The thread: Computed, not quoted — page 25
What the eye does
One session has a second valley
Every essay on older observers since the second session has read its fitted lens bands and macula shifts from one census: nine simulated observers, three designs, twenty noisy experiments each, every fit a damped descent from two starts. Two of forty experiments refitted in the last essay found a better fit than the census had. The prediction was that the census misses one fit in twenty, almost all where the band is small, and that the misses change no count it reported by more than a few hundredths. Refitted from three more starts, 66 of its 540 experiments have a better fit — 45 of 180 under one session, 9 and 12 under two and three. Sixty of the 66 better fits move the macula's peak more than two nanometres further short, most shrinking the band, and 31 lie lower by under one unit of χ²: a second valley, nearly level with the first. Two sessions' counts stand to within one or two experiments. One session's do not: with the best fits an unmoved macula is reported moved in eight experiments of twenty, where the census said one.
What the brain does
The rim was never most of the glow
On brightened paper a halftone tint keeps a hue turn a reader fully adapted to the sheet cannot remove, and five essays have chased the part of it that lives at the ink's thin rim — merging spots, clustering them, growing them — down to a rule: the clump count sets the residue, and only coarser patterns lower it. The proposal was a screen coarse in the pale tints and finer as the tint darkens, and the prediction that six clumps on the tile beat the 150-line screen by more than half a degree below twenty per cent. They beat it by 0.11 to 0.17. Scheduled six, thirteen and twenty-six clumps save a third of a degree at ten per cent and give back a fifth at sixty. The reason is the floor. One clump on the tile — the coarsest pattern it can hold, nearly all its rim gone — still leaves 7.48 degrees of an orange's 8.14 at five per cent and 5.30 of 5.65 at sixty. What screening can move is the rim's tenth; the other nine tenths is bare paper glowing beside ink that does not.
What it takes to deliver it
Three nodes at the cap's corner cut the dulling by a third
A four-ink table printing a dark saturated colour between its nodes prints it duller than asked, and at 65 nodes a fifteen-unit knee in the ink limit carries 35 per cent of that dulling on a deep-blue ray. The prediction was that the knee's added dulling sits in the two or three segments just below where the cap binds, and that three nodes placed there take its share back under a tenth. The first half holds: the knee's addition peaks one unit of lightness below the binding point and four fifths of it lies within seven. The second does not. Three nodes placed one, two and three units below the bind cut the hard cap's dulling by a third, 0.0097 to 0.0064, and the knee's by nearly two fifths, where three more evenly spaced nodes cut each by under a tenth — but the knee keeps a quarter of the dulling. The hard cap has a corner, which one node can catch; the knee has a bend spread over a dozen units, which three nodes split but do not straighten. A node at the binding point alone helps only the hard cap, and leaves the knee 41 per cent.
What a scene does
A long floor never loses its colour to the finish
A satin finish on a room's grey faces costs its painted floor a little chroma when the room is narrow, and the shorter side at which the cost turns into a gain fell as the floor grew longer — 0.78 for a square, 0.64 for a floor one unit long, 0.51 for two. The prediction was that it settles near 0.45 by four units, the limit of a slot whose far walls pipe nothing. Solved at six strips a side, floors three and four units long gain chroma for every one of seventy-two paints at every shorter side from 0.35 to 0.65: there is no turn to find. The floor two units long already loses only in a window, shorter sides from about 0.3 to just past 0.5, and gains on either side of it; by three units the window has closed, and a four-unit floor gains more than a three-unit one everywhere. The loss was never the slot's. It belonged to the far pair of walls, and moving them away removes it.
