Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Three numbers — page 4

An infinite-dimensional spectrum is projected onto three cone responses, and everything colour science can do — and every way it fails — follows from that single collapse.
What one change of light costs, surface by surface — daylight to tungsten. A rising curve of 125 points, one per surface in the test set, sorted from the surface this change of light costs least to the one it costs most, with the published mean drawn across it as a horizontal line. The published residual for daylight to tungsten is 1.635 ΔE₀₀. The curve runs from 4.4e-14 — 5 of the surfaces are flat greys, on which an adapted observer's gain is exactly right and the residual is exactly zero — to 3.058, which is 1.87 times the mean. The mean line crosses the curve about two thirds of the way along, so most surfaces cost less than the published number and a minority cost a great deal more. This is what a single published residual is a summary of. What a scene does

A mean has a set under it

Every adaptation number this collection publishes is an average over a hundred and twenty-five surfaces that were written down once, in one file, with no argument for how many there should be or how saturated. The average runs from exactly zero to twice itself across them, and the set has never been varied.

 The error on a gap is not the two rows' errors added. Two bars for each of the 13 adjacent pairs in the census ranking. The upper, shorter bar is the standard error of the gap taken as a paired difference — the same 125 surfaces score both rows, so a surface that is awkward under one change of light is usually awkward under the other and the difference is quieter than either. The lower bar is the two rows' own errors added in quadrature, which is what comparing error bars by eye amounts to. Pairing is worth a factor of 1.78 on average and 3.36 on the pair it helps most, and it is the difference between 6 adjacencies unordered and 4. The gain is largest where the two rows are two daylights or two tungstens, because then the surfaces they find awkward are nearly the same surfaces. Difference and uniformity

The error on a gap is not the errors at its ends

Comparing two rows of a table by looking at whether their error bars overlap is the wrong comparison, and here it is wrong by a factor of up to 3.4. The same 125 surfaces score both rows, so the difference between them is quieter than either — and how much quieter is a measurement of how alike the two rows are.

 Every census row under five constructions of the same test set. A slope chart with 5 columns — lattice, coarse, fine, uniform, natural — and one line per change of light in the census, each line joining that row's mean residual under each construction. Four of the five columns describe the same region of surfaces walked at different densities or against different measures; the last is the clamped, realistic family, which is not linear in its parameters and is therefore answering a slightly different question. The levels move: between the coarse and fine lattices every row shifts by seven to nine per cent, in the same direction, which is a common-mode factor no published residual here has ever carried. The order almost survives. Inside the region exactly one pair crosses, and it is the pair the standard error had already flagged; under the clamped set two more cross, including one the error separates by nearly nine standard errors. The crossing lines are drawn heavy. Where the model breaks

The instrument named the pair that moved

A standard error over a test set flagged four steps of the adaptation census as unresolved. Rebuilding the set three different ways reversed exactly one pair, and it was one of the four. Rebuilding it to a different rule reversed a pair the error separated by nearly nine standard errors — which is not a failure of the instrument but a statement of what it is about.

 What a fourth reflectance dimension costs the theorem that a change of light is a matrix. Four rising curves on axes of the fourth dimension's amplitude, left to right, against what is left of daylight to tungsten after the exact 3×3 change-of-light matrix has been applied, in ΔE₀₀. All four begin at exactly zero: on the three-dimensional family the matrix is solved rather than fitted and there is no remainder at all, which is the theorem this collection's adaptation argument is built on. Adding a fourth reflectance dimension breaks it, and how badly depends far more on the fourth function's shape than on its size — at five per cent amplitude the four shapes cost 0.329, 0.572, 0.063, 0.124 ΔE₀₀ respectively, a factor of 9.1 between the dearest and the cheapest. For scale, the smallest von Kries residual anywhere in the census is 0.26 ΔE₀₀, so the cheapest of the four is a quarter of it and the dearest is twice it. What a scene does

A theorem about a family

A change of light acts on the test surfaces used here as an exact 3×3 matrix with no residual whatsoever, and the whole adaptation argument is built on that being exact. It is exact because the surfaces span exactly three dimensions, and they span exactly three dimensions because three basis functions were written down.

