A corner moves both terms

A bounce is elementwise multiplication. Light that has reflected off a coloured wall carries that wall’s reflectance, so a patch in a corner is lit by something the middle of the room is not, and the essays that establish that compute exactly one consequence: the patch’s spectrum is different.

They are right about the spectrum and they are describing half the situation. A corner is also a place, and a place is a viewing condition.

How enclosed the corner is is the argument both terms depend on, and it can be moved.

The spectral half of the same construction can be moved the same way, and the light it is taken under is the obvious thing to vary next.

The enclosure is a continuum rather than a pair of cases, and two more settings of it are what make that visible rather than asserted.

The claim #

A corner moves the stimulus and the viewing condition at once, and only the first is what chromatic adaptation is for.

• At an enclosure of six tenths a patch is ΔE00 6.97 from its flat-wall reading, measured against the room’s white.
• Adapting to the corner’s own white removes 63 per cent of that, leaving 2.57. A corner is a change of illuminant and that is what a diagonal gain does.
• What is left is not a residue of that calculation. The corner is 1.40 times brighter, and that lift alone moves the appearance by 1.05 CAM16-UCS units with the white point held exactly still.
• And the residue that adaptation does leave is not zero. A von Kries gain is an approximation to a change of illuminant, not an identity, and at a deep enclosure the remainder is 4.25 units.

What a corner is #

Two walls and a floor meeting at a point form a partial cavity. A surface in one receives the lamp’s light directly and also light that has already bounced, and if the surrounding surfaces have reflectance ρ then the total incident is the geometric series — the lamp’s light divided by one minus the enclosure fraction times ρ, band by band.

That expression is a spectral multiplier: it changes the light arriving at the patch, wavelength by wavelength, in a direction set by the surrounding paint. If the walls are warm the corner’s light is warmer than the room’s, more so the deeper the corner.

Everything on this site about corners so far follows from that one expression: the metameric match a corner breaks, the interreflection gain, the wall that costs more than it looks. All of them read the patch against the room’s white, which is what a measurement made at the door does.

The half adaptation removes #

An observer standing in the corner does not read it against the room’s white. They read it against whatever is in front of them, which is the corner’s light, and the machinery for doing that is a diagonal gain in the cone basis.

That is not an approximation to the corner’s multiplier; it is the same kind of object. A change of illuminant multiplies a spectrum; a von Kries adaptation multiplies three cone responses. If the multiplier were a pure gain in the cone basis the cancellation would be exact.

It is not, and the difference is measurable. At an enclosure of six tenths the raw difference is 6.97 units and 2.57 remains after adapting to the corner’s own white — 63 per cent removed. At nine tenths, 11.69 becomes 4.25.

The remainder is the von Kries error, and it deserves a sentence of its own because it is the quantity the whole constancy literature is about. Chromatic adaptation is a hypothesis about what changed — it assumes the change was a light, and that a light acts as three multipliers on three cone responses. That is right about most changes of illuminant and exactly right about none of them, because a spectrum has eighty-one degrees of freedom here and a gain has three. What survives the transform is whatever part of the change did not fit into three numbers.

So the remainder is: the part of a spectral multiplication that no three-number gain can represent. It is the same quantity that shows up whenever this site adapts between two illuminants and asks how exact the transform was, and it is a property of the transform rather than of the corner.

assertAdaptationIsNotExact requires it to be non-zero at a deep enclosure and requires it to be exactly zero when there is no enclosure at all, because a computation that returned a small number for no corner would be reporting its own arithmetic.

The half it cannot touch #

The corner is brighter. The same lamp delivers 1.40 times the light there at an enclosure of six tenths, because light that would have left the scene comes back instead.

Adapting luminance is an argument to the appearance model, and it does something that no white-point transform does: raising it raises apparent colourfulness. That is the Hunt effect, it is derived on this site from the appearance model rather than quoted, and it is the term the corner adds that adaptation cannot remove.

Holding the white point fixed and moving only the adapting luminance by the corner’s lift gives 1.05 CAM16-UCS units at six tenths and 1.47 at nine tenths. That is not a large number and it is the wrong kind of number to be small: it is a change in appearance produced by geometry, with the stimulus’s chromaticity held exactly where it was.

So a patch in a corner is more colourful than the same patch on a flat wall, in a way that has nothing to do with the wall’s paint and would survive the walls being neutral.

