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

The Donaldson matrix — where it appears

The full description of a fluorescent surface — what leaves at each wavelength for light arriving at each — of which an ordinary reflectance is only the diagonal. Measuring one needs two monochromators rather than one, which is why almost every instrument reports the diagonal and folds the rest of the matrix into it at whatever weight its own lamp gave.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

A sample with two reflectance curves, and neither below one. The apparent reflectance of an optically brightened sample, measured under D65 and A. It exceeds 1 — the shaded band — which no reflector can do: more light leaves at these wavelengths than arrives at them, because the sample absorbs in the violet and re-emits in the blue. And the two curves differ, so the sample has no single reflectance to store. The effect drawn here is a floor: most of the excitation band lies below 380 nm, outside the range computed here.

Some paper is brighter than white

The reflectance model assumes light leaving at a wavelength arrived at that wavelength. A fluorescent sample breaks the assumption, has no reflectance curve at all, and is in almost every sheet of white paper sold.

light · Light A sample with two reflectance curves, and neither below one. The apparent reflectance of an optically brightened sample, measured under D65 and A. It exceeds 1 — the shaded band — which no reflector can do: more light leaves at these wavelengths than arrives at them, because the sample absorbs in the violet and re-emits in the blue. And the two curves differ, so the sample has no single reflectance to store. The effect drawn here is a floor: most of the excitation band lies below 380 nm, outside the range computed here.

A surface that is not a multiplication

Every argument here about what light does to a surface begins by multiplying two spectra together. A surface with a brightener in it takes light at one wavelength and returns it at another, so it is a full operator rather than a diagonal one — and it does not have a reflectance at all.

scene · Scene A fluorescent sample is a matrix, and a reflectance is only its diagonal. The Donaldson matrix of a brightened sheet: how much light leaves at each wavelength for light arriving at each wavelength. A reflecting surface has entries on the diagonal and nowhere else, which is exactly the statement that light leaving at 440 nanometres arrived at 440. The block off the diagonal is the fluorophore — it takes light between about 305 and 420 nanometres and returns it between 400 and 500, wherever in that band it was absorbed, which is why the block is a rectangle rather than a smear along the diagonal. A spectrophotometer that reports a reflectance is reporting the diagonal and folding the block into it at whatever weight its own lamp happened to give.

A reflectance is a diagonal

A reflecting surface returns light at the wavelength it arrived at, so its whole description is one number per wavelength. A fluorescent one returns it somewhere else, so its description is a square matrix — and the curve every instrument reports is that matrix's diagonal with the rest of it folded in at whatever weight the lamp happened to give.

light · Light Two sheets with the same reflectance and two different colours. A brightened sheet and a dyed one built to match it under an instrument with no ultraviolet. Under that instrument the pair agrees to ΔE00 0.00, which is a rounding and is true by construction — the dyed sheet's reflectance is the curve the brightened one measured. Under an instrument that includes the ultraviolet they are 7.1 apart, and under daylight 10.6. This is not ordinary metamerism: the two sheets do not differ in reflectance anywhere the eye can see, so no change of light puts them back together and no adaptation removes the difference. One of them is a curve and the other is an operator.

Two sheets that match until the window

Ordinary metamerism is two reflectances that agree under one light and not another, and it can always be undone by putting the first light back. A dyed sheet and a brightened one have the same reflectance everywhere an eye can see, agree exactly under any lamp with no ultraviolet, and separate by ten units under daylight — and no change of light puts them back together.

scene · Scene 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 A radiance factor, split into the part that was reflected and the part that was not. The two components of what leaves a coated press stock under M₁ — D50 including its ultraviolet. The lower band is the reflected component, which is what a reflectance curve means and is everything a reflectance-based model can hold. The band above it is light emitted at wavelengths it did not arrive at, and its total is decided by how much ultraviolet the source had rather than by anything about the sheet's colour. The line at one is the boundary a reflecting surface cannot cross; the sum reaches 1.15 at 430 nanometres.

The ultraviolet is half the product

A fluorescent sample's departure from the reflectance model is exactly proportional to the ultraviolet the lamp carries — to twelve figures, over a range that includes zero. It is the only one of the round's four departures that really is a product of two magnitudes, and the reason is a rule about how molecules relax rather than anything about arithmetic.

light · Light

Named alongside it

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

FluorescenceOptical brightenersReflectanceBispectralIlluminantRadiance factorAssertionMarginalisationUltravioletBidirectional reflectanceBilinearityIntegration

All concepts