Concept

Specular — where it appears

The mirror direction, in which a smooth surface returns its interface reflection. Whether an instrument collects it is a stated part of its geometry, and on a dark sample it more than doubles the reported reflectance.

Named by 27 essays across 5 fields — each of them below, with the objects they name alongside it.

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

Light reflected at the surface of a glossy material never entered it, so it never met a pigment. Fresnel's reflectance varies by 5.5% across the visible band and the paint underneath varies by a factor of twenty — which makes a highlight the one region of a photograph that reports what the light was rather than what the object is.

scene · Scene
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

The same sample measured with the specular component included and excluded returns two different numbers, and both are correct answers to different questions. Colour is one of four appearance attributes and the only one most instruments report — so a specification that names a colour has silently named a geometry too, and usually does not say which.

scene · Scene
Recovering the lamp from a highlight rather than from an assumption. A green scene under illuminant A. The faint curve is the true lamp; the solid one is what the interface component of the glossy surfaces recovers, which is 0.30° from it. Grey-world on the same scene is 42.1° and max-RGB 17.5°, because both are assumptions about the surfaces and a Fresnel reflection is not.

The highlight is the white balance

Every estimator a camera uses is an assumption about the scene wearing the costume of a measurement. The specular component of a glossy surface is not — it carries the illuminant to within a third of a degree on a scene where averaging is fifty times worse, and it does so without knowing what colour the paint underneath is.

imaging · Capture
The same twenty-four samples, measured two standard ways. How far apart a 45°/0° instrument and a sphere with its gloss port closed are, on samples running from three per cent reflectance to seventy. The whole of the difference is the interface reflection — four per cent of the light, returned without ever meeting a pigment, thrown away by one geometry and collected by the other. It is the same four points in every row, which is why the disagreement is a property of how dark the sample is rather than of what colour it is: ΔE00 8.7 on the darkest samples against 1.93 on the lightest.

An instrument has a geometry

Every reflectance here arrives through a model with a bandpass, a sampling interval and no position at all. Real instruments say where they were standing, and the two standard answers disagree by ΔE00 8.35 on a dark gloss sample — a difference that adds rather than multiplies, and that no adaptation removes.

matching · Gamut
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

An integrating sphere reads a surface's own reflectance exactly, and the exactness is a theorem rather than good engineering — under a hemisphere of constant radiance, reciprocity makes the reading the sample's directional-hemispherical reflectance whatever the surface is. Every room fails that condition, and a viewing booth and a window get the sign of the error wrong in opposite directions.

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
An aperture and a gloss lobe, apart and together. Six materials, each measured through a four-millimetre radius and each given a gloss lobe, alone and at the same time. The pale bar is what the two cost added together as if they were independent; the dark one is what they cost when both are present. Every material comes out below the sum, by between 0.8 and 3.6 ΔE₀₀. The two departures partly cancel: the aperture removes light that went into the material and came back out too far away, and the interface returns light that never went in at all. Measuring either one alone therefore overstates what both together do, which is the opposite of the way interacting errors are usually assumed to behave.

Two departures that partly cancel

A glossy translucent sample has two of this round's four departures at once, and the expectation was that they would compound. They do the opposite. An aperture takes light away that went into the material and came back too far out; an interface returns light that never went in at all — so measuring either alone overstates what both together do, on every material tested.

difference · Metric
The collection's adaptation census, with its surfaces departed. Each row is one of the fourteen changes of light in this site's adaptation census, and the bar is what a von Kries gain leaves behind. The open marks are the published numbers; the filled ones are the same computation with every one of the hundred and twenty-five test surfaces replaced by what an instrument with an aperture, or a room with a direction in it, actually reports. Nothing moves by more than 9 per cent. A departure that does not depend on the light is very largely absorbed by the observer's own gain, because it changes the reflectance and the gain is applied afterwards. The fourth departure is not on this chart and cannot be: a fluorescent sample has a different curve under every light, so there is no set of reflectances to hand the census at all.

The chart was measured, not photographed

A camera profile is fitted so that the camera's response maps onto the chart's tristimulus values, and those values came out of a spectrophotometer. So the profile's target carries the instrument's departures and the camera does not — a mildly translucent chart read through a four-millimetre aperture supplies targets 2.55 ΔE₀₀ from the truth, which is twice the profile's own fitting error.

imaging · Capture
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 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

A gloss wall makes the floor's return less colourful seen from the front of a room and more colourful seen from the coloured walls, which leaves open whether the lobe removes colour or only moves it. A ledger of every flux between the room's faces answers it. At an eggshell finish the room's reflected light gains 16 per cent in quantity and loses 8.3 per cent of its chroma, and the loss is nearly the same for blue, green and orange walls while the walls' own losses range from 13 to 25 per cent. Only the painted walls receive light as colourful as before.

scene · Scene
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
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

