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

The thread: Computed, not quoted — page 21

Every swatch begins as a spectral power distribution and is carried through the colour-matching functions as it is drawn. None is a hex code recalled from a table.
How many of 72 paints make a more colourful room, by where the finish goes. Six room shapes, each with five sets of painted faces and the rest mid grey, and 72 paints. Each cell is the number of paints for which a satin finish makes the room more colourful than matt, with the finish on every face, on the painted faces only, and on the grey faces only. With the finish on the painted faces, 29 of the 30 cells are zero; with it on the grey faces, 28 are not. The low room with its side walls painted — the room where a finish was found to add colour — gains for 21 paints with every face finished, none with only the walls finished, and 51 with only the grey faces finished. What a scene does

A finish adds colour only where it covers grey

One low wide room was found in which a satin finish makes the room more colourful rather than less, and the explanation offered was dilution: the more of a room is grey, the more a finish's glancing return stands out. Painted area turns out not to decide it. Across six room shapes and five ways of painting them, a finish on the painted faces alone never adds colour, and a finish on the grey faces alone almost always does. The low room gains because its grey faces are most of it, and the whole finish is close to the sum of its two halves.

 Where a sharpened E with hairline serifs is seen to err, 48 point. The capital E with hairline serifs, a tenth of a stem thick, set at 48 point on a 300-dot page, sharpened on stored values rather than on light, as the eye's three spatial channels leave the difference at forty centimetres: black type on the left, reversed on the right, on one scale. Circles mark the serif tips and the junctions where the middle arm meets the stem. In black type the error follows the ink and gathers in the junctions' inside angles, and the serif tips hold 2 per cent of it; reversed, it spreads into the dark ground and the tips hold 6 per cent. What it takes to deliver it

A hairline spills its error onto the paper

Sharpening a scanned page on its stored values errs on the dark side of every edge, which is why black type's serifs hold less than their share of the error at text sizes. A hairline serif, a tenth of a stem, was expected to change that at display sizes: its dark side is so thin that the halo would wrap its corners. It does not. From 12 to 48 point, black type's hairline tips hold a third to three fifths of their share, less than slab serifs' tips, and the share falls as the letter grows. What gathers at display sizes is the junction of bar and stem, with or without serifs — and where the scan lands the letter against the pixels moves a tip's share more than its size does.

 The cast segment by segment under three toes of width 20. The seventeen-node table's cast from where black begins to the floor, for the three toes and the kinked start. Above the toe's end at L 40 the shapes differ; below it the parabola and the smoothstep draw one line. Between the toe's end and L 20 the returned cast averages 1.94, 0.94, 1.94 thousandths for the parabola, the eased cubic and the smoothstep. What it takes to deliver it

The cast returns where black is steep

Below a black-generation toe a four-ink table's grey picks up a small cast again, up to a third of the three-ink cast at the widest toe, and the explanation offered was the jump in black's curvature where the toe ends. It is not. A toe whose curvature is nothing at its end returns exactly the same cast, segment for segment, and a kink put into straight black casts in its own segment once in twelve tries. What the returned cast follows is how steep black has to be once the toe is done — how far the chromatic inks climbed while black was held back, and how fast they then have to come down.

 A pale orange tint's turn on a brightened sheet, against the reader's adaptation. The hue turn of a fifth-coverage orange tint against its solid on brightened office paper in daylight, as the reader's adaptation moves from daylight's white to the sheet's, beside a sheet of the same colour that does not fluoresce, a white sheet with the same dot gain, a heavily brightened sheet, newsprint and a coated sheet. Unadapted the brightened sheet turns it -41.5 degrees, the coated sheet -8.0 and newsprint 12.7. Fully adapted, the non-glowing sheet of the same colour comes down to -3.2, beside the white sheet's -3.4; the brightened sheet stops at -10.0. What the brain does

Adapting to a brightened sheet leaves the glow

A pale tint on newsprint turns its hue one way and on coated paper the other, and a reader fully adapted to the paper sees newsprint's reversal vanish, because it was only the paper's colour. On a sheet with an optical brightener the tint turns the coated way and five times as far — 41.5 degrees for a fifth-coverage orange. A reader adapted to the sheet removes most of that, as expected, but not all: ten degrees stay. The sheet's colour adapts away like newsprint's. The brightener's glow does not, because the ink puts it out under the solid and leaves it under the tint, and an added light is not something adapting to a white can divide out.

