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

Structural choice — where it appears

A decision inside a calculation that has no magnitude to vary: a formula, a model form, a coordinate system. It is distinguished from a declared number, which can simply be multiplied and re-run. Auditing one needs a menu of published alternatives or a parameter that joins two of them, and where neither exists the choice is out of reach.

Named by 40 essays across 9 fields — each of them below, with the objects they name alongside it.

What six of this collection's published numbers do when the unit changes. Six quantities, from six calculations that share nothing: a change of light after an observer has adapted, a camera profile's error, the gap between the two standard observers, a metameric pair under the lamp that breaks it, the same image on two papers, and an observer two seconds into a new room. Each is recomputed under all six units and every unit is calibrated onto ΔE2000's scale first, so the bar is not a change of units in the ordinary sense. The bar is the ratio of the largest reading to the smallest, and it runs from 1.71 to 2.30. Five of the six are printed in ΔE2000 by the essays that report them; the sixth is printed in CAM16-UCS, because the model it comes out of defines that unit.

A choice with no magnitude

An audit can multiply a width by 1.25 and report an elasticity. It cannot multiply CIEDE2000 by anything. Auditing a structural choice needs a different instrument, and building one shows that six published numbers in this collection each carry a factor of about two of unit-choice — after the change of scale has been taken out.

limits · Limits What six of this collection's published numbers do when the unit changes. Six quantities, from six calculations that share nothing: a change of light after an observer has adapted, a camera profile's error, the gap between the two standard observers, a metameric pair under the lamp that breaks it, the same image on two papers, and an observer two seconds into a new room. Each is recomputed under all six units and every unit is calibrated onto ΔE2000's scale first, so the bar is not a change of units in the ordinary sense. The bar is the ratio of the largest reading to the smallest, and it runs from 1.71 to 2.30. Five of the six are printed in ΔE2000 by the essays that report them; the sixth is printed in CAM16-UCS, because the model it comes out of defines that unit.

Three choices reached

Two rounds ago this collection named three things it rested on and could not audit — a unit, a diagonal and a template. All three are now reached, and the interesting part is not the three answers but that four of the round's own predictions were refused by the arithmetic and one of its measurements was wrong in a way only a cross-check caught.

limits · Limits The six arguments a surface's response has, and the one this model keeps. A surface's response to light is a function of six arguments: the wavelength, direction and place the light arrives with, and the wavelength, direction and place it leaves with. The model every colour here is computed from keeps one number per wavelength, which means it takes the diagonal of the first pair, integrates the second away, and assumes the third pair equal. Each departure drawn here restores one of them. The fourth departure is not on the diagram: the wavelength grid is the range of the index that was kept rather than an index that was dropped, which is why it is the cheapest of the four to fix and was still not fixed.

The model has six arguments

Every colour computed here is an integral of a reflectance against a light against three curves, and a reflectance is one number per wavelength. A real surface's response is a function of six arguments — the wavelength, direction and place light arrives with, and the three it leaves with — so the model keeps one of them, takes a diagonal, integrates two away and assumes two more equal.

limits · Limits Eight conditions under which the model equation is exact, and how exact each one is. Each of the four departures vanishes if either of its two factors is empty, which is eight conditions. The axis is logarithmic in the residual that is left when the condition is imposed. Three of the eight are identities: the fluorophore's loading is zero so the emitted term is an empty sum, and a Lambertian surface or a uniform field makes the pairing's second argument identically zero. The other five are limits — a Gaussian excitation band has no edge, an opaque sample still has a kernel a few microns wide, a four-metre aperture is still finite, and the observer is small rather than absent at 380 nanometres. Each limit is drawn with the sequence its residual falls along as the condition is pushed, because a small number is not evidence of a limit and a falling sequence is.

