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

What a scene does

A reflectance measured on a patch is a patch. Put it in a room and the light bounces, and a bounce is a multiplication — so shadows carry their own illuminant, highlights carry the lamp, and two surfaces that matched go on matching only until the second bounce.

52 essays. Read in order: Scene.

A spectrum after 2 bounces off the same surface. The lamp's spectrum at the top, then the same spectrum multiplied by a reflectance peaking at 530 nm once for each bounce. Interreflection is elementwise multiplication, so light that reaches the eye by the long way round carries ρ raised to the number of surfaces it met. Each row's swatch is drawn at fixed luminance so only the chromaticity changes, and the distance from the D65 white point, printed at the right, rises from 0.000 to 0.235. The spectrum narrows every time, which is why a room painted in one colour is more saturated in its corners than on its walls.

A bounce is a multiplication

part 1 A metameric match that a corner breaks. Two reflectances with identical XYZ under D65 — metamers, matching to ΔE00 = 5.4e-14, which is the numerical floor rather than an approximation. On a flat wall the light meets one of them once and the two patches are the same colour. In a corner, a fraction 0.200 of what leaves the surface returns to it, so part of what reaches the eye carries ρ twice — and the match fails by ΔE00 = 1.53. A metameric match is an identity between three integrals that are linear in ρ, and there is nothing linear left after a second bounce. The geometry, not the light and not the paint, is what breaks it.

Two paints that stop matching

part 2 The two components of daylight, computed from one radiator and one scattering law. A 5800 K radiator through the atmosphere at air mass 1. The direct beam is the radiator times e^{−τm}; the sky is what that extinction removed, so the two are complements and neither needs its own model. The sky is steeply blue because τ ∝ λ⁻⁴, and a surface in shadow is lit by that component alone. Open ground and shadowed ground therefore sit under two illuminants differing by ΔE00 = 21.4 — which is why a photograph of snow has blue shadows and why no single white balance fixes both halves of it.

A shadow has its own illuminant

part 2 A glossy surface returns two spectra, and only one of them is the paint. The dichromatic reflection model, computed rather than assumed. Light that enters a dielectric binder, scatters off pigment and comes back carries the reflectance — the body component, ΔE00 = 33.7 from the lamp. Light reflected at the interface never entered, so it carries the lamp's spectrum with only Fresnel's slight dispersion on it: ΔE00 = 1.06. The interface term is computed from Fresnel's equations on a Cauchy index at 55°, so its near-neutrality is a result here rather than an assumption. This is why a highlight is the one region of a photograph that tells a white balancer what the light was, and why removing highlights removes the evidence.

The highlight is the lamp

part 2 How far three channels drift from eighty-one, per bounce. One room, one geometry, one reduction to three channels, and the only thing changing is how many bounces of the Neumann series are kept. At one bounce the two agree to 2.5e-13 — the only reflectance in that path is the floor's, which is flat, and a flat reflectance is one of the few three numbers carry exactly. Every bounce after it multiplies another non-flat reflectance into the spectrum, and three numbers cannot carry a product they were never given the factors of. The curve levels off at ΔE00 = 2.38 because the light has run out, not because the disagreement has.

Rendering in three numbers

part 3 The interreflection gain of a room, band by band. A closed cavity of reflectance ρ returns 1/(1 − ρ) times the light that entered it, and because that is computed per band it amplifies the wall's colour along with its brightness. The wall here is only 5% off neutral — a paint anybody would call white — and at albedo 0.9 the room's white point has moved by ΔE00 = 50.7 against the lamp it was lit with. The gain is a geometric series, so the last few percent of albedo cost far more than the first: 1.64× at ρ = 0.4 against 8.41× at ρ = 0.9.

What a white wall costs

part 3 The form-factor matrix of the box. F_ij is the fraction of everything leaving face i that arrives at face j. The diagonal is zero because a flat face sees none of itself; each row sums to exactly 1 because the cavity is closed; and A_i F_ij = A_j F_ji, which is reciprocity and is checked to 10⁻⁹. For a cube the opposite face takes 19.98% and each of the four adjacent faces 20.00%, and the near-equality of those two numbers is a coincidence of the cube rather than a rule.

A corner is not a wall

part 3 Every face of the solved room. The red room after the transport is solved in all 81 bands. Only the ceiling emits; the other five faces are lit entirely by what the ceiling and each other send them, so their colour is the lamp multiplied by every reflectance along every path that reached them. The ceiling itself comes out 1.05× brighter than it emits, because a closed room returns light to its own source. The two chromaticities under each swatch are the spectral solve and the three-channel one, and the faces furthest from the lamp — the ones the light reached by the most bounces — are where they disagree most.

