Optimisation — where it appears
Named by 28 essays across 9 fields — each of them below, with the objects they name alongside it.
The best axes are not receptors
If the axes that make a von Kries gain work were nearly the cone fundamentals, the published adaptation transforms would be cone-like and their departures would be slack in a fit. They are not. Minimise the residual over all nine free numbers and the winner sits further from the measured dichromat confusion points than any of them.
No basis is good at both
The same nine numbers decide how well a von Kries gain reproduces a change of light and how nearly a lightness–chroma space makes the discrimination ellipses circles. Minimise either one and the other collapses. The basis built from the receptors is best at neither and is the only entry in the table respectable at both.
A compression goes below the floor
Elsewhere this collection minimised the anisotropy of MacAdam's ellipses over every chromaticity diagram there is, found 2.02, and called the residual a property of the eye. It is a property of the eye seen through a projective picture. A cube root after the right basis reaches 1.61 on the same twenty-five ellipses.
The exponent was never the argument
A century of colour science has argued about whether the eye's response is a cube root, a square root or a logarithm. Minimise the anisotropy of MacAdam's ellipses over every basis, at each of eight exponents, and the floor moves by under three per cent between a cube root and a tenth root — while the basis moves it by a factor of two.
Everyone is beaten by the same wall
Eight candidate adaptation bases, fourteen changes of light, and seven of the eight have their worst row in the same place — not a lamp, but a green wall reflecting twice. The one that does not is the one that was fitted, and what its fit bought was permission to give up on that row.
Primaries chosen for their inverse
Moving a display's white point is a gain on its R, G and B, so a display adapts in the inverse of its own primary matrix — a basis chosen by committees for gamut coverage and phosphor availability. Pose the design problem properly and the answer costs one per cent of the gamut argument and reaches within two per cent of the best basis there is.
A sensor designed for its inverse
A camera's white balance is a gain in a basis made from its own dyes and the light in the room. Choose the dyes for that basis instead of for cost and quantum efficiency, hold the sensor within a stated distance of the Luther condition, and the design reaches the best adaptation figure any basis achieves — and then the room moves it.
A constraint costs what it points at
Three primary chromaticities remove three of the nine numbers in an adaptation basis and cost one per cent. Three dichromat confusion points remove six and cost seventy. Counting what a constraint removes predicts neither, because an optimum is a long bowl and what matters is which way the constraint points.
A notch a pigment cannot cut
The worst change of light a painted room can produce has no maximum inside the box the search was given, which the previous round reported as a family with no worst case. Bounded by what a molecule can actually do, it has one — at a band six nanometres wide, narrower than any pigment and narrower than the box.
The darkest wall anybody sells
Asked which property of a paint decides the worst change of light a room can produce, anybody would answer how saturated it is allowed to be. A ceiling on saturation never comes near binding, because the worst wall is dark rather than colourful — and the constraint that does bind is one nobody would nominate.
Downhill from a published matrix
Walking steepest descent from each adaptation transform in use closes between a quarter and nine tenths of its distance to the best one, and most of that in the first step. The direction it sets off in is eighty to eighty-seven degrees away from the answer, and that turns out not to be an artefact of the three directions nothing can see.
The objective nobody chose
Every camera profile here, and as far as can be told everywhere, is a linear least-squares solve in tristimulus space. That is an objective and it is on nobody's menu — it weights an error by how bright the patch is. Refitting the same matrix to minimise a real colour-difference formula improves the fit in all six, and moves the matrix, which means different pixels rather than a different report.
A proof cannot be tuned for readers who disagree
A soft proof matched exactly for the standard observer is five colour differences wrong for one reader in twenty. Giving up that exactness to tune the display's three drives for a population instead moves the ninety-fifth percentile reader by 4 to 14 per cent even when the tuning is done on the very readers it is scored against — because what readers see is mostly each other's disagreement, and three drives act on every reader at once.
A press makes the cheap route a search
Holding a gradient inside a display's gamut cost nothing measurable, and the essay that found it credited the gamut's convexity. A press's gamut is not convex: 216 of 630 straight lines between colours on its own hue ring leave it. Holding gradients inside the press still costs little at best, a median of a tenth of a per cent. But the best route depends on where the relaxation starts, the free path is the best start on one gradient of ten, and on half of them some start is trapped at more than twice the free length. On the display no start is trapped.
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.
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.
A lopsided band does not break the trade
A fourth emitter added to a three-emitter LED renders colours better and costs a slow pigment its reach, and the position that renders best and the position that costs least were thirty-five nanometres apart. The obvious suspect was the shape: every emitter searched was a symmetric bump, and a band with its skirt on one side might fill a gap without splitting it. Freed to be lopsided, it saves between one and seven of the solid's three hundred directions. No band renders like the amber emitter and costs like the green one, the lopsided bands that help lean the other way from the proposal, and what the fourth band renders was never the lamp's white.