What the brain does
A light ink can cancel the glow at one strength
A pale tint on brightened paper keeps a hue turn a reader fully adapted to the sheet cannot remove, and nine tenths of it survives every screen, because it is the tint's bare paper glowing beside ink that puts the glow out. A light ink — the same pigment diluted, printed over more of the paper — was the proposed lever, and the prediction that a quarter-strength ink lowers a five-per-cent orange's residue by a third. It does not lower it; it reverses it, from −7.2 degrees to +10.2. Somewhere between, the residue passes through nought, and for every one of eight pigments there is a strength that cancels it: 0.41 for the red, 0.54 for the orange, 0.80 for the blue and the cyan at a five-per-cent tint. The cancelling strength moves by under a tenth from five per cent to forty, so one ink per pigment keeps every pale tint within a degree and a quarter to twenty per cent, where full-strength ink leaves up to nine. The lever is real; it has a setting, and the light inks printers carry are diluted past it.
What it takes to deliver it
Shared nodes belong in the deep shadow
Three nodes placed just below where each dark colour's ink reaches the press's cap cut a four-ink table's dulling of that colour by a third — but a profile's nodes along lightness are shared by every colour, and the colours' caps bind anywhere from L* 15 to 31. The prediction was that one shared placement keeps two thirds of the own-placement gain on a deep-blue ray and all of it on a brown. Placed once, one to three units below the median binding point, the nodes keep 63 per cent of the blue's hard-cap gain and 92 of the brown's — and the averages hide a split. Every colour whose cap binds more than two and a half units below the shared nodes keeps under a twentieth of its own gain; every colour whose cap binds more than two units above them keeps more than all of it, up to four times. For those colours most of the dulling was never at the cap's corner: it lies in the deep shadow, where black keeps climbing under the cap, and the shared nodes sit there. Spreading the three nodes across the binding points buys nothing three even nodes would not.
Where the model breaks
A steep side splits mercury's lines at forty nanometres
Under a clear mercury lamp a pigment built from Gaussian absorption bands no narrower than forty nanometres falls short in fifteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, because a band's tail reaches the line it should pass. Real dye bands are often lopsided, and the prediction was that a band whose one side is half the other's, the steep side free to face either way, loses at forty nanometres about as many directions as a symmetric band thirty wide — nine — and helps only the directions that pass the green. It loses one. At thirty-five and thirty it loses none. In every one of the twenty-five fits the absorbing band turns its steep side to the line it must pass, and both kinds of direction gain: none of fourteen passing the green is lost, one of eleven passing the yellow. A single band forty nanometres wide with a steep side splits the two lines within 0.37 per cent; a symmetric band must narrow to about twenty-seven to do as well.
What a scene does
The window of loss closes from both ends
A satin finish on a room's grey faces costs a painted floor a little chroma only in narrow rooms near square, and a floor two units long loses it only in a window of shorter sides, from about 0.30 to 0.51, while a floor three units long never does. The prediction for the floor two and a half units long was that the window closes from below: a gain at a shorter side of 0.3 and a loss only near 0.4 to 0.5, because the lower edge is where the far walls' pipe is weakest. The floor gains at 0.3, loses at 0.4 — by under six thousandths of a per cent — and gains again at 0.5. The window closes from both ends at once, to 0.39–0.42, and its deepest point rises through nought near 2.6 units long. Every box is still all or nothing, every one of seventy-two paints with the same verdict, and every verdict has the sign of the finish's tilt across the floor's reflectance. The far walls' cost shrinks smoothly with their distance and centres on a shorter side of about 0.4.
Where the model breaks
A band must be nearly two to one before its steep side pays
Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide loses fifteen or sixteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, and the same bands with one side a third of the width and the other two thirds lose one. The prediction for milder skews was that the losses fall in step with a single band's error — to about eight at sides of 0.55 and 0.45, three at 0.6 and 0.4 — so that a three-to-two skew already halves them. It does not. The census loses thirteen at 0.55, ten at 0.6, five at 0.65 and one at two to one: three directions recovered per twentieth of skew at first, twelve at the last. The losses track one band's error against the one-per-cent line the census counts by, and a single band crosses that line only near a skew of 0.62. A dye's modest natural asymmetry buys a few directions; the steep side pays only when it is close to half the other.
