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

Declared input — where it appears

A number a model needs and no measurement supplies, written down with the reasoning beside it rather than buried in the code. It is the opposite of a fitted parameter: nothing determines it, so what can be reported instead is how far a conclusion moves when it does.

Named by 46 essays across 9 fields — each of them below, with the objects they name alongside it.

Every width at the wide end of its span, and at the narrow end. One line per published quantity, each spanning the value it takes when all four declared widths are read at the narrow end of their reported ranges to the value at the wide end, with a marker at the value as declared. The largest span is the deutan margin at a factor of 2.62; the smallest is 1.22. This is the reading the population model's own documentation promised for four phases and nothing ever took. It is not a confidence interval — the four ends are not quantiles and the widths are not independent draws — it is what a reader who distrusts all four at once sees.

A width nobody varied

Five numbers say how much people differ from one another, and every conclusion drawn here about a population rests on them. Each was written down with the range the literature reports beside it, so that a result could be re-read at the pessimistic end. Nothing ever was.

eye · Cones Which width carries the answer, and which carries the doubt. Two columns of bars over the four things that differ between two pairs of eyes. On the left, the share of the population's disagreement each one accounts for — the attribution quoted here since the population was built, which puts the lens first at 81%. On the right, how much of the doubt each one puts on everything published here, which is its elasticity multiplied by how badly the width itself is known. The macular pigment comes first there, at 0.65 against the lens's 0.60 — a lead of 8%. The two lists agree exactly below the top.

Which measurement is worth making

Four things about an eye differ between people, and they have been ranked here by how much of the answers they carry since the population was built. Ranking them by how much doubt they carry gives a different order, and ranking them by which one takes a published claim closest to failing gives a third.

eye · Cones How wrong the ellipses would have to be for a pair to change places. One bar per adjacent pair in the uniformity table: the relative error on each ellipse's own axes at which that pair changes places in one draw in twenty. No error on the data is quoted anywhere — the question is inverted, so what is reported is how large an error would have to be, and a reader with an opinion about MacAdam's experiment can compare it with their own number. The nearest pair goes at 0.171; 2 of the 7 pairs do not reverse under any error this search covers.

How wrong would the data have to be

Twenty-five ellipses measured on one observer in 1942 are the ruler every colour space here is judged against, and they have never been given an error. Rather than invent one, the question is turned round, and asks how large an error would have to be before the ranking changed.

difference · Metric A mid-grey's lightness across a continuum of rooms. The lightness a mid-grey is predicted to have, plotted along the continuous surround parameter running from an average room to a dark one. The three rooms the standard tabulates are marked on it: average at the left, dark at the right, and dim 61% of the way between them rather than halfway. The whole span is 9.71 units of lightness and the step from average to dim is 5.64 of it — 58% — so choosing one of the three rows is a decision worth most of the range.

The surround is three rows of a table

An appearance model takes the room as three constants, and the standard tabulates three rooms. Every appearance figure in this collection is drawn at one of them. The parameter they are three points of is continuous, and the middle row is not in the middle.

brain · Appearance Every adaptation number here assumes a complete adaptation. Three curves and their mean: the colour difference an adapted observer is left with after a change of light, against the degree of adaptation from zero — no adaptation at all — to one. Every adaptation figure in this collection is computed at one, the right-hand end. The appearance model's own formula puts the degree at 0.941 for an average surround at a hundred candelas, marked, where the residual is 2.21 ΔE00 rather than 1.27 — larger by a factor of 1.74. The left-hand end is exactly the unadapted change, which is not an approximation but an identity, and is what says the curve interpolates between the two things it claims to.

