Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Computed, not quoted — page 8

Every swatch begins as a spectral power distribution and is carried through the colour-matching functions as it is drawn. None is a hex code recalled from a table.
How much of a whiteness figure is the sheet, and how much is the lamp. Each bar is how many points of CIE whiteness a stock has above an unbrightened sheet of the same base, measured with an ultraviolet-included instrument. The dark part is what survives when the ultraviolet is removed — the part that is a property of the paper. On average 82 per cent of the scale is the pale part, which is a property of the instrument's lamp. A whiteness figure without a measurement condition beside it is therefore not a measurement of a sheet; it is a measurement of a sheet and a lamp, quoted as though it were the first. Difference and uniformity

Whiteness is mostly the lamp

The CIE whiteness formula ranks white samples the way people do, which is what it was built for and is not in question. What is in question is what it is a measurement of — take the ultraviolet out of the instrument and 82 per cent of the scale collapses, because the part that separates a premium sheet from an ordinary one was contributed by the lamp.

 The best possible 3×3, and the patches it makes worse. Each row is one patch printed on a brightened sheet, measured under both conditions. The pale bar is how far apart the two measurements are; the dark bar is what is left after the best least-squares 3×3 over the whole set has been applied. It leaves 23 per cent of the mean, and — the part a mean hides — it makes 4 patches worse than doing nothing. The solids are the ones it damages: the ink blocks the ultraviolet, so a solid barely disagrees between the two conditions and the correction has no business touching it. A matrix has no way to apply itself only where the paper is showing. Difference and uniformity

A tolerance cannot cross a condition

If two measurement conditions disagree by seven units, the obvious repair is a correction matrix fitted between them. The best least-squares 3×3 over seventeen printed patches leaves 23 per cent of the disagreement and makes five patches worse than doing nothing — because the term it is trying to remove is proportional to how much paper is showing, and no linear map on three numbers can express that.

 Two lamps of the same colour, and one sheet that is two colours under them. The same brightened sheet under a xenon flash, the same flash behind its cover glass, and a phosphor-converted white LED of nearly the same chromaticity. Each patch is computed relative to its own lamp's white, which is what a perfect white balance does — so everything a camera can see and correct has already been removed. What is left is ΔE00 11.8 between the first and the last, against 3.2 between the two lamps themselves. The LED has no emission below 380 nanometres at all, because its pump die is at 450, so the sheet simply does not fluoresce under it and nothing in the photograph records why. What a camera does

A camera cannot record the excitation

Two lamps of nearly the same chromaticity, one with ultraviolet and one with none, put a brightened sheet twelve units apart after a perfect white balance. Nothing in the camera measured the difference — the filter stack removed the band before the sensor saw it — so the correction that would fix the picture needs a quantity the file does not contain.

 The lamps the four conditions shine, where they differ. The short-wave half of what each measurement condition puts on the sample, plotted to 560 nanometres because past that the three are indistinguishable in shape. M₁ is D50 with its ultraviolet; M₂ is the same lamp behind a cut filter at 400 nanometres, and at 360 it is 0.0 per cent of what M₁ delivers; M₀ is a tungsten lamp, which has some ultraviolet, has less than daylight, and is not specified at all by the standard — so two M₀ instruments need not agree with each other. M₃ is not plotted because its lamp is M₂'s; what makes it a fourth condition is a polariser. What a camera does

The lamp that stopped emitting ultraviolet

A blue-pumped white LED has a die at 450 nanometres and emits nothing shorter. Between about 2005 and 2020 that lamp replaced almost every other indoor source, which removed the excitation supply from a great many rooms — so brightened materials stopped glowing indoors without a single one of them being reformulated.

 Outside the set of colours a reflecting surface can be. How far each stock sits from the boundary of the object-colour solid, as a fraction of the bound. The line at zero is the boundary: a perfect diffuser sits exactly on it, and every reflectance ever made sits to its left. The pale marker is the sheet measured with the ultraviolet excluded and the dark one with it included. Two of the six cross the line — they are brighter, in a direction that can be written down, than any reflecting surface of their colour could be. This is a proof rather than a hull: for each sample a direction is found in which the largest value any reflectance can reach is computed exactly, and the sample exceeds it. Where the model breaks

A white that is not a reflectance

The object-colour solid is the hardest boundary in colorimetry — the set of tristimulus values any reflecting surface can produce, with no assumption about pigments in it at all. A coated press stock under a measurement standard's own lamp sits 1.5 per cent outside it, and a heavily brightened one 4.0, and with the ultraviolet removed both come back inside.