 How far white turns each display colour's hue, in CIECAM16 and with its cone space changed. The twenty-four most saturated sRGB colours, each mixed with white of the same luminance down to a fifth of its purity, and the turn of its hue angle in Oklab, in CIECAM16, in CIECAM16 with its compression done in the Hunt–Pointer–Estévez space rather than CAT16's, and in CIECAM02's front end. At the display's blue Oklab turns 16.3 degrees and CIECAM16 3.6; compressed in Hunt–Pointer–Estévez it turns 15.6, and CIECAM02 16.1. Over all twenty-four the two lie 3.2 and 3.2 degrees from Oklab, root mean square, against CIECAM16's 5.3. What the brain does

The blue's hue turn was lost with a matrix

Add white to a saturated blue and its hue turns — sixteen degrees by the constant-hue data Oklab was fitted to, four by CIECAM16. The obvious suspect was the model's compressive response, acting on a very large blue signal. It is not: removing the compression's saturation moves no hue by half a degree, and a cube root instead of its exponent moves the blue by one. What decides it is the cone space the compression happens in. Computed in Hunt–Pointer–Estévez, the space CIECAM02 compressed in, the same model turns the blue 15.6 degrees; CIECAM02's own front end turns it 16.1. CIECAM16 merged CIECAM02's two spaces into one sharpened for adaptation, and that one gives a display's blue almost as much long-wave signal as white has.

 A local-dimming panel's backlight behind a small highlight on black. The backlight of each of a panel's 32 by 18 dimming zones behind a small highlight on black, on a logarithmic scale from one per cent to full, after a diffuser that spreads each zone's light with a standard deviation of 1 zone. The zones driven by the picture are the darkest or the brightest; the diffuser's halo fills 3.5 per cent of the screen with levels in between, and each zone's floor follows its backlight. Difference and uniformity

Two floors trade the screen for the halo

A local-dimming panel in a dark room has no single black: its floor varies twenty-six times across a scene, and one declared floor misreads ΔEITP's grey steps by a factor of 2.6. The proposal was two floors, one for dimmed zones and one for lit ones, on the reasoning that a picture's zones cluster at the two ends. They do cluster, and two floors still do not work. Placed for the worst zone they halve the error and misread almost the whole screen a little; placed on the two clusters they read most of the screen exactly and misread the halo between as badly as one floor. How many floors a panel needs is set by its diffuser, not by the picture — and the only declaration that works everywhere is the backlight map.

 How far each unit moves a colour's balance, at the most saturated colour BT.2020 allows. For every fifteen degrees of hue at lightness 25, 50 and 75, the most saturated base colour whose one-unit lightness and chroma pairs stay inside a BT.2020 display, and the factor by which each unit's ratio of the two pairs changes from a 1.5 to a 10,000 candela display. CAM16-UCS moves every one by ×0.76 to ×0.89. ΔEITP moves them from ×0.54, a yellow-green at hue 90, to ×1.64, a red at hue 30 — with the red-purples beside it and the dark blues well below. Difference and uniformity

At the gamut's edge the reds move as far as the violets

As a display brightens, ΔEITP shifts a saturated colour's balance between lightness and chroma by an amount set by its three quantised signals, and CAM16-UCS shifts every colour alike. The largest shifts found so far were a yellow-green at ×0.63 and a dark blue at ×1.62, both at the census's edge — and the blue turns out to be a colour no BT.2020 display can show. Walked out to the display's real boundary, the yellow-greens go lower, to ×0.54, and the dark blues never pass ×1.35, because BT.2020's blue edge is at low chroma. The top of the range goes to the reds and red-purples at ×1.6: at the edge the L-minus-M term grows to two thirds of the S term. Pairing the two extremes still needs a third of the observers the violet experiment needs.

 Where a lens edge 10 nanometres off goes in a one-session fit. A sixty-year-old observer whose lens edge sits 10 nanometres longer (upper bars) or shorter (lower bars) than the model's, fitted by the model from one daylight-against-LED session: the shift in each fitted parameter, in that parameter's standard errors. The macula takes the most, 2.4 of its standard errors; the weight moves by 0.59 of its own, 0.035 in absolute terms, against a prediction of under a tenth for an edge ten nanometres off. What the eye does

A lens of the wrong shape lands in the macula

One session of colour matches can fit an observer's rod signal, lens and macular pigment together — but it fits a lens of the model's shape, one exponential in wavelength scaled by age, and a real lens need not be that shape. Give a sixty-year-old a lens with the right density at 400 nm and its edge ten nanometres off, and the fit puts the error mostly in the macula, moves the rod signal's weight by six tenths of its standard error — six times the predicted tenth — and leaves a misfit the session cannot see. A second lamp pair makes it worse: 3.5 standard errors. Fitting the edge as a fourth number removes the bias, costs the weight three to fourteen per cent, and finds the edge.

