Colour Science & Perception
what the eye settles, and what it cannot
Theme

The thread: The instrument is the reader — page 16

This is the one subject where the page is displayed on the apparatus under discussion, and the reader's own eye is the measuring device. Several figures here are experiments rather than illustrations.
Even two sessions fall into the far valley, and the prior lifts them out. For the observer with its band at 415 nm and its macula truly three nanometres short, every two-session experiment's fitted shift at the best of five starts and under a prior six nanometres wide. Three best fits lie beyond 7.5 nm short — −11.84 drawn to −1.90, −7.73 drawn to −3.05, −11.00 drawn to −0.94 — and the rest move by a tenth or two; the mean goes from −3.89 to −2.57. What the eye does

A second session leaves the prior one job

Under one session of an older observer's colour matches, a prior on the macular pigment's peak shift trades the lens band against real moves of the macula, six nanometres wide at the knee of the trade. Two sessions already find the band in twelve to twenty experiments of twenty and real three-nanometre moves in twelve to eighteen, and still report an unmoved macula moved in two to three. The prediction was that the six-nanometre prior, applied to two sessions, removes those false moves and costs nothing else. It removes none of them — two, two and none remain, about the rate at which a test at twice the standard error calls a move by chance. It costs nothing at five observers of six. Beside 415 nm, with the macula moved short, it finds four fewer moves: three of the best fits there were in the far valley, eight to twelve nanometres short, and counted as moves because their sign was right. With two sessions the far valley is the prior's only remaining job.

 Seven channels keep every light within a unit through the noise. Every light of the enlarged census read a hundred times by sensors of three, five, seven and twelve channels whose signals each carry independent noise of a quarter, a half and one per cent, each light by the matrix fitted without it: how many lights are read beyond a ΔEITP unit at least once. 3 channels: 40, 42, 46; 5 channels: 7, 9, 22; 7 channels: 0, 0, 0; 12 channels: 2, 2, 2. 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.

 Wings raise every core-built ceiling's margin over the peak. The reversed tube with every line a 2.4 nm Gaussian core convolved with Lorentzian wings, read through slits of four, five and six nanometres, each window's ceiling built for an unnotched line of the core's width: the least ceiling over the true peak at any column depth to five, against the wings' width. 4 nm: −5.7%, +2.2%, +13.0%; 5 nm: −11.6%, −3.3%, +8.0%; 6 nm: +7.3%, +15.4%, +27.8%. What light is

Wings make a broad line easier to bound from its core

A spectral table read through a wide slit can still bound a reversed mercury tube's true peaks with a ceiling built for an unnotched line, so long as the slit is more than about two and a half times the lines' Gaussian core; at a ratio of 2.08 a five-nanometre table fails from a column depth of 3.5. A real discharge line has Lorentzian wings, which widen the width a table declares, and the prediction was that wings move that threshold out in proportion to the declared width, while a ceiling built on the declared width keeps it where it was. Both halves fail, the first in direction. With wings a tenth of a 2.4 nm core's width the five-nanometre table holds to 4.5, and with a quarter it holds at every depth, by eight per cent: wings take power from a line's peak and give it to the table's entry. A ceiling built on the wider declared width fails three of six winged tables that the core-built ceiling serves.

 Read by size, the steepest ramp triples the brown's error. The same cases read by the mean size of the chroma error, overshoot and shortfall alike. On the ramp from L 17 the brown's low binders err by 15.72 thousandths against 5.04 from L 20; the blue's error is 5.19 against 5.21. What it takes to deliver it

A steeper ramp cures the shadow by overshooting it

A four-ink press's total-ink cap dulls the deepest colours it binds, and easing the cap by a fifth on a straight ramp from L* 20 to the floor halves the deep blue's dulling. A ramp bent to take more ink above L* 20 dulled every group more, so the cure seemed to follow how steeply the cap falls below L* 20. The prediction was that a ramp steeper still — the whole fifth spent from L* 17 — cures more. Read as the census has read every ramp, by the mean chroma lost, it does: every group dulls less at each step from L* 20 to 18.5 to 17. But the brown's low binders cross nought: their prints come out more chromatic than the colour asked, by as much as 0.07, and read by the error's size the brown ray's mean error more than doubles. The measure that ranked every earlier ramp was signed, and a signed measure cannot see an overshoot. Eased by a tenth, the steep ramp is no worse anywhere and better on three cases of four.

