Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Computed, not quoted — page 20

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.
The lamp's white, six walls and the light arriving at each. On the 1976 chromaticity diagram, in the closed cube under daylight: the lamp's white (centre), six saturated walls with bands centred from 450 to 650 nm (open circles), and the light arriving at each wall from the rest of the room, lamp included (filled). Each ambient lies on the line from the white to its wall, a fraction of the way along it: 450 nm 10 per cent, 490 nm 12 per cent, 530 nm 19 per cent, 570 nm 19 per cent, 610 nm 13 per cent, 650 nm 5 per cent. What a scene does

A probe at the wall prices the finish

The light arriving at a painted wall from the rest of its room is what a gloss finish hands back, and a small probe held against the wall reads it. It lies on the line from the lamp's white to the wall's own colour, a fraction of the way along, and the fraction is set by how much light the wall returns, not by how colourful it is — as predicted. It is not the fixed fraction the prediction said: it runs from 2 to 46 per cent across seventy-two paints in one room. That variation is what makes it useful. Read in the matt room, it orders what a satin finish would cost more tightly than the paint's own lightness does, in every room tried.

 An orange ink's tints on three papers: which way the hue turns. The hue of each tint of an orange ink, from the solid to a tenth coverage, against the solid on the same paper, in degrees of Oklab hue: on a coated sheet, an unbrightened uncoated sheet and a newsprint. Solid lines are read against daylight's white, as an instrument reads them; dashed lines against the paper's own white, as a reader adapted to the page. At a tenth coverage, read against daylight, the coated sheet's tint has turned -8.7 degrees and the newsprint's 21.8; read against each paper's white, -8.7 and -1.1. What the brain does

Newsprint turns a tint with its colour, not its gain

A halftone tint turns its hue the way a mixture of light does, by less, because optical dot gain carries it part of the way towards a paint tint. The straight line through a coated sheet's gain predicted that an uncoated sheet's larger gain would carry the orange past the crossing at a factor of 3.2, so that the same ink would turn opposite ways on two papers. It does turn opposite ways — on newsprint the orange's pale tints swing 13 degrees one way where a coated sheet's swing 8 the other. But the gain is not what does it. The road towards paint bends and stops at 60 to 70 per cent of the way, the orange needs a factor of 5.4 to cross, and newsprint's reversal comes almost entirely from the paper being yellow. Read against the paper's own white, as a reader looking at the page is adapted, it goes away.

 The share of its colour each display hue loses to a quarter of its luminance in white. The twenty-four most saturated colours of an sRGB display, one every fifteen degrees of HSV hue, each scaled to a luminance of 20 and given white of a quarter that luminance: the share of its colourfulness lost, in CIECAM16, CIELAB and Oklab. All three put the fastest loss at an orange and the slowest at a blue. At the orange end they are close — CIECAM16 27 per cent, CIELAB 28 per cent, Oklab 21 per cent — and at the blue end CIECAM16's 1.7 per cent is a quarter of the others'. What the brain does

White drains the blue last in every model

CIECAM16 says a dab of white costs a deep red wall four times the colour it costs a dark blue one, and the experiment proposed to test it asked whether observers lose colour in that hue order. Held at equal luminance and given equal doses of white, twenty-four display hues are ordered alike by CIECAM16, CIELAB and Oklab — an orange loses fastest and a blue slowest in all three. So the order cannot tell the models apart. What does is how much slower the blue is: CIECAM16 has it losing a sixteenth of what the orange loses, CIELAB and Oklab about a quarter. That ratio is the number an observer study has to measure.

 The appearance model's lightness-to-chroma balance, as the room takes a share of the adaptation. The median lightness pair's reading over the median chroma pair's, for twelve base colours, against the display's white, in a room of 20 cd/m². Solid: CAM16-UCS, with the viewer taking all, three quarters, half, a quarter and none of the adaptation from the display — darker lines take more from the display. Dashed: ΔEITP, which no room enters. With the display alone the model's balance falls 23 per cent; with half from the room, 13; with none from the display it does not move. Difference and uniformity

A lit room brings the units' medians together

As a display brightens, CAM16-UCS says chroma differences gain a quarter on lightness differences and ΔEITP says a tenth. All of the appearance model's movement comes from what the viewer is adapted to, and every calculation had the viewer adapted to the display alone. Give the room its share of the adaptation and the model's movement shrinks at every step: with a 20 cd/m² room supplying two thirds of it, the two units' medians fall by the same amount. What does not shrink is their disagreement about direction. The model still moves every colour the same way, ΔEITP still moves violets, reds and cyan-blues the other way, and in a lit room that becomes the whole of what separates them.