The room is the illuminant

part 3 Mixing two paints and stacking two filters are different operations. The same two reflectances combined two ways. Stacking them as filters multiplies the transmittances, which is right for gels in front of a lamp and wrong for pigment stirred into pigment: a stirred mixture is one scattering layer, not two in series, and light meets whichever particle is nearest rather than passing through both. Kubelka–Munk handles it by moving to K/S = (1 − R)²/2R, in which absorption and scattering add by concentration, and inverting afterwards. The two answers differ by ΔE00 = 15.0, and the filter model is the darker of the two because it charges every photon for both pigments.

Paint is not a filter

part 4 Why blue and yellow make green. 7 mixtures between a blue and a yellow pigment, mixed in Kubelka–Munk — K/S summed by concentration and inverted back to reflectance — and plotted against the straight line joining the two endpoints. The path bows towards green by 0.099 in chromaticity, and the reason is in the spectra rather than in the eye: the blue reflects below about 520 nm and the yellow above about 500, so the only band both return is the overlap between them. Mixing lights adds spectra and lands on the chord; mixing pigments intersects them and does not.

Why blue and yellow make green

part 4 The same medium at six path lengths. Beer's law at 6 depths of one absorbing medium. The absorption coefficient is a single spectrum and the only thing changing is how far the light travelled, yet the patches differ in hue by 7.8° as well as in lightness — because absorption is exponential in depth and the observer is linear, so the bands that survive at d = 8 are not a scaled copy of the ones that survive at d = 0.25. Path length belongs to the geometry, not to the substance, which is why this sits in a field about scenes.

The colour is in the thickness

part 4 One film, five viewing angles. The same 340 nm film seen from 5 directions. Nothing about the object has changed — not the light, not the material, not the thickness — and the colour swings by ΔE00 = 42. A pigment's spectrum contains no path length and no angle, so it cannot do this; a film's contains both. This is the clean separation between structural and pigmentary colour, and it is geometric rather than chemical.

A colour that moves with the viewer

part 4 What no surface can be more colourful than. The MacAdam limits at 4 lightnesses under D65, each computed by sweeping two-transition reflectances over the whole band and keeping those that land at the target luminance factor. This is a physical bound rather than a gamut: a reflectance above 1 is a surface that emits, so no pigment anybody invents will ever put an object colour outside these curves. The boundary shrinks steeply as the surface lightens, from 0.310 at Y = 0.1 to 0.028 at Y = 0.9 — a very light surface has almost no room to be colourful, and that is physics rather than pigment chemistry. Drawn against it is sRGB at the same luminance factor rather than as a primary triangle, because a triangle is what a display can reach at some luminance and the bound is what a surface can reach at one; matched properly, sRGB covers 36% at Y = 0.1, 40% at Y = 0.3, 40% at Y = 0.6, 21% at Y = 0.9. The faint triangle is the familiar figure, kept only to show how much it misleads.

No surface can be that colourful

part 5 One white balance across a scene lit by two lamps. A neutral surface of albedo 0.6 under 7 mixtures of A and D65, corrected by one diagonal transform chosen for the middle of the run — which is what a camera does when it estimates a single illuminant. The middle patch comes out neutral to ΔE00 = 0.00 and both ends do not: 19.8 at the A end and 16.9 at the D65 one. The failure is structural rather than a matter of a better estimator: white balance is one transform for the whole image, and a scene with two lamps in it has no single answer for that transform to be. Every patch here is the same surface.

A scene has no white point

part 5 A glossy surface returns two spectra, and only one of them is the paint. The dichromatic reflection model, computed rather than assumed. Light that enters a dielectric binder, scatters off pigment and comes back carries the reflectance — the body component, ΔE00 = 33.7 from the lamp. Light reflected at the interface never entered, so it carries the lamp's spectrum with only Fresnel's slight dispersion on it: ΔE00 = 1.14. The interface term is computed from Fresnel's equations on a Cauchy index at 45°, so its near-neutrality is a result here rather than an assumption. This is why a highlight is the one region of a photograph that tells a white balancer what the light was, and why removing highlights removes the evidence.

Gloss changes the measurement

part 5 A corner moves the spectrum and the viewing condition at once. A coloured patch in a corner of coloured walls, against how enclosed the corner is. The top curve is what a colorimeter set up at the door reports: light that has bounced carries the surrounding reflectance again, so the patch is lit by something the room is not. The middle curve is what is left once the patch is read against the corner's own white — most of it goes, because a corner is a change of illuminant and that is what chromatic adaptation is for. The bottom curve is the other thing a corner is: a brighter place, 1.76 times the light, which moves the appearance through the Hunt effect with the white point held still and cannot be adapted away at all.