 A whiteness measurement has a date on it. A brightener is consumed by the ultraviolet that makes it glow: the molecule that absorbs a photon occasionally does something other than re-emit it, and what it does is break. So the loading falls with accumulated dose and the sheet's whiteness falls with it, from W 123 to 88 — most of the way back to the unbrightened base. The fall is steepest at the start because the absorption saturates, so the first molecules lost are the ones that were doing the least work and the curve is convex from the beginning. The dose axis is in arbitrary units whose half-life is stated; what is not arbitrary is the shape. Where the model breaks

The brightener is being used up

The molecule that absorbs an ultraviolet photon occasionally does something other than re-emit it, and what it does is break. So a sheet's whiteness has a half-life, the fall is steepest at the start, and a specification quoting a whiteness figure without a date is quoting a property of a sheet that no longer exists.

 How blue the sheet reads, and how much of that is the room. Chroma in an appearance model, with the observer adapted to the light the sheet is under. The three markers on each row are an average, a dim and a dark surround; the open marker at the left is the same sheet with the ultraviolet removed. Every stock is close to neutral without the excitation and carries real chroma with it, at a hue of about 295 degrees — blue-violet — so the colour is the fluorescence and not the substrate. And it falls by nearly half between a bright room and a dark one, which makes how blue a sheet looks a property of where it is being looked at. What the brain does

A brighter white still looks white

Colorimetry says a brightened sheet is nine units of b* from neutral, which sounds like a visible blue. An appearance model with the observer adapted to the room says it carries eleven units of chroma at a hue of 295 degrees — genuinely blue-violet — and that the number falls by nearly half between a bright room and a dark one. What it cannot say is why anybody calls the result white.

 One observer's matching functions, in three of the bases the matches leave free. The three colour-matching functions after a change of basis 0.00 of the way from Hunt–Pointer–Estévez towards the set built from the dichromat confusion points. Every one of these triples predicts exactly the same matches as every other, because a match is an equality and a matrix applied to both sides of an equality changes nothing. What moves is where the peaks are and whether the curves go negative — these ones do not, and going negative is what the 1931 committee constructed XYZ to avoid. What the eye does

The matches do not name the cones

Colour matching is the whole empirical basis of colorimetry, and it fixes the observer's three curves only up to a nonsingular 3×3 — nine numbers that no match, in any quantity, to any precision, can see. One particular choice of those nine is used throughout here, and it was made for a different purpose.

 Where a dichromat's confusions converge. Every pair of colours a protanope cannot tell apart lies on one of these lines, and all the lines meet at a single point — at (0.7465, 0.2535) for this class. The point is the chromaticity of the missing cone's own response direction, which is why it need not lie inside the diagram or correspond to any light at all. Two of the three do not. The three points between them carry six numbers, and six is two thirds of what the matching data leave undetermined. What the eye does

A confusion point is a missing pigment

The nine numbers colour matching leaves free are fixed by three points on a chromaticity diagram, each of them the place where everything one class of dichromat cannot tell apart converges. Two of the three lie outside the diagram entirely, which is not a defect — a direction in tristimulus space need not correspond to a light.

 The locus and the triangle, drawn on one of the diagrams. The spectral locus and the sRGB triangle in CIE xy (1931) — the default of the discipline, and the default used here. The triangle covers 33.6% of the enclosed area here. Nothing about the observer or the display has changed between this picture and any other in the family; the coordinates have, and the coordinates are what an area is measured in. Matching and measuring

The diagram has no area

A chromaticity diagram is a projective picture of a three-dimensional space, and the freedom colour matching leaves in the observer acts on it as a projective map. Straight lines and mixture ratios survive that; area, distance and angle do not — so half of what the diagram is used to say is a statement about the paper.

 What share of the diagram the sRGB triangle covers, in twelve published coordinate systems. Each row is a chromaticity diagram somebody has printed, and each bar is the fraction of the enclosed visible area that the sRGB triangle covers in it. Every row describes exactly the same observer and exactly the same gamut. The answer runs from 8.5% to 38.4%, a factor of 4.52, because area is not preserved by the projective maps that carry one of these diagrams to another. The familiar "about a third" is a fact about CIE xy. Matching and measuring

Two thirds is not a property of the eye

It has been said from the beginning here that about two thirds of the chromaticity diagram cannot be shown on a screen. The figure is right, on the diagram it was measured on, and across twelve published diagrams the same triangle covers anything from 8.5 to 38.4 per cent of the same locus. Counting stimuli instead gives an answer that does not move.