 Which fourth dimensions are expensive, and how fine is too fine to matter. Two curves on axes of how many half-cycles a cosine fourth basis function makes across the visible band, against what it costs the matrix theorem in ΔE₀₀, at a fixed ten per cent amplitude. Both curves touch zero at exactly one and two half-cycles: those are the family's own second and third basis functions, so a fourth coefficient along them adds no dimension and a change of light stays exactly a matrix. Between them the cost climbs, reaches a maximum, and — for the smooth source — falls away again, because structure finer than the scale on which three broad cone sensitivities differ integrates to nearly nothing. The two curves part company at the fine end. Under daylight-to-tungsten the cost has fallen by a factor of 2.2 from its peak; under daylight-to-a-triphosphor-tube it has barely fallen at all, because a source with three narrow emission lines has structure of its own at that scale for the surface's structure to beat against. The observer is identical in both curves. What light is

A fourth dimension has a shape

How much a fourth reflectance dimension costs spans a factor of nine across four equally plausible shapes at one amplitude, and the expensive ones are not the shapes a variance figure would identify. The band that hurts is set by the illuminant rather than by the eye, which is why a triphosphor tube and a tungsten lamp disagree about it.

 The census as the surfaces stop being three-dimensional. A slope chart with three columns — a test set with no fourth reflectance dimension, one with a fourth dimension at ten per cent amplitude, and one at twenty — and a line per change of light. Almost every line rises: a surface with structure the observer's three channels cannot follow is a surface an adaptation gain handles worse. Two lines are drawn heavy. daylight to a three-primary display rises fastest, by 90 per cent, because a source made of three narrow lines is precisely the instrument that cannot see a fourth reflectance dimension. And daylight to a triphosphor tube falls — the only row that does — because a triphosphor tube already samples the spectrum at three places, so extra structure in the surface is partly averaged away rather than added. The order of the middle of the table is not the same at the two ends; the extremes do not move. What light is

The row a fourth dimension improves

Giving the test surfaces one more degree of freedom makes almost every change of light harder for an adapted observer — but not all of them, and which one it helps depends entirely on what the extra dimension looks like. A triphosphor tube is improved by one shape and hurt more than anything else in the census by another.

 No one surface carries the answer, and the set is smaller than it looks. A falling bar chart of the 125 surfaces in the test set, ordered by how much each contributes to the published mean for daylight to tungsten. The tallest bar is 1.50 per cent of the total, so the mean is not a few awkward objects with a crowd behind them and a leave-one-out would move it by well under a per cent. The tail is the other half of the story: 5 surfaces contribute essentially nothing, because a flat grey is a surface an adaptation gain handles exactly. Counting the set by how evenly it contributes rather than by how many members it has gives 108.5 effective surfaces out of 125, which is what "a mean over a hundred and twenty-five surfaces" is really worth. What the eye does

The surfaces that answer nothing

Five of the hundred and twenty-five test surfaces contribute exactly zero to every number the adaptation census reports — not approximately, exactly — and the reason is the one fact about von Kries adaptation that makes it worth having at all. Counting the set by how much it contributes gives about a hundred members rather than a hundred and twenty-five.

 A confusion point is about the pigments that remain. Three groups of three bars. Each group is one dichromat's confusion point; each bar is how far that point moves in chromaticity when one of the three cone pigments has its absorption peak shifted by eight nanometres. In every group the bar for the pigment that dichromat is missing has length zero — exactly zero, to machine precision, not merely small. The protanope's point does not move when the L pigment moves, the deuteranope's does not move when the M pigment moves, and the tritanope's does not move when the S pigment moves. The reason is algebraic rather than physiological: a confusion point is the direction that excites only the missing cone, which is the null space of the other two receptors' rows, and rescaling a row does not move where the other two are zero. So the point at which a protanope's confusion lines meet is not a fact about the pigment a protanope lacks, which is why the claim about it restates in nanometres of the M pigment. What the eye does

A point about the pigments that remain

The chromaticity at which a protanope's confusion lines meet does not move at all when the long-wave pigment moves — not slightly, exactly not at all. It moves a great deal when the medium-wave pigment does. A dichromat's confusion point is a fact about the two receptors they have rather than about the one they lack.

