The thread: Computed, not quoted — page 23
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 a camera does
Radiators and LEDs are one family to the sensor
An ambient-light sensor's three narrow departures see the collection's six smooth radiators and forty-four white LEDs along one line: the radiators' own short line lies on the LEDs' long one. Estimating a camera's colour error family by family, the census had treated them as two, and every classifier that tried to tell them apart paid for it. Merged into one family of smooth broadband sources, found by the nearest other lamp as before, the two-stage estimate comes to ±0.127 over fifty-four lamps held out — below the ratio line's ±0.141, and below the ±0.133 the separate families managed when every lamp's family was simply given. The halophosphate tube is still caught. The radiators, which the prediction said the merge would cost a little, are estimated better inside it than inside their own family, because a three-coefficient line fitted over five radiators was worse for them than the census-wide ratio line.
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.
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.
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.
What the eye does
A macula three nanometres short is a lens band
Two colour-matching sessions can fit a band of extra absorption in an older lens, its size and its centre, and the test proposed for real older observers was to count those with a band more than twice its standard error. That test assumes a significant band can only be a band. The observer model's macula peaks at 460 nm; a real one is two carotenoids whose mix moves its peak by a few nanometres. Simulated with its macula peaking three nanometres short and no band at all, an observer at sixty is fitted with a lens band of 0.06 at 438 nm, three standard errors, and noisy sessions find it in nineteen of twenty experiments. Six nanometres short, the phantom is larger than the band the earlier essays searched for. The confusion runs both ways: a real band fitted with the macula's peak free reads as a macula two to three nanometres short. Neither fit alone can tell them apart. Comparing the two can, with two sessions — never preferring the macula for a real band, and preferring it in sixteen to nineteen experiments of twenty for a short macula.
What the brain does
Stochastic dots hold the glow at every tint
On brightened paper a halftone tint keeps a hue turn that a reader fully adapted to the sheet cannot remove, because the paper's glow survives under the ink where the solid puts it out, and a dot's thin edge lets more of it back. A dot's size at a given coverage is set by the screen. Across conventional rulings from 100 to 300 lines the residue rises at every coverage for every pigment, a finer screen making every dot more edge. A stochastic screen of twenty-micrometre dots holds it flat — the orange within a quarter of a degree from five per cent to sixty, starting where a 300-line screen starts — as predicted. What the prediction did not say is where the switch costs: not the pale tints, by under a degree, but the mid-tones, by three or four, because a conventional screen's mid-tone dots are large and mostly full film. On ten-micrometre dots, all edge, the ramp turns over and the mid-tone keeps more than the pale tint. At any one coverage the residue is set by the dot's diameter alone, whichever screen printed it.
Difference and uniformity
A warm room tints the black, not the halo
A local-dimming panel's black can be declared as two floors, the room's light reflected off the screen folded in, and the earlier censuses gave that light a hundredth of a candela, grey. Under a warm lamp it is not grey. The prediction was that a warm room would hide the halo's misreading in the long and middle cone signals, leave its short-wave part exposed, and tint the halo's dark greys towards blue. The tint is real and it is none of those things. With the floors declared in grey, a 2,700 K room putting a tenth of a candela on the screen tints every zone of a dark picture by more than one ΔEITP — up to 17.7 at the darkest grey — and the tint points towards the lamp's own orange, opposite blue. It is not the halo's: it is the whole black's, because a grey declaration has the floor's colour wrong everywhere. The prediction's conclusion holds by a different road. A grey step laid on a warm floor is partly a change of colour, so the warm room leaves seven tenths of the moon picture's grey steps misread where a grey room leaves none; and declaring the floors in the room's colour removes all of it but the halo's own.
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.
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.
