A third of the appearance box is no surface

A colour specification written in an appearance model’s coordinates names what something should look like — a lightness, a chroma, a hue — and leaves how to make it to the supplier. The model’s inverse is a closed formula, so every such specification turns into three tristimulus values without complaint. An appearance is not always a stimulus asked how often those three numbers are not a light at all, found 10.8 per cent of the space, and recorded that the answer was a lower bound, because the test it used was the easy one.

A third, not a tenth #

About a third of the space an appearance is specified in has no reflecting surface under it, and near the top of the lightness scale most of it does not.

• Over a lattice of 10,488 lightness, chroma and hue triples under daylight, 65.7 per cent are colours a surface can have. 23.5 per cent are lights no surface can return, and 10.9 per cent are not lights at all.
• The surface share peaks at 82 per cent between lightness 40 and 50 and falls away at both ends: 56 per cent at lightness 10, 38 at lightness 90, 23 at lightness 95.
• At lightness 50 a surface can reach a mean chroma of 94 around the hue circle and sRGB 67. At lightness 90, 41 and 25.
• The lamp moves it by a few points. The surface share is 65.7 per cent under daylight, 62.5 under tungsten, 62.3 under a three-emitter LED and 59.4 under a phosphor white LED.

What a surface cannot do #

A light can have any spectrum. A reflecting surface can only return a fraction of the light that falls on it, and that fraction is between nought and one at every wavelength. The set of tristimulus values reachable by some reflectance between nought and one, under a stated lamp and observer, is the object-colour solid, and no surface can be that colourful is about its boundary.

That boundary has a precise description. For any direction in tristimulus space, the reflectance that goes furthest in that direction is one at every wavelength where the observer’s response points that way and nought everywhere else. Those all-or-nothing reflectances are the optimal colours, and the largest value any surface reaches along a direction is the lamp’s power integrated over the wavelengths where that direction’s response is positive. A tristimulus value that exceeds it along even one direction is outside the solid — not approximately, but by proof.

The solid is convex, so the converse holds: a point that exceeds no direction’s bound is inside, and some surface has that colour. The test is therefore exact in both directions, and it is the test used here, over 2,400 directions refined locally wherever a point comes near the boundary. It is checked where the answer is known. The perfect white diffuser sits on the boundary to 6 × 10⁻¹⁶, a fifty per cent grey is well inside, and a white two per cent brighter than the diffuser is outside.

The census, row by row #

The lattice is the one the easier census used: lightness from 5 to 95 in fives, chroma from nought to 110 in fives, hue all round in fifteen-degree steps, inverted under daylight with the model’s observer adapted to it. Each point is sorted into one of three kinds.

Not a light at all means one of the three tristimulus values is negative, or the luminance exceeds the white’s. That is the easy test, and 10.9 per cent of the lattice fails it — 10.7 for a negative value and 0.2 for exceeding the white.

A light, but not a surface means the tristimulus value is a possible light and lies outside the object-colour solid. That is 23.5 per cent of the lattice, more than twice the first kind. A surface colour is the rest, 65.7 per cent.

So of the appearances the easy test passed, about a quarter — 23.5 out of 89.1 — still cannot be made from a reflecting material. A specification that checked only whether its coordinates correspond to a light would have accepted all of them.

Light at the top, darkness at the bottom #

The three kinds are not spread evenly over the lightness scale, and the shape says what each boundary is made of.

At the dark end the easy test does most of the rejecting. At lightness 5, 44 per cent of the row is not a light at all and only 14 per cent is a light that no surface can make. A very dark colour with high chroma needs one cone signal far below the others, and near black the inverse runs out of room and returns a negative value before it runs out of reflectance.

At the light end the surface bound does most of it. At lightness 90 only 2 per cent of the row is not a light, and 61 per cent is a light no surface can return. A very light colour needs almost all the light at almost every wavelength, and a surface that returns almost all the light at almost every wavelength is nearly white, so it cannot also be very colourful. Lightness and chroma trade against each other for surfaces in a way they do not for lights, and the trade is the shape of the solid.