 The same claim in nanometres of pigment, where no declared width can reach it. Five horizontal bars on a scale of nanometres, one per published chromatic-adaptation transform, each showing how far the medium-wave cone pigment's absorption peak would have to move for the receptors' own protan confusion point to land where that transform puts it. Zero is the measured peak. The bars run from -10.5 to 18.2 nanometres — in both directions, so two of the transforms want the pigment shorter and two want it longer. Drawn across them is the 25 nm separation between the L and M pigment peaks, which is the whole basis of red-green vision and is not a number this collection declared. The nearest transform asks for a displacement of 30 per cent of that separation, and the span across the table is 28.7 nanometres — larger than the separation itself. No population, cloud or standard deviation appears anywhere in the statement. What the eye does

The claim, in nanometres

For four rounds the claim here has been that every published adaptation transform puts the protanope's confusion point outside any real population of eyes, stated in standard deviations of a population whose widths were declared rather than measured. Restated as a pigment displacement it needs no population at all — and the nearest transform asks the medium-wave cone to move thirty per cent of the way to the long-wave one.

 The same claim in nanometres of pigment, where no declared width can reach it. Five horizontal bars on a scale of nanometres, one per published chromatic-adaptation transform, each showing how far the medium-wave cone pigment's absorption peak would have to move for the receptors' own protan confusion point to land where that transform puts it. Zero is the measured peak. The bars run from -10.5 to 18.2 nanometres — in both directions, so two of the transforms want the pigment shorter and two want it longer. Drawn across them is the 25 nm separation between the L and M pigment peaks, which is the whole basis of red-green vision and is not a number this collection declared. The nearest transform asks for a displacement of 30 per cent of that separation, and the span across the table is 28.7 nanometres — larger than the separation itself. No population, cloud or standard deviation appears anywhere in the statement. What the brain does

Five transforms and the space between them

Every appearance prediction here chooses one of five published adaptation transforms, and the five disagree about where a protanope's confusion lines meet by more than the distance between the two pigments the disagreement is about. That spread is itself a scale, and using it needs no population model at all.

 The census along the line from ΔE76 to ΔE94, and past it. ΔE94 is ΔE76 with two weighting constants in it, and at zero those constants make every weight exactly one, so the two formulae are joined by a line rather than separated by a choice. The horizontal axis is how much of the published weighting is applied: 0 is exactly ΔE76, 1 is exactly ΔE94, and 3 is three times more weighting than anybody has proposed. The falling curve is the census's mean elasticity to how saturated its test set is, which drops from 1.11 to 0.74 — most of the fall happening before the published value is reached. The other curve is Kendall's τ against ΔE2000's ranking, and it peaks at w = 0.5, not at 1: the weighting that best reproduces the published ordering is about half the published weighting. There is no value of this dial that reaches ΔE2000, whose rotation term is not on this line at all. Difference and uniformity

A dial through a discrete menu

ΔE*94 is ΔE*ab with two weighting constants in it, and at zero those constants make every weight exactly one — so the two ends of the oldest disagreement in colour difference are joined by a line rather than separated by a choice. Walking it gives a derivative where a menu gives only a spread, and the derivative says the published weighting is on the far side of the interesting part.

