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

Modelling assumption — where it appears

A structural choice inside a calculation — a basis, a grid, a functional form, a metric — as distinct from a value fed into it. It has no range to sweep, so a sensitivity analysis reaches it only by exhibiting it as the limit of some continuous family.

Named by 40 essays across 6 fields — each of them below, with the objects they name alongside it.

How far each census row moves when the test set's own description does. A grid of bars, one row per change of light in the census and one bar in each row per number that describes the region the test surfaces are drawn from: how saturated they are, how bright, and how far the two modulations may go together. A bar's length is the elasticity — the proportional change in the published residual for a proportional change in that number. Saturation runs from 0.49 to 0.91 and brightness averages 0.104, so a test set's chroma range is nearly everything and its lightness range is nearly nothing. For scale, the largest elasticity found anywhere among this collection's five declared population widths is about a half — and those at least have declared ranges, while these three numbers have never been quoted with one.

The input nobody declared

An audit that swept every declared width in this collection found the largest elasticity anywhere to be about a half. The most elastic input turns out to be one that was never declared, never quoted with a range and never varied — and being undeclared is exactly why it escaped the audit that was looking for it.

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

Every worst surface sits on a declaration

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

scene · Scene Every census row under five constructions of the same test set. A slope chart with 5 columns — lattice, coarse, fine, uniform, natural — and one line per change of light in the census, each line joining that row's mean residual under each construction. Four of the five columns describe the same region of surfaces walked at different densities or against different measures; the last is the clamped, realistic family, which is not linear in its parameters and is therefore answering a slightly different question. The levels move: between the coarse and fine lattices every row shifts by seven to nine per cent, in the same direction, which is a common-mode factor no published residual here has ever carried. The order almost survives. Inside the region exactly one pair crosses, and it is the pair the standard error had already flagged; under the clamped set two more cross, including one the error separates by nearly nine standard errors. The crossing lines are drawn heavy.

What the audit still cannot reach

Two rounds have now swept every declared width in this collection and one of its structural choices. Three structural choices remain, none of them has a multiplier to sweep, and the reason each resists is different — which makes the list a description of where this kind of audit ends rather than a queue of work.

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 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 Two slabs with one reflectance, and two colours through an aperture. Two constructed media whose bulk reflectance agrees at every wavelength to fifteen figures, and whose diffusion lengths differ by a factor of four. The upper curve is that shared reflectance — both slabs lie on it exactly. The two patches on the right are what a 4 millimetre radius returns from each, and they are 6.3 ΔE₀₀ apart. This is a metamerism with no observer in it: the two samples are the same colour to anybody under any light, and the instrument separates them because it is measuring a kernel through a hole rather than measuring a reflectance.

The appearance model has no slot for it

Two slabs with identical reflectance and different kernels come out of CIECAM16 with identical lightness, chroma and hue — to six decimal places, because the model's input is a stimulus and their stimuli are the same. Translucency is a recognised appearance attribute with no correlate in any appearance model, and the reason is structural rather than an oversight.

brain · Appearance 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 What three fill-in rules cost when a five-nanometre table is read at one. A five-nanometre table read at one nanometre by three rules — hold the value, straight lines, a cubic through four points — each compared with the same tenth-nanometre reference. The dashed rule is the answer obtained by summing the table as it stands, at 0.00000 ΔE₀₀. Two of the three interpolations are worse than not interpolating. That is not a paradox: the summation's error is already small because the normaliser cancels most of it, and an interpolator introduces a shape the original curve did not have, which the cancellation cannot touch. A finer grid is not more resolution when the table is not finer.

A finer reading of a coarser table

Interpolating a five-nanometre spectrum to one nanometre helps a daylight calculation by a factor of five and harms a three-emitter LED by a factor of a hundred and twenty thousand. Both are the same operation on the same table, and which one happens is decided by a property of the light nobody records.

light · Light How much the answer moves when the 5-nanometre grid is slid through one cell. Each bar is the spread of one light's colour across five grid origins, all at the same 5-nanometre step, in ΔE₀₀. A smooth light barely moves, and what movement it has is the end cells rather than the sampling. The fluorescent tube moves by 3.18 units and the laser projector by 35.0, because their emission lines are narrower than the step and whether a sample lands on one is a coincidence of arithmetic. This is the measurement that separates a quadrature error from an aliasing error, and no average over origins can substitute for it.