 The overshoot follows how fast the cap gives ink back. For each ramp and ease, the brown ray's deepest overshoot — the most its print exceeds the chroma asked — against the rate at which the eased cap returns ink below the ramp's start. A tenth from L 17 and a fifth from L 20 return ink at the same 7.3 points a unit and overshoot by 0.013 and 0.019; a fifth from 17, at twice the rate, by 0.070. What it takes to deliver it

Each dark ray wants its own ease

Easing a four-ink press's total-ink cap below L* 20 cures the dulling of the deepest colours it binds, and a steeper ramp — the whole ease spent from L* 17 — cures more on the signed mean, but eased by a fifth it prints the brown ray's low binders up to 0.07 more chromatic than asked and more than doubles the brown's error read by size. The prediction was that three twentieths from L* 17 is the ease the steep ramp wants: below every earlier setting on both rays. The brown still overshoots, by 0.039, and errs nearly twice as much as on the gentle ramp. The overshoot follows one number — how fast the eased cap gives ink back per unit of lightness — and the two rays want that number set differently. Hard, the blue errs least on three twentieths from L* 17 and the brown on three twentieths from L* 20; the brown's best costs the blue a fifteenth more error, the blue's best costs the brown four fifths more.

 A tilt moves every light's reading much the same way. Each of the fifty-one lights' shift in reported u′v′ when the seven-channel sensor's gains are tilted by +1 and −1 per cent. At +1 per cent every light moves by about 6.0 in u′ and 3.4 in v′, ten-thousandths, towards the long wavelengths, and no light lies more than 3.3 from that mean; −1 per cent mirrors it. 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.

 Even half a core of wings leaves the three-nanometre table short at the deepest column. The three-nanometre table's worst ceiling over the true peak at every column depth, built for the core's width, with the lines bare and with wings a tenth, a quarter and a half of the core's width. It fails from depths of 2, 2, 2.5, 5 as the wings widen; with wings a half it holds to 4.5 and falls 1.1 per cent short at five. What light is

Wings rescue a three-nanometre table only to its deepest column

Lorentzian wings on a reversed tube's lines make every four-, five- and six-nanometre table's ceiling, built for the lines' 2.4 nm core, safer by eight to twelve points of margin a step, so that wings a quarter of the core's width carry all three to every column depth. The three-nanometre table, at a slit-to-core ratio of 1.25, is the one a bare broad line failed earliest, from a depth of two. The prediction was that it still fails with a quarter's wings, from about three, and holds at every depth only with a half's, because a narrow slit averages less of a line's skirt. The first half holds, half a step early: a quarter's wings fail it from 2.5. The second fails by a hair: with wings half the core's width the three-nanometre table holds to a depth of 4.5 and falls 1.1 per cent short at five. Its margin climbs by 6.6, 9.0 and 13.2 points a step — the smallest lifts of any table, from the lowest start.

 The red phosphor bends nothing and tilts the warm whites. The 2,700 K LEDs' residual at each Duv under the eleven-term line, with the square of the Duv times the estimated red phosphor added, and with the Duv times the estimated red added instead. The bend term leaves ±0.006 at ±0.068 and ±0.088, against ±0.069 and ±0.087; the slope term takes them to ±0.046 and ±0.067. What a camera does

The red phosphor tilts the warm whites and bends nothing

An error line with a level, a slope and a curve in Duv for each colour temperature reads white LEDs at 4,000 and 6,500 K within ±0.04 ΔE₀₀ across and beyond a manufacturing bin, but warm whites at twice the bin's edge only within ±0.087. The prediction was that the red phosphor sets a warm white's bend, so that one term more a pump — the square of the Duv times the estimated red — brings them within ±0.05. It moves them by a thousandth. The red phosphor enters the slope instead: the Duv times the red takes them to ±0.046 below the locus and ±0.067 above. And the rest points upstream. The first stage reads a warm white's red phosphor squeezed toward the middle of its range, the warm misses follow that misreading times the Duv at r = 0.79, and given the red each LED was built with the warm whites come within ±0.05 — though the same substitution doubles the error at 4,000 K.

