Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Computed, not quoted — page 15

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 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. What it takes to deliver it

The instrument is one observer exactly

A spectrophotometer has no observer uncertainty. It computes through the 1931 tables to the last bit, reproducibly, forever — which is its whole value and is also why it cannot report the three colour differences of observer spread between itself and whoever is looking at the sample.

 The same white, matched at six primary widths. At every width the three primaries are solved to match D65 exactly for the reference member; the bands are what the population sees. A broad primary integrates the observer differences over a band and averages them away; a narrow one samples them at a point and passes them straight through. From 40 nm to 2 the ninety-fifth percentile rises from 11.3 to 17.9 ΔE00, monotonically, and the technology has been moving from left to right for thirty years. Matching and measuring

A narrow primary buys a disagreement

The observer audit decomposes what a display costs a population. Narrowing the primaries raises the pigment-peak departure monotonically, moving one raises or lowers the macular departure, and the two respond to different design variables — so a wide gamut and an observer-robust display are bought with the same money.

 Each departure over forty-two surfaces rather than one. The same six departures measured over a family of forty-two analytic reflectances — an absorption band of stated centre, width and depth — with the smallest, the median, the ninety-fifth percentile and the largest marked. Every one of them spans more than a factor of three, and the ranking between them is not stable across the family: what decides a departure's size is which sample it is asked about, because a departure is a pairing and the sample is one of the two factors. Quoting any single number for what an observer's age is worth is quoting a choice of example. What a camera does

The chart was measured by an observer too

A camera profile is fitted so that the camera's numbers reproduce the chart's measured tristimulus values. Those values were computed through the 1931 observer, so the fit inherits every departure in this round — and the fit's own residual, at 0.19 ΔE₀₀ on the chart, is fifteen times smaller than the term it cannot see.

 Each departure over forty-two surfaces rather than one. The same six departures measured over a family of forty-two analytic reflectances — an absorption band of stated centre, width and depth — with the smallest, the median, the ninety-fifth percentile and the largest marked. Every one of them spans more than a factor of three, and the ranking between them is not stable across the family: what decides a departure's size is which sample it is asked about, because a departure is a pairing and the sample is one of the two factors. Quoting any single number for what an observer's age is worth is quoting a choice of example. What the brain does

The census under another observer

This collection's largest computed result is an adaptation census — fourteen changes of light judged over a hundred and twenty-five constructed surfaces. Every number in it was computed through one observer, and the observer's own departures are between one and two and a half units on the same surfaces, which is the size of the effects the census reports.

 Every departure under every light. Six departures across six lights, each cell the difference between two observers in ΔE₀₀, drawn as a bar whose length is the number. The rows are not multiples of one another: the lens is worst under tungsten and the pigment peaks are worst under a three-emitter LED, because a departure is a pairing and which light is being paired with decides it. The laser projector's row is empty, and that is not a fact about lasers — on this collection's five-nanometre grid a three-line spectrum is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about exactly. What it takes to deliver it

The booth and the eye disagree

A viewing booth is specified to simulate daylight, and daylight is the light under which observers disagree least. So a booth is the condition in which a specification is least likely to be tested against the thing that breaks it, and the shop it will be judged in roughly doubles the term.

 Each departure over forty-two surfaces rather than one. The same six departures measured over a family of forty-two analytic reflectances — an absorption band of stated centre, width and depth — with the smallest, the median, the ninety-fifth percentile and the largest marked. Every one of them spans more than a factor of three, and the ranking between them is not stable across the family: what decides a departure's size is which sample it is asked about, because a departure is a pairing and the sample is one of the two factors. Quoting any single number for what an observer's age is worth is quoting a choice of example. What it takes to deliver it

A brand colour for a population

A brand colour is chosen once, specified in a number, and seen by everybody. The observer audit says two candidate colours with identical specifications can carry observer spreads differing by an order of magnitude, and that the spread is computable from an ink's spectrum at the moment the ink is chosen.

 What choosing a space to divide the white out in is worth. Three pairs of routes to the same colour, over forty-two surfaces: dividing the white out in tristimulus values, in a published cone space, and in the observer's own cones. The first two agree to 0.59 ΔE₀₀ at the median. Either of them differs from the observer's own cones by more than fifteen. That is why the two exact conditions in this round are exact only in the eye's own coordinates: the identity belongs to the receptors, and every published arithmetic works in a basis somebody else chose. What the brain does

The appearance model takes XYZ

CIECAM16 predicts how a colour looks, and its input is three tristimulus values computed through a standard observer. Everything this round measures happens before the model is called, so an appearance prediction inherits six observer departures and a choice of cone space before it begins.

 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. What the eye does

Two observers and one metamer

An observer departure is invisible on a single sample compared with nothing. It becomes a disagreement the moment two spectra are being asked to match, because a match is an identity between three integrals and a different observer takes different integrals. Everything in this round is a statement about pairs wearing a single sample's clothes.