Where the model breaks
A band is only as wide as its steep side
Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide must be skewed nearly two to one before it can split the green line from the yellow, and the prediction for bands five nanometres narrower was that a single band crosses one per cent of error near a skew of 0.57 and the census's losses fall to five or fewer by 0.6. The losses do fall to four by 0.6; the single band crosses one per cent at 0.555. But the forty-nanometre rule that read the losses off that one-per-cent line breaks at once, and what replaces it is simpler: a thirty-five-nanometre band at 0.6 and a forty-nanometre band at 0.65 have the same steep side, fourteen nanometres, and lose the same four directions by the same shortfalls to a hundredth of a point. Three widths and fourteen pigments fall on one staircase in the steep side alone. Searching every loss again from the milder skews' answers finds one direction the census missed, at both widths, and corrects the forty-nanometre counts to nine and four.
Difference and uniformity
Seven channels end the trade and lose the tube
A three-channel ambient sensor fitted to a room's forty-seven lights trades its error on LEDs against its error on lamps, and a fourth channel moved that trade without closing it. The prediction for more channels was that six, spread from 445 to 650 nm, bring every lamp, mixture and LED under a ΔEITP unit with the lamps within 0.0025 in the CIE 1960 plane, and that nothing short of eight reads the halophosphate tube. Six channels leave the 2,200 K lamp beyond a unit and the lamps at 0.0040; seven bring every lamp, mixture and LED in, with the LEDs and lamps both under 0.001, and the trade is gone — both families improve together. The halophosphate tube is read from six. But the triphosphor tube, held out of its own fit, is read only by three and four broad channels and lost by every finer sensor, its tint growing to thirty-three units at twelve channels. A sensor learns the lights it was fitted on, and a finer one extrapolates further from a light with no relative in the census.
Difference and uniformity
Relatives, not twins, teach a sensor a tube
An ambient sensor of seven evenly spread channels, fitted to a room's forty-seven lights, reads every lamp, mixture and LED within a ΔEITP unit with each held out of its own fit, and loses the triphosphor tube, which nothing else in the census resembles — thirty-three units out at twelve channels. The prediction was that four triphosphor siblings, the same phosphors mixed for 3,000 to 5,000 K, would bring the tube within a unit at seven and eight channels and stop its rise past eight. They do both: the tube reads at 0.04 with seven channels and 0.06 with twelve, and seven or eight channels then read all fifty-one lights. A twin is not what does it. Three siblings, every one warmer than the tube and the nearest sixty mireds from it, read it within a unit at every count from five. What the siblings cost is paid at the finest sensors: at twelve channels the Planckian lamps' worst error rises fivefold, while from five to eight it moves by under a quarter.
What the brain does
One light ink serves three papers within two degrees
On brightened office paper a pale tint keeps a hue turn even for a reader fully adapted to the sheet, and each pigment has one light-ink strength — the pigment diluted and printed over more of the paper — at which that residue vanishes. The prediction for a printer with several stocks was that the cancelling strength falls as the brightener loading rises, and that the office sheet's ink leaves a premium sheet's pale tints within two degrees and a lightly brightened sheet's within one. The strength does fall, at every step and for every pigment — by a fifth for the orange and the yellow, by under one per cent for the green. The office ink keeps the premium sheet within 1.8 degrees, as predicted, but the light sheet only within 1.8 too: it overshoots there, turning seven of eight pigments' residues past nought the other way, and falls short on the premium sheet for all eight. Either way it removes more than four fifths of what the full-strength ink leaves.
What the brain does
An ink for a fading sheet is set for its middle age
A light ink cancels a pale tint's residual hue turn on brightened paper at a strength that falls along a straight line in the sheet's glow, and a brightener bleaches with use, so a sheet's glow falls over its life. The prediction was that as the premium sheet's brightener bleaches, every pigment's cancelling strength rises back along that line, and that the fresh sheet's ink stays within two degrees until half the brightener is gone. Both hold. Bleached by a quarter, a half and three quarters, the premium sheet asks for strengths within 0.015 of the census line at its glow, always on or just below it; the fresh premium ink errs by 1.75, 1.80 and 2.35 degrees at worst. But the office sheet's ink, set for a glow the premium sheet passes through on its way down, keeps the bleaching sheet within 1.4 degrees at every stage after the first. An ink set for the sheet's middle age outlasts one set for its first day.
