Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Computed, not quoted — page 19

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.
Which way a tint turns a pigment's hue, made in paint and made in light. For eight pigments tinted nineteen parts white to one of colour, the Oklab hue turn from the pure pigment: of the tint made in paint (upper bar) and of the additive mixture with white at the same luminance (lower bar). For seven of the eight the two point in opposite directions. The orange turns +13.5° in paint and −13.9° in light. What the brain does

A tint in paint turns the other way

A pale colour can be made two ways: by stirring white pigment into a coloured one, or by adding white light to it — which is what a halftone on white paper and a display both do. At the same luminance the two tints are not the same colour. For seven of eight pigments they turn the hue in opposite directions, an orange by +13.5 degrees in paint and −13.9 in light, and the paint tint is the more colourful of the two every time. The cause is in the reflectance: diluting a pigment moves its absorption edge, and adding light does not.

 Twenty-three pairs at one ΔE₀₀, read in two units that know the light level. Twenty-three pairs of surface colours, each exactly one ΔE₀₀ apart, on a display whose white runs from 1.5 to 10,000 cd/m² across, with a background at a fifth of the white. ΔE₀₀ has no argument for the light and stays at one. The median ΔEITP rises from 1.01 to 2.75 and flattens near the top, and the median CAM16-UCS distance from 0.76 to 1.20. Difference and uniformity

Two units with a light level disagree about lightness

ΔE₀₀ has no argument for how bright a display is. Two colour differences do: CAM16-UCS takes the room's adapting luminance, and ΔEITP — the difference defined for high-dynamic-range television — takes the stimulus's own absolute luminance. Twenty-three pairs at exactly one ΔE₀₀ grow in both as the display brightens, 2.1 times in ΔEITP and 1.5 in CAM16-UCS from a 5 to a 5,000 cd/m² white. But in ΔEITP the lightness part grows fastest, 2.7 times, and in CAM16-UCS it does not grow at all.

 The same chart in two studios and in the room that holds both lamps. Each of the chart's twenty-four surfaces, with the pixels its slide needs at two standard deviations at high gain, on a logarithmic scale. Lit by a 3000 K radiator alone the worst surface needs 16,194; lit by daylight at 6500 K alone, 139,574. Lit by both, 50 and 50 per cent of the light, with a highlight of each lamp read together, the worst needs 10 and the median 6. The studios' hard surfaces sit at different places on the chart, and no surface is hard in both. What a camera does

Two lamps decide what one lamp could not

Under one lamp a handful of surfaces cannot tell a glossy highlight from a matt over-exposure without most of a frame, because there the two differ by a scale a ratio cannot see. In a room lit evenly by a 3000 K lamp and daylight, every surface on the chart is decided from ten pixels. A body can sit at one lamp's white, not at two, and the rescue holds only while the second lamp carries a fifth of the light and its white sits far enough from the first.

 How far each lamp's sensor reading is from what the camera predicts, with RGB + clear. Fourteen lamps, six smooth and eight structured, each scored by how far an ambient-light sensor with red, green, blue and clear channels reads from what the camera's white predicts through a map fitted on smooth radiators and daylights. The smooth lamps score at most 0.057 and the structured at least 0.133; the dashed line is the threshold at the gap's geometric middle, 0.087. The lights the map was fitted on score at most 0.0113. Every lamp falls on its own side of the line. What a camera does

Two sensors disagree about deep red, not lines

A phone's ambient-light sensor and its camera read the same lamp differently, and the difference sorts fourteen lamps into smooth and structured without a single mistake — where flicker made five. It is not reading their lines. Almost all of it comes from the sensor's clear channel collecting deep red the camera's infrared cut throws away, so daylight with its far red trimmed is called structured and a white LED with a far-red emitter is called smooth.