Where the grid starts

Holding the step at five nanometres and sliding the grid's origin through one cell moves a fluorescent tube's computed colour by 3.18 ΔE₀₀ and a laser projector's by 35.0. Refining the step does not fix it and averaging over origins hides it. It is the one tabulation fault with no smooth error to cancel against.

light · Light How far this collection's analytic observer is from the tabulated one. The construction every figure in this family uses is three pigment absorptances through one fitted 3×3, and this is the residual of that fit against the CIE's 1931 functions: root-mean-square error as a percentage of each curve's peak, and the colour difference it produces over forty-two surfaces. The short-wavelength function is the worst at 16.4 per cent, which is where a pigment template is weakest and where the ocular media are doing most of the work. The median colour difference is 1.42 ΔE₀₀, so this is an observer of the right shape rather than a copy of the table — and every departure in this family should be read beside that number rather than against zero.

The reference had to be built

A five-nanometre error cannot be measured with five-nanometre data. Interpolating the tables and integrating finely measures the interpolator, not the grid — so the audit of this collection's index had to be run against an observer made of formulae, and the price of that is a residual of 1.42 ΔE₀₀ that every number in the section is read beside.

limits · Limits 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 What this collection's grid does to its own observer audit. Two bars per light: the mean departure of the observer computed on this collection's five-nanometre grid, and the same computation on a quarter-nanometre one. For five of the six lights the two agree to two decimal places, which is what a well-sampled spectrum looks like. For the laser projector the coarse answer is exactly zero and the fine one is 2.34 — the largest in the table. On a five-nanometre grid a three-line spectrum with lines at 465, 532 and 638 nanometres is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about to the last bit. The two departures do not compound here; the first conceals the second.

The grid hid the observer

On this collection's five-nanometre grid a three-laser projector's observer disagreement is exactly zero. On a quarter-nanometre grid it is 2.34 ΔE₀₀ and the largest in the table. The two audits of this round meet here, and the first one does not compound with the second — it removes it.

limits · Limits 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 The directional solver reduces to the radiosity solver exactly. A solver with a new unknown in it is worth nothing until it reproduces the one it replaces. Setting every wall's bidirectional distribution to ρ/π collapses all thirty ordered-pair radiances onto their patch's radiosity divided by π, and the answer agrees with this collection's existing radiosity solution to 9.8e-16 relative — the floating-point floor. That is the check that makes every other number in this family a statement about lobes rather than about a new piece of arithmetic, and it is the reason the reduction is drawn rather than mentioned.

Thirty unknowns instead of six

A directional transport solver is worth nothing until it reproduces the one it replaces. Setting every wall's bidirectional distribution to ρ over π collapses thirty ordered-pair radiances onto six radiosities and reproduces this collection's existing answer to 9.8 × 10⁻¹⁶ relative — which is the only reason anything else it says can be believed.

scene · Scene What the floor receives, matt walls against walls of roughness 0.2. The spectral radiance leaving the floor towards the front of the room, computed twice. The matt curve peaks at 530 nanometres, where the walls' pigment is. The glossy curve is higher everywhere and higher by relatively more away from that peak, because the extra light is a Fresnel return and a Fresnel return has the lamp's spectrum rather than the paint's. That difference in shape is the desaturation, drawn before it is reduced to a number, and it is the reason the two lines cannot be brought together by any exposure change.

A lobe takes colour out of a bounce

A gloss wall sends more light to the floor and less colour. The extra light is a Fresnel reflection at the interface, it carries the lamp's spectrum rather than the paint's, and it arrives at the next surface white — so a green room in satin paint is less green than the same room in flat paint by nearly a colour difference of chroma.

scene · Scene Where the viewer stands, at a wall roughness of 0.2. The chroma of the light the floor sends towards each of the five other faces of the room, at one roughness. The spread is 2.00 ΔE₀₀ between the extremes. A radiosity solution assigns one radiosity to the floor and therefore cannot have a spread at all — the whole width of this chart is a quantity the method has no slot for, rather than one it approximates badly. The two side walls see the most because they are where the coloured light comes from, and the direction the lobe favours is the direction it came from.