 The squeeze belongs to the temperature, not the Duv. For the 2,700 K LEDs at a Duv of ±0.003 and ±0.006, the first stage's mean estimate of the red phosphor against the red each was built with, by three first stages; the dashed line is a perfect reading. With one Duv term, no red and a full red are read at 0.13 and 0.80; with the Duv times the departure and the mireds at 0.14 and 0.80; with a level and a departure slope for each temperature at 0.02 and 1.01. What a camera does

The line had already undone the squeeze

An ambient sensor's first stage reads a 2,700 K white LED's red phosphor squeezed about a third toward the middle of its range, and the camera-error line's warm misses outside the bin follow that misreading times the Duv. The prediction was that giving the first stage the Duv times its 600 nm departure and times the mireds would undo the squeeze. It changes nothing: the squeeze belongs to the temperature, and a level and a departure slope for each temperature read the warm red four times better, within ±0.035. But the eleven-term error line does not move — no warm LED by more than 0.006 — because at one temperature a squeeze is a rescaling its red and red-times-mireds terms had already undone. Only the red-slope line, whose one coefficient a pump spans three temperatures, gains: to ±0.034 below the locus, within 0.002 of the red known. And the cold whites pay for it.

 Split between two estimates, the line keeps both ends. The warm whites at ±0.006 and the cold whites at −0.003, read by the red-slope line on the squeezed red, on the un-squeezed red, and split — the eleven terms on the squeezed estimate, the red slope on the un-squeezed. The split line reads the warm whites at ±0.034 and ±0.057 and the cold at ±0.025; the squeezed line reads them at ±0.046, ±0.067 and ±0.021, the un-squeezed at ±0.034, ±0.057 and ±0.040. What a camera does

Two estimates of one red each read their own end

Two first stages read a white LED's red phosphor from an ambient sensor: one squeezes the warm whites' red a third toward the middle, the other reads it four times better, and a camera-error line with a red slope prefers the first for cold whites and the second for warm. The prediction was that a line handed both — the eleven-term line on the squeezed estimate, the red slope on the faithful one — keeps each end's gain. It does, within 0.004 at the cold end. But the second estimate is not needed: the squeezed red alone, its red slope multiplied by the mireds, reads every LED within a ten-thousandth of the split line. And the two estimates' difference is worth a term of its own. Added to the line, it reads the cold whites within ±0.016 at every Duv and on the locus within a thousandth, and moves the warm whites and the middle by nothing.

 A third session finds the band beside the lens's edge. For each observer, how many of twenty experiments find the lens band beyond twice its error and within 20 nm of where it was put: two sessions under the six-nanometre prior, and three sessions at their best of five starts. Beside 415 nm with the macula unmoved, two sessions find it in 12 and three in 18; beside 430 nm with the macula short, 20 and 16. What the eye does

A third session finds the edge band and counts bands misplaced

Two sessions of colour matches find an older observer's lens band in eighteen to twenty experiments of twenty at every place but one: a band at 415 nm, beside the lens's own short-wave edge, found in twelve. The prediction was that a third session — a phosphor LED set against a lamp with a cyan line — finds it in at least seventeen, and that the experiments losing it under two sessions fit the lens three or more years old. Three sessions find it in eighteen, and every loss under two did fit the lens older, but by under three years in five of eight. The count hid something else. It credits a band wherever a fit puts it, and up to twelve of 180 fits put it sixty to eighty nanometres from its place, on the macula's long flank. With three sessions the six-nanometre prior no longer closes that valley.