Why the two halves are worth separating #

Because they behave differently under every intervention anybody would try.

Paint the walls neutral and the spectral half goes to nothing: a neutral multiplier is a gain, adaptation removes it completely, and the corner’s colour cast disappears. The brightness half stays exactly where it was, because a neutral wall bounces just as much light as a coloured one of the same lightness.

Make the corner shallower and both halves shrink together, which is why they were never separated: every measurement anybody had varied the enclosure, and the enclosure moves both.

Move the observer — stand in the room and look into the corner rather than standing in it — and the adaptation half changes completely, because what the observer is adapted to is now the room. The spectral half does not change at all.

There is a fourth intervention and it is the one that shows the terms are genuinely independent. Change the lamp’s level without touching the geometry — turn the dimmer down until the corner’s light matches what the flat wall had before — and the brightness term is removed while the spectral one is untouched. The patch is then the same colour as it was and less colourful than it was, which is a combination neither term produces on its own.

That last pair is the useful test. The same patch, the same corner, the same lamp, and an observer in two places: the spectral term is identical and the appearance term is not, which is a difference no measurement made with an instrument could produce.

The four readings, in one table #

The first row is exactly zero in all four columns and that is a check rather than a result: with no enclosure there is nothing to bounce, so a computation returning anything but zero would be reporting its own error.

The third column is lower than the second at every enclosure past four tenths, which looks like the appearance model doing more than a plain adaptation does — though the two columns are not in the same unit, which a section below has to sort out. And the fourth column is a term with the stimulus held completely fixed.

Sixty-three per cent is a constant, not a reading at one depth #

The 63 per cent is quoted at one enclosure and reads as a number that happens to fall out there. Dividing the second column by the first at every depth gives 0.379, 0.374, 0.369, 0.366 and 0.364 — so adaptation removes between 62.1 and 63.6 per cent of a corner’s colour cast, and the share does not depend on how deep the corner is.

A constant to within one and a half percentage points across a fivefold range of raw difference is not a coincidence and it says what kind of quantity the von Kries residual is. Deepening a corner raises the interreflection multiplier towards a higher power of the wall’s reflectance, which makes the cast larger; it does not much change the multiplier’s shape, and the shape is what decides how much of a spectral multiplication three numbers can represent. A gain fits the part of the multiplier that looks like three scalars and misses the rest, and the proportion it misses is a property of the paint rather than of the geometry.

That has a practical form worth more than the number at one depth. Measure the cast once, at any enclosure, and the share adaptation will remove is fixed. A corner twice as deep is twice as far off and no more forgivable; a wall painted a different colour changes the share and a wall painted the same colour in a deeper corner does not. It also means the second column carries no information the first does not — the two are one measurement and a constant — so a report that gave both is giving one, and the quantity worth publishing beside the cast is the share, once.

And the fourth column’s share falls, from 0.233 of the raw difference at a shallow corner to 0.126 at a deep one. The brightness term grows in absolute terms and shrinks as a proportion, which is the opposite behaviour from the adaptation residue and is the clearest evidence in the table that the two are genuinely different quantities rather than two readings of one.

Two of these columns are not in the same unit as the other two #

There is a comparison drawn under the table that the table cannot support, and it is worth catching because this site’s own house rule forbids it elsewhere.

The first two columns are quoted in the text as ΔE00 — a patch is ΔE00 6.97 from its flat-wall reading, and 2.57 remains. The fourth is quoted as CAM16-UCS — 1.05 CAM16-UCS units. The third is an appearance-model reading and belongs with the fourth.

So the sentence observing that the third column is lower than the second past four tenths is comparing a CAM16-UCS distance with a ΔE00 distance, and the site’s own caution about exactly that says the two should not be compared directly. The crossover it reports is not established: two quantities in different units crossing marks where their scale factors happen to cancel and says nothing about the phenomena.

The explanation offered for the crossing — that the appearance model’s account of a brighter surround lands nearer the flat-wall reading — may well be right, and nothing here refutes it. It is not supported by the table as it stands, and supporting it needs both columns in one unit.

The repair is cheap and it is the same repair either way round. Report all four in CAM16-UCS, which is the unit the appearance half is native to and which the first two can be recomputed in without changing what they measure; or report all four in ΔE00 and lose the appearance model’s own account of the surround. The first is better, and the reason is that the fourth column has no ΔE00 version at all — a change of adapting luminance with the stimulus held fixed produces no change in CIELAB, so the term this essay exists to isolate is invisible in the unit the other half of the table is quoted in.