A corner moves both terms

part 6 The same wall, applied once and applied twice. A room lit by light that has bounced off its own walls is a change of illumination like any other, and a corner is the same change applied twice. Squaring a reflectance sharpens it, a sharper change of light is further from being a gain, and the residual an adapted observer is left with therefore grows faster than the change does: the second bounce is 1.33 times the change and 1.96 times the residual. This is the adaptation half of what a corner does to a metameric match.

The same wall applied twice

part 7 A sample with two reflectance curves, and neither below one. The apparent reflectance of an optically brightened sample, measured under D65 and A. It exceeds 1 — the shaded band — which no reflector can do: more light leaves at these wavelengths than arrives at them, because the sample absorbs in the violet and re-emits in the blue. And the two curves differ, so the sample has no single reflectance to store. The effect drawn here is a floor: most of the excitation band lies below 380 nm, outside the range computed here.

A surface that is not a multiplication

part 7 Two sheets with the same reflectance and two different colours. A brightened sheet and a dyed one built to match it under an instrument with no ultraviolet. Under that instrument the pair agrees to ΔE00 0.00, which is a rounding and is true by construction — the dyed sheet's reflectance is the curve the brightened one measured. Under an instrument that includes the ultraviolet they are 7.1 apart, and under daylight 10.6. This is not ordinary metamerism: the two sheets do not differ in reflectance anywhere the eye can see, so no change of light puts them back together and no adaptation removes the difference. One of them is a curve and the other is an operator.

Two sheets that match until the window

part 8 What a sheet of glass takes out of the band a brightener eats. The transmittance of four glazings across the short-wave band, with the brightener's own absorption shaded underneath. The overlap between a curve and the shading is what the sheet behind that glass has to work with. Ordinary window glass stops below about 310 nanometres and leaves most of the band; laminated glass has a plastic interlayer that was put there to hold the sheet together in a crash and happens to absorb almost to 380; a filter sold to protect a print removes the band entirely. The curves are logistic edges at stated wavelengths rather than measurements of particular products.

The window is part of the light

part 6 How short of determining the light a photograph is, as the scene grows. Each cell is the number of unknowns left over after every equation the image supplies: three sensors, a three-dimensional illuminant, and reflectances confined to a linear model of the dimension on the left. At one and two dimensions more surfaces close the gap. At three the gap never closes, because each further surface adds three equations and three unknowns; at four it widens. The count is arithmetic and has no algorithm in it.

An image does not determine the light

part 9 The same border signal, filled in with a boundary and without one. Two fields, each 6 degrees across. The signal is injected along a ring just inside a contour and varies around it, brightest on one side and dimmest on the other. On the left the signal diffuses and the contour is impermeable: the interior settles to 1.000 against a border mean of 1.000, which is the mean-value property of a harmonic function arriving as a prediction about appearance. On the right the same signal is handed to a Gaussian pool of 0.5°, which has no notion of inside: it reaches 0.040 at the centre, because a kernel weights the near rim more than the far one and a filled region does not.

A pool with an edge

part 10 A gap in the wall and a gap in the drive are not the same gap. What the centre of a region settles to under three conditions. With the contour closed it reaches its border's value exactly. Open a 16% hole in the barrier and leave the border signal unbroken and it still reaches it, to 1e-8 — a ring of driven cells encloses the centre whatever the wall outside it is doing, so nothing can escape. Break the signal too and it falls to 0.960. All of what a gap costs is the piece of border that stopped driving, and none of it is the hole.

A gap in the drive, not in the wall

part 10 The worst case is wherever the box stops. Four horizontal tracks, one per parameter of a painted wall. Each track spans the range an ordinary paint is allowed to occupy, with a second, wider range drawn behind it, and two markers show where the search for the worst change of light came to rest under each. Under the narrower box the answer sits on the wall in centre and width; under the wider one, in centre, width, base. The residual rises monotonically towards a narrower notch at a shorter wavelength on a darker wall, so there is no interior maximum to find. The worst change of light is 21.3 ΔE00 under one box and 28.4 under the other, and the census's own worst row is 3.37.

