Interreflection — where it appears
Named by 45 essays across 2 fields — each of them below, with the objects they name alongside it.
A bounce is a multiplication
Colorimetry multiplies an illuminant by a reflectance once and integrates. A surface in a room is lit by every other surface the lamp reached first, so the spectrum arriving at the eye has been multiplied several times — and the second multiplication is where the whole apparatus of matching starts to come apart.
Two paints that stop matching
A metameric match is an identity between three integrals that are linear in reflectance. A second bounce carries reflectance squared, and no linear identity survives being squared — so two paints certified identical on a flat chart come apart in a corner, by an amount the geometry decides and the colorimetry cannot express.
Rendering in three numbers
Almost every renderer ever shipped bounces red, green and blue rather than a spectrum. The error that costs is exactly zero at the first product and grows at every one after it — because three numbers cannot carry a product they were never given the factors of, and each bounce is another product.
What a white wall costs
A closed room returns 1/(1−ρ) times the light that entered it, computed band by band — so a paint that is five percent off neutral becomes a strongly coloured illuminant once the room has finished bouncing. The gain amplifies the tint along with the brightness, and the last few percent of albedo cost far more than the first.
A corner is not a wall
A form factor is the fraction of everything leaving one surface that arrives at another, and it is the only place geometry enters the colour of a room. It is also the one number here with a published closed form to check against — and the check turned out to converge at two different rates for two cases that look identical.
The room is the illuminant
Colour bleeding is usually described as an aesthetic phenomenon of rendered images. It is better described as a measurement. In a room with one lamp, five of six surfaces emit nothing at all, so their light is entirely a product of other surfaces' reflectances — and the bleeding saturates rather than running away, for a reason worth deriving.
A scene has no white point
White balance is one transform applied to a whole image, and a scene lit by two lamps has no single answer for that transform to be. The failure is structural rather than a matter of a better estimator — and every colour-managed workflow in existence takes exactly one white point as an input, with no field in which to say there were two.
A corner moves both terms
The interreflection essays compute what a corner does to a spectrum, which is one of the two things a corner does. It is also a brighter place with a differently coloured background — a viewing condition, not a stimulus — and adaptation removes most of the first and none of the second. At an enclosure of six tenths that is 63 per cent of seven units gone and a further unit arriving from the extra light alone.
The same wall applied twice
A bounce off a painted wall is a change of illumination, and adaptation handles it about as well as it handles a change of colour temperature. A corner applies the same reflectance twice, which sharpens it — and leaves an adapted observer with 1.96 times as much for a change only 1.33 times as large.
Everyone is beaten by the same wall
Eight candidate adaptation bases, fourteen changes of light, and seven of the eight have their worst row in the same place — not a lamp, but a green wall reflecting twice. The one that does not is the one that was fitted, and what its fit bought was permission to give up on that row.
The worst case is where the box stops
The worst change of light this collection quotes is two bounces off a green wall, and it is the worst of fourteen changes somebody wrote down. Searching the family those fourteen were drawn from reaches six times further — and does not stop, because the residual rises monotonically towards a narrower notch on a darker wall. There is no worst case in this family, and the number anybody quotes for one is a number about their own constraint.
A room bounds its own bounces
The adaptation census has one bounce and two bounces as separate rows, and a search over the family treats the count as a free integer it always takes to the largest value offered. A room offers no integer at all — it applies a geometric mixture of every number of bounces, and that mixture is bounded by the walls reflecting less than everything.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
What an adapted viewer loses is set by the wall
A satin finish takes about a tenth of a room's colour, measured on the room's light against its lamp's white. Read through an appearance model by a viewer adapted to the room, the same thirty-six rooms lose between 12.8 and 30.9 per cent — 1.4 to 2.4 times as much — and the wall colour decides the multiplier: a deep red room loses twice what a green one does, under every lamp. A test on the bare wall predicts it. Add a little of the lamp's white to the wall's own colour and ask the model what that costs: the answer orders the thirty-six multipliers at 0.95.
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.
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.
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.
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.
A finish adds colour only where it covers grey
One low wide room was found in which a satin finish makes the room more colourful rather than less, and the explanation offered was dilution: the more of a room is grey, the more a finish's glancing return stands out. Painted area turns out not to decide it. Across six room shapes and five ways of painting them, a finish on the painted faces alone never adds colour, and a finish on the grey faces alone almost always does. The low room gains because its grey faces are most of it, and the whole finish is close to the sum of its two halves.
