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

Every essay — page 27

Page 27 of 40, continuing through the fields in the same order.

What light is What the eye does Matching and measuring Difference and uniformity What the brain does What a scene does What a camera does Where the model breaks What it takes to deliver it

SeriesObserversNamed objectsRefutationsSearch

Matching and measuring

CIE XYZ, chromaticity, and gamut — a system for predicting when two lights match, frequently mistaken for a system that predicts how they look.

How much of a coated press is left at each margin inside its boundary. The share of a coated press's printable volume in CIELAB that lies at least a given distance inside its boundary, for margins from half a unit to 32. A margin of one unit keeps 88%, two keep 80%, four 66% and eight 45%. At every margin up to 30 what is left is a single connected piece; at 31 units, with 0.14% of the volume left, it first splits, into a core of 623 cells and 2 fragments of one or two cells — the deepest point is 32.7 units inside, so what splits is the last crumb of the core, not a waist.

A margin costs a press its corners

A gradient held inside a coated press by projection was predicted to tear if the press were first shrunk by a safety margin, at any pinch narrow enough for the shrinking to cut. The press has no such pinch: shrunk by any margin up to thirty CIELAB units it stays in one piece, and projected routes on it are neither trapped nor torn. What a margin costs is concentrated at the corners — the solid yellow moves nearly four times the margin to get inside, like the tip of a 31-degree spike.

7 figures

What a scene does

A reflectance measured on a patch is a patch. Put it in a room and the light bounces, and a bounce is a multiplication — so shadows carry their own illuminant, highlights carry the lamp, and two surfaces that matched go on matching only until the second bounce.

A spectrum after 2 bounces off the same surface. The lamp's spectrum at the top, then the same spectrum multiplied by a reflectance peaking at 530 nm once for each bounce. Interreflection is elementwise multiplication, so light that reaches the eye by the long way round carries ρ raised to the number of surfaces it met. Each row's swatch is drawn at fixed luminance so only the chromaticity changes, and the distance from the D65 white point, printed at the right, rises from 0.000 to 0.235. The spectrum narrows every time, which is why a room painted in one colour is more saturated in its corners than on its walls.

A bounce is a multiplication

Colorimetry multiplies an illuminant by a reflectance once and integrates. A surface in a room is lit by every other surface the lamp reached first, so the spectrum arriving at the eye has been multiplied several times — and the second multiplication is where the whole apparatus of matching starts to come apart.

8 figures A metameric match that a corner breaks. Two reflectances with identical XYZ under D65 — metamers, matching to ΔE00 = 5.4e-14, which is the numerical floor rather than an approximation. On a flat wall the light meets one of them once and the two patches are the same colour. In a corner, a fraction 0.200 of what leaves the surface returns to it, so part of what reaches the eye carries ρ twice — and the match fails by ΔE00 = 1.53. A metameric match is an identity between three integrals that are linear in ρ, and there is nothing linear left after a second bounce. The geometry, not the light and not the paint, is what breaks it.

Two paints that stop matching

A metameric match is an identity between three integrals that are linear in reflectance. A second bounce carries reflectance squared, and no linear identity survives being squared — so two paints certified identical on a flat chart come apart in a corner, by an amount the geometry decides and the colorimetry cannot express.

8 figures The two components of daylight, computed from one radiator and one scattering law. A 5800 K radiator through the atmosphere at air mass 1. The direct beam is the radiator times e^{−τm}; the sky is what that extinction removed, so the two are complements and neither needs its own model. The sky is steeply blue because τ ∝ λ⁻⁴, and a surface in shadow is lit by that component alone. Open ground and shadowed ground therefore sit under two illuminants differing by ΔE00 = 21.4 — which is why a photograph of snow has blue shadows and why no single white balance fixes both halves of it.

A shadow has its own illuminant

Outdoors there are two lights, not one. The direct beam is a radiator reddened by the atmosphere; the sky is precisely the power that reddening removed. A shadow is lit by the second alone, so shadowed ground and sunlit ground sit under illuminants 21 units of ΔE apart — before any surface, any eye or any opinion is involved.

9 figures A glossy surface returns two spectra, and only one of them is the paint. The dichromatic reflection model, computed rather than assumed. Light that enters a dielectric binder, scatters off pigment and comes back carries the reflectance — the body component, ΔE00 = 33.7 from the lamp. Light reflected at the interface never entered, so it carries the lamp's spectrum with only Fresnel's slight dispersion on it: ΔE00 = 1.06. The interface term is computed from Fresnel's equations on a Cauchy index at 55°, so its near-neutrality is a result here rather than an assumption. This is why a highlight is the one region of a photograph that tells a white balancer what the light was, and why removing highlights removes the evidence.

The highlight is the lamp

Light reflected at the surface of a glossy material never entered it, so it never met a pigment. Fresnel's reflectance varies by 5.5% across the visible band and the paint underneath varies by a factor of twenty — which makes a highlight the one region of a photograph that reports what the light was rather than what the object is.