 Which steps of the census ranking a change of unit reverses. Every adjacent pair in the published census ranking that at least one unit puts the other way round. The bar counts how many of the five other units reverse it. The marker on the left says whether the test set had already declared the pair unresolved — a gap smaller than twice its own paired standard error, which is a statement about sampling over 125 surfaces and shares no arithmetic with a change of ruler. The two pairs every unit reverses are both flagged, which is the agreement. The pair at the bottom is the disagreement: the test set resolves it at 9.1 standard errors and four of the five units reverse it anyway, because a sampling error cannot see a change of ruler and a change of ruler cannot see a sampling error. Where the model breaks

Two instruments and one ranking

A sampling error over a hundred and twenty-five surfaces and a change of colour-difference formula share no arithmetic at all, and they were asked the same question of the same table. Every adjacency the whole menu reverses had already been flagged as unresolved. And one the test set settles at nine standard errors is reversed by four of the five formulae, which is what makes them two instruments rather than one.

 What six of this collection's published numbers do when the unit changes. Six quantities, from six calculations that share nothing: a change of light after an observer has adapted, a camera profile's error, the gap between the two standard observers, a metameric pair under the lamp that breaks it, the same image on two papers, and an observer two seconds into a new room. Each is recomputed under all six units and every unit is calibrated onto ΔE2000's scale first, so the bar is not a change of units in the ordinary sense. The bar is the ratio of the largest reading to the smallest, and it runs from 1.71 to 2.30. Five of the six are printed in ΔE2000 by the essays that report them; the sixth is printed in CAM16-UCS, because the model it comes out of defines that unit. Matching and measuring

The observers differ by a unit's worth

The gap between the 1931 and 1964 standard observers is the one quantity in this collection's audit with no published number under it — nothing here reports it as a single figure over a stated set. It also has the second-largest dependence on which colour-difference formula is used, running from 1.60 to 3.55 across the menu.

 Two sensitivities from two libraries, under every unit. Two quantities that share no code, no test set and no physical question: how much the adaptation census's residual depends on how saturated its surfaces are, and how much a camera profile's reported error depends on how saturated its test chart is. The first is a mean over fourteen changes of light built from cosine combinations; the second is one number about one silicon sensor scored on Gaussian bumps. Under the published unit they sit at 0.687 and 0.656. Across the whole menu they move together, from about 0.5 under the appearance unit to about 1.15 under plain CIELAB, staying within 12 per cent of each other at the worst point. Two numbers agreeing once is a coincidence; two curves agreeing at six points across a factor of two and a half is a shared mechanism, and the mechanism is the compression the unit applies to a chroma difference. What a camera does

The coincidence was a mechanism

Two sensitivities from two libraries with no shared code came out two per cent apart, and the claim made about them was that they share a mechanism rather than a number. That claim has a colour-difference formula inside it, so it can be tested by changing the formula — and both curves move together across the whole menu, from 0.5 to 1.15.

 A camera matrix refitted to minimise each unit, rather than solved in XYZ. Every camera profile here, and as far as can be told every camera profile anybody ships, is a linear least-squares solve in XYZ. That is an objective and it is on nobody's menu: it weights a difference by how large the tristimulus values are. Each row here refits the same 3×3 by direct search to minimise one of the six units instead. The upper bar is how much better the fit gets in that unit; the lower is how far the matrix itself moves, as a relative Frobenius norm. Both matter and they do not agree: CAM16-UCS moves the matrix least, at 0.34 per cent, for the largest improvement of the six, while ΔE*94 moves it 6.8 times as far for less. A score that changes is a report changing; a matrix that changes is the camera rendering different pixels. What a camera does

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.

 What an observer is left with, by how much it is allowed to know about the room. Six ways of discounting a change of light, averaged over the fourteen changes in the adaptation census and 125 test surfaces each. The bar is what each leaves behind, on a logarithmic axis because the models span two orders of magnitude. The second line under each name is the count that matters: how many numbers about this room the model has to be given. Doing nothing leaves 15.7 ΔE₀₀. A single gain read off the two whites' luminances leaves 15.3. A matrix fitted across half the census and then applied everywhere, knowing nothing about the room at all, leaves 12.5. The published von Kries gain, which is told the white and nothing else, leaves 1.312 — and bolting a fixed correction onto it, at no cost in scene information, leaves 1.368, which is very slightly worse. The exact matrix leaves nothing and is not on the chart: its nine numbers are the change of light, which is the quantity being discounted. What a scene does

Three numbers the scene supplies

An adaptation model's parameters are not all the same kind of thing. Some are numbers an observer must estimate from the room it is standing in; others could have been settled once by evolution. Counting them separately turns the diagonal gain from a crude approximation into the only model of the set that gets a large answer from information the observer can actually have.