The worst case is where the box stops

part 11 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

part 9 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

part 10 What one change of light costs, surface by surface — daylight to tungsten. A rising curve of 125 points, one per surface in the test set, sorted from the surface this change of light costs least to the one it costs most, with the published mean drawn across it as a horizontal line. The published residual for daylight to tungsten is 1.635 ΔE₀₀. The curve runs from 4.4e-14 — 5 of the surfaces are flat greys, on which an adapted observer's gain is exactly right and the residual is exactly zero — to 3.058, which is 1.87 times the mean. The mean line crosses the curve about two thirds of the way along, so most surfaces cost less than the published number and a minority cost a great deal more. This is what a single published residual is a summary of.

A mean has a set under it

part 11 What a fourth reflectance dimension costs the theorem that a change of light is a matrix. Four rising curves on axes of the fourth dimension's amplitude, left to right, against what is left of daylight to tungsten after the exact 3×3 change-of-light matrix has been applied, in ΔE₀₀. All four begin at exactly zero: on the three-dimensional family the matrix is solved rather than fitted and there is no remainder at all, which is the theorem this collection's adaptation argument is built on. Adding a fourth reflectance dimension breaks it, and how badly depends far more on the fourth function's shape than on its size — at five per cent amplitude the four shapes cost 0.329, 0.572, 0.063, 0.124 ΔE₀₀ respectively, a factor of 9.1 between the dearest and the cheapest. For scale, the smallest von Kries residual anywhere in the census is 0.26 ΔE₀₀, so the cheapest of the four is a quarter of it and the dearest is twice it.

A theorem about a family

part 12 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

part 12 What an observer is left with, by how much it is allowed to know about the room. Six ways of discounting a change of light, averaged over the fourteen changes in the adaptation census and 125 test surfaces each. The bar is what each leaves behind, on a logarithmic axis because the models span two orders of magnitude. The second line under each name is the count that matters: how many numbers about this room the model has to be given. Doing nothing leaves 15.7 ΔE₀₀. A single gain read off the two whites' luminances leaves 15.3. A matrix fitted across half the census and then applied everywhere, knowing nothing about the room at all, leaves 12.5. The published von Kries gain, which is told the white and nothing else, leaves 1.312 — and bolting a fixed correction onto it, at no cost in scene information, leaves 1.368, which is very slightly worse. The exact matrix leaves nothing and is not on the chart: its nine numbers are the change of light, which is the quantity being discounted.

Three numbers the scene supplies

part 12 How much of the residual a partial correction removes. Between the diagonal gain and the exact matrix there is a line: apply the correction that would make a row exact, but only a fraction of it. The horizontal axis is that fraction and the vertical is the share of the row's residual it removes, for all fourteen census rows. The straight diagonal is where a correction worth exactly its fraction would fall, and in the published unit every curve lies on it to within 2.2 percentage points. The lower band of curves is the same interpolation measured in CAM16-UCS, which departs by up to 17 points — because its distance is a power of the Euclidean one and a power is not homogeneous along a ray, where every ordinary norm is. The straight line is therefore a property of the ruler rather than of the correction, and the exception is what says so.

A partial correction is worth its fraction

part 13 A correction an observer could have been born with, fitted on half the census and tested on the other. The same six models, each scored twice: on the seven census rows the fixed matrices were fitted to, and on the seven they were not. The split alternates by position so both halves contain daylight changes and discharge lamps. The upper bar is in sample and the lower is out, on a logarithmic axis. For the four models with nothing fitted the two bars differ only because the halves are different questions. For the two fitted ones the gap is the finding, and it is largest where it matters least: bolting a fixed correction onto the von Kries gain takes it from 1.2724 to 1.2592 on the rows it was fitted to, and from 1.3511 to 1.3679 — worse — on the rows it was not. There is no correction to the diagonal that an observer could arrive with.

A model is a claim about what can be known

part 13 How much light comes back at each distance from where it went in. The diffuse reflectance kernel of 3 materials at 550 nanometres, computed from the dipole approximation to the diffusion equation. Both axes are logarithmic. The horizontal axis is the distance from the point the light entered, in millimetres; the vertical is how much comes back out per unit area there. Each curve's own diffusion length is marked with a tick. Coated paper returns almost everything within a fifth of a millimetre; marble is still returning light at ten. The reflectance the model wants is the whole of each curve, integrated over the plane, and what an instrument reads is only the part inside its aperture.

A surface has a kernel

part 12 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

part 12 Five fields, by how much light arrives from each elevation. The radiance arriving at a surface from each direction in one vertical plane, for five ways of lighting it. The vertical axis is logarithmic, spanning the three decades between a sun and the sky around it. The number beside each name is the share of the light that would have to be moved to make the field uniform: zero for the overcast sky, 0.93 for a lamp on a stand. A uniform field is the condition under which a reading is the sample's own reflectance, and the only place it exists is inside an instrument.