 The two tabulation choices over forty-two surfaces, under a 6500 K thermal radiator. Each column is one choice, measured over a family of forty-two analytic reflectances rather than on a single example: an absorption band of stated centre, width and depth. The four marks are the smallest, the median, the ninety-fifth percentile and the largest cost in ΔE₀₀, logarithmically. Under a smooth light the range is worth 9.1 times the step at the median, so a collection wanting one repair should widen its range rather than refine its step — and under a fluorescent tube the ranking reverses outright. What the brain does

The grid under the census

The adaptation census is computed on eighty-one wavelengths from 380 to 780 nanometres. Under the daylight and blackbody sources it uses, the range is worth about half a colour difference on ordinary surfaces and the step about a twentieth — so the census carries a tabulation term as well as an observer one, and they are not the same size.

 The share of each light outside 380–780 nanometres, as power and as visual product. Two measurements of the same truncation. The upper bar is the fraction of the light's radiant power that lies outside the range; the lower is the fraction of the product of light, sample and observer — which is what a colour is made of. A thermal radiator puts a fifth of its power outside and about a thousandth of its colour, because the observer is zero where most of that power is. The gap between the two bars is the observer's own tails doing their job, and reading the upper number as though it were the lower is how a range gets argued about without being measured. What light is

A lamp is two audits at once

A lamp's ultraviolet content decides what its tabulation costs and its blue content decides what its observer costs, and the two run in opposite directions with colour temperature. So no lamp is good for both audits and no lamp is bad for both, and a single figure of merit for either is a figure of merit for one property of a spectrum.

 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. Where the model breaks

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.

 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. Where the model breaks

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.

 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. Where the model breaks

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.

 The arguments a standard observer does not have. Seven choices inside a set of colour-matching functions, each with the shape it takes and what it is worth in ΔE₀₀ on a red pigment under a 6500 K radiator. Six are measurements: a field size, an age, a macular density, a cone optical density, three peak wavelengths and a rod contribution. The seventh is not — a change of basis is a change of curves and not a change of observer, and its entry is exactly zero because the space an experiment measures is what an observer is. Printing that zero beside the others is the clearest statement of what the other six are measurements of. What a camera does

A sensor has no lens

A camera is a fourth observer and it is the only one with none of the six arguments this round measured. It does not age, it has no macular pigment, its response does not broaden with density and its peaks do not vary — so it is perfectly reproducible and is not any of the observers it is fitted to reproduce.

 One surface dimmed sixteen times, and the two things that happen to it. A single surface, dimmed by successive halvings, with the lens age departure measured on it at every level. The tristimulus deviation falls by exactly the dimming factor — 16 times over the sweep, to the last bit, because the colour integral is linear in the stimulus. What a unit of that deviation is worth rises by 8.2 times over the same sweep. The colour difference the audit reports is the product of the two, and it falls by only 1.95. Matching and measuring

A deviation is not a difference

The previous round priced twenty departures in colour differences and treated each number as a property of the thing that departed. Every one of them is a product of two factors — how far the reading moved, which is linear and belongs to the departure, and what a unit of that movement is worth where it landed, which is not linear and belongs to the colour. Dimming one surface sixteen times scales the first by exactly sixteen and the second by eight.

 The angle between two departures, which nothing in the audit records. Every pair of the six departures on every surface — 2520 pairs — binned by the angle between their two deviations in the local metric. The distribution reaches both ends: 440 pairs sit under thirty degrees and point almost the same way, and 615 sit above a hundred and fifty and point almost opposite. On 1095 of the 2520 the two together cost less than the larger of them alone. A table of magnitudes cannot say which case it is in. Matching and measuring

A size is not a direction

An audit that reports magnitudes cannot say what two of them cost together. Over two and a half thousand pairs of observer departures the angle between them in the local metric runs from one degree to a hundred and seventy-nine, and on forty-three per cent of them the two together cost less than the larger of the two alone. Adding the angle predicts the composition to one and a quarter per cent; Pythagoras is out by twenty-eight.

 The straight piece under the cube root, and where it stops. CIELAB's lightness against relative luminance, over the bottom 5.0 per cent of the range. Below Y = 0.008856 it is a straight line of slope 7.787; above it, a cube root. The two meet at L* 8 in value and in slope, exactly — the CIE's two constants are chosen to make that true. The dashed curve is the pure cube root, which reaches negative lightness before it reaches zero luminance and has an infinite slope there. The break is marked, and the axis runs to Y = 0.050. Matching and measuring

The straight piece under the cube root

CIELAB's lightness is described everywhere as a cube root and below a stated luminance it is a straight line, spliced on with two constants chosen so the join is exact in value and in slope. Every black a delivery chain reaches is inside that straight piece — a press black at L* 2.4, a projected black at 1.1 — where the compression does not compress, the price of a deviation is flat to two parts in a thousand, and the second derivative the composition of two departures needs does not exist.

