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

Constraint — where it appears

A condition a searched-over parameter must satisfy, which decides where an optimum can be. An active constraint is one the answer sits on, so the answer reports the constraint; an inactive one leaves the answer reporting the objective instead.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

What a constraint costs is how far it pushes, in the directions that are seen. A scatter of every constraint imposed here on the nine free numbers. The horizontal axis is the length of the displacement from the optimum measured only in the six directions the objective can see; the vertical, on a logarithmic scale, is the excess cost that displacement actually carries. Requiring the basis to be the inverse of three realisable display primaries sits at the bottom left, at 0.068 and 0.022 ΔE00 — it removes three degrees of freedom and moves the answer almost nowhere. Requiring it to hit the three dichromat confusion points removes six and pushes 13 times as far, for 0.68. The vertical spread at similar horizontal positions is the part a count of parameters cannot predict.

A constraint is a direction and a distance

Four restrictions on the same nine numbers cost nothing, nothing, two per cent and seventy. How many parameters each removes predicts none of it. What does is the quadratic form evaluated along the displacement — and showing that it does means walking in towards the optimum rather than arguing at the edge, because at the edge the prediction is out by a factor of three.

limits · Limits Every worst surface sits on a number somebody typed. The region the test surfaces are drawn from, in its own two modulation coordinates: a square of allowed depths with a diamond inscribed in it, the diamond being the requirement that the two depths sum to no more than 0.7. The 14 marked points are the worst surface for each change of light in the adaptation census, found by search over the whole region. Every one of them lies exactly on the diamond, and every one is also at the brightest level the region allows — both declared constraints active, on all 14 rows, with no interior maximum anywhere. That is the opposite of what bounding the wall gave: there the worst case turned over at a band width of six nanometres because a narrow band returns too little light, which is physics. Here the worst case is a reading of two numbers. The one constraint that is about the world — a paint's excitation purity may not exceed 0.6 — is slack everywhere: the most saturated surface the region admits reaches 0.459.

Every worst surface sits on a declaration

Bounding the wall in a painted room produced a real worst case — the residual turns over at a band six nanometres wide because a narrower band returns too little light. Bounding the surfaces the residual is averaged over produces nothing of the kind, because all fourteen answers sit exactly on two numbers somebody typed and the one constraint that comes from the world never binds at all.

scene · Scene What a gamut charges a gradient, against what the relaxation's noise is. For each gradient, how much longer the best path that stays inside sRGB is than the free shortest path, as a percentage of the free one. The shaded band is the relaxation's own noise, measured by relaxing the same free path from two different starting points: 0.42 per cent at its worst. Every excess is inside it. So holding a gradient inside a display's gamut costs nothing measurable in the metric's own units, on any of these pairs — including the ones whose free path is outside at most of its points.

A gamut charges a gradient nothing

On red to green the shortest path happened to stay inside sRGB where the straight line did not, and that was called a coincidence of mechanism. Made into a measurement it is not a coincidence: constraining a path to stay inside the gamut costs less of the metric's own length than the relaxation's own noise, on every gradient tested — including the ones whose free shortest path is outside at fifteen of its twenty-one steps. The gamut's boundary and the metric's cheap region point the same way, and the price of the constraint is nothing.

matching · Gamut How steep an edge a band draws, and what it costs. A Gaussian absorption band of stated width produces a reflectance edge whose own width depends on how deep the band is, because the exponential saturates: where the absorbance is large the reflectance is already nought and the edge is over. A forty-nanometre band at an absorbance of 3 draws an edge 32 nanometres wide; at 12 it draws one 21 nanometres wide. Below an absorbance of 2.3 the band never reaches a reflectance of a tenth at all and has no edge in this sense. The dashed lines are each band's own width, which is the number a slope limit would have been given.

A sharp edge is bought with depth

A slope limit on reflectance was introduced as the weakest honest statement of a pigment's bluntness, with a band-shape limit named as the stronger version to be written later. Written, it is not stronger. Three absorption bands none narrower than forty nanometres reach further than a forty-nanometre slope limit in 94 of 154 directions of the object-colour solid, because a band's width and the width of the reflectance edge it draws are different quantities — and what converts one into the other is how much colorant is in the film.

limits · Limits The same gradients, priced by a penalty and by a projection. Each row is one gradient held inside a coated press from six starts. The pale dots are the penalised relaxation — a free step, with a price for leaving the press — and the dark ones are the projected relaxation, which takes the free step and then moves each point to the nearest printable colour. Across is how much longer than the free path each result is, logarithmic. On the 10 gradients whose straight line leaves the press, the penalty leaves 9 starts at more than twice the free length and the projection leaves none. The projection's best route costs a median 0.02% against the penalty's 0.13%, and its spread across starts is 0.95% against 132.5%.

A projection has no reason to detour

Holding a gradient inside a press by penalising the excursion turned the gamut's price from a number into a search: on five of ten crossing gradients some starting point leaves the relaxation trapped at more than twice the free length, and the spread across six starts runs to 271 per cent of the free path. Replacing the penalty with a projection — take the free step, then move each point to the nearest printable colour — leaves no trap on any gradient and a spread of 0.06 to 2.4 per cent.

matching · Gamut 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.

matching · Gamut

Named alongside it

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

CIELABColour differenceGamutGamut mappingTrade-offBoundCensusChromatic adaptationOptimisationProcess inksReflectanceAbsorption

All concepts