That last observation is the finding rather than the complaint. The unit the first two columns are in cannot express the fourth column at all, which is a sharper way of saying what the essay says about adaptation: a colorimetric difference has no term for how much light there is, so the half of a corner that adaptation cannot remove is also the half a colour-difference formula cannot see.

What was computed, and how #

The corner is a cavity of stated enclosure with surrounding surfaces of a stated reflectance, and the incident light is the closed-form series. The patch is a coloured surface and the walls are a differently coloured one, so the multiplier is chromatic and the effect has something to act on.

Four readings are taken. Against the room’s white, which is a colorimeter at the door. Against the corner’s own white, which is a von Kries adaptation. Through the appearance model with the corner’s own white and its own adapting luminance, which is an observer standing in it. And through the appearance model with the white held at the room’s and only the luminance lifted, which isolates the brightness term.

The background is held constant across the four, deliberately. A real corner also changes what is around the patch — the walls are its background, and their colour and lightness are arguments to the appearance model too — so the numbers here are a floor rather than an estimate, and the background term is left out because putting it in would need a decision about how much of the field the walls occupy that nothing in the geometry supplies.

A deep corner under a tungsten lamp is the case that puts both changes at their largest at once.

The same argument, at a different scale #

A corner is the small case. The large one is a room, and it has the same two terms with the same division between them.

A white room raises the light level everywhere by the cavity gain, so everything in it is more colourful than the same objects under the same lamp outdoors — the brightness term, applied to a whole interior. A room painted a colour tints everything in it, and an occupant adapts to the tint and mostly stops seeing it — the spectral term, mostly removed.

That is why a coloured room is described as feeling a certain way rather than as making objects look wrong: the part an occupant would notice as an error has been adapted away, and what is left is a change in the apparent colourfulness of everything, which has no natural description as an error.

The corner is the useful case for a computation because both terms are large there and both are computable from one geometry. The room is the case anybody has an intuition about.

Where it stops #

The appearance model has no spatial structure, so the corner is treated as a viewing condition with a single adapting luminance and a single background rather than as a scene with a gradient in it. A real corner is darker at the crease and brighter away from it, and an observer’s adaptation is pooled over a region rather than taken at a point — which is a spatial adaptation question this site now has machinery for elsewhere and does not join here.

The Hunt term is computed with the model’s own dependence of colourfulness on adapting luminance, which this site has verified reproduces the effect at 2.24× over four decades. A lift of 1.40 is a small step on that curve, so the term is at the shallow end of a relation that is much better established at large ratios than at small ones.

And nothing here is a measurement of people in corners. It is what two models say when both are asked, and their disagreement about which term dominates is the finding rather than either number being a prediction about anybody’s report.

Who found it, and when #

Colour bleeding — the coloured light a wall throws on a neighbouring surface — was the effect that made radiosity worth computing in the 1980s, and it is still the demonstration image every renderer produces. That literature is entirely about the stimulus: solve for the radiance, render it, and the picture is right.

Colour appearance modelling grew up alongside it and never met it. Appearance models take a viewing condition as an input, and in every application of them the viewing condition is a room, a booth or a display — a thing a person sets up, not a thing that varies from one part of a scene to another.

Between them sits the observation that a scene has a field of viewing conditions: every point in it has its own local adapting luminance, its own local background, and its own local white. Rendering supplies all three and appearance modelling consumes all three, and the two are almost never connected, because a renderer’s output goes to a display and the display is where the viewing condition is applied.

Where the ladder goes next #

The obvious extension is the background term, and it is the larger of the two missing pieces. A patch in a corner is surrounded by the walls, so its background is their lightness rather than the room’s average, and the appearance model’s background parameter is one of the three that decides apparent contrast. Supplying it needs a solid angle rather than a form factor, which is a small addition to the geometry and would probably double the appearance term.

The sharper computation is the field. Solve a room, then compute a map of adapting luminances and local whites over its surfaces, and ask the appearance model what a patch looks like from each of several observer positions. That produces something no renderer currently outputs and no appearance model currently consumes: a picture whose colours depend on where in it the observer is standing, which is a fair description of what a room actually does.