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

Degrees of freedom — where it appears

How many independent numbers a model carries, as against how many its data determine. The difference between the two is the dimension of the family of answers that fit equally well, and it is the quantity a residual is guaranteed to be silent about.

Named by 23 essays across 8 fields — each of them below, with the objects they name alongside it.

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.

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.

eye · Cones 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.

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.

eye · Cones 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.

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.

scene · Scene 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.

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.

imaging · Capture 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.

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.

limits · Limits 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.

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.

eye · Cones The same optimum, along its narrowest direction and its widest. The adaptation objective along two straight lines through its own minimum, both of unit length in the nine coefficients. Along one of them the cost rises steeply; along the other the same step costs 8.0 times less, and a design constrained to move that way gives up almost nothing. That is why restricting the nine numbers to be the inverse of three realisable primaries — three degrees of freedom gone — costs about one per cent, while requiring them to hit the three dichromat confusion points costs seventy. Counting what a constraint removes predicts neither number; what matters is which way it points.

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.

limits · Limits The points a ratio needs are proportional to the ratio. A scatter of 133 points on logarithmic axes, one per MacAdam ellipse under each of six coordinate systems. The horizontal position is that ellipse's true axis ratio; the vertical is the smallest sample size, from a sequence of doublings, at which the sampled ratio comes within one per cent and stays there. A line of slope 0.94 runs through them, against a predicted 1 — the minimum's notch is 0.88 σ₂/σ₁ radians wide, so resolving it takes a number of points proportional to σ₁/σ₂, and nothing about the basis or the ellipse enters beyond that. An ellipse with a ratio of two needs seventeen points and one with a ratio of twenty-six needs a hundred and ninety-two.

An extremum is not a sample

Three separate measurements in this collection took a maximum or a minimum over a sample of a set — forty-eight points round an ellipse, twenty-four directions out of an optimum, fourteen changes of light off a list. All three are wrong, all three are wrong in the same direction, and the error in each grows with the very quantity being measured.

limits · Limits Nine eigenvalues, six of which exist. Nine points on a logarithmic vertical axis: the eigenvalues of the Hessian of the adaptation residual at its own optimum, largest to smallest. The first six run from 6.8×10² down to 7.6×10⁻¹, a condition number of 890. Then the axis drops: the seventh is 1.9×10⁻⁴, and the last three are separated from the sixth by a factor of 4.0×10³. Those three are not small curvatures. They are the finite-difference truncation error on directions along which the objective is exactly constant, and a shaded band marks them as the numbers the objective does not have.

The rank is the invariance

A von Kries gain cannot see the scale of a row of its basis. That is an identity, proved in a line, and it can be measured instead — as the rank of a second-derivative matrix. Both objectives this collection minimises over the observer's nine free numbers have a Hessian of rank exactly six, and the three directions they cannot see are the three scalings, to a hundredth of a degree.

eye · Cones The bowl the eigenvalues describe and the bowl a sample found. Six points on a logarithmic vertical axis — the distance from the optimum of the adaptation residual to a 5 per cent rise along each of the six directions the objective can see — with a shaded band behind them showing the whole range 24 random directions reported. The eigen-radii run from 1.2e-2 to 3.6e-1, a factor of 29.8. The band runs from 2.2e-2 to 1.8e-1, a factor of 8.0, and sits entirely inside the ends of the true range: a random direction in nine dimensions carries a share of every eigenvector and so reports the middle of the bowl, never an end of it.

How long is the bowl

The optimum of an adaptation objective is twenty-nine times longer one way than another, and the two ends have names — the stiffest direction is almost entirely the short-wavelength row of the basis, the flattest almost entirely the long-wavelength one. Twenty-four random directions reported a factor of eight, and no number of them would have done better.

matching · Gamut 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 width at the wide end of its span, and at the narrow end. One line per published quantity, each spanning the value it takes when all four declared widths are read at the narrow end of their reported ranges to the value at the wide end, with a marker at the value as declared. The largest span is the deutan margin at a factor of 2.62; the smallest is 1.22. This is the reading the population model's own documentation promised for four phases and nothing ever took. It is not a confidence interval — the four ends are not quantiles and the widths are not independent draws — it is what a reader who distrusts all four at once sees.

A width nobody varied

Five numbers say how much people differ from one another, and every conclusion drawn here about a population rests on them. Each was written down with the range the literature reports beside it, so that a result could be re-read at the pessimistic end. Nothing ever was.

eye · Cones Which width carries the answer, and which carries the doubt. Two columns of bars over the four things that differ between two pairs of eyes. On the left, the share of the population's disagreement each one accounts for — the attribution quoted here since the population was built, which puts the lens first at 81%. On the right, how much of the doubt each one puts on everything published here, which is its elasticity multiplied by how badly the width itself is known. The macular pigment comes first there, at 0.65 against the lens's 0.60 — a lead of 8%. The two lists agree exactly below the top.

Which measurement is worth making

Four things about an eye differ between people, and they have been ranked here by how much of the answers they carry since the population was built. Ranking them by how much doubt they carry gives a different order, and ranking them by which one takes a published claim closest to failing gives a third.

eye · Cones Twenty-five ellipses is a sample, and the score has an error bar. One row per colour space this collection ranks: the mean axis ratio its ellipses come out at, with the standard error of that mean over the twenty-five ellipses it was computed from. No literature is quoted — a mean of twenty-five numbers has a standard error those twenty-five numbers determine. The bars are far from equal: the best space carries ± 0.07 and the worst ± 1.56, because a space that makes the ellipses nearly circular makes all of them nearly circular and one that does not is dominated by whichever ellipse it handles worst.