 Where one session puts a band in the lens, by the band's wavelength. A band of a tenth of an optical density in a sixty-year-old's lens, centred from 410 to 460 nm, fitted from one daylight-against-LED session by the model with weight, lens age, macula and the lens edge free: each parameter's shift in its own standard errors. Near 410 to 430 nm the band is read as lens; from 440 it is read as macula; the edge takes little of it anywhere; and the rod weight moves at every centre, by 0.37 to 1.39 of its standard errors. What the eye does

A band in the lens reads as age or as macula

A lens whose absorption edge is steeper or shallower than the model's biases a matching session's fit, and fitting the edge's position repairs it. An ageing lens also grows bands: yellow pigments that add a shoulder of absorption just past 400 nanometres. The edge parameter takes none of one. A tenth of an optical density centred at 410 to 430 nm is read as four to six years of extra lens age; centred at 440 to 460, as three to ten hundredths of macular pigment. Either way the rod signal's fitted weight moves, by up to 1.4 of its standard errors in one session and 1.8 in two, and the session's misfit stays within its noise. The lens model has to carry the band, or the experiment has to state what it does not know.

 What straightening the 570 nm channel buys an estimate of the camera's error. The root-mean-square and worst held-out errors of linear estimates of the camera's error, over fifty-four lamps with the three-emitter source kept in: from the 570 nm channel as read and through three transforms, from 450 nm alone, and from 450 nm with a log-transformed 570 nm channel or with 500 nm. Every transform rescues 570 nm from worse than the mean; none reaches 450 nm; and beside 450 nm it adds nothing, where 500 nm does. What a camera does

A straightened channel repeats the one beside it

A phone's narrow sensor channel at 570 nm ranks a camera's colour error under a lamp almost as well as the one at 450, and estimates it worse than guessing the mean, because one lamp reads ten times higher than any other. Read through a logarithm, the channel is rescued: its estimate improves from ±0.57 to ±0.23. It still does not reach the 450 nm channel's ±0.16, and put beside that channel it adds nothing at all, where a 500 nm channel cuts the error to ±0.14. Straightened, 570 nm sorts the lamps as 450 nm already does; and the two lamps it reads highest have the most different errors, which no transform of one reading can tell apart.

 Which way each narrow channel departs from what the camera predicts. Each lamp's two narrow channels, their departures from the camera's prediction kept signed. 37 of the fifty-four lamps, every structured one among them, lie in the lower right: more light than predicted at 450 nm, less at 500 — a peak and a trough. 8 lie in the upper left, the opposite pattern, most of them violet-pumped LEDs whose pump misses 450 nm. In both the departure of the two readings' ratio is the sum of the two distances. The 7 in the upper right are the most heavily cyan-filled LEDs, whose fill puts light at 500 nm, and two daylights near the origin. What a camera does

One ratio reads a peak and a trough

Two narrow sensor channels, at 450 and 500 nm, estimate a camera's colour error under a lamp to ±0.14 ΔE₀₀; the proposal was to read their ratio as one number instead. Taken as one channel's departure over the other's, the ratio is worthless — no better than the mean, because for smooth lamps both departures are nearly nothing. Taken as the departure of the two readings' ratio from what the camera predicts for it, one number estimates the error to ±0.141, as well as the two channels as two. The reason is a sign: under every structured lamp the channels depart in opposite directions, a peak at 450 and a trough at 500, and the ratio adds them; where a cyan fill lifts 500, it subtracts them, as the error falls. And no five per cent calibration error moves any of these estimates by a hundredth.

 A correction measured under one tube, carried to seven other lamps. For each lamp, the mean error over sixty-eight notch filters of the separable table (upper bar), of the separable table corrected by the ratio of product to separable colour measured under the census tube (middle), and of the product measured under the lamp itself (lower). Under the census tube the correction is the product exactly; under tubes with other line widths or a moved phosphor bed it removes nine tenths of the error; under a tube with its lines re-mixed, 63 per cent; under the laser and the LEDs it makes things worse. What light is

A correction travels with its lines

Keeping a sample's reflectance and a lamp's spectrum as separate tables costs fifteen times the error of measuring their product under a fluorescent tube. The middle course proposed was to measure the product once under a representative tube, store the ratio of product to separable colour with the sample, and apply it under any tube. Under tubes whose lines are narrower or wider, or whose phosphors sit elsewhere, it removes nine tenths of the error. Under a tube with the same lines in a different mix it removes under two thirds, and less the further the mix moves. Under an LED or a laser it makes things worse — a white LED five times worse than no correction. A class of lamps, for this purpose, is a mix of lines.