The floor is a different colour from the door

With a lobe on the walls the floor sends chroma 18.76 towards the front of the room and 20.94 towards the side walls, a spread of two colour differences. A radiosity solution assigns the floor one number, so the whole of that spread is a quantity the method has no slot for rather than one it estimates badly.

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 Where a point-sampled patch stops resolving a lobe. The horizontal axis is the wall's roughness, logarithmic; the vertical is how far the answer moves when the cone quadrature is refined from eight directions to sixteen, also logarithmic. A patch in a cube subtends a cone of angular radius 25.8° at the face opposite, and a microfacet lobe of roughness a is about a radians wide, so a lobe below about 0.15 is narrower than the quadrature that samples it. The movement at 0.05 is 21.2 ΔE₀₀ and at 0.3 it is 0.033. This is where the method stops, not where the paint does: matt, eggshell and satin finishes are inside it and a high-gloss varnish is not.

Where a patch stops being a point

A patch in a cube subtends a cone of 25.8° at the face opposite. A microfacet lobe of roughness 0.05 is about three degrees wide. Point-sampling the second inside the first returned a chroma of thirty-four thousand and a negative lightness, and the boundary between working and not working is measured rather than declared.

limits · Limits The lobe's share of what leaves a surface of body reflectance 0.5. For light arriving at 45°, the fraction of what leaves the surface that is the interface's Fresnel return rather than the pigment's. It runs from about 9.1 per cent at an eggshell finish down to 4.2 at a matt one. That is a small share, and it is the whole of the effect: a tenth of the return arriving white is enough to move the room's colour by units of ΔE₀₀, because the bounce is what a room's colour is made of and every bounce is multiplied by the next.

A tenth of the return arriving white

Nine per cent of what leaves a satin wall is a Fresnel reflection carrying no pigment. That nine per cent moves the room's colour by 4.89 ΔE₀₀ and its chroma by five per cent, because an interreflection multiplies and a small contribution with a different spectrum compounds into a large one.

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

The limits assume a pigment that switches instantly

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

limits · Limits How far CIECAM16 moves a monochromatic hue when the light is brightened thirtyfold. A monochromatic stimulus at a relative luminance of 2 and of 60, read through CIECAM16 in one room, and the difference between its two hue angles. The model was never fitted to this effect and has it anyway: the shift is positive at the short end, negative through the greens, positive again in the yellows and reds, and crosses zero at 459, 495, 502, 570 nanometres. The marks are the invariant wavelengths the literature reports — 474, 506, 571.

The model has a hue shift it was never given

A monochromatic light changes hue as it is brightened, except at three wavelengths that do not move — an effect measured since the nineteenth century and not among the things CIECAM16 was fitted to. The model has it anyway: brightening a stimulus thirtyfold moves its hue angle, and the places where the movement crosses zero land at 459, 495, 502 and 570 nanometres against the reported 474, 506 and 571. The sizes are another matter.

brain · Appearance What the choice of adopted white is worth, in a room lit by two lights. A room lit half by daylight and half by an incandescent lamp. The light on the surfaces is fixed; what varies is the white the model is told the observer has adapted to, running from the lamp on the left to the window on the right. The median of twelve surfaces moves 32.6 CAM16-UCS units if the lamp is adopted and 24.5 if the window is, against the room's own mixture in the middle. The model offers no way to choose, and its degree of adaptation is 0.94 at every point of the dial.

A room with two lights has no white

An appearance model takes one adapting white. A desk beside a window has two, and the mixture falling on the paper is not the same thing as the white the person reading it has adapted to. Mixing the lights is arithmetic. Choosing the white is not, and the choice is worth sixty units of appearance — most of which is a cast, and not all of which is.

brain · Appearance One grey scale, three backgrounds. CIECAM16's lightness against the luminance factor of a neutral sample, with only the background changed. A grey reflecting 19% reads 46.7 on a near-black background and 32.8 on a near-white one. The three curves are not three shapes: each is the same curve raised to a different power, because the background reaches lightness only through the exponent z, which runs 1.621 to 2.374 across the three.

