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

Subsurface scattering — where it appears

Light entering a material, scattering many times inside it, and leaving at a different place from where it arrived. It makes a surface's response a kernel over the plane rather than a number at a point, which is why a translucent sample reads differently through different apertures.

Named by 12 essays across 8 fields — each of them below, with the objects they name alongside it.

The six arguments a surface's response has, and the one this model keeps. A surface's response to light is a function of six arguments: the wavelength, direction and place the light arrives with, and the wavelength, direction and place it leaves with. The model every colour here is computed from keeps one number per wavelength, which means it takes the diagonal of the first pair, integrates the second away, and assumes the third pair equal. Each departure drawn here restores one of them. The fourth departure is not on the diagram: the wavelength grid is the range of the index that was kept rather than an index that was dropped, which is why it is the cheapest of the four to fix and was still not fixed.

The model has six arguments

Every colour computed here is an integral of a reflectance against a light against three curves, and a reflectance is one number per wavelength. A real surface's response is a function of six arguments — the wavelength, direction and place light arrives with, and the three it leaves with — so the model keeps one of them, takes a diagonal, integrates two away and assumes two more equal.

limits · Limits Eight conditions under which the model equation is exact, and how exact each one is. Each of the four departures vanishes if either of its two factors is empty, which is eight conditions. The axis is logarithmic in the residual that is left when the condition is imposed. Three of the eight are identities: the fluorophore's loading is zero so the emitted term is an empty sum, and a Lambertian surface or a uniform field makes the pairing's second argument identically zero. The other five are limits — a Gaussian excitation band has no edge, an opaque sample still has a kernel a few microns wide, a four-metre aperture is still finite, and the observer is small rather than absent at 380 nanometres. Each limit is drawn with the sequence its residual falls along as the condition is pushed, because a small number is not evidence of a limit and a falling sequence is.

Either factor being zero

Four different departures from the colour integral turn out to have one algebraic form — each is an inner product of something the sample does that the model has no slot for with something the light does that the model assumed away. Either factor being zero makes the departure exactly zero, and the sizes of the two factors decide neither how large it is nor which way it goes.

limits · Limits 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

Light that enters a translucent material does not come back where it went in. It scatters some thousands of times and leaves a few millimetres away, so what the surface has is not a reflectance but a function of distance — and the reflectance the model wants is that function's integral over the whole plane, which no instrument ever collects.

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 4-millimetre radius, which is about what a hand-held spectrophotometer has. At that aperture coated paper reads 98 per cent of its own reflectance and candle wax reads 48. 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.

An aperture is a filter

A measuring aperture throws away the light that came back outside it, and the light that comes back furthest is the light at the wavelengths the sample absorbs least. So an aperture does not attenuate a translucent sample evenly — it attenuates its peaks more than its troughs, which is a filter whose transmission curve the sample itself decides.

applied · Delivery 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

A piece of pale marble measured through a wide aperture is faintly yellow. Measured through a narrow one it is faintly blue, and between the two there is an aperture at which it is exactly neutral. Nothing about the stone changes; the aperture is a filter with a colour, and what decides that colour is the size of the particles rather than the pigment between them.

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 What is read at each distance from the edge of a lit region. Three materials under a half-plane of light, with the boundary at the centre of the horizontal axis and the lit side on the right. The vertical axis is the radiance leaving the surface as a share of what it leaves far inside the lit region. On the unlit side the sample is emitting light while receiving none, so the ratio the model calls a reflectance has a zero denominator there. The distance over which the curve runs from a tenth to nine tenths is 0.21 millimetres on coated paper and 5.1 on pale marble — which is the width of the neighbourhood a point's colour is decided by.

A pixel has an aperture too

A camera photographing a translucent object has the same two discs a spectrophotometer has — one lit, one looked at — and gets them the other way round. Its illumination covers the whole scene, so the flat colour of a translucent surface comes out exactly right at any magnification, and the error moves entirely into the edges.

imaging · Capture 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 Two slabs with one reflectance, and two colours through an aperture. Two constructed media whose bulk reflectance agrees at every wavelength to fifteen figures, and whose diffusion lengths differ by a factor of four. The upper curve is that shared reflectance — both slabs lie on it exactly. The two patches on the right are what a 4 millimetre radius returns from each, and they are 6.3 ΔE₀₀ apart. This is a metamerism with no observer in it: the two samples are the same colour to anybody under any light, and the instrument separates them because it is measuring a kernel through a hole rather than measuring a reflectance.

A pair the aperture separates

Two constructed slabs with the same reflectance at every wavelength, to fifteen figures — the same colour to any observer under any light — and 6.25 ΔE₀₀ apart when measured through a four-millimetre aperture. It is a metamerism with no observer in it, no illuminant in it, and no spectral difference to construct it from.

matching · Gamut What is read at each distance from the edge of a lit region. Three materials under a half-plane of light, with the boundary at the centre of the horizontal axis and the lit side on the right. The vertical axis is the radiance leaving the surface as a share of what it leaves far inside the lit region. On the unlit side the sample is emitting light while receiving none, so the ratio the model calls a reflectance has a zero denominator there. The distance over which the curve runs from a tenth to nine tenths is 0.21 millimetres on coated paper and 5.1 on pale marble — which is the width of the neighbourhood a point's colour is decided by.

Two blurs before the eye's own

A translucent object arrives at the eye already blurred, by a kernel that is a property of the material rather than of the optics. On every material in this collection's table that blur is coarser at ordinary reading distance than the finest detail the eye can resolve — so the softness of marble or skin or wax is not a failure of vision, it is the object.

eye · Cones Two slabs with one reflectance, and two colours through an aperture. Two constructed media whose bulk reflectance agrees at every wavelength to fifteen figures, and whose diffusion lengths differ by a factor of four. The upper curve is that shared reflectance — both slabs lie on it exactly. The two patches on the right are what a 4 millimetre radius returns from each, and they are 6.3 ΔE₀₀ apart. This is a metamerism with no observer in it: the two samples are the same colour to anybody under any light, and the instrument separates them because it is measuring a kernel through a hole rather than measuring a reflectance.

The appearance model has no slot for it

Two slabs with identical reflectance and different kernels come out of CIECAM16 with identical lightness, chroma and hue — to six decimal places, because the model's input is a stimulus and their stimuli are the same. Translucency is a recognised appearance attribute with no correlate in any appearance model, and the reason is structural rather than an oversight.

brain · Appearance 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

Named alongside it

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

ApertureMarginalisationTranslucencyDiffusion lengthPoint spread functionBilinearityEdge lossMeasurement conditionMeasurement errorBidirectional reflectanceChromaThe Donaldson matrix

All concepts