A discount nobody measured

Every adaptation number in this collection assumes an observer who adapts completely. The appearance model's own formula says they do not — it puts the degree at 0.94 in an ordinary room — and the difference is not a rounding. It is a factor of 1.7 on the residual every one of those figures reports.

brain · Appearance A quadratic is believed least far at the one place anybody takes one. One bar per basis: the radius, in the nine coefficients, within which the second-order model predicts the objective to within ten per cent in every one of eighteen directions. The shortest bar is the objective's own optimum, at 2.3×10⁻², and the longest is XYZ scaling at 1.1×10⁻¹ — several times further. The reason is not that the model is worse at a minimum but that it has less to do there: away from one the linear term is exact and carries most of the change, so a ten per cent error in the prediction takes longer to accumulate. It does not make a Hessian at a minimum wrong; it says the picture drawn from it describes the smallest neighbourhood in the table.

How far a quadratic can be believed

A second-order model has a radius inside which it describes a surface and outside which it does not, and that radius can be measured. Measured at eight places on one objective, it is smallest at the optimum — the one place anybody ever takes a Hessian.

limits · Limits Room is not safety: two orderings of the same three claims. Three pairs of bars, one pair per published statement about the confusion points. The upper bar in each pair is the margin — how far the measured number is from the threshold that makes the statement true, as a ratio. The lower bar is the headroom — the factor by which one declared width of the population model would have to be wrong for the statement to fail. Both start at one, which is the line. Ordered by margin the three read the protan margin, the tritan margin, the deutan margin; ordered by headroom they read the protan margin, the deutan margin, the tritan margin, and the middle two change places. Every one of the three is inside a factor of two of failing, which the margins do not say.

What would have to be wrong

A great many statements here have thresholds written into them, which turns out to make an audit possible — for each one, the smallest change in a declared input that would stop it holding. Most are unreachable. One is inside a factor of one and a third.

limits · Limits The family does have a worst case, at a band no pigment can cut. The worst change of light a painted wall can produce, at each band width, with the wall's centre wavelength, depth and base optimised at every point. The horizontal axis is logarithmic in the width. The curve rises as the band narrows, turns over at about 6.02 nanometres, and falls again — a band that narrow returns too little light to move the white much. The previous round's search reported no worst case because its box stopped at ten nanometres, marked, which is on the wrong side of the turn. The peak is 28.54 ΔE00 against 28.38 at that floor, which is 0.6 per cent higher: wrong in principle, right in practice to a fraction of a per cent.

A notch a pigment cannot cut

The worst change of light a painted room can produce has no maximum inside the box the search was given, which the previous round reported as a family with no worst case. Bounded by what a molecule can actually do, it has one — at a band six nanometres wide, narrower than any pigment and narrower than the box.

light · Light The winner survives the census's own construction; the middle of it does not. One row per perturbation of a constant the adaptation census is built from — the imaginary wall's centre wavelength, its width, its depth, its base, the macular filter's density and the two lens ages — each moved by an amount plausible for that quantity in its own units, up and down, and then all of them together. Each row shows where the five published transforms rank under it. Bradford holds the first column in all 14 rows. The second and third columns, which the table as built separates by six parts in a thousand, change places in 2 of them — so that ordering was never a fact about the transforms.

The census is a construction too

Five of the fourteen changes of light this collection scores adaptation transforms against are not measurements of anything — they are a wall somebody invented, at a wavelength somebody chose. Moving those constants by amounts plausible in their own units moves the mean residual by two fifths and never changes which transform wins.

light · Light A camera's dye widths are free under one requirement and not under another. Three panels, one per dye. In each, a pair of bars per requirement: how far that dye's centre wavelength and its bandwidth can move before the requirement gets five per cent worse. Under the adaptation objective — the one the previous round measured — every width has far more room than its centre, which is the finding that put a tolerance budget on the centres. Throughput and the colour matrix's noise gain, the two requirements that objective was said to be silent about, reach the edge of the search in every direction and hold nothing. What tightens the widths is the Luther residual, which was in the model already. The bottom pair in each panel is what survives all four.