The tall room gains once its walls are divided
The room census solves each face as one patch, and it says a satin finish on the grey faces of a tall room with painted side walls makes the room more colourful for none of seventy-two paints. The true box disagreed: rays from points on the grey end walls said the finish sends the lamp's light towards the paint. Dividing every face into patches settles it. The end wall's upper patches, which see the lamp at a grazing slant, send its light across to the painted walls; its lower patches send it to the floor. The room gains for 0, 4, 8, 11, 15 and 15 paints as the faces are divided one to six times a side — the light, broad paints — and the narrow slot beside it goes from 22 to all 72.
Divided faces keep the census's count and lose its rule
The painted-room census, solved with one patch a face, found that a satin finish on a room's grey faces adds colour in 28 of 30 rooms, and a later essay found one geometric number whose sign sorted which rooms gain. Re-solved with every face divided into twenty-five or thirty-six patches, the count survives exactly — 28 rooms, though two of them trade places — and the finish colours every paint in 17 rooms instead of 13. The rule does not: the send number, now matching the true box's own to a rank correlation of 0.96, orders the divided counts at 0.63, and rooms on the wrong side of its zero gain for every paint. Nine of the thirty counts move by twenty paints or more.
The paint's own light does not order the rooms
A satin finish on a room's grey faces colours some paint in 28 of the painted census's 30 rooms, and the number meant to say which rooms gain most — how much of the lamp's light the finish sends to the paint — orders the rooms solved on divided faces at only 0.63. The rooms it misplaced return mostly the paint's own light, so the proposal was to count that instead. Counted, the paint's light orders the rooms at −0.07. In every room with its floor painted the finish sends the floor's light away from the floor, and every paint gains; with the ceiling painted the paint is the lamp and the two numbers are the same number. Even the room's own response, read off the solve with the paint in place, orders the counts only at 0.71. The census has to be solved room by room.
The paint itself gains all or nothing
The painted census counts how many of seventy-two paints a satin finish on a room's grey faces makes more colourful, reading the room's colour as all of its light, and eleven of its thirty rooms sit part-way. Read on the painted faces alone — the light a viewer looking at the paint receives — no room sits part-way: twenty-eight gain for every paint and two for none. The change on the paint is never more than a per cent and a half, where the room's reading swings from −18 to +57 per cent, so the census's middle ground was the grey faces all along. A box with its floor painted and the same box with its ceiling painted gain the same colour to a hundredth of a point while the light they gain differs by up to eighty-two points; neither send number orders the size of the gain, and the paint's own brightness does.
The slot's loss survives every finer division
Read on the painted faces alone, a satin finish on a room's grey faces makes the paint more colourful in twenty-eight of the painted census's thirty rooms and less colourful in two: the tall narrow slot with its floor painted and with its ceiling painted, where the paint loses about a twentieth of a per cent of its chroma. A loss that small could belong to how finely the slot's faces were divided, and the prediction was that it does not — that the sign holds at seven and eight strips a side. It holds. Solved from one strip a side to eight, the slot's paint gains at one, two and three strips and loses from four on, every one of seventy-two paints crossing at the same division; the loss deepens at every finer division, to −0.055 and −0.059 per cent at eight strips, and run out to no strip width it settles near −0.058 and −0.065. The coarse census had the sign wrong; the census at six strips had it right. The bright, broad paints lose most, as they gained most everywhere else.
The finish stops costing the paint six tenths deep
In the tall narrow slot a satin finish on the grey faces costs a painted floor or ceiling a little of its chroma; in the tall room, the same box a full unit deep, it adds some. Solved at eight depths between, the sign turns at 0.62 of the box's width with the floor painted and at 0.63 with the ceiling — later than the prediction of 0.45 to 0.55, and far enough along that the loss belongs to a range of shallow boxes rather than to the slot alone. A finer division moves the turn a little deeper, not back. Every one of seventy-two paints turns at the same depth, and each paint's share of the change is the same at every depth, loss or gain. Brightness does not decide it: the finish brightens the painted floor by between half and more than double while hardly touching the painted ceiling, yet the two turn together. What decides it is which reflectance the finish favours — whether it multiplies the light that has been back to the paint by more or by less than the light that has never touched it.