10 figures How far three channels drift from eighty-one, per bounce. One room, one geometry, one reduction to three channels, and the only thing changing is how many bounces of the Neumann series are kept. At one bounce the two agree to 2.5e-13 — the only reflectance in that path is the floor's, which is flat, and a flat reflectance is one of the few three numbers carry exactly. Every bounce after it multiplies another non-flat reflectance into the spectrum, and three numbers cannot carry a product they were never given the factors of. The curve levels off at ΔE00 = 2.38 because the light has run out, not because the disagreement has.

Rendering in three numbers

Almost every renderer ever shipped bounces red, green and blue rather than a spectrum. The error that costs is exactly zero at the first product and grows at every one after it — because three numbers cannot carry a product they were never given the factors of, and each bounce is another product.

9 figures The interreflection gain of a room, band by band. A closed cavity of reflectance ρ returns 1/(1 − ρ) times the light that entered it, and because that is computed per band it amplifies the wall's colour along with its brightness. The wall here is only 5% off neutral — a paint anybody would call white — and at albedo 0.9 the room's white point has moved by ΔE00 = 50.7 against the lamp it was lit with. The gain is a geometric series, so the last few percent of albedo cost far more than the first: 1.64× at ρ = 0.4 against 8.41× at ρ = 0.9.

What a white wall costs

A closed room returns 1/(1−ρ) times the light that entered it, computed band by band — so a paint that is five percent off neutral becomes a strongly coloured illuminant once the room has finished bouncing. The gain amplifies the tint along with the brightness, and the last few percent of albedo cost far more than the first.

7 figures 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.

8 figures Every face of the solved room. The red room after the transport is solved in all 81 bands. Only the ceiling emits; the other five faces are lit entirely by what the ceiling and each other send them, so their colour is the lamp multiplied by every reflectance along every path that reached them. The ceiling itself comes out 1.05× brighter than it emits, because a closed room returns light to its own source. The two chromaticities under each swatch are the spectral solve and the three-channel one, and the faces furthest from the lamp — the ones the light reached by the most bounces — are where they disagree most.

The room is the illuminant

Colour bleeding is usually described as an aesthetic phenomenon of rendered images. It is better described as a measurement. In a room with one lamp, five of six surfaces emit nothing at all, so their light is entirely a product of other surfaces' reflectances — and the bleeding saturates rather than running away, for a reason worth deriving.

7 figures Mixing two paints and stacking two filters are different operations. The same two reflectances combined two ways. Stacking them as filters multiplies the transmittances, which is right for gels in front of a lamp and wrong for pigment stirred into pigment: a stirred mixture is one scattering layer, not two in series, and light meets whichever particle is nearest rather than passing through both. Kubelka–Munk handles it by moving to K/S = (1 − R)²/2R, in which absorption and scattering add by concentration, and inverting afterwards. The two answers differ by ΔE00 = 15.0, and the filter model is the darker of the two because it charges every photon for both pigments.

Paint is not a filter

Stacking two filters multiplies their transmittances. Stirring two pigments together does not multiply their reflectances, because a mixture is one scattering layer rather than two in series — light meets whichever particle is nearest. Treating the two as the same operation is a 15-unit error, and which way it errs turns out to depend on whether the comparison holds the amount of pigment fixed.

6 figures Why blue and yellow make green. 7 mixtures between a blue and a yellow pigment, mixed in Kubelka–Munk — K/S summed by concentration and inverted back to reflectance — and plotted against the straight line joining the two endpoints. The path bows towards green by 0.099 in chromaticity, and the reason is in the spectra rather than in the eye: the blue reflects below about 520 nm and the yellow above about 500, so the only band both return is the overlap between them. Mixing lights adds spectra and lands on the chord; mixing pigments intersects them and does not.

Why blue and yellow make green

The oldest fact in colour, and the usual explanations are wrong. It is not because green sits between blue and yellow, and it is not a fact about the eye at all — it is that the only band both pigments return is their overlap, and the overlap of a blue and a yellow reflectance is green. Computed, the mixing path bows away from the straight line by a measurable amount.

7 figures The same medium at six path lengths. Beer's law at 6 depths of one absorbing medium. The absorption coefficient is a single spectrum and the only thing changing is how far the light travelled, yet the patches differ in hue by 7.8° as well as in lightness — because absorption is exponential in depth and the observer is linear, so the bands that survive at d = 8 are not a scaled copy of the ones that survive at d = 0.25. Path length belongs to the geometry, not to the substance, which is why this sits in a field about scenes.

The colour is in the thickness

Transmittance is exponential in path length and the observer is linear, so doubling the depth of an absorbing medium squares the transmittance rather than halving the colour. A translucent object therefore has no one colour — its thin edge and its thick middle are different spectra of the same substance, and the hue moves between them.

9 figures