue corner in four units, as the blue primary moves towards the violet. For displays with red and green primaries at 630 and 530 nm and a monochromatic blue primary from 440 to 490 nm, the blue corner's move per unit of margin in CIELAB, ΔE₀₀, CAM16-UCS and ΔEITP. ΔEITP's blue is sharpest at 485 nm, ×4.37, and falls to ×1.23 at 440; CIELAB's runs the other way, from ×1.92 at 480 nm to ×11.8 at 440, and ΔE₀₀ follows it; CAM16-UCS stays near a face. Matching and measuring

A violet primary is no spike to the quantiser

ΔEITP makes a display's blue its sharpest corner, about three times a margin, and the account of why predicted that a shorter-wavelength blue would be sharper still — past four times by 450 nm. Moving a monochromatic blue primary from 490 nm to 440 does the opposite. ΔEITP's blue corner is sharpest at 485 nm, ×4.37, and falls at every step to ×1.23 at 440, where CIELAB's runs the other way, from ×1.92 at 480 to ×11.8, and ΔE₀₀ follows CIELAB. The spike needs the two primaries that can be added to the blue to move it in nearly the same direction. Adding red always raises the blue's L signal by the larger fraction. Adding green does so only while the blue carries enough M signal of its own; a violet primary carries almost none, so adding green raises its M by the larger fraction and the two steps split, from 37 degrees apart to 111. No cone response rescues it — a logarithm blunts the violet further — and a brighter display dulls it while sharpening the longer blues.

 A wider knee leaves the capped shadow quieter. For nine light-black separations under caps of 200 to 260 per cent on a seventeen-node table, the mean cast over every segment the cap reaches, by cap, and the mean cast of the segment where the cap is first reached, with the cap met hard and through knees of five, ten and fifteen units of lightness. The capped shadow's cast falls from 11.5 thousandths to 3.0; the binding segment's falls faster. What it takes to deliver it

A knee in the ink limit quiets the shadow

A cap on total ink coverage quiets a four-ink table's shadow, and the one segment where the cap is first reached is quiet or loud depending on where the binding point falls among the table's nodes: inside a segment it is quiet, within a quarter of a segment of a node it is not. A real cap is approached through a knee. The prediction was that a knee as wide as a segment would make the binding segment quiet wherever it fell, and that a knee much wider would turn the cap into a steep straight black that casts like the heavy-black strategy. The first half holds only in part: a five-unit knee halves the binding segment's cast and quiets three of the nine loud ones, but those whose binding point sits within a quarter of a segment of a node stay louder than a neighbour. The second half is wrong in sign. A wider knee leaves the whole capped shadow quieter at every step — its mean cast falls from 11.5 thousandths to 3.0 at fifteen units — because a knee spreads the chromatic inks' forced descent, and a slower descent is one the table's straight lines between nodes can follow.

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