 How well the table's entries around the 546 nm line fit each width of line. For the tube's green mercury line, 1.2 nm wide, the seven table entries around it fitted by a line of each width from 0.2 to 12 nm through the same slit on a sloping background, centre, height and background free: the best fit's miss. Through a one-nanometre slit the miss has a sharp minimum at the true width and only widths near it pass half a per cent; through two, a broader dip; through three and five the curve is flat from the narrowest width tried up to about two, and any of them passes. What light is

A narrow table declares its own lines

A bound on a colour table's bandpass error needs one number from the lamp's maker: its narrowest feature, 1.2 nm for a fluorescent tube. Through a slit about as narrow as that feature, the table's entries around each line carry its width. Measured to half a per cent, a one-nanometre table's entries admit only line widths within four per cent of the truth, and a bound built on the narrowest of them is within two per cent of the bound on the maker's number. Through two nanometres the table still declares a usable width. Through three it cannot rule out a line five times narrower, and the bound it supports is the global one. The failure is as sharp as predicted, and never unsafe.

 Directions each pigment model loses, under three broadband lamps and a lamp of lines. For each lamp, how many directions of the object-colour solid a pigment model cannot reach within one per cent of the ideal: a slope limit, absorption bands no narrower than forty nanometres, and bands whose minimum width is fixed in energy. The order is the same under every lamp. The broadband lamps cost every model far more than the three narrow emitters: under daylight 89, 73 and 70 against 31, 23 and 20. Where the model breaks

The lamp outweighs the pigment model

Three models of what a real colorant can do — a slope limit, absorption bands forty nanometres wide, and bands whose width is fixed in energy — were ranked under lamps of narrow emitters, slope dearest and energy cheapest. Under daylight, tungsten and a white LED the ranking holds, every time. What changes is everything else: each model loses two and a half to three and a half times as many directions of the object-colour solid under a broadband lamp as under three narrow emitters, and the difference between the models is smaller than the difference between the lamps. Under tungsten, whose power rises smoothly to the red, the three models are nearly one.

 What keeps the solid yellow sharp in ΔE₀₀, and what rounds it. The solid yellow's corner read as a cone: the opening a margin of two ΔE₀₀ found, the opening a cone about the tongue's measured axis would have once rescaled by ΔE₀₀'s local lengths, and the same with each of three reasons changed — the axis along chroma, the lightness weight as at mid-lightness, the hue function at one. The tilt narrows the corner from 71 to 49 degrees; the two weights both widen it. Matching and measuring

The yellow stays sharp because it leans

A safety margin stated in ΔE₀₀ rounds almost every corner of a coated press, and leaves the solid yellow nearly as sharp as a cube's corner. The explanation offered was that the yellow's tongue points up towards lightness as well as out in chroma, so ΔE₀₀ — which shortens chroma at yellow five times and lightness one and a half — shortens it less than a pure chroma spike. Measured from the press's own cells, the tongue leans 23 degrees, less than predicted, and a cone about that axis rescaled by ΔE₀₀ accounts for the yellow's opening to within ten degrees; about a chroma axis it would open twenty-two degrees wider. ΔE₀₀'s own low hue weight at yellow, offered as the alternative, works the other way: it rounds the yellow. And the same account fails at every other corner.

 Where a grey finish sends the lamp's light orders which rooms it colours. The thirty rooms of the census, each placed by one number computed from its box and lamp alone — how much more of the lamp's light a satin finish on the grey faces sends to the painted faces than their matt return did — and by how many of 72 paints a finish on the grey faces makes more colourful. Over the twenty-four rooms with a wall or the floor painted the rank correlation is 0.92, and the vertical line at zero sorts them: right of it every room gains for 29 paints or more, left of it for 22 or fewer. The six rooms with the ceiling painted, drawn hollow, all sit left of the line and five of them gain at every paint. What a scene does

A grey finish mirrors the lamp onto the paint

A satin finish on a room's grey faces makes the room more colourful in almost every arrangement, and the explanation offered was that it re-weights each grey face towards the paint it sees at a slant. That number, computed from the geometry alone, orders the rooms at a rank correlation of 0.24. The number that orders them is the same slant pointed the other way and lit: how much of the lamp's light the finish sends to the paint. It orders the rooms with a wall or floor painted at 0.92, and its sign sorts them without an exception.

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