 Every pair of slits, over 68 notches. The colour error, in ΔE₀₀ from the truth, for every pair of slit widths — the lamp's table blurred through the width down the side, the sample's through the width across — on 68 notches, the mean over all of them under a fluorescent tube. Circle area follows the error. The best pair is 5 nm on the lamp and 1 nm on the sample, at 0.26; the best single slit, on the diagonal, is 5 nm at 0.36. One slit on the reflected light, at 5 nm, averages 0.016. What light is

A second slit buys a quarter

A line lamp wants a five-nanometre slit and a notched sample a one-nanometre slit, so an instrument with a slit for each should do much better than one with a single compromise. Over sixty-eight notches under a fluorescent tube, it does better by 28 per cent. One slit on the reflected light does fifteen times better than the best pair, and an oracle choosing the best pair for every notch is still six times worse. The error was never that the factors were flattened; it was that they were flattened separately.

 Three bounds against the error they bound, over 68 notches under a fluorescent tube. Each notch placed across by its actual colour error from blurring the lamp and the sample separately, and up by a bound on that error, both on logarithmic scales; the dashed diagonal is where a bound equals the error, and a valid bound sits above it. Cauchy–Schwarz with the true window variances is above the diagonal on every notch, a median 14.7 times the error. Estimated from the blurred tables it falls below on 6 of 68, as low as 0.45 of the error. The Bhatia–Davis bound from the tables and declared ranges is above on every notch and a median 196 times the error. What light is

The tables cannot bound what they discarded

A colour computed from a lamp's blurred table and a sample's blurred table is wrong by the covariance the two blurs threw away, and Cauchy–Schwarz bounds a covariance by two variances. With the true variances the bound always holds and sits fifteen times above the error. With variances read from the tables it fails on six of sixty-eight notches under a fluorescent tube and twenty-two under a laser projector — on the line, where the error is largest. A blurred table does not carry the width of a line, and the covariance depends on it.

 Three uniform spaces, three answers: red to blue. The gradient from above, in the a and b plane: CIELAB's straight line, and the shortest paths under ΔE₀₀'s local metric, under CAM16-UCS's and under Oklab's, whose shortest path is its own straight line carried back into CIELAB. They bow from CIELAB's line by 14.4, 42.5 and 42.6 units. ΔE₀₀'s and CAM16-UCS's paths run 28.3 apart at their furthest, Oklab's runs 29.0 from ΔE₀₀'s and 5.6 from CAM16-UCS's, and the space furthest from the other two here is ΔE₀₀. Matching and measuring

A third space breaks the tie only once

ΔE₀₀ and CAM16-UCS disagree about which colours lie between two colours. Oklab was the obvious tie-breaker, and it breaks the tie on one gradient of five: on red to blue its path runs within six CIELAB units of CAM16-UCS's and twenty-nine from ΔE₀₀'s. On red to green and cyan to magenta it keeps to CIELAB's straight line while both others bow, and on blue to yellow it bows further than either. Each of the three spaces is the odd one out somewhere. Asking also exposed a CAM16-UCS path that had never converged.

 Six starts for every held gradient, and where each one lands. Each row is one gradient held inside a gamut, relaxed from six starts: the straight line, the free shortest path, and the straight line bent towards and away from the neutral axis and up and down in lightness. Each dot is how much longer than the free shortest path that start's result is, on a logarithmic scale from a hundredth of a per cent to a thousand; the ring marks the best. The first 10 rows are gradients between colours on a coated CMYK press's boundary whose straight line leaves the press: their best routes cost a median of 0.12% and at most 1.1%, and on 5 of them some start lands at more than twice the free length. The next 3 are the display gradients held inside sRGB, whose best cost at most 1.3% and whose starts spread by at most 4.2%. The last 6 are press gradients that never leave, where no start is trapped. Matching and measuring

A press makes the cheap route a search

Holding a gradient inside a display's gamut cost nothing measurable, and the essay that found it credited the gamut's convexity. A press's gamut is not convex: 216 of 630 straight lines between colours on its own hue ring leave it. Holding gradients inside the press still costs little at best, a median of a tenth of a per cent. But the best route depends on where the relaxation starts, the free path is the best start on one gradient of ten, and on half of them some start is trapped at more than twice the free length. On the display no start is trapped.

