The thread: Three numbers — page 13
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 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 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.
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.
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.
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 the eye does
A bound on the macula trades false moves for hidden ones
Fitted thoroughly, one session of an older observer's colour matches finds a second valley in a quarter of its experiments — the macula's peak moved ten or more nanometres from the model's, where no real macula sits — and reports a moved macula where there is none in up to eight experiments of twenty. The prediction was that holding the macula's shift within six nanometres either way, the range real pigment mixtures span, restores every observer's one-session counts to within two of the census's and leaves about one fit in four pressed against the bound. The false moves vanish: an unmoved macula is reported moved in one, two and one experiments of twenty for the three bands, against eight, six and two unbounded. Seven of nine observers keep their count of moved maculas within two of the census's. But only one fit in six ends on the bound, and beside the band at 445 nm the bound hides real moves: a macula truly three nanometres short is found in three experiments of twenty, where the census found seven. Two sessions barely notice the bound.
What the eye does
A soft prior on the macula frees the lens band
One session of an older observer's colour matches, fitted thoroughly, can drift into a valley where the macula's peak sits ten or more nanometres from any real eye's, and a hard bound at ±6 nm removes those false moves at the cost of hiding some real ones — beside a lens band at 445 nm, a macula truly three nanometres short is found in three experiments of twenty where the census found seven. The prediction was that a Gaussian prior four nanometres wide, which lets a fit go further when the matches insist, keeps the false moves away and restores those real ones. It keeps the false moves away — none, one and one for the three bands. It does not restore the real moves: it finds them as seldom as the bound or more seldom, two of twenty beside 445 nm, because it pulls every fitted shift towards nought, to a mean of 1.2 to 1.9 nanometres for moves of three. What it buys instead is the lens band. With the macula held near the standard, the band beside it is found in ten to twelve experiments of twenty, where the census found three to seven.
Difference and uniformity
Seven channels carry their noise as lightly as three
An ambient-light sensor of seven narrow channels, fitted to a room's forty-seven lights and four triphosphor siblings, reads every one of the fifty-one within a ΔEITP unit of a dim moon picture's tint, each light held out of its own fit. A fine sensor was expected to read chromaticity from small differences between neighbouring channels, with large weights of both signs that multiply every channel's noise. The prediction was that seven channels stay within a unit at a quarter of a per cent of channel noise and are no better than five at one per cent. The first half holds. The second fails completely: at one per cent seven channels still read every light's every one of a hundred readings within a unit, while five lose twenty-two lights. The premise was wrong. Seven channels' matrix barely cancels — its contributions sum with one sign — and per unit of noise it moves the reported chromaticity no more than three broad channels do.
Difference and uniformity
A tilt moves every light's reading alike
An ambient-light sensor of seven narrow channels reads all fifty-one of a room's lights within a ΔEITP unit of a dim moon picture's tint and keeps them there through one per cent of independent noise on every channel, because its matrix barely cancels. A drift shared by the channels does not average out: as a photodiode warms its response tilts, rising at long wavelengths and falling at short. The prediction was that a tilt reaching ±1 per cent at the ends of the range leaves the seven-channel sensor within a unit for every light and that the twelve-channel sensor, whose Z row cancels, loses more lights than seven at every tilt. Seven channels hold to ±2 per cent, the worst light at 0.56. Twelve loses two lights at every tilt — the two it misreads with no tilt at all — and none to the tilt. Every sensor's reported chromaticity moves by about seven ten-thousandths in u′v′ per per cent of tilt, every light nearly the same way.
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