Where the model breaks
At one half-width a flank can lose nothing or nearly everything
Under a clear mercury lamp, a pigment of Gaussian absorption bands splits the green line from the yellow according to its steep side's half-width alone, and a Gaussian flank is fixed by its half-width — so the census could not say whether it measures the half-width or the absorbance left in the flank's foot at the line it should spare. The prediction was that the foot decides: at a half-width of fourteen nanometres, a Lorentzian steep side loses most of the twenty-five splitting directions and a sharper-footed one none. It does. A power-four Gaussian loses none, the Gaussian four, a squared hyperbolic secant nineteen and a Lorentzian twenty-two, in the order of their single bands' errors. But a single band's error is not the whole account: three profiles drawn at the half-widths that give a single band the same error — 7.73, fourteen and 39.1 nanometres — lose two, four and seven directions, and the widest of them loses directions the others keep, because its shoulder starts absorbing nearer the green.
What a camera does
Beside the locus the converter still gives the red away
On the Planckian locus, a white LED's red phosphor pushes its converter to a shorter peak, so an ambient sensor's 600 nm departure, which reads the converter, reads the red phosphor too — within ±0.12 of a spread of ±0.36 — and through it the camera's colour error within ±0.074 ΔE₀₀. Real LEDs are binned a few thousandths off the locus, and the prediction was that a Duv of 0.003 moves the converter about as far as a fifth of the red phosphor does, so that the line learned on the locus misreads off-locus LEDs by a fifth of the range and their error falls back to ±0.12. The converter moves under two nanometres. The line reads the off-locus LEDs' red within ±0.11 and ±0.12, as well as it reads the locus's own held out — with a bias of about a twentieth that follows the Duv's sign. The error reads a third to a half worse, ±0.096 and ±0.107, and most of that is the camera's error itself moving off the locus with nothing in the recipe changed.
What a camera does
Across a bin, the camera's error bends with the temperature
White LEDs a Duv of 0.003 off the Planckian locus have their red phosphor read by an ambient sensor as well as LEDs on it, and their camera colour error read a third to a half worse, because the error itself moves with the Duv. The prediction was that one Duv term in the error's line, learned over all eighty-nine LEDs, brings those beside the locus back to the locus's ±0.074 ΔE₀₀ and costs the locus LEDs nothing. One term buys little: ±0.109 below the locus and ±0.085 above. The error turns across the locus one way at 2,700 K and the other at 4,000 and 6,500 K, and bends most where it is warmest, so the Duv has to enter as a slope and a curve for each temperature — ten numbers a pump, not five — before both sides come back, to ±0.056 and ±0.048. That line then reads the locus LEDs at ±0.073, worse than the ±0.062 a line with no Duv in it manages once it is learned across the bin.
What light is
A 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.
Where the model breaks
A hundredth of Lorentzian undoes a Gaussian flank
Under a clear mercury lamp, a band pigment whose steep side is a Gaussian fourteen nanometres wide at half depth loses four of the twenty-five directions that split the green line from the yellow, and the same side drawn as a Lorentzian loses twenty-two. A real absorption band is usually a Voigt profile, a Gaussian core with a Lorentzian share in its wings, and the prediction was that a twentieth of Lorentzian doubles the Gaussian's losses and a fifth reaches nineteen. The direction is right and the scale is not. A hundredth of Lorentzian takes the pigment from four lost directions to eleven; a fiftieth to fourteen; a twentieth to eighteen; a fifth to twenty. The best-placed band leaves the passed line more than three half-widths from its centre, out in the wing, and there a wing one per cent Lorentzian carries twice the absorbance of the whole Gaussian. The count saturates near twenty while the worst direction's shortfall keeps growing, from 2.8 per cent at a hundredth to 51 with a full Lorentzian.