 How much of the residual a partial correction removes. Between the diagonal gain and the exact matrix there is a line: apply the correction that would make a row exact, but only a fraction of it. The horizontal axis is that fraction and the vertical is the share of the row's residual it removes, for all fourteen census rows. The straight diagonal is where a correction worth exactly its fraction would fall, and in the published unit every curve lies on it to within 2.2 percentage points. The lower band of curves is the same interpolation measured in CAM16-UCS, which departs by up to 17 points — because its distance is a power of the Euclidean one and a power is not homogeneous along a ray, where every ordinary norm is. The straight line is therefore a property of the ruler rather than of the correction, and the exception is what says so. What a scene does

A partial correction is worth its fraction

Between a diagonal gain and the exact matrix there is a line, and a bounded observer's natural hope is that the first part of it is worth a disproportionate share. It is not. On all fourteen changes of light, at every setting, the share of the residual removed matches the share of the correction applied to within 2.2 percentage points — which closes the last way the gap could have been cheap.

 A correction an observer could have been born with, fitted on half the census and tested on the other. The same six models, each scored twice: on the seven census rows the fixed matrices were fitted to, and on the seven they were not. The split alternates by position so both halves contain daylight changes and discharge lamps. The upper bar is in sample and the lower is out, on a logarithmic axis. For the four models with nothing fitted the two bars differ only because the halves are different questions. For the two fitted ones the gap is the finding, and it is largest where it matters least: bolting a fixed correction onto the von Kries gain takes it from 1.2724 to 1.2592 on the rows it was fitted to, and from 1.3511 to 1.3679 — worse — on the rows it was not. There is no correction to the diagonal that an observer could arrive with. What a scene does

A model is a claim about what can be known

The exact answer to chromatic adaptation is nine numbers, and the nine numbers are the change of light itself. A model whose parameters are quantities the observer cannot obtain is not a worse model of the same thing — it is a model of something else, and counting parameters without asking where they come from hides the difference.

 How far this site's median observer sits from the 1931 standard, by template. Twenty-four natural reflectances under D65, each given a tristimulus value twice: once by the 1931 colour-matching functions and once by this site's median member, with each judged against its own white. The bar is the mean difference, which is the residual this collection bounds and calls inescapable. It is inescapable, and it is smallest for the simpler template: Lamb's 1995 nomogram gives 0.9006 against Govardovskii's 0.9522, and removing Govardovskii's secondary band brings it down again to 0.9354. Neither is an argument for changing template — a nomogram is fitted to measurements of individual receptors, not to colour matches, so agreement with the standard observer is not what either was trying to achieve. What it says is that the residual is a mismatch between two kinds of observer rather than a shortfall a better pigment model would close. The number after each bar is the template's tail ratio: how far the L cone's half-maximum reaches below its peak against how far it reaches above. What the eye does

The population rests on a template

Two hundred observers here are built from one formula fitted to microspectrophotometry in 2000. The obvious alternative — the tabulated cone fundamentals — is not available, and the reason is the finding. A tabulated fundamental has no peak wavelength to move, so the moment it is used the population collapses to a single observer.

 A template's asymmetry against what its observer costs. The horizontal axis is the tail ratio of the L cone's pigment absorbance — how far the curve reaches below its peak at half maximum against how far it reaches above — and the vertical is how far the observer built from that template sits from the 1931 standard. A real visual pigment has a long short-wavelength tail, so the three curves derived from a published nomogram sit above 1.1 and the two Gaussians sit below. The four are matched in width, so nothing here is about size. The ordering is the point: the two caricatures cost between two and four times what either nomogram does, and the axis they are separated on is the one feature the caricatures do not have. What the eye does

A template is mostly its tail

Four pigment templates matched to the same width at the same peak, differing only in which side of the peak their half-maximum reaches further. Ordered by that one number, the observers they build are ordered by how far they sit from the standard one — and the two with the tail on the wrong side cost two and four times what either real nomogram costs.

