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

Bound — where it appears

A stated limit on what a parameter may be, which turns an unbounded search into an answerable one. A bound that the answer sits on is doing the work; a bound the answer never approaches is indistinguishable from no bound at all unless somebody checks.

Named by 12 essays across 5 fields — each of them below, with the objects they name alongside it.

A published residual is a mean, and the worst object in the room costs twice it. Three bars for each of the 14 changes of light in the adaptation census, ordered by how uneven the change is across surfaces. The first bar is the published mean residual. The second is the worst single surface in the audit's published test set. The third is the worst surface anywhere in the region that set is drawn from, found by search rather than by reading a maximum off a lattice. The mean-to-worst ratio runs from 1.90 to 4.02 and averages 2.43, so every published adaptation number has a worst case about twice it that no essay had ever quoted. The gap between the second and third bars is the other finding: a maximum over 125 sampled points understates the region's own maximum by up to 34 per cent.

A mean is not a worst case

Every adaptation number this collection publishes is an average over objects, and the reader asking whether adaptation will fail them is asking about the object it fails on. That object costs between 1.9 and 4.0 times the published figure, and how uneven a change of light is across objects turns out to be a property of the change rather than a constant.

limits · Limits A published residual is a mean, and the worst object in the room costs twice it. Three bars for each of the 14 changes of light in the adaptation census, ordered by how uneven the change is across surfaces. The first bar is the published mean residual. The second is the worst single surface in the audit's published test set. The third is the worst surface anywhere in the region that set is drawn from, found by search rather than by reading a maximum off a lattice. The mean-to-worst ratio runs from 1.90 to 4.02 and averages 2.43, so every published adaptation number has a worst case about twice it that no essay had ever quoted. The gap between the second and third bars is the other finding: a maximum over 125 sampled points understates the region's own maximum by up to 34 per cent.

An extremum is still not a sample

Two rounds ago three measurements turned up that took a maximum over a sample of a set and were short by up to a factor of two. The same error was live in a fourth place the whole time, on the set of surfaces every adaptation number is averaged over, and it is short by up to a third.

matching · Gamut Every worst surface sits on a number somebody typed. The region the test surfaces are drawn from, in its own two modulation coordinates: a square of allowed depths with a diamond inscribed in it, the diamond being the requirement that the two depths sum to no more than 0.7. The 14 marked points are the worst surface for each change of light in the adaptation census, found by search over the whole region. Every one of them lies exactly on the diamond, and every one is also at the brightest level the region allows — both declared constraints active, on all 14 rows, with no interior maximum anywhere. That is the opposite of what bounding the wall gave: there the worst case turned over at a band width of six nanometres because a narrow band returns too little light, which is physics. Here the worst case is a reading of two numbers. The one constraint that is about the world — a paint's excitation purity may not exceed 0.6 — is slack everywhere: the most saturated surface the region admits reaches 0.459.

Every worst surface sits on a declaration

Bounding the wall in a painted room produced a real worst case — the residual turns over at a band six nanometres wide because a narrower band returns too little light. Bounding the surfaces the residual is averaged over produces nothing of the kind, because all fourteen answers sit exactly on two numbers somebody typed and the one constraint that comes from the world never binds at all.

scene · Scene What a code lattice costs, and where. Twelve thousand colours quantised to 8 bits per channel through the sRGB transfer function and read back, with lightness across the bottom and the colour difference the rounding cost up the side. The mean is 0.191 and the worst case is 1.15, a factor of 6.0. The bars are band means, and they rise: the encoding spends its codes in the shadows, so the top of the ramp is where the lattice is coarsest against a metric that does not compress as hard.

A lattice has no derivative

Every departure priced here was priced by perturbing something and reading the answer, which requires the thing being perturbed to have a derivative. A file written on a code lattice does not have one — its output is flat almost everywhere and jumps on a set of measure zero — so quantisation can be bounded and never propagated. The bound is 1.15 colour differences at eight bits per channel against a mean of 0.19, and it is worst where the encoding spends fewest codes.

matching · Gamut What a slope limit costs the object-colour solid, direction by direction. For each transition width, how much of its ideal reach the solid keeps: the median direction, the tenth percentile, and the worst. At twenty nanometres the median keeps 0.997 and the tenth percentile 0.985; at eighty they keep 0.946 and 0.766, and at 160 0.826 and 0.417. The worst direction falls from 1.00 at five nanometres to 0.20 at 160, with directions reaching under five units beyond black set aside. The cost is in a corner only at widths sharper than an ordinary pigment's.