The widths were free because nothing else was asked

A camera's three dye bandwidths carry almost all of the flattest direction of the adaptation objective, so that objective says a tolerance budget belongs on the centre wavelengths. The two requirements it was said to be silent about turn out not to bind either — and the one that does was in the model already.

imaging · Capture What a camera's dye 1 is allowed to be, under four requirements. The plane a colour-filter dye is designed in: its centre wavelength across, its bandwidth up, both in nanometres, so the two axes are comparable and the shapes mean something. Four outlines, one per requirement, each the set of dyes within five per cent of the designed one on that requirement; the shaded region is where all four hold. Two of the four — throughput and the colour matrix's noise gain — reach the edge of the search in every direction and are invisible as boundaries. The intersection is ±6.8 nanometres of centre and ±13.3 of width, against the adaptation objective's own ±20.6 in width alone.

Two tolerances do not meet in a tolerance

A specification lists requirements separately and a manufacturer has to satisfy them together. Where two long thin regions cross at an angle, what is left is much smaller than either, its longest direction is neither of theirs, and no list of tolerances describes it.

imaging · Capture A room applies its wall a different number of times at each wavelength. The mean number of bounces the surviving light has made, wavelength by wavelength, in a closed room whose walls are the green paint the adaptation census uses. It runs from 0.33 in the band the wall absorbs to 5.67 in the band it reflects — a factor of 17.00 — because the light that survives many bounces is the light the wall was reflecting all along. The census has one bounce and two bounces as separate rows and a search treats the count as a free integer; a room has neither, and what it has is bounded by the walls reflecting less than everything.

A room bounds its own bounces

The adaptation census has one bounce and two bounces as separate rows, and a search over the family treats the count as a free integer it always takes to the largest value offered. A room offers no integer at all — it applies a geometric mixture of every number of bounces, and that mixture is bounded by the walls reflecting less than everything.

scene · Scene The three worst walls, drawn as the reflectances they are. Three reflectance curves, one per bound: the wall each search settled on. All three are dark over most of the spectrum with a single band near the short-wavelength end — the arithmetic bound's is 10 nanometres wide, the physical one's 40, and a paint somebody sells the same. None of them is a saturated colour: their excitation purities are 0.18, 0.52, 0.52 against a ceiling of 0.6, which is why the purity constraint never bites. What breaks an adapted observer is a wall that takes most of the light away, not one that is a strong colour.

The darkest wall anybody sells

Asked which property of a paint decides the worst change of light a room can produce, anybody would answer how saturated it is allowed to be. A ceiling on saturation never comes near binding, because the worst wall is dark rather than colourful — and the constraint that does bind is one nobody would nominate.

scene · Scene What a display's red primary is allowed to be, under four requirements at once. A close view of the chromaticity plane around one designed primary, 0.101 units across. Four outlines: the set of positions the primary can take before each of four requirements gets one per cent worse — how well a gain in the display's own basis undoes a change of light, how much of the diagram the three primaries enclose, how many real surfaces fall inside them, and whether a light of that colour exists at all. The shaded region is where all four hold. It is 5% of the smallest outline's area, because the outlines are long and thin and cross at an angle rather than nesting. adaptation holds 42% of its boundary, gamut holds 10% of its boundary, realisable holds 48% of its boundary.

A primary is chosen for four things

A display's primaries have to adapt well, cover the diagram, hold the surfaces anybody photographs, and be colours a light can actually have. Drawing all four tolerance regions around one primary shows that no single requirement decides where it can go, and that one of the four never decides anything.

matching · Gamut The same tolerance, in the two numbers somebody actually sets. The plane a maker of a single-peak emitter works in: peak wavelength across, full width at half maximum up. Each marker is a candidate emitter whose chromaticity falls inside the colorimetric tolerance drawn for this display's green primary. They occupy a narrow band — peaks from 528 to 535 nanometres, a span of 7, against widths from 25 to 45 — so a tolerance stated as a region in chromaticity becomes ±3.5 nanometres of peak and a great deal of latitude in width. 2.0% of the 2501 candidates land inside at all: most of a region drawn in chromaticity is a colour no single-peak emitter makes.