Either factor being zero

Four different departures from the colour integral turn out to have one algebraic form — each is an inner product of something the sample does that the model has no slot for with something the light does that the model assumed away. Either factor being zero makes the departure exactly zero, and the sizes of the two factors decide neither how large it is nor which way it goes.

limits · Limits Six functions of wavelength, and the six different places they stop. Every table this collection integrates against, drawn over the range the body that published it defined it on. The scale is logarithmic so that the ultraviolet and the near infrared both fit. The bottom row is the range used here before the infrared band was added, and it is the intersection of the two rows that matter for an eye looking at a reflector — which is the right answer only while everything in the integral is being multiplied together. The daylight basis runs 80 nanometres further down than that intersection, and it was published that way because the ultraviolet in daylight is what makes a brightened sheet of paper glow. The analytic row is drawn to the edge of the plot because it has no edge: Planck's law is a formula and is exact at every wavelength, which is why illuminant A needs no table at all.

The grid is a range, not an index

Three of the four departures in this round restore an argument the model dropped. The fourth does not — the 380-to-780-nanometre grid is the range of the one argument the model kept, chosen in the first weeks and never revisited. It costs 6.70 ΔE₀₀ on a coated printing paper, it is the cheapest of the four to fix, and it is the one still unfixed.

light · Light 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.

A departure is not a unit

The previous round audited six published quantities by swapping the unit they were quoted in — a function applied at the end of each computation. Nothing of that shape works here. A departure changes the object at the start, so every stage in between has to accept it, and only two of the same six quantities can take one at all. The fourth cannot even be expressed in the interface.

limits · Limits The six arguments a surface's response has, and the one this model keeps. A surface's response to light is a function of six arguments: the wavelength, direction and place the light arrives with, and the wavelength, direction and place it leaves with. The model every colour here is computed from keeps one number per wavelength, which means it takes the diagonal of the first pair, integrates the second away, and assumes the third pair equal. Each departure drawn here restores one of them. The fourth departure is not on the diagram: the wavelength grid is the range of the index that was kept rather than an index that was dropped, which is why it is the cheapest of the four to fix and was still not fixed.

The audit that changed the object

Three rounds audited this collection's numbers, its sets and its conventions, and each one ended by naming the same thing it could not reach — that a surface is a reflectance. This round reached it, found that four departures share one algebraic form, had two of its predictions refused, and discovered that its own hemisphere quadrature had been passing a convergence check by luck.

limits · Limits a fluorescent tube, mercury lines on a phosphor bed, with a 5-nanometre grid marked on it. The light drawn at a fifth of a nanometre, with the 5-nanometre tabulation points marked beneath. 38 of the 53 points carry more than a twentieth of the peak. What a summation over those points computes is not an approximation to the area under this curve; on a spectrum with features narrower than the spacing it is a different quantity, and the difference depends on where the points fall rather than on how many there are.

The index is a choice too

Every colour in this collection is a sum over eighty-one numbers running from 380 to 780 nanometres in steps of five. That index is not a property of the eye, the light or the sample — it is a tabulation, and it holds three separable decisions that behave completely differently from one another.

light · Light The two tabulation choices over forty-two surfaces, under a tungsten lamp at 2856 K. 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 6.3 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.

A neutral has no grid

A perfectly flat reflectance computes to exactly the same colour on every wavelength grid, through every slit, at every origin, and for every observer — not nearly, but to the last bit of a floating-point number. The condition is an identity rather than a limit, and what makes it one is the white point.

difference · Metric What each end of the 380–780 nanometre range costs, by light. Two bars per light, on a logarithmic axis: the upper is what extending the range down to 300 nanometres moves the answer, the lower what extending it up to 830 does. The asymmetry is the whole figure. A thermal source has about a fifth of its power outside this collection's range and almost all of it at the long end, where the observer is already zero; what costs money is the short end, where the observer is small but not zero and daylight is still strong. A light with no ultraviolet — an LED lamp, a laser — pays nothing at either end, which is the pairing again: a range only costs what the light puts in it.