 What a fourth emitter costs, by where it is put. A three-emitter LED with one more emitter added at each position from 470 to 610 nanometres, and for each lamp the number of the object-colour solid's directions that lose more than a per cent of their reach to a 40-nanometre transition limit. The three-emitter lamp itself costs 62 of 312. A fourth emitter in the middle of the blue-to-green gap, at 490 nanometres, costs 172; one at 540, beside the green emitter, costs 65. The two dips sit on the existing lines and the two peaks sit between them. Where the model breaks

A fourth emitter spends the gap it fills

A three-emitter LED lets a pigment that takes forty nanometres to switch reach most of its ideal solid, because the lamp is dark where the pigment is slow. Adding a fourth emitter to render better takes that darkness back — but only if it is put in the middle of a gap. At 490 nanometres it costs 172 of the solid's 312 directions against the three-emitter lamp's 62, and renders two points worse. At 540 it costs three directions and renders two and a half points better.

 The rescue is spent on light between the lines, not on width. The three emitters of a narrow-band LED broadened together, from their nominal widths up to six times them, plotted against the light left in the darkest of the lamp's two gaps as a share of its peak. Up is the share of the object-colour solid's directions that lose more than a per cent of their reach, at three transition limits, with each limit's cost under daylight marked at the right. At forty nanometres half of the rescue is gone by a floor of 7.4 per cent — emitters only 1.30 times their nominal width — and all of it by about a fifth. At twenty nanometres and at eighty there is little to lose either way. Where the model breaks

The gap has to be dark, not the line narrow

A lamp whose light sits in three narrow emitters lets a blunt pigment reach most of its ideal solid, and the reason was given as the spacing of the lines. Broadening those emitters without moving them says otherwise. At 1.3 times their nominal width the lamp still looks like a line spectrum, its closest spacing has not changed at all, and half the rescue is gone — because the darkest point of the narrow gap has risen from one per cent of the lamp's peak to seven.

 How steep an edge a band draws, and what it costs. A Gaussian absorption band of stated width produces a reflectance edge whose own width depends on how deep the band is, because the exponential saturates: where the absorbance is large the reflectance is already nought and the edge is over. A forty-nanometre band at an absorbance of 3 draws an edge 32 nanometres wide; at 12 it draws one 21 nanometres wide. Below an absorbance of 2.3 the band never reaches a reflectance of a tenth at all and has no edge in this sense. The dashed lines are each band's own width, which is the number a slope limit would have been given. Where the model breaks

A sharp edge is bought with depth

A slope limit on reflectance was introduced as the weakest honest statement of a pigment's bluntness, with a band-shape limit named as the stronger version to be written later. Written, it is not stronger. Three absorption bands none narrower than forty nanometres reach further than a forty-nanometre slope limit in 94 of 154 directions of the object-colour solid, because a band's width and the width of the reflectance edge it draws are different quantities — and what converts one into the other is how much colorant is in the film.

 What declaring a narrowest feature buys, and where it stops being true. The median looseness of a Cauchy–Schwarz bound whose lamp variance is bounded by a declared narrowest feature, against the width declared, for a fluorescent tube and a three-laser projector. Each lamp's own Bhatia–Davis bound — the peak declared and nothing else — is the upper dashed line, and the bound with the true variances is the lower one. The marks are the width each lamp's lines actually have. Declaring it truly takes the tube from ×196 to ×86 and the projector from ×30 to ×14. The open circles are declarations the lamp does not meet, where the bound falls below the error. What light is

A declared width buys a factor of two

A colour engine given two separately blurred spectral tables cannot bound its own error from them, and the bound that always holds — the peak declared and nothing else — sits a median 196 times above the error under a fluorescent tube. Adding one number, the width of the lamp's narrowest feature, brings that to 86. It never fails on any declaration the lamp truly meets, it fails on 47 of 68 notches on one it does not, and its rank correlation with the error it bounds is 0.27.