 How wrong a declared veil makes the unit, for four true veils. On a 100 cd/m² display, the worst error over grey steps from L 2 to L 90 — the size of the natural logarithm of the declared reading over the true one — against the veil declared, for rooms putting 0.1, 0.3, 1 and 3 cd/m² on the screen. Each curve reaches nought at its own true veil and rises on both sides. The flat stretch at the left is declaring almost nothing, which is declaring none: 0.22 for a true veil of 0.1, 0.54 for a true veil of 0.3, 1.14 for a true veil of 1, 1.89 for a true veil of 3. Difference and uniformity

A guessed veil halves the error

A colour difference that takes a display's absolute luminance leaves out the light a room reflects off the screen, and on an ordinary display in an ordinary room that makes it wrong about the darkest greys by a factor of three. Giving the unit the veil as a declared argument fixes that when the veil is known. The worry was that it never would be — that a guessed argument is no better than none. It is better: any declared veil up to about twice the true one beats declaring none, and one middling guess for every room halves the worst error. What a guess cannot do is reach ten per cent; that needs the veil known within a sixth, which is what a luminance meter aimed at a black screen gives.

 How far a mesopic match moves as the S weight opens: two rooms against one field. For every pair of the five lamps, the median over forty-two surfaces of how far a match moves as the rod signal's weight into the S channel goes from nothing to equal: pale for the two-room match, each half adapted to its own lamp; dark for a bipartite field whose two halves share one adaptation. The field's signal is larger for every pair, by ×1.5 to ×7.7. What the eye does

One field keeps what two rooms divide out

An asymmetric match can measure how strongly the rods feed the blue-yellow pathway, but set with the observer adapted to each lamp in turn it needs seventy-five settings on the best pair of lamps and hours of waiting between them. Putting the two lamps on the two halves of one field was proposed as the quick version, at the cost of a weaker signal. The signal is not weaker. Under one shared adaptation it is three times stronger for daylight against a white LED, and the best pair needs six settings. The adaptation that makes the slow version slow is also what was dividing the rods' contribution out of each half.

 How much of a coated press is left at each margin inside its boundary. The share of a coated press's printable volume in CIELAB that lies at least a given distance inside its boundary, for margins from half a unit to 32. A margin of one unit keeps 88%, two keep 80%, four 66% and eight 45%. At every margin up to 30 what is left is a single connected piece; at 31 units, with 0.14% of the volume left, it first splits, into a core of 623 cells and 2 fragments of one or two cells — the deepest point is 32.7 units inside, so what splits is the last crumb of the core, not a waist. Matching and measuring

A margin costs a press its corners

A gradient held inside a coated press by projection was predicted to tear if the press were first shrunk by a safety margin, at any pinch narrow enough for the shrinking to cut. The press has no such pinch: shrunk by any margin up to thirty CIELAB units it stays in one piece, and projected routes on it are neither trapped nor torn. What a margin costs is concentrated at the corners — the solid yellow moves nearly four times the margin to get inside, like the tip of a 31-degree spike.

 Narrowing the slit makes the declared-width bound looser, not tighter. The median ratio of the declared-width bound to the error it bounds, over sixty-eight notches under each lamp, with the slit and the grid narrowed together from five nanometres to one and each lamp declaring its true line width; the dotted lines are the bound with both variances known. Under the tube the declared bound goes from ×86 at five nanometres to ×253 at one, while the true-variance bound stays between ×12 and ×15. Under the projector the declared bound lies on its true-variance bound at every slit. What light is

A narrow slit shrinks the error, not the bound

A colour engine given two separately blurred spectral tables can bound its own error if the lamp declares its narrowest feature, and under a fluorescent tube that bound sits 86 times above the error at a five-nanometre slit. Narrowing the slit was supposed to close the gap, because a comb of narrow features stops fitting in a narrow window. It does the opposite. At one nanometre the error falls 27-fold and the bound only sevenfold, so the bound is 253 times loose and certifies no notch within a colour difference. The looseness was never the comb. It is the lamp's one declared peak, charged to every window.