A taller box must be deeper before the finish adds colour
In a box one unit wide and two and a half tall, a satin finish on the grey faces costs a painted floor a little of its chroma when the box is shallower than 0.62 of its width and adds some when it is deeper. Whether that turn is set by the width or by the height was the open question, and the prediction was that it scales with the height, because the satin walls pipe the lamp down to the floor and a taller box is a longer pipe. It does, and faster than in proportion. A box 1.5 tall gains at every depth from 0.35 and turns shallower than that; a box 4 tall loses at every depth to half again its width and turns near 1.67. As a share of the height the turn rises from about 0.17 to 0.25 to 0.42 — close to a depth growing as the square of the height. The taller box pipes far more of the lamp to the floor, tripling its light at 0.35 deep, and at every depth of every box the paint's colour still follows which reflectance the finish favours. Six strips a side keep both new boxes' signs.
The width moves the turn only when it is the narrower side
In a box one unit wide and two and a half tall, a satin finish on the grey faces costs a painted floor a little chroma when the box is shallower than about 0.64 and adds some when it is deeper. The prediction was that the width hardly matters — that the two end walls a depth apart form the pipe that carries the lamp down, and wider walls only add more of it — so the turn would move by under a tenth between widths. Solved at six strips a side, a box two units wide turns between 0.5 and 0.65, within the tenth or so predicted. A box half a unit wide still loses at 1.5 deep, and its quarter-turned twin says it still loses at 2: its turn is more than a unit and a quarter deeper. With only the floor painted a box w wide and d deep is the box d wide and w deep, so depth and width are one question, and the answer is that the floor's shorter side decides: under about half a unit the finish greys the paint, past about 0.65 it colours it. The scan's first pass, at four strips, said otherwise, and got four of seven signs wrong in the long, thin boxes.
A square floor needs more room than a long one
A satin finish on a room's grey faces costs its painted floor a little chroma when the room is narrow and adds some when it is wide, and in a box two and a half tall the turn was found on the floor's shorter side — 0.64 when the longer side is one unit, about 0.51 when it is two. The prediction for a square floor, with no longer side to speak of, was a later turn, near 0.7, because both pairs of walls are then equally close and both carry the lamp down. Solved at six strips a side, the square floor loses for every one of seventy-two paints at 0.5, 0.6 and 0.7 on a side and gains for every one from 0.8, turning near 0.78. A floor 0.7 by 0.7 loses where a floor 0.7 by one gains. The finish brightens the square's darkest floor by more than an oblong's at every shorter side — by a hundred per cent against forty-six at half a unit — and the turn falls steadily as the longer side grows: 0.78, 0.64, 0.51. The longer side matters, and the light-pipe account says why.
A long floor never loses its colour to the finish
A satin finish on a room's grey faces costs its painted floor a little chroma when the room is narrow, and the shorter side at which the cost turns into a gain fell as the floor grew longer — 0.78 for a square, 0.64 for a floor one unit long, 0.51 for two. The prediction was that it settles near 0.45 by four units, the limit of a slot whose far walls pipe nothing. Solved at six strips a side, floors three and four units long gain chroma for every one of seventy-two paints at every shorter side from 0.35 to 0.65: there is no turn to find. The floor two units long already loses only in a window, shorter sides from about 0.3 to just past 0.5, and gains on either side of it; by three units the window has closed, and a four-unit floor gains more than a three-unit one everywhere. The loss was never the slot's. It belonged to the far pair of walls, and moving them away removes it.
The window of loss closes from both ends
A satin finish on a room's grey faces costs a painted floor a little chroma only in narrow rooms near square, and a floor two units long loses it only in a window of shorter sides, from about 0.30 to 0.51, while a floor three units long never does. The prediction for the floor two and a half units long was that the window closes from below: a gain at a shorter side of 0.3 and a loss only near 0.4 to 0.5, because the lower edge is where the far walls' pipe is weakest. The floor gains at 0.3, loses at 0.4 — by under six thousandths of a per cent — and gains again at 0.5. The window closes from both ends at once, to 0.39–0.42, and its deepest point rises through nought near 2.6 units long. Every box is still all or nothing, every one of seventy-two paints with the same verdict, and every verdict has the sign of the finish's tilt across the floor's reflectance. The far walls' cost shrinks smoothly with their distance and centres on a shorter side of about 0.4.