 Which two lamps to stand a mesopic match between. Every pair of the five lights, by how far a match made under one and set under the other moves as the rod signal's weight into the S channel goes from nothing to equal — the median over forty-two surfaces, which is the signal an experiment has to resolve. The best pair is daylight against phosphor LED at 0.58 ΔE₀₀; the worst is tungsten against fluorescent tube at 0.09, a factor of 6. The count at the right is how many settings it takes to resolve the weight to a tenth at half a colour difference of scatter per setting. What the eye does

The reference lamp must not move

To measure an uncertain weight, use the condition in which the answer depends on it most. That is right about half of an asymmetric colour match and exactly wrong about the other half: a match measures a difference of two displacements, and a reference field that also moves with the weight cancels the signal the test field carries. Daylight is the least sensitive of five lamps and belongs in every one of the three best pairs — 75 settings against a phosphor LED, 2,804 against the pair of lamps the principle as stated would have chosen.

 Two ways to ask how much rod signal an older eye has. The rod signal's catch of each lamp divided by the S cones' own catch, against the observer's age, for five lamps — drawn twice. The upper curves take the rod signal at a fixed absolute size, and every one of them roughly doubles from twenty to seventy-five: an older lens cuts the blue before the S cones see it and the rods, peaking further into the green, lose much less. The lower curves take the rod signal at a tenth of each cone's own peak absorptance, which is the model's own definition, and they barely move at all. Nothing about the retina differs between the two; only the normalisation does. What the eye does

A rod signal has no natural size

An older lens absorbs where the S cones are sensitive, so it should make the uncertain rod-to-S-cone weight cheaper. Measured, the rod signal's catch of a phosphor LED as a share of the S cones' own catch more than doubles from twenty to seventy-five — and the share the model actually uses falls by a fifth. The two differ by the S cone's peak absorptance, which the lens takes 60 per cent of, and which entered the model as a normalisation rather than as a claim.

 What survives holding the lightness still. For each of five groups of rooms sorted by how much light the wall returns, the rank correlation of the finish's share with three properties of the paint, taken with the wall's luminance factor held. The wall's chroma runs from -0.90 among the darkest walls to 0.81 among the palest, crossing zero in the middle — the reversal. The band's width, which was the predicted mechanism, never leaves the range -0.15 to 0.22, and the census already contains four families whose bands are all the same width. What a scene does

The finish adds the room's own colour

Among dark walls a more saturated paint loses a smaller share of its colour to a gloss finish, and among pale walls a larger one. The mechanism proposed for that reversal was spectral concentration — a narrow tall band — and the census that found it already contained four families of paints whose bands are all the same width. Holding lightness still, the band's width orders the losses at a rank correlation of 0.01. What does order them is that the light a finish adds has already bounced off the walls.

 How each unit balances lightness against chroma, as a display brightens. Twelve pairs built to differ in lightness alone and twelve built to differ in chroma alone, each at exactly one ΔE₀₀, read in both units at nine display levels. Up is the median lightness pair's reading divided by the median chroma pair's, so a falling curve means chroma differences becoming relatively more visible. Both fall. ΔEITP's median runs from 0.91 at a 1.5-candela white to 0.81 at 10,000, a fall of 11 per cent; CAM16-UCS's from 1.13 to 0.87, a fall of 23 per cent. Neither turns: both units say chroma differences gain on lightness differences as a display brightens, and CAM16-UCS says it twice as strongly. Difference and uniformity

The units part by hue, not by light level

Two colour differences with a light level in them were compared on a display running from 5 to 5,000 cd/m², and a prediction was drawn from how each divided a difference between lightness and chroma. Built into pairs that differ in lightness alone and in chroma alone, both units say the same thing about light level: as a display brightens, chroma differences gain on lightness differences — ΔEITP by a tenth, CAM16-UCS by a quarter. Where they part is hue. The appearance model moves every colour's balance by the same factor; ΔEITP moves the violets, reds and cyan-blues the other way, and at nine of twelve base colours the two units disagree about the direction.