 The ceiling a table reads for each window, against the lamp it came from. The fluorescent tube between 525 and 590 nanometres, with the green and yellow mercury lines, and for each window of a one- and a five-nanometre slit the highest the lamp could be there according to the table's own entries. The dashed line is the tube's peak, which the global declaration charges to every window. At one nanometre the ceiling follows the lamp, a median 2.05 times each window's true peak; at five it sits on the global peak for twenty nanometres either side of each line. What light is

A narrow table can vouch for its own peaks

The bound a colour engine can put on its bandpass error is loose because a lamp declares one peak for its whole spectrum and the bound charges it to every window. A table measured through a slit about as narrow as the lamp's lines can do better: no feature of the declared width can be blurred below a computable fraction of its height, so the table's own entries cap how tall the lamp can be in each window. At one nanometre that ceiling is never below the truth and sits twice above it, where the global peak sat twenty times above. The bound halves, and for the first time it certifies notches within a colour difference. At five nanometres it buys nothing at all.

 A finer grid shrinks both arrangements, and the gap between them stays. The mean error over sixty-eight notches of the best fixed pair of separate slits and of one slit on the reflected light at the grid's own step, on grids of one, two and five nanometres, under the tube and the projector. The tube's best pair falls from 0.256 to 0.019 and reaches the product's five-nanometre error of 0.016 without passing it; the product on the same fine grid is 0.0007. Under the projector the pair at one nanometre is 0.135. What light is

A finer grid keeps the price of separating

A sample's reflectance and a lamp's spectrum measured through separate slits can be reused under any other lamp; measured together through one slit on the reflected light, they are fifteen times more accurate and belong to that one lamp. A finer grid was expected to close the gap, because a one-nanometre table barely blurs anything. It closes most of the distance to the old answer and none of the distance to the new one. On a one-nanometre grid the best separate pair under a fluorescent tube is as good as one slit on the product used to be at five, and one slit on the product on the same fine grid is twenty-nine times better again.

 Every symmetric and every skewed fourth emitter, and the best of each at every cost. Each fourth emitter added to the three-emitter lamp, as its fidelity index against the number of the solid's directions a 40-nanometre slope limit costs more than a per cent: 315 symmetric bands and 1260 skewed ones over centre, widths and power. The two step lines are the best rendering each family reaches at or below each cost. They run a few directions apart and never further: rendering at 88 costs 103 directions with a symmetric band and 97 with a skewed one. The ringed point is the three emitters alone, Rf 78.6 at 62. Where the model breaks

A lopsided band does not break the trade

A fourth emitter added to a three-emitter LED renders colours better and costs a slow pigment its reach, and the position that renders best and the position that costs least were thirty-five nanometres apart. The obvious suspect was the shape: every emitter searched was a symmetric bump, and a band with its skirt on one side might fill a gap without splitting it. Freed to be lopsided, it saves between one and seven of the solid's three hundred directions. No band renders like the amber emitter and costs like the green one, the lopsided bands that help lean the other way from the proposal, and what the fourth band renders was never the lamp's white.

 The same lamps, charged by two kinds of slow pigment. Each lamp — the three emitters, a fourth band at seven positions, and the cheapest symmetric and lopsided bands rendering at 88 — as its fidelity index against the directions of the solid a slow pigment loses more than a per cent of its reach in, for a forty-nanometre slope limit (open) and for three absorption bands none narrower than forty nanometres (filled), joined by a line. Every line runs leftward: the band-shape pigment loses fewer directions under every lamp, and the lamps rendering best gain the most. At 88 the slope limit loses 50 and the band-shape pigment 33. Where the model breaks

The pigment decides the trade, not the band

A four-emitter lamp that renders well costs a slow pigment part of its reach, and reshaping the fourth band barely changes that. Changing the pigment changes it a great deal. Modelled as three absorption bands none narrower than forty nanometres, rather than as a forty-nanometre slope limit, a slow pigment loses a third fewer of the solid's directions under the best-rendering lamp, and the price of going from the three-emitter lamp to a fidelity of 88 halves. The lopsided band that helped a slope-limited pigment does nothing at all for this one. And at 590 nanometres a fourth band renders at 86 almost for free.