The slope limit was exact all along
Every count of what a slow pigment cannot reach compares a slope limit, solved exactly on 161 levels of reflectance, with two band models found by searching. Searched harder, the band models lost a sixth of their losses, and the open question was whether the slope limit was being flattered by its own grid — whether 321 or 641 levels would lower its counts too. The prediction was a fall of under a twentieth. There is no fall at all: on every finer grid, under all nine lamps, every direction reaches exactly what it reached, and 463 directions are lost as before. The reason is the shape of the answer. The best reflectance that may change by at most an eighth per five-nanometre band is a staircase whose every step is a whole eighth, because a weighted sum is maximised at a corner of the allowed set and every corner is built from full steps off nought or one. A grid of only the nine eighths gives every answer to the last digit. The census's count is therefore not an estimate on a grid but an exact statement about every pigment that slow, and the slope limit's 1.38-fold lead over the searched band model is a floor that further search can only raise.
A band pigment must be narrower than the gap it splits
Under a clear mercury lamp a pigment built from absorption bands no narrower than forty nanometres falls short in fifteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse — the count an earlier search left short however deep it cut its cliff. The slope-limited pigment's losses turned out to be the lines' 31.5 nm spacing against its swing, and the prediction was that the band pigment's are the same arithmetic, falling to nought near a band thirty nanometres wide. Searched on a one-nanometre grid with the lines narrowed, the count falls at every narrower band — fifteen at forty, eleven at thirty-five, nine at thirty — and reaches nought at twenty-five, not thirty. Every loss already splits the pair and falls short on both lines at once: a Gaussian band deep enough to absorb one line keeps absorbing a little of the other. One band of depth twelve, placed as well as it can be, errs by 3.2 per cent at forty nanometres, 0.8 at thirty and 0.2 at twenty-five; doubling its depth buys less than narrowing it by five nanometres.
A steep side splits mercury's lines at forty nanometres
Under a clear mercury lamp a pigment built from Gaussian absorption bands no narrower than forty nanometres falls short in fifteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, because a band's tail reaches the line it should pass. Real dye bands are often lopsided, and the prediction was that a band whose one side is half the other's, the steep side free to face either way, loses at forty nanometres about as many directions as a symmetric band thirty wide — nine — and helps only the directions that pass the green. It loses one. At thirty-five and thirty it loses none. In every one of the twenty-five fits the absorbing band turns its steep side to the line it must pass, and both kinds of direction gain: none of fourteen passing the green is lost, one of eleven passing the yellow. A single band forty nanometres wide with a steep side splits the two lines within 0.37 per cent; a symmetric band must narrow to about twenty-seven to do as well.
A band must be nearly two to one before its steep side pays
Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide loses fifteen or sixteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, and the same bands with one side a third of the width and the other two thirds lose one. The prediction for milder skews was that the losses fall in step with a single band's error — to about eight at sides of 0.55 and 0.45, three at 0.6 and 0.4 — so that a three-to-two skew already halves them. It does not. The census loses thirteen at 0.55, ten at 0.6, five at 0.65 and one at two to one: three directions recovered per twentieth of skew at first, twelve at the last. The losses track one band's error against the one-per-cent line the census counts by, and a single band crosses that line only near a skew of 0.62. A dye's modest natural asymmetry buys a few directions; the steep side pays only when it is close to half the other.
A band is only as wide as its steep side
Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide must be skewed nearly two to one before it can split the green line from the yellow, and the prediction for bands five nanometres narrower was that a single band crosses one per cent of error near a skew of 0.57 and the census's losses fall to five or fewer by 0.6. The losses do fall to four by 0.6; the single band crosses one per cent at 0.555. But the forty-nanometre rule that read the losses off that one-per-cent line breaks at once, and what replaces it is simpler: a thirty-five-nanometre band at 0.6 and a forty-nanometre band at 0.65 have the same steep side, fourteen nanometres, and lose the same four directions by the same shortfalls to a hundredth of a point. Three widths and fourteen pigments fall on one staircase in the steep side alone. Searching every loss again from the milder skews' answers finds one direction the census missed, at both widths, and corrects the forty-nanometre counts to nine and four.