 The count falls by one and then by seven. The forty-nanometre pigment's lost splitting directions at a twentieth of Lorentzian against the half-width of its steep side, at the best of every search, and the count each half-width would lose if every direction fell short by its seven-nanometre constant times one band's error. Searched again, 14 nm loses 18, 10 nm loses 13, 7 nm loses 12, 5 nm loses 5; the constants predict 16 at 14, 13 at 10, 12 at 7, 9 at 6, 5 at 5, 3 at 4.5, 0 at 4. Where the model breaks

Each direction falls short by one band's ratio

A band pigment asked to reflect mercury's green line and absorb its yellow, with a twentieth of Lorentzian in its steep side, loses eighteen of twenty-five splitting directions at a steep side fourteen nanometres wide, thirteen at ten, twelve at seven and five at five. A second search had lowered the five-nanometre count, and the prediction was that searching the seven- and ten-nanometre pigments again lowers each by one or two, leaving a count nearly linear in the half-width. Searched again, both lose exactly what they lost — no direction gains six tenths of a point. The uneven fall is real, and the reason is plainer than any search: every direction's shortfall falls with the half-width by the same factor as a single band's error, so the count is a fixed ranking of constants, one a direction, read against a line that moves.

 The ranking reads six nanometres to the direction. The forty-nanometre pigment's lost splitting directions at a twentieth of Lorentzian against the steep side's half-width, at every half-width searched, beside the count the ranking of seven-nanometre constants predicts. At 6 nm the ranking predicted 9 and the search found 9; at 4.5 nm the ranking predicted 3 and the search found 1; at 4 nm the ranking predicted 0 and the search found 0. Where the model breaks

The ranking reads six nanometres to the direction

Every direction a band pigment loses under mercury's lines falls short, as its steep side narrows, by the same factor as a single band's error — so each direction carries a constant, and the count at any half-width can be read off one band with no search at all. Read that way, a twentieth of Lorentzian loses nine directions at a six-nanometre steep side, three at four and a half and none at four. Searched, six nanometres loses nine: the same nine, every direction within 6.2 per cent of where the constants put it. Four loses none. Four and a half loses one, not three — the two the ranking added sit 0.005 and 0.011 of a point under the one-per-cent line. And at the narrowest widths the gentle directions begin to fall short by more than the ranking says, by up to 29 per cent, which is where the single band stops being the whole story.

 A longer glow moves the cool inks; a shorter one moves them all. For each pigment on the premium stock at the office sheet's loading, the change in its cancelling strength at a ten-per-cent target when the brightener's emission moves from 435 nm to 450 and to 420, the same photons absorbed and emitted. At 450 nm the blue moves by +0.176 and the violet by −0.058, the orange by +0.0011; at 420 nm the orange by +0.038 and the violet by +0.077. What the brain does

The glow's colour sets the blue ink and spares the orange

A light ink printed on brightened paper turns in hue, because the paper's glow passes through the ink's open area differently from the ink's own film, and a cancelling strength undoes the turn. Every sheet so far glowed with one brightener, and its ink was set by the glow's amount alone. The prediction was that a brightener emitting fifteen nanometres longer, nearer the blue and violet inks' hues, moves their strengths by more than a hundredth and the orange's and red's by under a fifth of that. It does: the blue's strength rises from 0.81 to 0.98, because a glow at the blue's own hue barely turns a blue tint, the violet's falls by 0.06, and the orange moves by a thousandth though the glow's share of the sheet's luminance nearly doubles. A brightener emitting fifteen nanometres shorter moves every ink, the orange by 0.04, and no curve in the glow's luminance reads any of them.