 Every lamp, charged by a slope limit and by bands in nanometres and in energy. For each lamp, ordered by fidelity index, the directions of the solid lost by a forty-nanometre slope limit, by three absorption bands none narrower than forty nanometres, and by three none narrower than 0.158 electronvolts. The energy convention loses fewest under every lamp: 20 against 23 and 31 under the three emitters, and 26 against 33 and 51 under the best-rendering lamp. Where the model breaks

Written in energy, the bands spare the green

A slow pigment modelled as absorption bands pays half what a slope limit pays for a four-emitter lamp's rendering, and a fourth band at 590 nanometres costs it almost nothing. That result assumed every band is at least forty nanometres wide, and a real absorption band has a width in energy, not in nanometres. Written that way it is narrower in the blue and wider in the red, and the fear was that the red would take back the amber. It does not. The band limit written in energy loses fewer directions under every lamp, the fourth band at 590 still costs six, and the price of rendering at 88 falls from ten directions to six — because the directions it wins back are green, not amber.

 The hue disagreement with the quantiser, and without it. For twelve base colours of CIELAB lightness 50 and chroma 40, the factor by which ΔEITP's reading of a lightness pair over its reading of a chroma pair changes from a 1.5 to a 10,000 candela white. With the perceptual quantiser, it runs from ×0.75 at the yellow-greens to ×1.23 at the violets. With the quantiser replaced by a pure power law and nothing else changed, it is ×1 at every hue. Difference and uniformity

The hue disagreement is the quantiser's

As a display brightens, the appearance model shifts every colour's balance between lightness and chroma by one factor, and ΔEITP shifts violets towards lightness and yellow-greens towards chroma. The proposed source was the perceptual quantiser acting on three cone-like signals at unequal levels. Replacing the quantiser with a pure power law and changing nothing else removes the disagreement exactly: every colour moves by a factor of one. With the quantiser, how far a colour moves follows the spread of its three signals, and which way follows where its S signal sits — above the other two, towards lightness; below, towards chroma.

 Three panels in three rooms: the veil alone, and the best single floor. On a 1,000 cd/m² display, for each panel and room, the worst error over five-per-cent grey steps from 0.005 cd/m² upwards of declaring only the room's veil (upper bar) and of the best single declared floor (lower bar). For an LCD without dimming and for an OLED the best floor is the display's black plus the veil, and it is exact. For the LCD it is also necessary: the veil alone misreads by 2.8 in a dark room. For a local-dimming LCD no single floor is exact, and in a dark room the best errs by 0.96. Difference and uniformity

A dimming panel has no single black

ΔEITP reads a display's shadows correctly only when it is told the floor under every pixel, and that floor has two parts: the room's light reflected off the screen and the display's own black. They fold into one declared number exactly, and on a bright LCD the display's part is the one that matters — leaving it out misreads the deepest shadows seventeenfold. A local-dimming panel breaks the folding, because its black changes with the picture: in a dark room its floors vary twenty-six times across one scene, and no single declaration comes within a factor of two and a half. A lit room squeezes them back into one number.

 CIECAM16's colourfulness loss by hue, with each suspect removed. The share of colourfulness each of the twenty-four most saturated sRGB colours loses when white of 25 per cent of its luminance is added, all at the same luminance, in CIECAM16 as published, with its eccentricity factor held at one, and with its cone-signal denominator replaced by its achromatic signal; CIELAB chroma for reference. The blue at 240 degrees loses 1.7 per cent as published and 1.6 with the hue factor removed, and 5.4 with the denominator replaced — the curve then lies close to CIELAB's. What the brain does

The blue resists through the model's denominator

CIECAM16 says a saturated blue is nearly impervious to added white: at equal luminance and an equal dose it loses a sixteenth of what an orange loses, where CIELAB and Oklab say a quarter. The model has two obvious places for that to come from — a hue factor fitted to make colourfulness equal across hues, and the sum of cone signals it divides its opponent signals by. Setting the hue factor to one changes nothing. Replacing the sum with the model's own achromatic signal makes the blue lose 0.233 of the orange's loss, beside CIELAB's 0.221 and Oklab's 0.223. The disagreement is one denominator.