 What a lit room takes, and where it takes it. On a display whose white is 1,000 cd/m², how much of the ΔEITP a one-unit lightness step is given survives a veiling luminance reflected off the screen, against where on the lightness scale the step sits. At L 2 — a deep shadow — one candela of reflected light removes 22 per cent of the difference and three candelas remove 46. At L 90 the same veils remove nothing measurable. The shadows the unit counts most are the ones a room removes first. Difference and uniformity

The shadows a unit counts are the ones a room removes

ΔEITP's growth with display brightness is largest in the dark greys, and dark greys are where a lit room's light reflected off the screen sits. One candela a square metre of veiling luminance removes 22 per cent of the difference the unit gives a step at the bottom of the scale on a 1,000-candela display and nothing measurable at the top. The same veil raises the growth the unit reports across display levels from a factor of six to a factor of seventeen, because it destroys a dim display's shadows first.

 A halftone tint, between the two ideals and nearer one of them. For each pigment, how far the hue turns when it is tinted to the same lightness three ways: mixed with white pigment, mixed with white light, and printed as a halftone at a Yule–Nielsen factor of two. Right is towards a longer wavelength. 7 of the eight pigments' two ideals turn the hue opposite ways. The halftone mark sits between them at every pigment and between 26 and 44 per cent of the way from the light ideal to the paint one — so it keeps the additive tint's direction and loses about a third of its size. What the brain does

Printing does not change the sign

A tint mixed in paint and a tint mixed in light turn seven of eight pigments' hues opposite ways. A halftone sits between the two because light entering the paper between dots emerges under a dot, and the question was where. At the optical gain a coated sheet is fitted at, a halftone has travelled between 26 and 44 per cent of the way from the light ideal to the paint one — the same fraction for every ink — and only one pigment in eight changes direction.

 The same gradients, priced by a penalty and by a projection. Each row is one gradient held inside a coated press from six starts. The pale dots are the penalised relaxation — a free step, with a price for leaving the press — and the dark ones are the projected relaxation, which takes the free step and then moves each point to the nearest printable colour. Across is how much longer than the free path each result is, logarithmic. On the 10 gradients whose straight line leaves the press, the penalty leaves 9 starts at more than twice the free length and the projection leaves none. The projection's best route costs a median 0.02% against the penalty's 0.13%, and its spread across starts is 0.95% against 132.5%. Matching and measuring

A projection has no reason to detour

Holding a gradient inside a press by penalising the excursion turned the gamut's price from a number into a search: on five of ten crossing gradients some starting point leaves the relaxation trapped at more than twice the free length, and the spread across six starts runs to 271 per cent of the free path. Replacing the penalty with a projection — take the free step, then move each point to the nearest printable colour — leaves no trap on any gradient and a spread of 0.06 to 2.4 per cent.

 Where a sharpened E errs, per pixel and as seen: black type on paper, 10 point. The letter E at 10 point on a 300-dot-an-inch page, black type on paper, sharpened by an unsharp mask on stored values and on light; the two maps show the colour difference between the results, pixel by pixel on the left and after the eye's three spatial channels at 40 centimetres on the right, on one scale. The thin line is the letter's outline and the dashed circles mark two pixels round each corner — 8 outer corners and stroke ends, 4 inside angles. The circled zones are 31% of the outline and hold 28% of the per-pixel error and 25% of the seen error. What it takes to deliver it

Sharpened type errs on its dark side

A sharpened square hides its halo along its edges and shows it at its corners, so a sharpened page of type was predicted to show its error at the corners of its letters, several times out of proportion to their length. Black type does the opposite: its outer corners and stroke ends hold between two thirds and nine tenths of their share. As the eye leaves it, the error lies on the dark side of every edge, and a corner holds as much of it as it has dark ground around it — a quarter of a disc where a black corner points out into paper, three quarters where black wraps round a pale one.