A tolerance in the wrong coordinates

A display primary's tolerance is written as a region in chromaticity, because that is where the colorimetry lives. Nobody has a knob for chromaticity. What a maker of an emitter sets is a peak wavelength and a bandwidth, and the map between the two is so anisotropic that on the red primary its condition number is over eleven thousand.

matching · Gamut Exactly flat everywhere, and eigenvectors at one point only. Two columns over the same nine places. On the left, how far from zero the objective's second derivative is along a row-scaling direction, on a logarithmic axis — it is between 10⁻⁹ and 10⁻⁶ of the largest eigenvalue at every one of them, which is a numerical zero. Scaling a row of the basis is a straight line along which the cost does not change, and that is true at every point, not only at the optimum. On the right, the angle between those three directions and the Hessian's own three smallest eigenvectors: 0.025 degrees at the optimum and up to 88 away from it. An invariance is a property of the function; being an eigenvector is a property of the function at a minimum, and the two coincide only where everybody computes.

Only the flat directions keep their names

Three of the nine numbers a colour match leaves free do nothing, and they do nothing everywhere — exactly, at every basis in this collection's table. They are the objective's own principal directions at one point only, and everywhere else the directions carrying the curvature have turned by tens of degrees.

matching · Gamut At a published matrix the slope arrives long before the bowl. One row per basis in this collection's table. Each row is a logarithmic axis of distance in the nine coefficients, with two markers: the radius at which the objective's curvature becomes as large as its slope, and the distance from that basis to the optimum. The first is between 3.2 and 108 per cent of the second. So over almost the whole journey from a published matrix to the best one, the surface is a slope and not a bowl — and a table of eigenvalues taken there describes a neighbourhood the optimum is nowhere near. XYZ scaling is the exception, at 1.08 of the distance, because its slope is the steepest in the table.

The slope arrives before the bowl

The adaptation transforms colour management actually uses are not optima of anything. At every one of them the objective has a slope, and the slope is the larger term over almost the whole distance to the best matrix — so a table of curvatures taken there describes a bowl nobody meets on the way anywhere.

applied · Delivery Where a delivered colour's error actually comes from. Three bars and two totals. The profile's interpolation between its nodes contributes 0.13 ΔE00; the rendering intent, at the colorimetric setting, moves nothing that was already printable and contributes 0.00; and looking at the result in a dim room rather than the booth it was proofed in contributes 4.30, which is 97 per cent of the total. The two totals are the three added — 4.43 — and combined in quadrature — 4.30. The gap between those two rules is 0.127, about the size of the entire profile stage, and no standard says which rule to use.

A delivery tolerance is three tolerances

A colour reaches a reader through a profile, a gamut mapping and a room, and each has a number somebody chose. Each has an essay of its own here, and none of those essays had put the three costs in one column. Put there, the stage nobody controls carries ninety-seven per cent of the total.

applied · Delivery Where a sample's colour goes as the aperture closes. The a and b of three translucent materials as the measuring aperture narrows from forty millimetres to one. Each track starts at the open circle, which is the colour the model says the sample has, and ends at the filled one. The axes cross at the neutral point. pale marble passes through neutral at a radius of 5.32 millimetres and comes out on the other side; candle wax passes through neutral at a radius of 7.07 millimetres and comes out on the other side; skin passes through neutral at a radius of 0.76 millimetres and comes out on the other side. Nothing about the sample changed: the aperture is a filter with a colour of its own, and the colour is decided by how the sample scatters rather than by what it absorbs.

The hue the hole decides

A piece of pale marble measured through a wide aperture is faintly yellow. Measured through a narrow one it is faintly blue, and between the two there is an aperture at which it is exactly neutral. Nothing about the stone changes; the aperture is a filter with a colour, and what decides that colour is the size of the particles rather than the pigment between them.

scene · Scene What the interface does to a reflectance, and the straight line it is taken for. The Saunderson relation between the reflectance inside a pigment layer and the reflectance an instrument reads off it, for a boundary of refractive index 1.50. The curve is the real map; the dashed line joins its two endpoints, which is the straight relation an additive pedestal assumes. They are 0.216 of a reflectance unit apart at their widest, which is 5 times the pedestal itself. The curvature comes from the k₂ term — light reflected back down into the layer from underneath the boundary — which is 0.60 where the outward reflection is 0.04.