Two ends and one is empty

Extending this collection's wavelength range down to 300 nanometres moves a red pigment under daylight by 0.502 ΔE₀₀. Extending it up to 830 moves the same colour by 0.00015. A fifth of a thermal source's power lies outside the range and almost none of its colour does, and confusing those two shares is how a range gets argued about instead of measured.

light · Light What the normaliser cancels, per light. Two bars per light, logarithmic. The upper is the colour error a 5-nanometre sum makes when the white it is divided by is computed finely; the lower is the same sum divided by the white computed on the same coarse grid, which is what every colorimetric calculation actually does. The ratio is between 1.3 and 4.1. The grid appears twice in a tristimulus value and the two errors are the same error, so most of it divides out — which is why five nanometres has been good enough for a century without anybody having to be careful about it.

The normaliser carries the error too

A five-nanometre sum gets a red pigment's tristimulus value wrong by two hundredths of a per cent and its colour wrong by six hundredths of a unit. Those two numbers are not the same size because the grid appears twice in a colour — once in the sample and once in the white — and the two errors are largely the same error.

difference · Metric The rectangle sum against the trapezoid sum, under a 6500 K thermal radiator. Colorimetry's summation is the rectangle rule at the tabulated points. The trapezoid rule differs from it by exactly one thing — half a cell at each end of the range — and the gap between these two lines is therefore that term and nothing else. At five nanometres it is a factor of 14.3, which means the number everybody calls a sampling error is mostly a truncation error wearing the step's clothes. On a light whose lines are narrower than the step the two rules agree to three decimal places, because there the error really is the sampling.

The endpoint term has a name

The five-nanometre error on a smooth light falls linearly with the step, which is not what a sampling error does. It is the half-cell at each end of a truncated range, it is first order where the sampling is second, and halving two weights removes fourteen fifteenths of it for nothing.

light · Light What a tabulation step costs, by light, on paper. The horizontal axis is the tabulation step in nanometres, from one to twenty; the vertical is how far the resulting colour is from the same integral taken at a tenth of a nanometre over the same range, in ΔE₀₀, on a logarithmic scale. Each line is one light. The three with no feature narrower than the step fall smoothly and stay below a tenth of a unit at five nanometres, which is the grid used throughout. The fluorescent tube and the laser projector do not fall at all: their lines are narrower than any step drawn here, so the answer depends on where the samples land rather than on how many there are. The sample is held at paper throughout.

A grid is not a resolution

Ten essays into this collection there is one sentence about wavelength sampling, and it is that five nanometres is enough. Ten measurements later there are three decisions, three mechanisms, three repairs and two rankings, and the word resolution names none of them.

light · Light 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.

The third factor is a construction

Every figure in this collection names its observer, which was the whole point. None of them says what an observer is made of. Three curves are not a measurement of the eye; they are a projection of one, with a field size, an age, a macular density, three peak wavelengths and a luminance constraint inside them.

eye · Cones 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.

A neutral is everyone's colour

Two eyes differing by fifty years of lens yellowing, by a factor of three in macular pigment and by six nanometres of long-wavelength peak agree about a grey card to four parts in ten thousand billion. The agreement is an identity rather than a coincidence, and it says exactly what an observer disagreement is a disagreement about.

eye · Cones 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.

The identity is in the eye's own coordinates

Two conditions in this round are exact — a gain on each cone is not a different observer, and three curves for one space are one observer. Imposed in a published cone space rather than the eye's own they leave 13.0 and 8.11 ΔE₀₀ standing. The identities belong to the physiology and every arithmetic in use works somewhere else.

eye · Cones The three cone absorptances at two settings of the field size. Solid and dashed are the same construction at the two ends of the CIE's own second observer. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 16.0 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

A field size is two changes

The CIE publishes two standard observers and the difference between them is usually described as a field size. What actually differs is a macular pigment the light no longer passes through and a cone outer segment the light no longer travels the length of — two changes, in two places, with different signs and different sample dependence.

eye · Cones The three cone absorptances at two settings of the rods. Solid and dashed are the same construction at the two ends of a tenth of the cone response, which is a dim room. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 9.1 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

The rods are a fourth curve

Colorimetry has three numbers and a rod signal is a fourth. It cannot be absorbed by a gain, it cannot be removed by a white point, and adding a tenth of one to a three-curve observer costs 1.60 ΔE₀₀ — the narrowest distribution of the six departures, because an addition behaves quite differently from a filter.

eye · Cones 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.