What the eye does
A second session keeps a band and the macula apart
A lens band and a macula peaking a few nanometres short look alike to a colour-matching fit that has only one of them, and an older eye may have both. Fitted with all six numbers — rod weight, lens age, macular density, the band's amplitude and centre, and the macula's peak — nine simulated observers at sixty, each with a band at 415, 430 or 445 nm and a macula moved three nanometres either way or not at all, test whether the matches can carry both. The prediction was that the band at 445 and a short macula would trade almost completely, a correlation beyond 0.95 and the band's standard error tripled. That is what one session does: correlations of 0.91 to 0.95 at 430 and 445 nm, and a band found in at most twelve experiments of twenty. Two sessions do not. Their correlations stay between 0.42 and 0.83, the band is found in twelve to twenty experiments for every observer, and a moved macula in twelve to nineteen — while an unmoved one is found moved in at most three. The price of the extra number is paid by the band nearest the macula, whose standard error rises by three quarters; the rod signal pays almost nothing.
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.
What the brain does
Merged spots give the mid-tones half the glow back
On brightened paper a halftone tint keeps a hue turn a reader adapted to the sheet cannot remove, and a stochastic screen's small dots, being mostly thin edge, keep it high along the whole ramp — three times the 150-line screen's cost in the mid-tones. That census kept every dot separate. A real stochastic screen's spots touch and merge, and a shared edge between two spots is no edge of the ink. The prediction was that a merged screen would follow the separate-dot curve to about thirty per cent and then fall towards the conventional screens, giving back half the mid-tone cost by sixty. It does, almost exactly. Square spots twenty micrometres on a side, switched on in a dispersed order, stay apart up to a fifth coverage and start touching near thirty per cent; by sixty, nine in ten touch, the rim has fallen from 84 per cent of the ink to 50, and the merged screen keeps the orange's residue at 6.9 degrees against the separate spots' 8.6 and the 150-line screen's 5.7 — giving back 58 per cent of the cost. Every pigment gets back the same share, 55 to 58 per cent, because the share is set by the geometry of the ink, not by its colour.
What light is
A stricter fit declares a wider line
A two-nanometre table of a mercury lamp whose every line is self-reversed admits every reversed line as a Gaussian within half a per cent, and declares a width its own ceiling cannot stand on from a depth of two and a half. The obvious repair is to fit harder. The prediction was that at a tenth of a per cent the table refuses the reversed blue and green lines, admits the least-reversed yellow, declares unsafely from it between — and that a threshold that strict would refuse even unreversed lines on a real instrument's noisy table. The refusals come where predicted: the blue from a depth of three, the green from four. But the stricter threshold makes the declaration less safe, not more. It narrows the range of widths each line admits from below, so the yellow, never refused, declares a wider width at every depth, and the ceiling built on it falls below the true peak from a depth of two instead of two and a half. And a tenth of a per cent survives a quiet instrument: no unreversed line is refused until the table's noise reaches half a per cent — and then the first line refused is the yellow, the one that declares.
What a scene does
The slot's loss survives every finer division
Read on the painted faces alone, a satin finish on a room's grey faces makes the paint more colourful in twenty-eight of the painted census's thirty rooms and less colourful in two: the tall narrow slot with its floor painted and with its ceiling painted, where the paint loses about a twentieth of a per cent of its chroma. A loss that small could belong to how finely the slot's faces were divided, and the prediction was that it does not — that the sign holds at seven and eight strips a side. It holds. Solved from one strip a side to eight, the slot's paint gains at one, two and three strips and loses from four on, every one of seventy-two paints crossing at the same division; the loss deepens at every finer division, to −0.055 and −0.059 per cent at eight strips, and run out to no strip width it settles near −0.058 and −0.065. The coarse census had the sign wrong; the census at six strips had it right. The bright, broad paints lose most, as they gained most everywhere else.