Between the two, from lightness 30 to 60, both bounds are relaxed and a surface can have four fifths of the row.

The top rows are where paper lives #

At lightness 95 only 23 per cent of the row is a colour a surface can have, and that row is not a corner of the space nobody uses. It is close to the lightness of a good printing paper and of a white wall, and every pale tint printed on a paper that is the white point of its own reproduction lives in the top rows.

A pale tint asks a surface to return nearly all the light at nearly every wavelength and to withhold a little somewhere, and the solid says how little can be withheld while the surface stays that light. A pastel is the kind of specification most likely to name a colour no surface can have, and the reason is not the quality of any ink but the lightness, which leaves the surface almost nothing to withhold. An ink on paper does worse than the solid, since its reflectance cannot exceed the paper’s own.

The materials that escape the bound at the top do it by not being reflectors. Some paper is brighter than white because an optical brightener absorbs ultraviolet and re-emits it as visible blue, returning more light at those wavelengths than arrived there. That puts a fluorescent sheet outside the solid by construction, and it is the commonest way a very light, visibly tinted colour gets past the bound.

Three ceilings #

The row shares hide where round the hue circle the bounds bite, and the difference between hues is large.

At lightness 50 the largest chroma that is a light at all is off the scale through the blues and purples and about 80 through the oranges and yellows. A surface reaches a mean of 94 around the circle — about 120 in the reds and about 75 in the blues and purples, where the light ceiling is far above it. sRGB reaches a mean of 67, at or below the surface ceiling at every hue.

That last fact is worth stating plainly because it is not obvious. Every colour sRGB can show at a given lightness is a colour some surface can have — the displayable share of the lattice is 43.8 per cent, and every one of those points passes the surface test. A wider-gamut display is a different matter, and a laser projector’s primaries lie outside the solid.

At lightness 90 the picture changes shape. A light with that lightness can still be very colourful through the blues and reds. A surface cannot: its mean ceiling is 41, and through the blues and purples it is under 20. The exception is around hue 120, among the yellow-greens, where a surface still reaches a chroma of about 100, because a very light colour there needs to withhold only a little light, from the ends of the spectrum the eye weights least. Pale purple is the opposite case and close to a contradiction in a material: a surface that returns nearly all the light cannot also withhold the middle of the spectrum, which is what purple is. A colour order system’s lightest pages are where that runs out first.

The lamp moves it a little #

The solid depends on the lamp, because the lamp is the light a surface returns a fraction of, and the model’s observer adapts to the lamp too. So the census has to be repeated under each lamp to say how much of the result belongs to daylight.

Under a tungsten lamp the surface share is 62.5 per cent, against 65.7 under daylight, and more of the dark end fails the easy test — 15.2 per cent of the lattice is not a light at all. At lightness 90 a surface can have 35 per cent of the row, against 38.

Across four lamps the surface share runs from 59.4 per cent under a phosphor white LED to 65.7 under daylight. The two LEDs, with their spiky spectra, give surfaces less room than the two smooth sources, because a lamp with holes in its spectrum gives a surface fewer wavelengths to return and the solid is correspondingly thinner. But the spread is six points on a bound that removes thirty-four. The bound belongs to reflecting material, not to any lamp.

A screen that shows what no surface is #

Every colour sRGB shows is a colour some surface can have, so an approval on an sRGB screen is at worst an approval of something makeable. A display with narrower primaries loses that guarantee, and the reason sits on the solid’s edge.

The only surface colours with the chromaticity of a single wavelength are those of zero luminance. To return one wavelength and nothing else a reflectance must be a spike of no width, and a spike of no width returns no light. So a laser primary, at any luminance a display uses, is outside the solid, and so is a band of colours around it. A proof on a display with narrow primaries can show a saturated green that passes the easy test, lies well inside the display’s own gamut, and has no reflecting surface under it anywhere.