A gloss finish takes 8.3 per cent of the chroma out of a green room's reflected light, and a viewer adapted to the room should discount a loss that affects everything alike. The appearance model says the opposite. Adaptation removes the colour the whole room shares, leaves the colour that differs from face to face, and the finish takes as large a share of that as of anything — so to a viewer standing in the room the faces lose 13.9 per cent of their chroma, not 8.6.

scene · Scene
The statistic a converter would read, under each model. The log ratio of the two channels that are still open, against how bright the surface is, under the two models of what a highlight is. A matt surface over-exposed keeps its own ratio exactly — the line is flat, and it must be, because scaling every channel by the same amount leaves a ratio alone. A glossy surface carries a reflection of the lamp on top of its body colour, so its ratio slides towards the lamp's as the reflection strengthens: -0.086 of a log unit between a quarter of full scale and nine tenths. That slide is the whole of the evidence a converter has for choosing between them.

The converter can choose except where it matters

A converter rebuilding a clipped highlight has to say whether the surface was matt and over-exposed or glossy and carrying a reflection, and the evidence is whether its raw chromaticity slides towards the lamp's on the way up. Read through the sensor's own noise, the slide is clear on most of the chart from the few tens of pixels a specular highlight holds — and on the surfaces where it takes half a frame, confusing the two models costs more than the median. A ratio cannot see a common scale, and an exposure supplies one.

imaging · Capture
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 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

Measured against daylight's white, a satin finish under six lamps and six wall colours takes anything from −6 to 42 per cent of a room's colour, and one room reads as more colourful glossy than matt. Measured against the white of the lamp each room is actually lit by, the same thirty-six rooms lose between 7.7 and 12.9 per cent and none gains. The whole spread was the lamp's own colour, and the finish does one simple thing to every room: it pulls the room's colour a tenth of the way towards the lamp's white.

scene · Scene
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

A satin finish takes about a tenth of a room's colour, and the tenth varies. The natural explanation is that a weak colour loses a larger share of itself to the white light a glossy surface adds. Over seventy-two paints under one lamp that explanation orders nothing: the loss runs from 2.5 to 15.5 per cent, it follows how much light the wall returns at a rank correlation of −0.89, and it follows how colourful the wall is at −0.04. Two rooms equally colourful matt lose shares nearly six times apart, and the one that loses more is the darker.

scene · Scene
The same chart in two studios and in the room that holds both lamps. Each of the chart's twenty-four surfaces, with the pixels its slide needs at two standard deviations at high gain, on a logarithmic scale. Lit by a 3000 K radiator alone the worst surface needs 16,194; lit by daylight at 6500 K alone, 139,574. Lit by both, 50 and 50 per cent of the light, with a highlight of each lamp read together, the worst needs 10 and the median 6. The studios' hard surfaces sit at different places on the chart, and no surface is hard in both.

Two lamps decide what one lamp could not

Under one lamp a handful of surfaces cannot tell a glossy highlight from a matt over-exposure without most of a frame, because there the two differ by a scale a ratio cannot see. In a room lit evenly by a 3000 K lamp and daylight, every surface on the chart is decided from ten pixels. A body can sit at one lamp's white, not at two, and the rescue holds only while the second lamp carries a fifth of the light and its white sits far enough from the first.

imaging · Capture
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

Among dark walls a more saturated paint loses a smaller share of its colour to a gloss finish, and among pale walls a larger one. The mechanism proposed for that reversal was spectral concentration — a narrow tall band — and the census that found it already contained four families of paints whose bands are all the same width. Holding lightness still, the band's width orders the losses at a rank correlation of 0.01. What does order them is that the light a finish adds has already bounced off the walls.

scene · Scene
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

A gloss finish takes the least colour from a saturated dark wall and the most from a saturated pale one, and in a closed cube the sign changes at a wall returning about a third of the light. The prediction was that a less enclosed room would move that point up the lightness scale and that a room with a window would have no pale end. Opening one wall moves it down, from a luminance factor of 0.36 to 0.29, and opening a second to 0.24; stretching the room into a corridor or flattening it nudges it slightly up. The ambient does whiten, as predicted. A whiter ambient does not delay the colour term — it weakens it, so the other term takes over sooner.

scene · Scene
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

The light arriving at a painted wall from the rest of its room is what a gloss finish hands back, and a small probe held against the wall reads it. It lies on the line from the lamp's white to the wall's own colour, a fraction of the way along, and the fraction is set by how much light the wall returns, not by how colourful it is — as predicted. It is not the fixed fraction the prediction said: it runs from 2 to 46 per cent across seventy-two paints in one room. That variation is what makes it useful. Read in the matt room, it orders what a satin finish would cost more tightly than the paint's own lightness does, in every room tried.

scene · Scene

Named alongside it

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

InterreflectionRadiosityBidirectional reflectanceFresnelModelling assumptionChromaReflectanceDichromatic reflectionGlossAlbedoMeasuring geometryColour bleeding

All concepts