 How much of the turn a reader sees, adapted part of the way to newsprint. The hue turn of a fifth-coverage tint against its solid ink on newsprint, for five pigments, as the reader's adaptation moves from daylight's white to the paper's. Open circles mark where each tint stops turning the opposite way from the same tint on a coated sheet: orange at 0.97, red at 0.89, magenta at 0.98, cyan at 0.90, green at 0.93. Half-way, the orange still turns 7.3 degrees. What the brain does

A reader sees the reversal until almost adapted

An orange halftone tint on newsprint turns its hue one way and the same tint on coated paper turns it the other, and the reversal is the newsprint's yellow. An instrument sees it in full; a reader fully adapted to the paper does not see it at all. A real reader is somewhere between, adapted to a mixture of the paper and whatever surrounds it. Half-adapted, the reader still sees more than half of the instrument's turn, and the reversal lasts until the reader is adapted nine-tenths or more of the way to the paper. On a paler uncoated sheet it is gone by just past half-way. A coated sheet does not care.

 What a weight fitted under the wrong reading does across ages. If the retina's rod signal has a fixed absolute size and the weight is fitted with the model's scaled reading, the fitted weight is the true one times the ratio of the two readings' signals: ×0.80 at twenty, ×1 at thirty-two, ×1.62 at seventy-five. The other way about, the reciprocal. A cohort of observers sharing one weight would show it as a trend with age. What the eye does

Age can size the rod signal, if the lens is known

A rod signal leaking into the colour pathways has no natural size in the model: read as a fraction of each cone's own peak it shrinks as the lens yellows, read as a fixed amount it grows relative to the cones, and the two readings differ by a factor of two over a lifetime. An asymmetric match on observers of different ages can tell them apart, because they predict opposite trends: one says a weight fitted at seventy-five will look sixty per cent larger than at thirty-two, the other says it will not move. A handful of observers per age would show it. But a lens mistaken by ten years looks like forty per cent of the weight, and a tenth of an optical density of macular pigment as much again, so the experiment is only as good as what it knows of each eye's lens and macula.

 What each pair of lamps pays to fit the lens and the macula as well. For each pair of lamps, the standard error of the rod signal's S weight fitted from one session — with the observer's lens and macula known (top bar), with the lens fitted too (middle) and with both fitted (bottom). Daylight against the LED is the best pair with everything known, ±0.026, and pays most of its advantage for the macula; tungsten against the LED pays almost nothing for either. What the eye does

The matches carry their own lens

A lens mistaken by ten years moves a colour match as far as half the rod signal being measured, and a tenth of an optical density of macular pigment as far again, so an experiment on the rod signal looked as if it needed a densitometer for every observer. It does not. Across a family of surfaces the three move the matches in different patterns, and one session of forty-two settings fits all three together — the lens to within a year and a quarter, better than a densitometer was asked for. The price is the weight's precision, and a second pair of lamps nearly removes it.

 What the classifier misses, against what the camera gets wrong. Every lamp as the pair of channels reads it against the camera's mean colour error under it, with its two-matrix profile blended at the lamp's colour temperature. The band is the range of the six smooth census lamps, 1.14 to 1.62. Left of the line the classifier calls a lamp smooth; no lamp there is above the band. The violet-pumped LEDs it misses leave the camera at least as accurate as a smooth lamp does, two of them more so. What a camera does

The lamps two channels miss are smooth to the camera

A narrow channel at 450 nm tells a phone which lamps are structured, and a white LED pumped in the violet might hide from it; a channel at 500 nm does the same job, and an LED with a cyan emitter should blind it. Reading both and calling a lamp structured if either departs was supposed to cover both families. It covers the cyan family only because the 450 nm channel already did, and half the violet-pumped LEDs escape both channels. Every one that escapes leaves the camera's colours as accurate as a smooth lamp does. What the classifier misses is what the camera does not need to be warned about.