An observer is a contract

Six departures, all above a delivery tolerance, none dominant, and no reader matching the standard on any of them. That would be a demolition if the standard observer were a description of an eye. It is not — it is an agreement about what a number means, and an agreement is not falsified by being unlike anybody.

limits · Limits What the Lambertian assumption costs a room, against how rough its walls are. The horizontal axis is the roughness of the two coloured walls; the right-hand end is nearly matt, which is what a radiosity calculation assumes. One line is the distance in ΔE₀₀ between the floor's colour and what radiosity gives for the same room — 4.89 at an eggshell finish, falling to 0.97 at the matt end. The other is the chroma of the bounce, which falls as the walls get glossier: what an interface returns is a Fresnel reflection and carries no pigment, so the fraction of the return that goes into the lobe is a fraction that arrives at the floor white. The lobe is taken out of the body term rather than added beside it, which is what a real finish does.

The solver had no slot for gloss

Every scene result in this collection is computed by radiosity, and radiosity is not an approximation that could be made more accurate. Its unknown is one number per surface, and a surface that returns light differently in different directions does not have one. A missing slot cannot be wrong by a small amount.

scene · Scene Two ways of putting a lobe on a wall, and the sign they disagree about. The chroma of the floor's return against the wall's roughness, computed twice. In one the interface's return is taken out of the body term — light reflected at the boundary never reaches the pigment, which is what a real finish does. In the other it is added beside the body term, which is what a microfacet model does if nobody couples the two. The first says a gloss wall makes the room less coloured and the second says more, and the gap at the glossiest end is 2.87 units of chroma. Neither is a numerical error; the difference is a modelling decision that is usually made by omission.

Two ways to put a lobe on a wall

Take the interface's return out of the body term and a gloss wall makes the room less colourful. Add it beside the body term and the same wall makes the room more colourful. Same solver, same room, one line of energy accounting, and the two answers differ by nearly three units of chroma at the glossy end.

scene · Scene What the Lambertian assumption costs a room, against how rough its walls are. The horizontal axis is the roughness of the two coloured walls; the right-hand end is nearly matt, which is what a radiosity calculation assumes. One line is the distance in ΔE₀₀ between the floor's colour and what radiosity gives for the same room — 4.89 at an eggshell finish, falling to 0.97 at the matt end. The other is the chroma of the bounce, which falls as the walls get glossier: what an interface returns is a Fresnel reflection and carries no pigment, so the fraction of the return that goes into the lobe is a fraction that arrives at the floor white. The lobe is taken out of the body term rather than added beside it, which is what a real finish does.

Every scene in this collection was matt

The green wall, the corner, the bounce series and the metamer separation are all computed on Lambertian surfaces, because the solver that produced them requires it. Each would move by between one and five colour differences on an ordinary satin finish, and none of those essays says what finish it means.

scene · Scene 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.

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.

brain · Appearance 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.

limits · Limits 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.

limits · Limits 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.

limits · Limits 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.

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.

matching · Gamut 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.

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.

imaging · Capture 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.

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.

imaging · Capture 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.

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.

imaging · Capture 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.

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.

imaging · Capture The same highlight, clipped in two places. A ramp running from inside the sensor's range to 1.6 times over it, clipped at the sensor and clipped after the matrix. Below the ceiling the two are identical to the floating-point floor. Above it they part, reaching 21.0 colour differences and 78 degrees of hue. Clipping late keeps a highlight neutral and clipping early keeps its hue, and converters do both.