 The secondary band, dialled from nothing to twice what Govardovskii published. Govardovskii's template has a second, smaller absorption band below 400 nm, published at 0.26 of the α-band's peak. It is the one coefficient in the whole template whose contribution is somewhere else in the spectrum than the peak, and it is the part of the template a reader is least likely to have heard of. Sweeping it from nothing to twice the published value moves the median observer's distance from the 1931 standard from 0.9354 to 0.9790 ΔE₀₀, monotonically upwards. That is a small effect — about a fiftieth of the residual — and its being monotone is the interesting part: there is no interior optimum, so nothing here recommends the published value over any other, and a coefficient whose measured value is not the one that best fits an unrelated agreement is a coefficient to leave where the measurement put it. What the eye does

The band below four hundred

The pigment template every observer here is built from carries a second, smaller absorption band in the ultraviolet, published at 0.26 of the main one. Dialling it from nothing to twice that moves the median observer monotonically away from the standard one, with no interior optimum — which is what a physical constant looks like when nothing downstream is pulling it anywhere.

 The share of a sample's reflectance an aperture recovers, by how wide it is. Six materials, and the fraction of each one's true reflectance that a measurement recovers through an aperture of the stated radius. The horizontal axis is logarithmic in millimetres; the vertical is a share, so 1.0 is the whole of it. The dashed line is a 8-millimetre radius, which is about what a hand-held spectrophotometer has. At that aperture coated paper reads 99 per cent of its own reflectance and candle wax reads 66. Every curve approaches one from below and none of them reaches it: the kernel's tail is what is being cut, and it falls as one over the aperture rather than exponentially. What it takes to deliver it

Either disc can be the wide one

Every standard on translucent samples says to illuminate a larger area than is measured, and explains it by saying that light leaks out of the lit spot. That is true and it is not the reason, because the instruction works equally well the other way round — measuring a larger area than is lit gives a reading just as exact. The error is a product of two apertures and either one being wide kills it.

 Four departures from the model equation, each at an ordinary strength. What each of the four assumptions inside a colour integral costs, in ΔE₀₀, on a stated sample under a stated light. The wavelength index is a coated printing paper measured with and without the ultraviolet of D50; the range is the same paper integrated from 300 nanometres and from 380; the place index is a pigmented plastic through a four-millimetre radius; the direction index is an eggshell paint beside a window. The spread is a factor of 7.0. This is a ranking of four examples rather than of four departures — each of them can be made larger by choosing a more extreme sample, and the marble in the same collection of materials reaches 12.7 on the index that comes third here. What it takes to deliver it

Which index to buy an instrument for

Four departures from the colour integral, ranked by what they cost and by what it would take to remove each one. The ranking by cost and the ranking by price are almost exactly reversed — the two largest are removed by specifying a lamp and by widening a table, and the two that need new hardware are the two smallest.

 What is read at each distance from the edge of a lit region. Three materials under a half-plane of light, with the boundary at the centre of the horizontal axis and the lit side on the right. The vertical axis is the radiance leaving the surface as a share of what it leaves far inside the lit region. On the unlit side the sample is emitting light while receiving none, so the ratio the model calls a reflectance has a zero denominator there. The distance over which the curve runs from a tenth to nine tenths is 0.21 millimetres on coated paper and 5.1 on pale marble — which is the width of the neighbourhood a point's colour is decided by. What a camera does

A pixel has an aperture too

A camera photographing a translucent object has the same two discs a spectrophotometer has — one lit, one looked at — and gets them the other way round. Its illumination covers the whole scene, so the flat colour of a translucent surface comes out exactly right at any magnification, and the error moves entirely into the edges.

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