 The camera's error, and the two-channel estimate of it. Each of the fifty-four lamps as the camera's true mean colour error against the estimate from the 450 and 500 nm channels, fitted with that lamp left out. The held-out error is ±0.14 ΔE00 against ±0.16 from the 450 nm channel alone and ±0.30 from the mean. The dashed lines at 1.5 mark the repair decision: points right of the vertical and below the horizontal are lamps that need a repair and are not given one. What a camera does

Two channels estimate what one can only sort

A narrow sensor channel was fitted to call lamps smooth or structured, and the label turned out to be a proxy for the thing a camera cares about: its own colour error under the lamp. Scored against that error directly, the best channel is still the one at 450 nm — the band where camera and observer differ most comes close and no closer. Read as an estimate rather than a verdict, a second channel at 500 nm, useless as a vote, earns its place: together they predict the camera's error to within 0.14 ΔE00 over fifty-four lamps, and deciding when to repair from that estimate raises a quarter of the false alarms the label raises.

 Where a sharpened serif letter's error is seen, in black type and reversed, 10 point. The serifed letter E at 10 point, sharpened on stored values rather than on light, as the eye's three spatial channels leave the difference at forty centimetres, in black type on paper and in paper-coloured type on black, on one scale. Dashed circles mark the serifs' corners: their tips and their brackets against the stems. In black type the error lies along the strokes' dark side and the serifs' corners hold little of it; reversed, it gathers round the serif tips, which hold 11 per cent of the letter's seen error against 3 in black type. What it takes to deliver it

A serif moves the error and adds little

When a scanned page is sharpened on its stored values, the error the eye sees lies on the dark side of every edge, so black sans-serif type does not gather it at its corners. The worry was that a serif changes that: every stroke end gains a bar and every bar an inside angle, the configuration where large letters did gather error. It does not. Black serif type's brackets hold three quarters of their share of the error and its serif tips half. Reversed, the serif tips hold two and a half times theirs. Across both, serifs lengthen the outline by a tenth and the error by a few per cent — they move it, and in black type away from the corners.

 The table's cast along the grey axis, with and without a toe. A seventeen-node four-ink inverse table filled from a strategy that begins black at L* 60: the mean chroma of the grey it prints, segment by segment. Above the start every strategy is the same three inks and the cast is about 0.0126. Below it the cast falls under a tenth of that 2, 7, 12, 12 units below the start for the kinked start and toes of 10, 20 and 30 — later with a wider toe, but by less than the toe's width, and nowhere near where black takes over. What it takes to deliver it

A little black ends the cast

A four-ink inverse table errs by a warm cast along the grey axis wherever the separation strategy prints only three inks, and almost only in lightness wherever it prints black. Real strategies start black gently, and the gentle start was expected to keep the cast until black carried much of the grey. It does not. The cast falls to a tenth of its three-ink size once black supplies a tenth to a sixth of the grey's darkening, at every toe from nothing to thirty units wide — because inside the toe the table's chord runs along the trade of black against the other three inks, which errs in lightness. What the toe does cost is the segment where black begins, which goes from the table's best to its worst.

 Each end colour's move into the eroded press, over the margin, in CIELAB and in ΔE₀₀. The 9 end colours of the census gradients, each moved to the nearest colour of the press eroded by a margin, for margins of one, two and four. Squares: the margin in CIELAB units and the move in CIELAB. Circles: the margin in ΔE₀₀ and the move in ΔE₀₀. The dashed line at one is a flat face and the one at 1.73 a cube's corner. In CIELAB every end colour moves further than a cube's corner at two units, the solid yellow ×4.6, ×3.9, ×3.8. In ΔE₀₀ six of them move between 1.06 and 1.34 times the margin, and only the solid yellow, at ×1.91, ×2.04, ×2.10, is still sharper than a cube's corner. Matching and measuring

A tolerance's own unit rounds the corners

Shrinking a coated press by a margin in CIELAB units made its corners pay several times the margin — the solid yellow nearly four. A print buyer states a margin in ΔE₀₀, and in that unit the corners pay about half as much: six of nine saturated end colours move within a third of the margin, and only the solid yellow is still sharper than a cube's corner. The prediction that the dark corners would stay sharp was wrong; they round too. What the tolerance's unit costs instead is volume — two ΔE₀₀ keep two thirds of the press where two CIELAB units kept four fifths.

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