Two converters and one highlight

The clip is the only step in a raw pipeline that destroys information rather than moving it, and it is the step whose position varies most between converters. Below the sensor's ceiling its position changes nothing at all, exactly. Above it, clipping at the sensor and clipping after the matrix land twenty-one colour differences and seventy-eight degrees of hue apart, and which hues are affected is a property of the camera's own dyes.

imaging · Capture Where a stated appearance has no stimulus under it. The chroma the inverse can still return a light for, at lightness 50, all the way round the hue circle. Below the curve a stated appearance corresponds to a stimulus; above it the formula still returns three numbers and one of them is negative. Over a lattice of 10488 appearances spanning the whole space a specification is written in, 10.8 per cent are of that kind, and only 44 per cent are colours a display could show.

An appearance is not always a stimulus

CIECAM16's inverse is a closed form that returns three numbers for every lightness, chroma and hue it is handed. Whether those three numbers are a light it does not ask, and over a lattice spanning the space a specification is written in, 10.8 per cent of them are not — one tristimulus value negative, or a relative luminance above the white's. Only 43.8 per cent are colours a display could show at all.

brain · Appearance The model's hue scale, and the four numbers it is built from. Hue quadrature against hue angle. The four anchors are the unique hues, each with its own weight, and between them the scale is a hyperbolic interpolation rather than a straight line. The quadrant from blue back to red spans 143 degrees of hue angle for its hundred units of quadrature, against 70 for red to yellow.

The hue scale has four corners

CIECAM16 reports hue twice — as an angle in its own opponent plane, and as a quadrature interpolated through four unique hues with four fitted weights. The second is the one hue tolerances and hue-preserving mappings are written in, and it is piecewise — its slope jumps at each of the four anchors, by a factor of 1.74 at green and by 0.41 at blue, and it runs 4.1 times faster near yellow than near blue.

brain · Appearance The same blend, taken on the stored values and on the light. Six pairs blended at 50 per cent, once by averaging the values as they are stored and once by averaging the light they stand for. Every resize, every antialiased edge and every transparency composite in an ordinary pipeline does the first. The two land 15.8 colour differences apart at the mean and 19.3 on a red against a green, and the stored-value blend is the darker on all six, by up to 23 units of lightness.

An average on the stored values

A resize, an antialiased edge, a transparency composite and a chroma subsample are all averages, and in almost every pipeline they are taken on the numbers as stored. The numbers as stored are encoded, the encoding is a compression, and a half-and-half blend of black and white taken that way lands 18.7 colour differences from the half-and-half blend of the light — 22.7 units of lightness darker, on every pair, always in the same direction.

applied · Delivery The same colour, delivered from two documents. A ramp of chroma, mapped into a press's gamut under the perceptual intent from two source solids: the whole of sRGB, and the same solid with its chroma limited, which is what a document containing only muted colours amounts to. Under a colorimetric intent the two answers are identical to the floating-point floor. Under the perceptual one they are 0.69 apart at the mean and 2.64 at the worst, on colours the destination could hold either way.

An intent is not a function of the colour

A rendering intent is chosen per job and applied per pixel, which suggests it is a function of the colour. The colorimetric ones are, exactly — the same colour from two documents comes out identical to the floating-point floor. The perceptual one is not — it compresses the whole source solid into the destination, so the same colour delivered from a wide document and from a muted one lands 2.6 colour differences apart, on colours the destination could hold either way.

applied · Delivery A black level slightly wrong, through the balance, under daylight. A grey ramp from half a per cent to seventy-two per cent reflectance, with a pedestal error of a tenth, three tenths and one per cent of white left in all three raw channels before the white balance, against the same ramp with none. The horizontal axis is the grey's reflectance, logarithmic. At a three-tenths error a two per cent grey is 2.4 colour differences off, most of it chroma, and a seventy-two per cent grey 0.21. An equal offset in the raw channels is not equal after three different gains.