 The glow's pull reads every ink its luminance cannot. For each pigment, the largest miss over the thirty sheets, each left out, of lines in the glow's share of luminance, the share and the chromaticity y, the pull's Oklab a and b, and the pull with its squares and product. The last misses by at most 0.0139; the share alone by up to 0.249, the blue's. What the brain does

The glow's pull on the white sets every ink

A light ink on brightened paper is diluted to cancel the hue turn the paper's glow gives its tints, and on sheets that all glow with one brightener its strength is a single curve in the glow's share of luminance. Move the brightener's emission and that curve fails by up to 0.21. The prediction was that two numbers — the share and the glow's chromaticity — read every ink within a hundredth. Across thirty sheets, eight of them a grid of emission peaks from 420 to 455 nm, they miss by up to 0.07, and the share alone misses the blue by a quarter. The two numbers that work are the glow's pull: how far, in Oklab, it moves the paper's own reflection to the white an adapted reader sees. Read with their squares, they set every ink within 0.014 — and the warm inks need least strength just short of where that pull is strongest.

 Held near the lens, the band is found where it was put. For each observer, how many of twenty three-session experiments find the lens band beyond twice its error and within 20 nm of its place, with its centre free to range from 390 to 500 nm and held to 400–460. Beside 415 nm the bounded fits find it in 19, 20, 19 with the macula short, unmoved and long, against 16, 18, 15 free. What the eye does

A band held near the lens closes the valley the prior could not

Three sessions of colour matches find an older observer's lens band beside the lens's own edge as predicted, but twelve of 180 fits place the band sixty to eighty nanometres away on the macula's long flank, each with the macula fitted far short — a valley the six-nanometre prior on the macula's shift no longer closes. The prediction was that holding the band's centre to 400–460 nm, where a lens band can be, empties that valley, finds the 415 nm band with the macula short in at least eighteen experiments, and raises the moves found beside 430 nm from nine to twelve. The first two hold: no fit is left in the valley, and the band is found in nineteen. The third does not: 430 nm stays at ten. And the valley turns out to have been nearly level with the truth — holding the band costs ten of the twelve fits under four units of the fit's chi-squared, and every one's macula comes home.

 Holding ICtCp's hue and lightening the chroma still turns the sky. Each mixture mapped by BT.2390 on ICtCp intensity on the 100 cd/m² display and brought into its gamut three ways — clipped channel by channel, scaled until the largest drive is full, and by its ICtCp chroma alone at fixed intensity and hue — read as the change in CIECAM16 hue quadrature from the master's. The chroma-only return moves −16.8, −14.4, −12.6, −17.0, −9.2 units at no green to half green; scaled into gamut, −1.3, −1.6, −1.8, −1.6, −2.2. Matching and measuring

Lowering only the chroma turns the sky white and its hue with it

A tone curve applied to ICtCp's intensity keeps a high-dynamic-range sky's hue, but a saturated blue's intensity lies under the curve's knee, so the colour it asks for lies outside every display's gamut. The prediction was that bringing it back by lowering its chroma alone — intensity and ICtCp hue held — keeps the CIECAM16 hue within three units, lifts the lightness by ten and halves the chroma: a paler blue of the right hue. The ICtCp hue and intensity are kept to a hundredth. Everything else fails. The chroma must fall to a fifth or less on a 100 cd/m² display, the sky rises from a lightness of 22 to 76 and higher, and its hue quadrature moves eight to seventeen units toward green — a pure blue's by 17. Holding ICtCp's hue while lowering chroma does not hold the hue a viewer sees, and the intensity the curve chose was never a blue's.