 What one tolerance accepts, around four colours. The surface in tristimulus values that ΔE₀₀ 1.0 draws around four colours, each outline scaled to its own size so the shapes can be compared. The volumes they enclose differ by a factor of 1.0e+5 across the sRGB cube, and the longest axis of one shell is between 3.3 and 29.9 times its shortest. A tolerance is written as one number and is a different set of samples at every colour it is applied to. Matching and measuring

What one number accepts

A delivery tolerance is written as a single colour difference and it acts on three tristimulus values, so what it actually accepts is a closed surface. Measured over sixty-four colours in the sRGB cube, the volume inside that surface varies by a factor of a hundred thousand and its longest axis is between three and thirty times its shortest. The same contract, applied to a dark colour and a light one, is two different requirements.

 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. Matching and measuring

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.

 Two primaries mixed, and the line a reader assumes they take. The additive mixture of two sRGB primaries, walked in twenty steps, plotted in the a and b of CIELAB. The filled points are where the light actually goes, which is exactly straight in tristimulus values because that is Grassmann's second law. The open points are the straight line between the two readings. They part company by 26.7 ΔE₀₀ at their furthest, at 60 per cent of the way along, and the half-and-half mixture misses the midpoint by 21.1. Matching and measuring

The mixture line bows

Grassmann's second law says an additive mixture is exactly linear in tristimulus values, and tested here it is exact to floating point. Nothing downstream of the three numbers preserves it. The physical half-and-half mixture of two colours sits a median of 5.5 colour differences from the midpoint of their two readings and up to thirty; on a green and a blue display primary it is twenty-one, which is a quarter of the distance between them.

 Three exchanges, two of which move the colour. The documented pipeline is a white balance, a colour matrix, a tone curve and a clip. Each bar is what happens when two neighbours change places, over 30 surfaces the modelled sensor captures: the filled bar is the mean and the tick is the worst patch. Exchanging the balance and the matrix costs 9.2 colour differences at the mean and 13.2 at the worst. Exchanging the curve and the clip costs exactly nothing, and that is a theorem rather than a small number: a monotone curve onto the unit interval commutes with clamping to it. What a camera does

The order is not in the documentation

A raw converter performs a white balance, a colour matrix, a tone curve and a clip, and every account of the process lists them in that order without saying the order decides anything. Exchanging the first two moves the picture by nine colour differences at the mean and thirteen at the worst patch. The twenty-four arrangements collapse into five outcomes running out to sixty-five, and nothing a converter ships says which of them it is.

 Where the mosaic is filled in, along one row through an edge. A Bayer row across a step from 0.9 to 0.08, in units of the sensor's own ceiling, reconstructed in linear light and reconstructed after the tone curve, with the second undone so the two are compared at the same point in the chain. Away from the edge they agree to 8.3e-14, because a constant interpolates to itself under any curve. At the edge they differ by 5.78 colour differences. Interpolating encoded values pulls an edge towards its dark side. What a camera does

One step has no choice

Four of a raw converter's operations can be arranged twenty-four ways. The reconstruction cannot be arranged at all — a colour matrix needs three numbers and a mosaic site has one, so filling in the mosaic is forced to the front by arithmetic rather than by convention. What is not forced is whether it happens in linear light or after the curve, and that decision costs 5.8 colour differences at an ordinary edge and nothing at all four sites away from it.

 A hue circle through a per-channel curve. 28 colours on a circle of constant lightness 55 and chroma 38, each put through the tone curve one channel at a time and read back. The curve is a function of a single number and has no idea what hue is, and it rotates the circle by up to 4.4 degrees — largest at hue 260 — while raising chroma by a factor of 1.27 and lightness by about 1 units. What a camera does

A contrast control is three controls

A tone curve is a function of one number at a time and knows nothing about hue. Applied to each channel separately it rotates a hue circle by up to seventeen degrees, raises chroma by a factor of 1.27 at ordinary strength, and lifts lightness — so a photographer who moves a contrast slider has moved three things and the interface names one of them. All three scale with the curve's strength, monotonically, and the hue rotation depends on which hue it is.

 A stop taken in raw, and the same lightness reached afterwards. Each row is a stop of exposure applied to the raw values, against a gain applied after the whole pipeline and solved so that an eighteen per cent grey comes out at the same lightness. The two are then the same brightness by construction and differ by 4.2 colour differences at the mean and 11.1 at the worst patch. A stop is a scalar in front of the curve and is not a scalar behind it. What a camera does

A stop is not a stop afterwards

Doubling the light is exactly a factor of two in raw values and in tristimulus values, which is the one thing about exposure everybody is sure of. A stop taken after the tone curve is a factor of something else, and matching the two on an eighteen per cent grey leaves the rest of the frame between three and four colour differences apart at the mean and up to eleven at the worst patch. The gain that matches one stop is 2.47 rather than 2.

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