The limits assume a pigment that switches instantly

The hardest boundary in colorimetry is reached by reflectances that jump between nought and one at a wavelength, and no material does that. Constrain the jump to take twenty nanometres — a sharp dye — and the median direction of the object-colour solid loses under half a per cent of its reach. Constrain it to eighty, an ordinary pigment, and the median loses five per cent, the tenth percentile nearly a quarter, and seven directions in ten lose more than one. The cost is in a corner only for chemistry sharper than paint.

limits · Limits How many directions a slope limit costs, under four lamps. For each transition width and each lamp, the share of the solid's directions that lose more than one per cent of their reach — each lamp's solid against its own ideal. Daylight, a tungsten lamp and a phosphor LED run close together. The three-emitter LED, whose power sits in lines at 455, 530, 625 nanometres, costs 62 directions at forty nanometres where daylight costs 183, and by eighty — about the spacing of its lines — it costs 214 against 218.

Three lines spare a slow pigment

A pigment that cannot switch faster than forty nanometres loses more than a per cent of its reach in 183 of the object-colour solid's 305 directions under daylight. Under an LED whose light sits in three narrow lines, it loses that much in 62. Between the lines almost nothing is measured, so a slow reflectance can do its changing there — until its transitions are as wide as the lines are far apart, at which point the lamp stops helping and the count jumps to daylight's.

limits · Limits A 40-nanometre limit written in nanometres and written in energy. The transition width a reflectance is allowed, across the spectrum, for two ways of stating the same sharpness. Written in nanometres it is 40 everywhere. Written as a fixed spread of photon energy, which is how an absorption band's width is set, it is 40 at 550 nanometres and grows as the square of the wavelength: 21 at 400 and 67 at 700. The steps are the quantisation the calculation actually imposes.

A limit written in energy charges the reds

A slope limit on reflectance is usually written in nanometres and applied the same way across the spectrum. An absorption band's width is closer to a fixed spread of photon energy, which is nearly twice as many nanometres at 700 as at 500. Written that way, a limit that is forty nanometres at 550 is kinder to the object-colour solid overall — 489 of 913 directions lose a per cent rather than 544 — and it charges the reds more. Which directions pay is decided by one thing: whether their optimal edges fall above or below the reference wavelength.

limits · Limits Two tables, two directions of error: 5-node ramps of four inks. For each ink, the mean signed lightness error between the nodes: of the ordinary forward table, which errs light, and of four ways to fill the inverse table, all of which err dark except the one refitted against its own objective. On cyan the forward table errs by +0.206; an inverse filled from the press by -0.317; one inverted from the ordinary table by -0.554, because it inherits the forward table's error on top of its own; one inverted from the refitted table by -0.367; and one refitted in its own right by -0.002.

The inverse table errs dark

A profile's forward table predicts a print lighter than the press makes, and refitting its nodes removes the bias. The table a colour engine actually uses to separate an image is the other one — the inverse, from colour to ink — and it errs the opposite way: filled exactly from the press it asks for too much ink and prints dark, and built by inverting the ordinary forward table it prints darker still. Inverting the refitted forward table removes the inherited part and leaves the inverse's own. The round trip through both tables improves only when each is refitted against its own error, and then the two are no longer each other's inverse.

applied · Delivery 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.

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.

limits · Limits 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.

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.

limits · Limits 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.

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.

limits · Limits 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.

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.

light · Light

Named alongside it

The objects these essays reach for when they reach for this one.

ReflectanceObject-colour solidOptimal coloursPigmentSpectral power distributionWavelength gridWorst caseModelling assumptionChromatic adaptationIlluminantMacadam limitsNarrow band displays

All concepts