Difference and uniformity
Adapting to a warm room deepens the shadows' tint
A warm room's light reflected off a local-dimming screen tints the shadows of every dark picture when the screen's black is declared in grey — by up to 17.7 ΔEITP — and ΔEITP has no viewer in it. A viewer who has sat in a 2,700 K room all evening is adapted partly to the room, and adaptation to a warm white discounts warm light. The prediction was that a viewer adapted a third of the way to the room would see under half the tint, and one adapted two thirds of the way would see the darkest greys turn blue. Read through CIECAM16, the opposite happens in a dark picture. The tint the viewer sees grows at every step of adaptation: in the moon picture's darkest grey from 2.7 CAM16-UCS units for a viewer adapted to the display to 3.3, 3.9 and 4.4. The darkest greys never turn blue; they stay orange and fade. What turns blue is the mid-greys, whose light is the display's own and now looks cool. In the bright window picture the tint shrinks, and declaring the floor in the room's colour still removes it for every viewer.
What a scene does
The finish stops costing the paint six tenths deep
In the tall narrow slot a satin finish on the grey faces costs a painted floor or ceiling a little of its chroma; in the tall room, the same box a full unit deep, it adds some. Solved at eight depths between, the sign turns at 0.62 of the box's width with the floor painted and at 0.63 with the ceiling — later than the prediction of 0.45 to 0.55, and far enough along that the loss belongs to a range of shallow boxes rather than to the slot alone. A finer division moves the turn a little deeper, not back. Every one of seventy-two paints turns at the same depth, and each paint's share of the change is the same at every depth, loss or gain. Brightness does not decide it: the finish brightens the painted floor by between half and more than double while hardly touching the painted ceiling, yet the two turn together. What decides it is which reflectance the finish favours — whether it multiplies the light that has been back to the paint by more or by less than the light that has never touched it.
What a camera does
LEDs varied in their phosphors fall off every line
The merged smooth family estimates a camera's colour error from an ambient-light sensor's departures to ±0.127, and five new white LEDs, varied in their phosphors where the census's forty-four vary their pump, missed by ±0.196 because the family's line had learned almost nothing about the sensor's 640 nm channel. Rebuilt on forty-eight LEDs varied in pump, converter and red phosphor, the line learns it: its 640 nm weight goes from −0.13 to −0.74, most of the way to the −0.85 the new LEDs implied. The prediction was that the new LEDs would then land within ±0.13. They miss by ±0.229, worse than before, crossing to the other side of the line, and the census's own held-out error more than doubles, to ±0.294. No pool of LEDs restores it, and no line in the sensor's departures reads the grid's own LEDs better than ±0.29. Across the grid the camera's error is a V about the Planckian locus, rising away from it on the green side and on the pink — and every one of the census's LEDs sat on the green side.
What light is
Two nanometres hide the yellow's notch from any fit
A plain Gaussian fit admits the reversed tube's 578 nm yellow line at every threshold on a two-nanometre table, and the ceiling built on its width fails from a column depth of two and a half. A fit with the notch in it — an emitted Gaussian with an absorbed one taken out of its centre — might see what the plain fit cannot. The prediction was that two nanometres leave too few entries to determine the notch, and one nanometre finds it from a depth of two. Two nanometres hide it completely: the best notch fit carries a negative notch, a spike, at every depth, and the readings a two-standard-error rule calls reversed are the same noise draws at every depth, each fitting a notch that erases the whole line. One nanometre finds it sooner than predicted — in every reading from a depth of one and half of them at a half — and its fitted depth climbs with the column. But its standard error cannot be trusted with seven entries and four numbers: the rule calls a third or more of unreversed readings reversed. The notch's fitted depth itself separates them cleanly.
Where the model breaks
A harder search pulls the band pigments away from the slope limit
Every count in the pigment-limit censuses — how many directions of the object-colour solid a slow pigment cannot reach within one per cent of the ideal — compares a slope limit solved exactly with two band models searched from a few structured starts. Under the mercury lamp a search from 680 starts had already found the band model reaching further in seventeen of twenty-five directions. Searched from 594 starts under all nine lamps, both band models reach further in between three fifths and nineteen twentieths of the directions they lost, and their counts fall by a sixth: 395 to 335 for bands in nanometres, 370 to 311 for bands in energy, while the slope limit stays at 463. The rankings the censuses reported hold where they were stated. What moves is everything measured against the slope limit: its lead over the band models grows from a sixth to more than a third, the coating's odd ranking — bands losing more than the slope limit — disappears, the triphosphor tube passes the three emitters for bands in nanometres, and under the mercury lamp the band models' deep losses halve.