A dark background moves every difference and no match

CIECAM16's background is one number, and it reaches lightness as one exponent. That is enough to change what a grey looks like and not enough to change which of two greys is lighter — so a match survives the background exactly, a corresponding colour is invariant to it, and a tolerance is not. The effect the background is usually invoked to explain is absent from the model entirely.

brain · Appearance A crispening term puts the peak where the background is. How much lightness the model returns for a small change in the sample's level, against the sample's level measured as a log ratio to the background's — so that all three backgrounds share one axis and a peak at the background is a peak at zero. The pale curves are CIECAM16 as it stands, which has no peak anywhere: they rise slowly and monotonically because a background that enters as four constants fixed before the sample arrives cannot know where the sample sits relative to it. The solid curves are the same model with a term of amplitude 6 and width 0.5 added to lightness. Each peaks at zero to within 0.000 of a log unit, and the peak's height is 12.00 lightness units per log unit of level — the amplitude divided by the width, exactly.

The cancellation is exact and cheap to lose

CIECAM16 has no crispening, and adding one was expected to be expensive: the background's exactness in a corresponding colour comes from its being a common exponent, and a function of the sample's own level is not one. It is expensive in kind and not in size. A term that raises a straddling pair's lightness difference by half moves a corresponding colour by five thousandths of a tristimulus unit — a thousandth of what stating the background differently at the two ends already costs.

brain · Appearance The pairs that change places in chroma are a wedge with two straight edges. Every pair of samples, plotted by the log ratio of the two samples' background-free responses across and the log ratio of their chroma at a background of 2 up. A pair changes places between that background and one of 80 exactly when its chroma ratio and its response ratio point in opposite directions and the chroma ratio is the smaller — which is the wedge between the horizontal axis and a line of slope -0.2598, half the change in the lightness exponent. 754 of 14028 pairs are inside it, and the condition names every one of them and nothing else: 0 disagreements between the line and the model. The pale dots are one pair in eleven of those outside; the filled ones are every pair inside.

The reversals have a straight edge

Twenty-one of 276 pairs change places in chroma between a dark background and a light one, and the reason given was that chroma is a product of a term carrying the exponent and a term that does not. That is true and it is not a description of which pairs. Chroma at one background is a single common factor times a power of lightness at another, so a pair reverses exactly when its chroma ratio and its lightness ratio point opposite ways and the first is the smaller — a wedge with two straight edges, which names every reversal and nothing else.

brain · Appearance The glossiest finish the solver can report, against what it costs to report it. Six quadratures, each with the roughness at which its answer stops being stable, on logarithmic axes. The line is a fit and its slope is -0.350: the reachable roughness falls as the cost to the power of about a third, so reaching a finish twice as glossy costs about 7 times the work. The solver used here sits at 108 directions and reports down to a roughness of 0.145, which is where its own note put the boundary by inspection.

The boundary belongs to the quadrature

The directional solver stops at a roughness of about 0.15, and below that its answers are not imprecise but unphysical. The boundary is where the lobe stops being resolved by the sampling, so it belongs to the discretisation rather than to the room — and moving it is a purchase. Measured across six quadratures the reachable roughness falls as the cost to the power of a third, so a finish twice as glossy costs seven times the work and a polished varnish costs two hundred and thirty-six times.

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

A limit written in energy charges the reds

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

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

The inverse table errs dark

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

applied · Delivery What the rods cost a match, by where their signal enters and under which lamp. For five lights, the median colour difference over forty-two surfaces between the reference observer and the same observer with a rod signal a tenth of each cone's peak added — into all three cone channels, into the long- and middle-wavelength channels only, or into the short-wavelength channel only. Under daylight the first two are 1.03 and 0.90: whether the rods reach the S pathway hardly matters. Under a phosphor white LED they are 1.24 and 0.45, a factor of 2.75, and the S-only route alone costs 0.96.

