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

Every essay — page 22

Page 22 of 40, continuing through the fields in the same order.

What light is What the eye does Matching and measuring Difference and uniformity What the brain does What a scene does What a camera does Where the model breaks What it takes to deliver it

SeriesObserversNamed objectsRefutationsSearch

Where the model breaks

Seventeen observers in 1931, an error in the blue that was never fully repaired, colour vision deficiency, and the display this page is being read on.

The pairing against the direct computation, for each departure that admits both. Each departure can be computed twice: directly, by taking the difference between the fuller model and the integral one, and as a pairing — an inner product of the sample's deviation with the light's. The bar is how far apart the two answers are, relative to the answer, on a logarithmic axis. The three directional rows agree to a part in a thousand billion, which is the arithmetic of one shared quadrature. The lateral row agrees to three parts in a hundred thousand, and the gap there is the radial quadrature rather than the identity: the two integrals are taken over different grids. The pairing is not an approximation to the departure. It is the departure, written so that its two factors are separate.

Three audits and one shape

The tabulation, the observer and the scene solver have nothing in common as subjects. Each turned out to hold a departure that is a pairing of two deviations, vanishes exactly when either is empty, and had been invisible because its parameter had no call site. Three subjects, one shape, and the shape is the round's result.

5 figures The conditions under which an observer's departure is exactly zero. A departure of the observer is the pairing of something belonging to the observer with something belonging to the stimulus, so emptying either factor empties the product. The axis is logarithmic in what is left when the condition is imposed. Six rows empty the stimulus's factor — a perfectly neutral sample is the same colour for every observer, at any age and any field size — and two empty the observer's, since a gain on each cone and a change of basis are both absorbed exactly. All eight are identities rather than small numbers. The last two are the same two conditions imposed in a published cone space rather than in the observer's own, and they are worth eight and thirteen units: the identity is about the eye, and the arithmetic everybody uses is in somebody else's coordinates.

The conditions are the result

The round measured about twenty departures and established ten conditions. The departures are numbers that depend on a sample, a light and a construction; the conditions are exact, they hold under every substitution tried, and they are what a reader can act on. A size is a measurement and a condition is a mechanism.

5 figures Which of the collection's published quantities a departure can be pushed through. The six quantities the previous round recomputed under six different colour-difference units, and whether the same treatment works for a departure. Two do: the adaptation census and the metameric pair both take reflectances and a light, which is what a departure acts on. Four do not, and the reasons are different in each case rather than a single obstacle. A unit is a function applied to the answers, so it can be swapped at the end of any computation; a departure changes the object at the start, so it has to be accepted by every stage in between. That is the practical difference between auditing a convention and auditing a structure.

What this round could not reach

Three audits, about twenty departures and ten conditions, and a longer list of things that were named and not measured. Every item on it is specific, most of them are an afternoon's work, and the reason none was done is the same in every case — the round ran out of round.

5 figures 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.

7 figures 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.

6 figures 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.

7 figures 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.

6 figures 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.

5 figures 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.

6 figures

Matching and measuring

CIE XYZ, chromaticity, and gamut — a system for predicting when two lights match, frequently mistaken for a system that predicts how they look.

The CIE 1931 chromaticity diagram with its unreachable region marked. The spectral locus encloses every chromaticity a human eye can see. Cells inside the sRGB triangle are drawn in their own colour; the 85 per cent outside it are hatched, because no value this display accepts is the colour belonging there.

Most of this diagram cannot be shown

The chromaticity horseshoe is the canonical illustration of colour science, and nearly every printed copy is filled edge to edge with colours the page cannot produce. The honest version marks them, and the marking covers most of the picture.

8 figures Two identical grey patches on different surrounds. Both inner squares are #868686. The one on the dark field looks lighter. The values are checked to be equal before the figure is drawn, so the claim is a fact about the drawing rather than a promise.

Matching is not appearance

CIE XYZ predicts when two lights will look the same under identical viewing conditions. It was never a model of how anything looks, and most of the confusion in applied colour comes from using it as one.

8 figures sRGB, Display P3 and Rec. 2020 compared on the chromaticity diagram. Three nested triangles inside the horseshoe. sRGB covers 74 per cent of the area P3 covers. Rec. 2020's red and green primaries sit on the spectral locus itself, within 0.000 and 0.002 of it, meaning they are monochromatic.

What a gamut costs

Three primaries reach a triangle and the visible region is not a triangle, so something has to give. Widening the primaries helps, has a price in precision and compatibility, and runs into a limit that is geometric rather than technological.

5 figures