What the eye does
A line on the macula's flank buys less than a repeated session
Fitted with six numbers, two sessions of colour matches keep a lens band and a moved macula apart, but the band at 445 nm pays for it: correlated with the macula's shift at 0.82, its standard error 1.8 times its value when the macula is not fitted. A moved macula changes its long flank past 470 nm, where no lens band reaches, so a third session — the LED against a lamp with a narrow line at 487 nm — was predicted to bring the correlation under 0.6 and the band's error within a quarter of its band-only value. The correlation falls to 0.80 and the error to 1.5 times. A third session that simply repeats the second does better on both errors. The flank line pins how much macular pigment there is, not where it peaks, and it ties the band's height to its centre. Moved to 465 nm, onto the macula's peak, the same session brings the band within about a fifth of its band-only error — but no position of the line brings the correlation below 0.78. The band at 445 nm and the macula's peak are told apart by their short flanks, and a third session adds matches rather than a new kind of evidence.
What a scene does
A taller box must be deeper before the finish adds colour
In a box one unit wide and two and a half tall, a satin finish on the grey faces costs a painted floor a little of its chroma when the box is shallower than 0.62 of its width and adds some when it is deeper. Whether that turn is set by the width or by the height was the open question, and the prediction was that it scales with the height, because the satin walls pipe the lamp down to the floor and a taller box is a longer pipe. It does, and faster than in proportion. A box 1.5 tall gains at every depth from 0.35 and turns shallower than that; a box 4 tall loses at every depth to half again its width and turns near 1.67. As a share of the height the turn rises from about 0.17 to 0.25 to 0.42 — close to a depth growing as the square of the height. The taller box pipes far more of the lamp to the floor, tripling its light at 0.35 deep, and at every depth of every box the paint's colour still follows which reflectance the finish favours. Six strips a side keep both new boxes' signs.
What the eye does
Knowing the macula's density frees two sessions, not the line on its peak
Fitted with six numbers, an older observer's colour matches cannot fully split a lens band at 445 nm from a macula whose peak has moved: their correlation stays above 0.78 under every third session tried. The proposal was to measure the macular pigment's density from outside the matches, leaving five numbers, and the prediction was that a line on the macula's peak would then split the pair below 0.6 while two plain sessions stayed near 0.8. It runs the other way. With the density known exactly, two sessions fall from 0.83 to 0.62 and the line at 465 nm only from 0.86 to 0.74. The density was the road the band and the peak traded along — correlated with each at about −0.75 in two sessions' fit — and the designs that had already narrowed that road gain least from closing it. A realistic outside measurement closes almost none of it: the matches already know the density to 0.027, so a flicker measurement good to 0.05 moves the correlation by three hundredths. And far from the macula, at 415 nm, a known density binds the band to the peak more tightly than before.
Matching and measuring
The green primary decides how sharp the blue is
ΔEITP makes a display's blue its sharpest corner when the blue sits near 485 nm and nearly a face when it sits at 440, and the turn between them is where adding green stops raising the blue's L signal by the larger fraction. The prediction was that the turn follows the green — near 445 nm with a 545 nm green, near 470 with a 515 nm one. It follows the green, by less than half as much at the short end: 461 nm with a 515 nm green, 446 with a 545 nm one, and with a 555 nm green there is no turn left in the range at all. What the prediction did not ask is larger. The green sets how sharp the blue can be: with the blue at 485 nm the corner moves 2.4 times a margin against a 505 nm green and 8.2 times against a 555 nm one, while the sharpest blue stays at 485 nm for every green. The red primary, moved from 615 to 690 nm, leaves the turn where it was to a hundredth of a nanometre. The appearance model follows the green far more gently, under half of ΔEITP's move at every green.
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