At one half-width a flank can lose nothing or nearly everything
Under a clear mercury lamp, a pigment of Gaussian absorption bands splits the green line from the yellow according to its steep side's half-width alone, and a Gaussian flank is fixed by its half-width — so the census could not say whether it measures the half-width or the absorbance left in the flank's foot at the line it should spare. The prediction was that the foot decides: at a half-width of fourteen nanometres, a Lorentzian steep side loses most of the twenty-five splitting directions and a sharper-footed one none. It does. A power-four Gaussian loses none, the Gaussian four, a squared hyperbolic secant nineteen and a Lorentzian twenty-two, in the order of their single bands' errors. But a single band's error is not the whole account: three profiles drawn at the half-widths that give a single band the same error — 7.73, fourteen and 39.1 nanometres — lose two, four and seven directions, and the widest of them loses directions the others keep, because its shoulder starts absorbing nearer the green.
A hundredth of Lorentzian undoes a Gaussian flank
Under a clear mercury lamp, a band pigment whose steep side is a Gaussian fourteen nanometres wide at half depth loses four of the twenty-five directions that split the green line from the yellow, and the same side drawn as a Lorentzian loses twenty-two. A real absorption band is usually a Voigt profile, a Gaussian core with a Lorentzian share in its wings, and the prediction was that a twentieth of Lorentzian doubles the Gaussian's losses and a fifth reaches nineteen. The direction is right and the scale is not. A hundredth of Lorentzian takes the pigment from four lost directions to eleven; a fiftieth to fourteen; a twentieth to eighteen; a fifth to twenty. The best-placed band leaves the passed line more than three half-widths from its centre, out in the wing, and there a wing one per cent Lorentzian carries twice the absorbance of the whole Gaussian. The count saturates near twenty while the worst direction's shortfall keeps growing, from 2.8 per cent at a hundredth to 51 with a full Lorentzian.
A narrower flank moves its wing in with it
Under a clear mercury lamp, a band pigment whose steep side is a Gaussian fourteen nanometres wide loses four of the twenty-five directions that split the green line from the yellow, and with a twentieth of Lorentzian in that side it loses eighteen, because the wing three half-widths out out-absorbs the whole Gaussian there. A Lorentzian wing falls as the square of the distance over its half-width, and the prediction was that narrowing the side to seven nanometres brings the twentieth-Lorentzian pigment back to four losses and to five nanometres to none. It loses thirteen at ten nanometres, twelve at seven and seven at five, while the Gaussian loses nothing at all three. The wing still decides: the narrower band does not stay where the wider one sat but moves fourteen nanometres in towards the line it must pass, so its wing there falls fourfold rather than eightfold.
A five-nanometre flank carries a fiftieth of Lorentzian
A band pigment splitting mercury's green and yellow lines with a steep side five nanometres wide loses none of the twenty-five splitting directions as a Gaussian and seven with a twentieth of Lorentzian in its flank, because the band slides in towards the passed line and carries its wing there. The prediction was that at five nanometres the count follows the single band's error, which grows in proportion to the share: about two lost at a fiftieth and none at a hundredth. A hundredth loses none, as predicted. The first search lost one direction at a fiftieth — and searched again from the best pigments found at the neighbouring shares, that direction reaches within two thousandths of the ideal: a fiftieth loses none either. The same second search recovers two of the twentieth's seven, leaving five. A sharp enough flank tolerates a fiftieth of Lorentzian where a fourteen-nanometre one lost fourteen directions to it.
Each direction falls short by one band's ratio
A band pigment asked to reflect mercury's green line and absorb its yellow, with a twentieth of Lorentzian in its steep side, loses eighteen of twenty-five splitting directions at a steep side fourteen nanometres wide, thirteen at ten, twelve at seven and five at five. A second search had lowered the five-nanometre count, and the prediction was that searching the seven- and ten-nanometre pigments again lowers each by one or two, leaving a count nearly linear in the half-width. Searched again, both lose exactly what they lost — no direction gains six tenths of a point. The uneven fall is real, and the reason is plainer than any search: every direction's shortfall falls with the half-width by the same factor as a single band's error, so the count is a fixed ranking of constants, one a direction, read against a line that moves.
The ranking reads six nanometres to the direction
Every direction a band pigment loses under mercury's lines falls short, as its steep side narrows, by the same factor as a single band's error — so each direction carries a constant, and the count at any half-width can be read off one band with no search at all. Read that way, a twentieth of Lorentzian loses nine directions at a six-nanometre steep side, three at four and a half and none at four. Searched, six nanometres loses nine: the same nine, every direction within 6.2 per cent of where the constants put it. Four loses none. Four and a half loses one, not three — the two the ranking added sit 0.005 and 0.011 of a point under the one-per-cent line. And at the narrowest widths the gentle directions begin to fall short by more than the ranking says, by up to 29 per cent, which is where the single band stops being the whole story.
Named alongside it
The objects these essays reach for when they reach for this one.
PigmentObject-colour solidOptimal coloursAbsorptionChromatic adaptationBasisWavelength gridTrade-offThe von Kries transformCIELABCAT16Identifiability