 Holding the true hue saves one or two units of seventeen. Each mixture's CAM16-UCS distance from the master on the 100 cd/m² display: mapped on intensity and returned by ICtCp chroma, returned along CIECAM16's hue at the same lightness, scaled into the gamut, and the master's drives shown unmapped. The CIECAM16 return lies 16.7, 17.2, 17.3, 16.8, 15.7 from the master, against 18.9, 18.7, 18.5, 17.6, 16.8 for ICtCp's; the master's own drives 4.14, 4.10, 4.07, 4.02, 4.05. Matching and measuring

The sky a dim display can show best is the master's own

A tone curve applied to ICtCp's intensity maps a saturated high-dynamic-range sky to a colour no display can show, and bringing it back by lowering only its ICtCp chroma turns it pale and toward green. The prediction was that a return along CIECAM16's own hue line, at the lightness the curve chose, holds the hue by construction, needs a deeper chroma cut, and still lies at least twice as far from the master as simply scaling the drives. It holds the hue to a hundredth, keeps more chroma than ICtCp's return rather than less, and lies two and a half to four times as far as scaling — because holding the true hue removes only one or two of seventeen units of difference. The rest is the lightness the curve held. And the nearest a dimmer display can come to the master's appearance turns out to be the master's drives, shown unmapped.

 An offset draws every light toward one colour. The seven-channel sensor's reading of each of the fifty-one lights in CIE 1960 uv with no offset and with a five-per-cent offset, beside the colour the sensor reports for equal signals in every channel, at 0.210, 0.316. Every light moves straight toward it, by 4.9 to 11.3 per cent of the way. Difference and uniformity

A dark offset pulls every light toward one colour

An ambient sensor of seven narrow channels reads all fifty-one of a room's lights within a ΔEITP unit of a dim picture's tint through independent noise and a gain tilt, because its matrix barely cancels. A dark current is different again: it adds the same signal to every channel whatever the light, and at the dim end of a room's lighting it is a larger share of a weak channel than of a strong one. The prediction was that an offset of a quarter of a per cent of the strongest channel costs the seven-channel sensor lights, while three broad channels move least. Seven channels hold every light to two per cent. Three move furthest. And the offset does something simpler than either sensor's design suggests: it draws every light's reading straight toward the one colour the sensor reports for equal signals in every channel.

 Built for the notch, the three-nanometre table holds at every depth. The three-nanometre table's worst ceiling over the true peak at every column depth, built for the unnotched core and for each window's notched line, with the lines bare and with wings half the core's width. Built for the notch, the bare table's least margin is +0.3% at a depth of 1.5 and +8.1% at five; built for the core it was −29.9% at five. What light is

A ceiling built for the notch holds the three-nanometre table

A table of a reversed discharge lamp read through a three-nanometre slit cannot see the notch a cool column of vapour cuts into each line's core, and a ceiling built for an unnotched core — the bound on how high the lamp can peak inside each window — fails from a column depth of two, and even with broad wings at five. The prediction was that a ceiling built for each window's notched line holds at every depth with any wing, by more than two per cent. It holds at every depth with any wing. With bare lines it holds by three tenths of a per cent at its tightest, at a moderate column, beside mercury's violet lines. Knowing the notch costs the shallow columns and pays the deep ones, because a shallow notch flattens a line's top and a deep one splits it.

 The further from the equal-signal colour, the sooner a light goes. For each of the 19 lights the seven-channel sensor loses by an eight-per-cent offset, the offset at which it crosses a unit against its reading's distance from the colour the sensor reports for equal signals. The correlation is −0.93; the nearest lost light, triphosphor 3500 at 0.035, crosses at 7.9%, the furthest, planck 2200 at 0.091, at 2.1%. Difference and uniformity

A light's distance from the equal-signal colour sets when it goes

An uncorrected dark offset draws every light an ambient sensor reads straight toward the one colour its matrix reports for equal signals in every channel, and a seven-channel sensor holds the room's fifty-one lights within a ΔEITP unit to an offset of two per cent of each light's strongest channel. The prediction was that the first light goes near three per cent and is the warmest, that each light's crossing offset falls with its distance from the equal-signal colour, and that two readings a light predict its crossing within a fifth. The warmest goes first — the 2,200 K radiator — but at 2.1 per cent. The twelve 2,700 K LEDs go together, between 3.9 and 4.4. Crossing follows distance at r = −0.93, and a straight line through each light's tints at one and two per cent predicts its crossing within 8 per cent, always early. Below half a per cent nothing moves at all.

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