The rods' route is priced by the lamp

A rod signal in a dim room disturbs a colour match, and how much depends on which of the cone pathways it reaches — a weight the physiology leaves uncertain, especially for the blue–yellow pathway. Under daylight the uncertainty is nearly free: a rod signal that skips the S pathway costs 0.90 at the median surface against 1.03 for one that enters all three. Under a phosphor white LED it is worth a factor of 2.75, 0.45 against 1.24. What decides it is one number per lamp: how large the rod signal is compared with each cone class's own catch of the light.

eye · Cones How far each model turns a display colour's hue as white is added. The twenty-four most saturated colours an sRGB display makes, one every 15° of HSV hue across, each mixed with the display's white at the same luminance down to a fifth of its purity. Up, how far that mixture's hue angle turns in CIECAM16, CIELAB and Oklab. From red to a fifth of its purity CIECAM16 turns −9.4° and Oklab −10.2°; CIELAB −18.1°. From the display's blue, Oklab turns +16.3°, CIECAM16 +3.6° and CIELAB −8.9°.

White turns a hue, and the models part at blue

Mix a saturated light with white and its hue changes as well as its saturation — the Abney effect, and the reason lines of constant perceived hue curve in a chromaticity diagram. Asked what adding white does to the twenty-four most saturated colours a display makes, CIECAM16, CIELAB and Oklab all turn the hue. About reds they agree: CIECAM16 turns within a degree of Oklab, whose hue was fitted to observers' constant-hue judgements. About the display's blue they do not: Oklab turns sixteen degrees, CIECAM16 four, and CIELAB nine the other way.

brain · Appearance Which two lamps to stand a mesopic match between. Every pair of the five lights, by how far a match made under one and set under the other moves as the rod signal's weight into the S channel goes from nothing to equal — the median over forty-two surfaces, which is the signal an experiment has to resolve. The best pair is daylight against phosphor LED at 0.58 ΔE₀₀; the worst is tungsten against fluorescent tube at 0.09, a factor of 6. The count at the right is how many settings it takes to resolve the weight to a tenth at half a colour difference of scatter per setting.

The reference lamp must not move

To measure an uncertain weight, use the condition in which the answer depends on it most. That is right about half of an asymmetric colour match and exactly wrong about the other half: a match measures a difference of two displacements, and a reference field that also moves with the weight cancels the signal the test field carries. Daylight is the least sensitive of five lamps and belongs in every one of the three best pairs — 75 settings against a phosphor LED, 2,804 against the pair of lamps the principle as stated would have chosen.

eye · Cones Two ways to ask how much rod signal an older eye has. The rod signal's catch of each lamp divided by the S cones' own catch, against the observer's age, for five lamps — drawn twice. The upper curves take the rod signal at a fixed absolute size, and every one of them roughly doubles from twenty to seventy-five: an older lens cuts the blue before the S cones see it and the rods, peaking further into the green, lose much less. The lower curves take the rod signal at a tenth of each cone's own peak absorptance, which is the model's own definition, and they barely move at all. Nothing about the retina differs between the two; only the normalisation does.

A rod signal has no natural size

An older lens absorbs where the S cones are sensitive, so it should make the uncertain rod-to-S-cone weight cheaper. Measured, the rod signal's catch of a phosphor LED as a share of the S cones' own catch more than doubles from twenty to seventy-five — and the share the model actually uses falls by a fifth. The two differ by the S cone's peak absorptance, which the lens takes 60 per cent of, and which entered the model as a normalisation rather than as a claim.

eye · Cones How far a mesopic match moves as the S weight opens: two rooms against one field. For every pair of the five lamps, the median over forty-two surfaces of how far a match moves as the rod signal's weight into the S channel goes from nothing to equal: pale for the two-room match, each half adapted to its own lamp; dark for a bipartite field whose two halves share one adaptation. The field's signal is larger for every pair, by ×1.5 to ×7.7.

One field keeps what two rooms divide out

An asymmetric match can measure how strongly the rods feed the blue-yellow pathway, but set with the observer adapted to each lamp in turn it needs seventy-five settings on the best pair of lamps and hours of waiting between them. Putting the two lamps on the two halves of one field was proposed as the quick version, at the cost of a weaker signal. The signal is not weaker. Under one shared adaptation it is three times stronger for daylight against a white LED, and the best pair needs six settings. The adaptation that makes the slow version slow is also what was dividing the rods' contribution out of each half.

eye · Cones

Named alongside it

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

AuditStructural choiceSpecificationSpecularBidirectional reflectanceInterreflectionRadiosityTest setCIECAM16Colour appearanceConvergenceMeasurement error

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