A black level is multiplied by the balance

Every raw value carries a pedestal that is subtracted before anything else, and a white balance is then a different gain in each channel. Subtracting a constant and multiplying by one commute only when the constant is zero or the gains are equal. So a pedestal left three thousandths of white too high becomes 2.4 colour differences in a two per cent grey, most of it chroma, in the colour the lamp starves — and pushing that shadow four stops in editing makes it 8.9. An offset in proportion to the lamp's own white is the exception, and it is exactly grey.

imaging · Capture How far the chain falls short of its sum, and how much of that is the unit. For each rendering intent, three ratios of the chain's end-to-end error to the sum of its four stages. The lowest bar is the published one, in the power-corrected unit. The middle bar is what that unit reports for a chain whose stages point the same way and add exactly — the exponent on its own. The top bar is the same chain measured in the model's own Euclidean space. Under the colorimetric intent the published ratio is 0.66, the exponent alone gives 0.74 and the chain in a space that can add 0.84: 77 per cent of the shortfall is the unit.

A chain measured in a unit that cannot add

A delivery chain's four stages, measured in the power-corrected appearance difference, sum to 7.66 while the chain end to end measures 5.05, and the shortfall was read as the stages partly cancelling. A chain whose four stages lay in a straight line and added exactly would still read 0.74 of its sum in that unit, because a distance raised to the power 0.63 cannot add. Measured in the model's own Euclidean space the same chain reaches 0.84 of its sum. Three quarters of the published shortfall was the exponent.

applied · Delivery Noise clipped at zero, averaged over a shadow, under tungsten. A grey ramp from black to ten per cent reflectance under tungsten, captured at three illustrative noise levels, with every negative raw reading set to zero before the readings are averaged over an area. Each line is the colour difference between that average and the noiseless grey. At high gain a half per cent grey is 1.04 off and a black frame 0.79; at very high gain the worst is 2.95, at 1.0 per cent. The same readings averaged before any clip come back exactly, at every level. The tint is gone once every channel sits several deviations above zero.

Clipped noise does not average away

Noise on a raw reading is as likely to fall below the true value as above it, which is why averaging an area removes it. A converter that sets negative readings to zero keeps the upper half and throws the lower away, and the mean of what is left is the signal plus a pedestal. With no black level error anywhere, a half per cent grey under a tungsten lamp comes out 1.04 colour differences off at high gain, 8.98 after a four-stop push — and a blur that removes every trace of the noise leaves the tint where it was.

imaging · Capture Four ways to fill in a clipped highlight: a glossy surface with a reflection of the lamp, under tungsten. Twenty-four chart surfaces under tungsten, as a glossy surface with a reflection of the lamp, taken up a ramp until their raw channels reach the sensor's ceiling. Each line is the mean colour difference, at equal lightness, between the true colour and what one response to the clipped reading makes of it: clipping to white, carrying the clipped values through, filling the clipped channel from the surface's own ratio, and filling it from the surface's colour plus the lamp's. At 0.4, where most surfaces have one channel clipped, the four leave 5.87, 4.37, 5.56, 0.00; at 2, 2.90, 26.97, 8.18, 8.18.

Filling in a highlight is a claim about the surface

A converter that rebuilds a clipped channel has to say what the highlight was. A matt surface over-exposed keeps its own colour, and filling the lost channel from that colour is exact. A glossy highlight is the surface's colour plus a reflection of the lamp, and the same fill leaves it 7.6 colour differences too colourful — while a fill that solves for surface and lamp is exact. Neither works once two channels are clipped, and a tungsten lamp keeps a highlight in the one-channel band more than twice as long as daylight does.

imaging · Capture

Named alongside it

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

SpecificationDeclared inputAuditModelling assumptionStandard observerTest setMeasurement errorColour managementInvarianceMarginalisationQuadratureWavelength grid

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