A taller room keeps its floor losing in proportion
A satin finish on a room's grey faces nearly always adds a little colour to a painted floor, and in a box 2.5 units tall it takes some away only in a pocket of proportions: floors shorter than about 2.6 units, with a shorter side near 0.4. The prediction was that the pocket scales with the room's height — that a box four units tall keeps losing until its floor is about 4.2 units long, 1.04 times its height as in the lower box. It loses until 4.04 units, 1.01 times its height: the crossing scales, a little short of the prediction. The pocket moves too. At a shorter side of 0.4, where the lower box lost, the taller box's floor gains at every length; its losses sit at 0.6 — at three and four units long — and at 0.8 when three units long, and its deepest loss lies near a sixth of its height, as the lower box's did. Every box is still all or nothing across seventy-two paints, and every box's change carries the sign of its finish's tilt across reflectance.
A room scaled whole loses its floor to the last digit
A satin finish on a room's grey faces costs a painted floor a little of its chroma only in a pocket of proportions, and a box four units tall was found losing in proportion to a box 2.5 tall — its crossing at 1.01 times its height against the lower box's 1.04, its pocket near a sixth of the height in both. The prediction was that the answer is the room's proportions alone: that the four-tall box built as the 2.5-tall box multiplied by 1.6 gives the lower box's loss to the solver's precision, and that the floor four units long loses from about 0.55 to 0.65 wide. The scaled box gives the lower box's numbers to eleven significant figures, and a pair the census had already solved agrees to twelve. The window sits higher than predicted at both edges, from 0.59 to 0.70. And the gap between the two rooms' crossings was the census's sampling: read at the pocket, the taller room crosses at 1.029 times its height against 1.036.
The wedge of loss comes to a point
A satin finish on a room's grey walls and ceiling adds colour to a painted floor in almost every proportion of room, and costs it a few thousandths of a per cent of its chroma in one narrow wedge of proportions: floors about as long as the room is tall and a sixth of its height wide. The wedge was known to close between floors 1.0 and 1.2 times the height long. The prediction was that a floor 1.04 times the height long still loses at a sixth of the height and one 1.08 times as long gains at every width near it, placing the tip between them a little above the coarse crossing of 1.036. The longer floor gains everywhere, as predicted. The 1.04 floor sits on the tip itself: its change in chroma is two parts in ten million, the row crosses nought at 1.0399, and there — and in no other box of the census — the seventy-two paints part company, fifty-five gaining and seventeen losing.
The pale paints cross the tip last
A satin finish on a room's grey faces costs a painted floor its chroma only in a narrow wedge of proportions, and at the wedge's tip — a floor 1.04 times the room's height long and a sixth of it wide — the mean change is nought and, alone in the census, the seventy-two paints part company: seventeen lose, fifty-five gain. The prediction was that the losers are the paints with the narrowest reflectance bands, whose chroma depends most on how much grey light mixes in, and that they lose most below the tip and gain least above it. The opposite holds. No narrow band loses; half the broadest do, and fourteen of the seventeen have the highest peak — the pale paints. They lose most below the tip and gain most above it: they are the paints the finish moves most, up to three times the mean, and each paint has its own tip, from 1.0387 to 1.0413 times the height, the palest last.
The pocket has a step where its bottom should be
Along a floor as long as the room is tall, a satin finish on the grey faces costs the floor a few thousandths of a per cent of its chroma in a pocket around a sixth of the height wide, deepest at 0.16 of the height by three samples a hundredth of the height apart. The prediction was that boxes between those samples trace one smooth dip with its bottom between 0.39 and 0.41 units, every box from 0.375 to 0.425 losing. The loss does deepen smoothly, at every step from 0.375 to 0.40. Then, a hundredth of a unit wider, at 0.41, every one of the seventy-two paints gains; at 0.42 they split; at 0.4375 every paint loses again. The row changes sign three times in a tenth of a unit of width, and the one step from 0.40 to 0.41 rises three times as fast as the pocket falls. The finish's brightening, which the chroma change rides on, falls smoothly straight through. What the census has called the pocket's bottom is a step.
Named alongside it
The objects these essays reach for when they reach for this one.
RadiositySpecularChromaModelling assumptionConvergenceReflectanceForm factorColour bleedingBidirectional reflectanceAlbedoCensusRank correlation