A mixture in the variable nobody named

Kubelka–Munk works because absorption and scattering add over a mixture and reflectance does not. What adds is the absorption of the pigment layer, and what an instrument reports is that layer seen through an interface — related by a Möbius function rather than by a constant. Mixing in the reported variable instead of the internal one costs between three and eight ΔE₀₀, and no source this collection quotes says which variable its curves are in.

scene · Scene 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 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 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.

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.

brain · Appearance 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.

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.

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 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 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.

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.

matching · Gamut 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 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 Where averaging the model's answers is and is not averaging its argument. Each row takes a spread of situations, averages the model's predictions across them, and compares that against the model's prediction for the average situation. The bar is the gap as a share of the spread itself. Over the differences between observers it is 1.4 per cent — the model is very nearly linear there. Over the range of adapting luminance one room covers in a day it is 59 per cent, and from indoors to outdoors 74.

Where the model's curve does not matter

This round has been about what a nonlinearity does to an average, and CIECAM16 is the most nonlinear thing in the collection. Over the spread a population of observers produces, the average of its predictions is its prediction for the average to within 1.4 per cent — the nonlinearity is there and the excursion is too small to reach it. Over the range of adapting luminance one room covers in a day, the same gap is 59 per cent of the spread, and from indoors to outdoors 74.

brain · Appearance The audit's six departures, read twice. Each departure priced in the matching unit it was published in and in the appearance unit an appearance prediction would be judged by. The model does not scale them by one factor: it amplifies the smallest by 1.58 and the largest by 1.03, so the range between them narrows from 3.34 to 2.17. The mean ranking is unchanged and 14 of the 42 surfaces reorder.

The audit, read as appearances

The previous round priced six observer departures in a matching unit and left them there. Read in the appearance unit an appearance prediction would actually be judged by, they are not the same six numbers scaled by a constant — the model amplifies the smallest by 1.58 and the largest by 1.03, so the range between the largest and smallest term narrows from 3.34 to 2.17. The published ranking survives in the mean and changes on fourteen of the forty-two surfaces it was averaged over.

brain · Appearance What a stated lightness pins down, and where. A lightness quoted to 0.05 of a unit, inverted, and the luminance it fixes. Read as a fraction of the colour's own luminance the requirement is 0.90 per cent at J 10 and 0.097 at J 95, a factor of 9.3. Read in absolute luminance it is the other way round, by a factor of 6.5. Both readings are true and they answer different questions.

A stated lightness is two requirements

A specification quotes an appearance to a stated precision — a lightness to a tenth of a unit, say — and the same precision everywhere. Inverted, a twentieth of a unit of lightness fixes the luminance to nine tenths of one per cent at the bottom of the scale and to a tenth of one per cent at the top, a factor of 9.3. Read in absolute luminance it is the other way round by a factor of 6.5, and both readings are true.

brain · Appearance The budget's three numbers, and the units they are in. The published three-stage budget's own figures, with each one's unit named, beside the same stage re-measured in a single unit over the same colours. Two of the three are colour differences between stimuli and the third is a distance between appearances, and the budget adds them. The fourth row is a stage the budget has no entry for: the colours the separation cannot reach even after the mapping has moved them, which comes to 1.45.

The budget adds two units

This collection publishes a three-stage error budget for a colour-management chain and prints its sum. Two of the three stages are colour differences between stimuli and the third is a distance between appearances, and the conversion between those is not a constant. Re-measured in one unit over the same colours the chain has four stages rather than three, its end-to-end error is 5.05 against a sum of 7.66, and the profile's contribution where a job actually lands is 3.2 times its published average.

applied · Delivery 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 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.

Structural choiceSpecificationChromatic adaptationSensitivityAuditCIECAM16Measurement errorColour differenceColour managementLightnessTransfer functionAnisotropy

All concepts