 MacAdam's ellipses, drawn on one diagram. The twenty-five measured discrimination ellipses at 10× actual size, on CIE xy (1931). Mean axis ratio 2.95 — one would mean every contour is a circle — and a size spread of 10.42 between the largest and the smallest. Both numbers depend on the plane, which is why the 1976 revision existed; neither can be taken to one, which is why the revision did not finish the job. Difference and uniformity

No diagram makes them circles

Every chromaticity diagram is a projective picture of the same measurement, so how badly MacAdam's ellipses fail to be circles can be minimised over the whole family of them. The best plane there is still leaves the average ellipse twice as long as it is wide — which makes the residual a fact about the eye rather than about anybody's choice of primaries.

 The same formula, applied after six different changes of basis. CIELAB's arithmetic — divide by a white, take a cube root, difference the results — run on six of the bases the matching data leave free. A linear change of basis leaves every match alone; a cube root does not commute with one, so the space, and therefore every colour difference computed in it, depends on which basis was in place before the nonlinearity. CIELAB's own choice gives an axis ratio of 3.44 and the best row here is LMS (confusion points) at 2.60. Difference and uniformity

A difference needs a basis too

A linear change of coordinates leaves every colour match exactly where it was. A cube root does not commute with one — so a lightness–chroma space, and every colour difference computed in it, is a property of the basis that happened to be in place before the nonlinearity. CIELAB's basis was chosen in 1931 for reasons that had nothing to do with difference.

 How short of determining the light a photograph is, as the scene grows. Each cell is the number of unknowns left over after every equation the image supplies: three sensors, a three-dimensional illuminant, and reflectances confined to a linear model of the dimension on the left. At one and two dimensions more surfaces close the gap. At three the gap never closes, because each further surface adds three equations and three unknowns; at four it widens. The count is arithmetic and has no algorithm in it. What a scene does

An image does not determine the light

A photograph of a scene under one illuminant gives three numbers per surface and asks for the illuminant plus three numbers per surface. The count closes only if reflectances lie in a two-dimensional model, and no number of surfaces helps — at three dimensions the alternative scenes can be written down, and they reproduce every sensor response exactly.

 What a camera matrix reports on its own chart, and what it delivers off it. The same camera fitted on charts of increasing chromatic range. The left bar of each pair is the mean error on the chart the matrix was fitted to, which is the number a profile comes with; the right bar is the error on a saturated set it never saw. At the thinnest chart the fit reports 0.19 ΔE00 and delivers 1.65, a factor of 8.6. The gap closes as the chart widens, and it closes because the chart improves rather than because the camera does. What a camera does

The chart decides the profile

A camera's colour matrix is nine numbers fitted to a set of patches somebody chose, and the number that comes with it is the error on those patches. On a chart with no chromatic range that number is 0.19 ΔE00 and the matrix delivers 1.65 — and a second matrix, indistinguishable on the chart, delivers 2.03.

 A camera matrix fitted under each light, used under each light. Mean ΔE00 over the same surfaces, with the matrix fitted under the row's light and the scene under the column's. The diagonal is what a profile's data sheet quotes and is between 1.0 and 1.2 everywhere. Off it the numbers rise steeply: the matrix fitted under illuminant A reports 1.17 there and delivers 9.34 under a 9000 K daylight, a factor of 8.0. Nothing about the camera changes between cells. What a camera does

A matrix is fitted under one light

A camera's colour matrix is nine numbers determined by a chart photographed under a particular illuminant, and the error it quotes is the error under that illuminant. Fitted under a tungsten lamp and used under a cold sky it delivers eight times as much — and the two-matrix scheme every real profile uses turns out not to be a compromise at all.

 An instrument, as the only thing it really is. The 3 filters a bank of that size puts across the visible range, each drawn against wavelength. Everything the instrument can report about a spectrum is 3 numbers — the integral of the light against each of these — so the set of spectra it cannot tell apart is everything orthogonal to all 3 of them, which is 78 dimensions of the 81 this site works in. Three of these is a colorimeter in spirit; the eye is three of them too. What light is

Three numbers cannot see a line

An instrument that returns three filtered readings of a spectrum determines a three-dimensional projection of it and is exactly blind to the other seventy-eight. On daylight that costs almost nothing; on a fluorescent tube, three quarters of the lamp lies in the part no reading reaches, and adding filters recovers it slowly.