A room is not a sphere

part 12 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

part 12 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

part 13 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

part 14 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

part 14 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

part 14 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

part 15 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

part 15 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

part 15 The room's reflected light and its colour, against how glossy the walls are. Four changes against the matt room as the coloured walls are made glossier, from roughness 0.8 on the left to 0.15 on the right. The room's reflected light rises by up to 19 per cent and its chroma falls by up to 9.1 per cent. The walls' own outgoing chroma falls fastest, by 14.6 per cent, and the floor's follows the room's. A lobe does not move colour from one face to another: the room as a whole has less of it.

A gloss finish takes colour out of the whole room

part 16 A gloss finish's loss of colour, read by the light and by a viewer in the room. Four changes against the matt room as the coloured walls are made glossier, from roughness 0.8 to 0.15: the chroma of the room's reflected light as a colorimeter reads it; the mean chroma of the six faces as CIECAM16 sees them adapted to the lamp and adapted to the room's own average light; and how far the faces sit from that average in the model's uniform space. At roughness 0.2 the light loses 8.3 per cent, the faces 8.6 per cent to the lamp-adapted viewer and 13.9 to the room-adapted one, and the spread 11.0 per cent against 11.2 read against the lamp.

A gloss room looks less colourful than it measures

part 17 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

part 18 A satin finish pulls each room towards the lamp's white — a 3000 K radiator. Six rooms lit by a 3000 K radiator, on the ab plane of an instrument referenced to daylight, whose own white is the cross at the centre. Each open circle is a matt room and the arrow runs to the same room with satin walls. The filled diamond is the lamp's own white on that plane. Every arrow points within 9.2 degrees of the diamond and covers between 8.1 and 11.5 per cent of the distance to it. Whether the daylight instrument then reads more chroma or less depends only on whether the room was nearer the cross than the diamond is.

A finish adds colour only to a daylight meter

part 18 Thirty-six rooms as a viewer adapted to each would see them. Each cell is one room at a satin finish: wall colour down, lamp across. The large number is the share of the faces' chroma a viewer adapted to the room's own light loses, read through CIECAM16; the small number under it is what the room's light loses against the lamp's white. The viewer loses between 12.8 and 30.9 per cent, always more than the light, and the rows differ far more than the columns: a deep red room loses about twice what a green one does under every lamp.

What an adapted viewer loses is set by the wall

part 19 The share a finish takes falls with how much light the wall returns. Seventy-two rooms under daylight, each with a different paint on two opposite walls — six hues, four band widths, three peak reflectances — at a satin finish. Across is the wall's luminance factor, the share of the lamp's light it returns; up is the share of the room's chroma the finish takes. The losses fall with the luminance factor at a rank correlation of −0.89, from 15.5 per cent on the darkest walls to 2.5 on the palest. The two ringed paints make rooms of the same matt chroma and lose 14.7 and 2.5 per cent.

A dark wall pays for a finish

part 19 What survives holding the lightness still. For each of five groups of rooms sorted by how much light the wall returns, the rank correlation of the finish's share with three properties of the paint, taken with the wall's luminance factor held. The wall's chroma runs from -0.90 among the darkest walls to 0.81 among the palest, crossing zero in the middle — the reversal. The band's width, which was the predicted mechanism, never leaves the range -0.15 to 0.22, and the census already contains four families whose bands are all the same width.

The finish adds the room's own colour

part 20 Where chroma stops protecting a wall, in five rooms. The seventy-two paints in each of five rooms, sorted by how much light the wall returns and read in overlapping windows of eighteen: the rank correlation of the share of colour a satin finish takes with the wall's chroma, lightness held. Below zero a more saturated paint loses less; above, more. The dots are where each room's curve crosses: cube at 0.36, corridor at 0.37, low room at 0.38, one wall open at 0.29, two walls open at 0.24. Solid lines are closed rooms of three shapes; dashed are the cube with one and two walls opened.

An open room hands over sooner

part 21 The lamp's white, six walls and the light arriving at each. On the 1976 chromaticity diagram, in the closed cube under daylight: the lamp's white (centre), six saturated walls with bands centred from 450 to 650 nm (open circles), and the light arriving at each wall from the rest of the room, lamp included (filled). Each ambient lies on the line from the white to its wall, a fraction of the way along it: 450 nm 10 per cent, 490 nm 12 per cent, 530 nm 19 per cent, 570 nm 19 per cent, 610 nm 13 per cent, 650 nm 5 per cent.

A probe at the wall prices the finish

part 21

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