 A four-ink inverse table's error along the grey axis, 9 nodes, black from L* 60. The colour difference between a grey asked for and the grey printed, from L 92 to 14, for a 9-node inverse table whose separation strategy begins black generation at L 60 (dashed line). Three tables: filled exactly from the strategy, mean 0.073; refitted with each node sliding along the strategy, 0.050; refitted freely, 0.036. To the left of the dashed line the strategy-bound refit lies close to the filled table; to the right it lies close to the free refit. Every table is exact at its nodes, marked on the axis. What it takes to deliver it

A strategy keeps the refit only where black prints

Refitting an inverse table's nodes against their own error halves it, and in a four-ink table the refit has to leave the separation strategy's choice of black alone. Held to the strategy, it keeps the whole of its gain wherever the strategy prints black and almost none of it where the strategy does not — and the share it keeps equals the share of the grey ramp printed with black, to within three hundredths. The start of black generation, predicted to be where the refit would fail, is where the table needs it least.

 The calibration error a narrow fourth channel survives, by where it is placed. For a red, green and blue ambient sensor with one more channel 10 nm wide, centred from 450 to 640 nm: the first calibration error at which some lamp of fourteen is misclassified. Dashed: the channel pooled into the whole residual, as the ambient sensor's disagreement was read; its best is ±1.50 per cent. Solid: the channel read on its own, as its departure from what the camera's white predicts for it; its best is ±4.0 per cent, at 450 nm. Crosses mark centres where the channel read alone does not separate the lamps at all. The two horizontal lines are the red, green and blue design (±1 per cent) and the design with a clear channel (±4 per cent). What a camera does

A narrow channel has to be read on its own

An ambient sensor with a clear channel tells structured lamps from smooth ones by reading deep red, and one without it reads structure but breaks at one per cent of calibration error. A narrow fourth channel between the camera's peaks was proposed to have both. Pooled into the sensor's disagreement with the camera, it reads structure and breaks at one and a quarter per cent — no better than the design it was meant to rescue. Read on its own, as its departure from what the camera predicts for it, a channel at 450 nm holds to four per cent, the clear channel's figure, and far red does not move it.

 Four ways to correct the corner, under each of four lamps, at 25°. The mean colour error left on the chart's coloured patches at the corner of the frame, under each lamp, after: the red row fitted under that lamp; one red row fitted on the chart under all four lamps with the grey held under daylight; a grey-card gain map calibrated under daylight; and the worst of the other lamps' rows used by mistake. daylight: 0.93, 1.33, 1.86, 4.91; tungsten: 0.99, 1.45, 1.87, 3.26; white LED: 0.36, 1.39, 3.28, 1.77; fluorescent: 0.37, 1.56, 3.09, 2.45. The pooled row holds every lamp between 1.33 and 1.56. What a camera does

One row for every lamp costs the lamps that lose least

A corner correction fitted under one lamp is right under that lamp and can be badly wrong under another, so a converter unsure of its lamp was offered a single row fitted under several at once. Pooled over four lamps, one row holds every lamp's corner between 1.33 and 1.56 colour differences — no lamp worse than a grey-card map, and none near the 4.91 a wrong row can leave. The price falls on the white LED and the fluorescent tube, which it leaves four times worse than their own rows, because they lose the least red at the corner and the pooled row is built to repair the lamps that lose the most. Two rows, one per class of lamp, keep almost all of both.

 Where chroma stops protecting a wall, in five rooms. The seventy-two paints in each of five rooms, sorted by how much light the wall returns and read in overlapping windows of eighteen: the rank correlation of the share of colour a satin finish takes with the wall's chroma, lightness held. Below zero a more saturated paint loses less; above, more. The dots are where each room's curve crosses: cube at 0.36, corridor at 0.37, low room at 0.38, one wall open at 0.29, two walls open at 0.24. Solid lines are closed rooms of three shapes; dashed are the cube with one and two walls opened. What a scene does

An open room hands over sooner

A gloss finish takes the least colour from a saturated dark wall and the most from a saturated pale one, and in a closed cube the sign changes at a wall returning about a third of the light. The prediction was that a less enclosed room would move that point up the lightness scale and that a room with a window would have no pale end. Opening one wall moves it down, from a luminance factor of 0.36 to 0.29, and opening a second to 0.24; stretching the room into a corridor or flattening it nudges it slightly up. The ambient does whiten, as predicted. A whiter ambient does not delay the colour term — it weakens it, so the other term takes over sooner.

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