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

Reciprocity — where it appears

Helmholtz's result that a surface's response is unchanged when the incoming and outgoing directions are exchanged. It is what makes an integrating sphere's reading a property of the sample rather than of the instrument, and it is a theorem rather than an approximation.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

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

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.

scene · Scene 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 The share of a sample's reflectance an aperture recovers, by how wide it is. Six materials, and the fraction of each one's true reflectance that a measurement recovers through an aperture of the stated radius. The horizontal axis is logarithmic in millimetres; the vertical is a share, so 1.0 is the whole of it. The dashed line is a 8-millimetre radius, which is about what a hand-held spectrophotometer has. At that aperture coated paper reads 99 per cent of its own reflectance and candle wax reads 66. Every curve approaches one from below and none of them reaches it: the kernel's tail is what is being cut, and it falls as one over the aperture rather than exponentially.

Either disc can be the wide one

Every standard on translucent samples says to illuminate a larger area than is measured, and explains it by saying that light leaks out of the lit spot. That is true and it is not the reason, because the instruction works equally well the other way round — measuring a larger area than is lit gives a reading just as exact. The error is a product of two apertures and either one being wide kills it.

applied · Delivery 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 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 grey finish sends the lamp's light orders which rooms it colours. The thirty rooms of the census, each placed by one number computed from its box and lamp alone — how much more of the lamp's light a satin finish on the grey faces sends to the painted faces than their matt return did — and by how many of 72 paints a finish on the grey faces makes more colourful. Over the twenty-four rooms with a wall or the floor painted the rank correlation is 0.92, and the vertical line at zero sorts them: right of it every room gains for 29 paints or more, left of it for 22 or fewer. The six rooms with the ceiling painted, drawn hollow, all sit left of the line and five of them gain at every paint.

A grey finish mirrors the lamp onto the paint

A satin finish on a room's grey faces makes the room more colourful in almost every arrangement, and the explanation offered was that it re-weights each grey face towards the paint it sees at a slant. That number, computed from the geometry alone, orders the rooms at a rank correlation of 0.24. The number that orders them is the same slant pointed the other way and lit: how much of the lamp's light the finish sends to the paint. It orders the rooms with a wall or floor painted at 0.92, and its sign sorts them without an exception.

scene · Scene The painted census's counts with one patch a face and with the faces divided. For each of the thirty rooms — six boxes, each painted five ways — how many of 72 paints a satin finish on the grey faces makes more colourful with one patch a face, as the census solved it, and with every face divided five times a side (six for the two tall boxes). 12 counts do not move; 9 move by twenty paints or more, shaded — darker where the divided room gains more. Some paint gains in 28 rooms both ways, and every paint in 13 with one patch and 17 divided.

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.

scene · Scene

Named alongside it

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

RadiositySpecularBidirectional reflectanceForm factorInterreflectionModelling assumptionQuadratureAlbedoCensusConvergenceFresnelLambertian

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