What light is
A 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.
Matching and measuring
A tone-mapped blue turns where it lands
In ΔEITP a display's blue corner, and the turn at which adding green to it changes which side of ICtCp's red–green axis it moves the colour to, both shift with the display's peak, because the quantiser reads absolute levels. A high-dynamic-range picture graded on a bright master and shown on a dimmer panel is tone-mapped, its blues delivered below the levels they were graded at. The prediction was that a blue at a tenth of full drive on a 1,000 cd/m² panel turns where a 100 cd/m² panel's full blue does, and that the corner's move differs, because the dimmed blue no longer sits at a vertex. The turn agrees within a hundredth of a nanometre. So does the corner, within half a per cent: the steps both are read from are directions at one absolute signal, whatever drive delivered it. The tone map's whole effect on the blue is where it puts the blue. Through ITU-R BT.2390's curve, a master's bright blues are gathered near a dim display's peak and share one turn there — two nanometres from where the master turned them — while blues under the curve's knee keep their turns exactly.
Matching and measuring
A tone curve per channel turns a sky towards cyan
A pure blue tone-mapped from a bright master to a dim display turns, in ΔEITP, wherever its delivered level puts it, and any tone curve that delivers it to the same level gives it the same turn. A picture's blue is not pure. The prediction for a blue mixed with a little green was that a tone curve applied to each channel separately raises the green's share by about half — the blue channel compressed above the curve's knee, the dimmer green left near it — turning the mixture's hue several degrees towards cyan and its turn a nanometre or more from where a curve that scales all three channels together puts it. The turn does move, by 1.6 to 3 nanometres on a 100 cd/m² display and up to 6.5 on a 300. But the share is not raised by half: through BT.2390 to a 100 cd/m² display, a blue graded with a tenth of green arrives with its green at 69 per cent of its blue, nearly seven times as much, and its ICtCp hue turns 23 degrees towards the green. The scaled curve keeps every mixture's hue within three degrees and its turn within half a nanometre of the pure blue's.
What it takes to deliver it
Easing the cap reaches the deep shadow and no higher
A four-ink table prints dark saturated colours duller than asked between its nodes, and for colours whose ink reaches the press's total-ink cap high up the dulling lies mostly in the deep shadow below L* 20, where black climbs under a held total. The proposal was to let the total fall below the cap there, and the prediction was that easing the cap by a tenth or a fifth between L* 20 and the floor halves those colours' dulling on even nodes — more than three shared nodes do — at a cost of a unit or two of L* in the darkest colour the press can reach. There is no cost: every eased table still prints the floor at L* 14. Eased by a fifth, the deep-blue ray's dulling halves and the brown's falls to an eighth. But the eight colours that bind above L* 26 keep three fifths of theirs: the ease removes two thirds of their deep-shadow dulling and none of what lies between their binding point and L* 20. And the eased cap does not let black rise less steeply — it makes black rise faster.
What it takes to deliver it
A higher ramp trades one shadow for another
Easing a four-ink table's total-ink cap by a fifth on a ramp from L* 20 to the floor halves the dulling of dark colours whose cap binds low, and leaves the colours that bind high with three fifths of theirs, because most of what is left lies between their binding point and L* 20, above the ramp. The prediction was that a ramp starting at L* 32 halves the high binders' dulling, at a small cost to the low binders. It does neither. Started at L* 26, the same fifth's ease does cut the high binders' upper dulling by more than half — the cap was its cause after all — but their deep-shadow dulling comes back to more than twice what the L* 20 ramp left, and their total is unchanged at 0.0052. The low binders give back nearly everything: on the brown ray from almost nothing to 0.0084 with the ramp at L* 32. Every ray, knee and group does best, or within a fiftieth of best, with the ramp at L* 20.
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