Twenty-five is a sample of the diagram

A colour space's uniformity score is the mean of twenty-five numbers, and a mean of twenty-five numbers has a standard error those twenty-five numbers determine. Nothing has to be quoted to compute it, and three of the seven adjacent pairs in this collection's ranking survive it.

difference · Metric A quadratic is believed least far at the one place anybody takes one. One bar per basis: the radius, in the nine coefficients, within which the second-order model predicts the objective to within ten per cent in every one of eighteen directions. The shortest bar is the objective's own optimum, at 2.3×10⁻², and the longest is XYZ scaling at 1.1×10⁻¹ — several times further. The reason is not that the model is worse at a minimum but that it has less to do there: away from one the linear term is exact and carries most of the change, so a ten per cent error in the prediction takes longer to accumulate. It does not make a Hessian at a minimum wrong; it says the picture drawn from it describes the smallest neighbourhood in the table.

How far a quadratic can be believed

A second-order model has a radius inside which it describes a surface and outside which it does not, and that radius can be measured. Measured at eight places on one objective, it is smallest at the optimum — the one place anybody ever takes a Hessian.

limits · Limits Room is not safety: two orderings of the same three claims. Three pairs of bars, one pair per published statement about the confusion points. The upper bar in each pair is the margin — how far the measured number is from the threshold that makes the statement true, as a ratio. The lower bar is the headroom — the factor by which one declared width of the population model would have to be wrong for the statement to fail. Both start at one, which is the line. Ordered by margin the three read the protan margin, the tritan margin, the deutan margin; ordered by headroom they read the protan margin, the deutan margin, the tritan margin, and the middle two change places. Every one of the three is inside a factor of two of failing, which the margins do not say.

What would have to be wrong

A great many statements here have thresholds written into them, which turns out to make an audit possible — for each one, the smallest change in a declared input that would stop it holding. Most are unreachable. One is inside a factor of one and a third.

limits · Limits Exactly flat everywhere, and eigenvectors at one point only. Two columns over the same nine places. On the left, how far from zero the objective's second derivative is along a row-scaling direction, on a logarithmic axis — it is between 10⁻⁹ and 10⁻⁶ of the largest eigenvalue at every one of them, which is a numerical zero. Scaling a row of the basis is a straight line along which the cost does not change, and that is true at every point, not only at the optimum. On the right, the angle between those three directions and the Hessian's own three smallest eigenvectors: 0.025 degrees at the optimum and up to 88 away from it. An invariance is a property of the function; being an eigenvector is a property of the function at a minimum, and the two coincide only where everybody computes.

Only the flat directions keep their names

Three of the nine numbers a colour match leaves free do nothing, and they do nothing everywhere — exactly, at every basis in this collection's table. They are the objective's own principal directions at one point only, and everywhere else the directions carrying the curvature have turned by tens of degrees.

matching · Gamut Downhill from every published matrix, one step at a time. Each curve is a steepest-descent walk from one of this collection's published bases, plotted as the objective against the distance walked in the nine coefficients. The horizontal line is the optimum. The first step of each walk is the long one — XYZ scaling closes 41 per cent of its whole gap in one — and every walk then flattens without reaching the line, because the valley floor is nearly flat and the steepest direction is nearly across it. Bradford starts closest and closes least: it is already in the flat part.

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.

applied · Delivery 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.

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.

scene · Scene How far each census row moves when the test set's own description does. A grid of bars, one row per change of light in the census and one bar in each row per number that describes the region the test surfaces are drawn from: how saturated they are, how bright, and how far the two modulations may go together. A bar's length is the elasticity — the proportional change in the published residual for a proportional change in that number. Saturation runs from 0.49 to 0.91 and brightness averages 0.104, so a test set's chroma range is nearly everything and its lightness range is nearly nothing. For scale, the largest elasticity found anywhere among this collection's five declared population widths is about a half — and those at least have declared ranges, while these three numbers have never been quoted with one.

The input nobody declared

An audit that swept every declared width in this collection found the largest elasticity anywhere to be about a half. The most elastic input turns out to be one that was never declared, never quoted with a range and never varied — and being undeclared is exactly why it escaped the audit that was looking for it.

limits · Limits 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.

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.

eye · Cones 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.

What the audit still cannot reach

Two rounds have now swept every declared width in this collection and one of its structural choices. Three structural choices remain, none of them has a multiplier to sweep, and the reason each resists is different — which makes the list a description of where this kind of audit ends rather than a queue of work.

limits · Limits One lightness scale, drawn as a contour across the rooms. Three surrounds up the page and the background's luminance factor across it, on a square-root scale so that the model's own exponent base is linear in the axis. Each curve joins the rooms whose lightness exponent is the same, and every room on one curve returns the same lightness for every sample. The marked curve is the one through a television in a lit living room against a mid grey: it also passes through a print on a desk against a background of 2.8 and a projection in a dark room against 47.0. A curve that leaves the plot has no member in that surround, because each surround multiplies a base that runs only from 1.48 to 2.48.

Two rooms with one lightness scale

CIECAM16's surround and its background both reach lightness, and they reach it through one product. So the rooms fall into classes: a television in a lit living room against a mid grey returns exactly the lightness a print on a desk against a background of 2.8 does, for every sample, to the last bit of a double. It returns 0.76 of its chroma and 0.81 of its brightness. Two of the model's four viewing-condition parameters are one parameter, and only colour tells them apart.

brain · Appearance

Named alongside it

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

Chromatic adaptationIdentifiabilityBasisEigenvalueSamplingCone fundamentalsThe von Kries transformCondition numberConvergenceDeclared inputQuadratic formAnisotropy

All concepts