 The colour is right long before the spectrum is. The colour error of the projection, against the number of readings. At twelve readings the fluorescent tube's colour is right to 0.48 ΔE00 while 66% of its spectrum is still unmeasured. That is the trap in one line: a reconstruction good enough to pass any colorimetric check will predict a match under a second illuminant that does not happen, because the part it got wrong is exactly the part a different lamp weights differently. What light is

The colour is right first

A reconstruction of a lamp from twelve filtered readings gets its colour right to half a unit while two thirds of its spectrum is still unmeasured. That combination is not a partial success — it is the exact condition under which a spectral prediction made from the reconstruction will be confidently wrong.

 What each fitted thing in these essays carries, what its data fix, and what is left. Three columns per row: how many numbers the model has, how many the stated data determine, and the difference — the dimension of the family that fits equally well. The third column is the one nobody publishes. A zero there does not mean the model is right; it means it is determined, which is a much weaker property and is compatible with being determined badly, as the camera row is. Where the model breaks

A fit can be exact and empty

Every fitted object here reports one number, the residual on the data it was fitted to, and every one of them has two more that nobody publishes — how many of its parameters the data actually determine, and how large the family of equally good answers is. The third column is where the failures live.

 Which of this collection's own claims survive a change of basis, and which are about the paper. Nine sentences this site says, sorted by whether they mean the same thing after the observer's three curves are replaced by a nonsingular combination of themselves. 5 of the nine do. The four that do not are not thereby wrong — they are statements about a chosen set of coordinates, and they are true of those coordinates. What they cannot be is statements about the eye, which is how every one of them is usually read. Where the model breaks

Which of these is a convention

Nine ordinary sentences from this collection, put through one test — do they mean the same thing after the observer's three curves are replaced by a nonsingular combination of themselves? Five survive and four do not, and none of the four is wrong, because each is a statement about a set of coordinates being read as a statement about an eye.

 What the next thousand patches buy a printer profile. A profile is a table, exact at its nodes and interpolated everywhere else. Each mark is a grid: the horizontal axis is how many patches somebody had to print and measure, the vertical is the worst error found between the nodes. Going from 135 patches to 3645 — 27.0× the work — buys 8.4× the accuracy. The error falls with the square of the spacing and the count rises with its cube. What it takes to deliver it

A profile is a fit between its nodes

A printer profile is a table, exact at every patch that was printed and interpolated everywhere else — and everywhere else is where every job lives. Going from a hundred and thirty-five patches to three and a half thousand is twenty-seven times the work for eight times the accuracy, and the exponents say that is as good as it gets.

 Where each published matrix puts the confusion points, whether or not it meant to. Every matrix from tristimulus values to cone responses commits itself to three confusion points, because the point is the direction the other two rows annihilate. The first row is the construction from the measured points and returns them exactly. The rest were chosen for other reasons and land elsewhere — Hunt–Pointer–Estévez, which this collection uses everywhere, misses the deuteranope's point by 1.28 in chromaticity. The worst here is 4.09. What the brain does

The cones an appearance model uses

CIECAM16 adapts in three axes whose rows are labelled L, M and S, and they were fitted to corresponding-colour experiments rather than measured on receptors. Run the dichromat construction backwards on them and they commit to a deuteranope confusion point 1.45 away in chromaticity from the measured one — which is a test the axes were never asked to pass.

 Four different bases, one adaptation model, one number. The middle row of the basis built from the confusion points multiplied by 0.21, 1, 3.7 and 11 in turn, with the resulting adaptation residual drawn as a bar in each case. The four bars are the same height to 9e-16 of a ΔE00, because the row's scale cancels exactly between the gain and the inverse. Three of the nine numbers a colour match leaves free are invisible to an adaptation model, which is why the six the dichromat data supply determine it outright with nothing left to fit. What the eye does

The three numbers a gain cannot see

Colour matching leaves nine numbers free. Three dichromat confusion points fix six of them and three choices of unit fix the rest — and it turns out that a von Kries gain is exactly blind to those last three. So the dichromat data do not merely constrain an adaptation basis. They determine it, with nothing left over to fit.

 How cone-like a basis is, against how well it adapts. Each basis placed by how far its own implied deuteranope confusion point falls from the measured one (horizontal) and by how much an adapted observer is left with in it (vertical). The construction from the confusion points sits at zero on the horizontal by definition and near the top on the vertical. Nothing near the left of the picture is near the bottom: the closer a basis is to the receptors, the more a von Kries gain leaves behind. The unconstrained winner sits at 1.63 on the horizontal, further from the measurement than any published transform except CAT02 and Bradford. What the eye does

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.

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