That is a second way a narrow-primary proof misleads, and it is independent of the first. A soft proof is exact for one reader is about readers disagreeing over a colour both media can show. This is a colour one medium shows and the other cannot make, for every reader alike. The first is reduced by proofing on broader primaries; the second can be caught before anybody looks, by running the support-function test on the proof’s own colours.

What a specification should check #

Three consequences, and the third is the one that saves money.

A specification for a material should be tested against the solid, not only against the inverse. A coordinate triple that inverts to a positive light has passed a test that almost every surface specification passes and that a quarter of impossible specifications also pass. The support-function test is one loop over directions and one comparison, and it proves impossibility when it fails.

A specification written near the top of the lightness scale is the most likely to be impossible. At lightness 90, 61 per cent of the chroma–hue plane is a light no surface can return. A brand colour specified as a pale, saturated purple can be displayed on a wide-gamut screen, approved there, and turn out to be unmakeable in ink, paint or dye — and nothing in the coordinates says so.

And an approval on a screen does not certify a material. sRGB sits inside the solid, so an sRGB approval is at least a makeable colour. A wider-gamut display does not, and a white that is not a reflectance already found real materials outside the solid for the opposite reason — fluorescence, which returns more light at some wavelengths than arrives there. The solid bounds reflecting materials only.

What was computed, and how #

The lattice is inverted through CIECAM16 with the white set to the lamp’s own white on the 1931 observer, an adapting luminance of 100 candelas a square metre and a twenty per cent background. Each resulting tristimulus value is tested first for negativity and for luminance above the white’s, then against the object-colour solid under the same lamp.

The solid’s support function along a direction is the lamp’s power times the positive part of the direction’s dot product with the observer’s three functions, summed over wavelength and normalised so the white has luminance 100. A point is outside if its dot product with any direction exceeds that bound. The directions are 2,400 points spread evenly over a sphere; a point that comes within three per cent of a bound has the direction refined by a local search before it is called a surface, because a finite set of directions can only understate how far outside a point is.

The ceilings are found by bisection on chroma at each hue, against each of the three tests.

Where the measurement stops #

The solid assumes a reflectance can be any function between nought and one, including the all-or-nothing spectra at its boundary that no real pigment has. Real colorants are smooth, so the solid of makeable colours is smaller than this one, and the third measured here is a lower bound on the share of the space no real material can reach.

Fluorescent materials are outside the model and outside the bound; so are retroreflective, metallic and structural colours, whose reflectance depends on geometry.

And the model’s inverse is being asked about appearances far outside the data it was fitted to. Very light, very colourful appearances are not in the corresponding-colour data; the census counts where the formula lands, not what an observer would report.

The habit #

The habit is about a necessary condition standing in for a sufficient one.

A test that is easy — is the number positive, is it below the maximum — is often the only one run, and it rejects the obviously impossible. The condition that actually decides feasibility is usually harder, and the gap between the two is not a detail: here it is more than twice the size of what the easy test found.

The move is to find the constraint the physical object obeys and test against it directly. For a reflecting surface the constraint is that reflectance lies between nought and one at every wavelength, and it has an exact test.

The failure mode is to report the easy test’s result as the answer. A lower bound is not a small number; it is a number that says nothing about how much larger the truth is.

Who noticed it first #

The object-colour solid and its optimal colours are due to Schrödinger, who described them in 1919, and to MacAdam, who computed their boundaries in 1935. The support-function characterisation — that the furthest reflectance in any direction is all-or-nothing — is the same result stated as a convexity argument.

That an appearance model’s coordinate space is much larger than the set of surface colours is implicit in every discussion of gamut boundaries in appearance space. The share, measured exactly under several lamps on the lattice a specification would use, does not appear in the sources consulted here.

Still open: the solid of smooth reflectances #

The bound above admits spectra no material has. A bound for smooth reflectances — spectra with a limited slope, or spanned by a few basis functions as measured reflectances are — would be tighter, and the share of the appearance space inside it is the number a paint or ink specification actually needs. The same test takes any convex set with a computable support function; a smooth-reflectance set is convex, and its support function is an optimisation rather than a sum.