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

Eigenvalue — where it appears

A number saying how much a matrix stretches along one of its own special directions, the eigenvector, which it does not rotate. For the second-derivative matrix of an objective the eigenvalues are curvatures, so a zero one is a direction along which the objective does not change at all and the model cannot see.

Named by 13 essays across 6 fields — each of them below, with the objects they name alongside it.

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 The cheapest direction to give ground in is the flattest one. Six bars, one per direction the adaptation objective can see, showing how much of the other objective a fixed budget of adaptation buys if it is spent along that direction. The rate is the slope of the second objective divided by the square root of the first's curvature, so it rewards a direction the second objective wants and punishes one the first is stiff in. The flattest direction wins at 10.68 against 3.29 for the next best and 0.54 for the stiffest — a factor of 20. Spending 1 per cent of the adaptation optimum there moves the anisotropy from 7.70 to 5.02.

The trade only runs one way

Standing at the basis that adapts best, one per cent of adaptation buys forty-four per cent of the way to the discrimination floor. Standing at the basis that discriminates best, the same one per cent buys under two. The scatter that shows two objectives pulling apart looks symmetric and is not, and the asymmetry is what a committee choosing between them would most want to know.

brain · Appearance Which of a camera's three dyes each direction moves. A grid with one column per direction — stiffest on the left, flattest on the right — and one row per parameter of a camera's three dyes. Each cell's bar length is that parameter's share of that direction, so a column with one long bar is a direction that moves one thing. The stiffest column is dominated by blue centre, at a weight of 0.96. The flattest column is spread across blue width, red width, green width — a combination rather than any single number, which is why a specification listing one tolerance per parameter cannot express it.

Where a camera is blind to itself

A colour filter array is six numbers — three dye centres and three bandwidths — and how well the resulting sensor adapts is far more sensitive to some combinations than to others. The stiffest direction is almost entirely where the blue dye sits. The flattest is all three bandwidths at once, and the design can move twenty-five times further along it for the same cost.

imaging · Capture A primary's tolerance is a shape, and part of it is unreachable. The CIE chromaticity diagram with the spectral locus drawn, and three closed regions marking where each primary of a display designed for its own inverse can sit while the adaptation cost stays within 1 per cent of its best. None of them is round: the widest runs 15.1 times further one way than another, so a single tolerance figure for a primary is the average of a shape the shape never takes. 47 of the 144 boundary directions leave the region a real primary can occupy, which is a second constraint the objective knows nothing about — the cost does not rise there, and the primary cannot go there.

A tolerance is a region

How far a display's red primary can move before its adaptation behaviour costs anything is not a distance. It is a closed region on the chromaticity diagram, fifteen times longer one way than another, and about half of it lies outside the area a real primary can occupy — where the objective does not rise and the primary cannot go.

applied · Delivery 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 A camera's dye widths are free under one requirement and not under another. Three panels, one per dye. In each, a pair of bars per requirement: how far that dye's centre wavelength and its bandwidth can move before the requirement gets five per cent worse. Under the adaptation objective — the one the previous round measured — every width has far more room than its centre, which is the finding that put a tolerance budget on the centres. Throughput and the colour matrix's noise gain, the two requirements that objective was said to be silent about, reach the edge of the search in every direction and hold nothing. What tightens the widths is the Luther residual, which was in the model already. The bottom pair in each panel is what survives all four.

The widths were free because nothing else was asked

A camera's three dye bandwidths carry almost all of the flattest direction of the adaptation objective, so that objective says a tolerance budget belongs on the centre wavelengths. The two requirements it was said to be silent about turn out not to bind either — and the one that does was in the model already.

imaging · Capture What a camera's dye 1 is allowed to be, under four requirements. The plane a colour-filter dye is designed in: its centre wavelength across, its bandwidth up, both in nanometres, so the two axes are comparable and the shapes mean something. Four outlines, one per requirement, each the set of dyes within five per cent of the designed one on that requirement; the shaded region is where all four hold. Two of the four — throughput and the colour matrix's noise gain — reach the edge of the search in every direction and are invisible as boundaries. The intersection is ±6.8 nanometres of centre and ±13.3 of width, against the adaptation objective's own ±20.6 in width alone.

Two tolerances do not meet in a tolerance

A specification lists requirements separately and a manufacturer has to satisfy them together. Where two long thin regions cross at an angle, what is left is much smaller than either, its longest direction is neither of theirs, and no list of tolerances describes it.

imaging · Capture 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 At a published matrix the slope arrives long before the bowl. One row per basis in this collection's table. Each row is a logarithmic axis of distance in the nine coefficients, with two markers: the radius at which the objective's curvature becomes as large as its slope, and the distance from that basis to the optimum. The first is between 3.2 and 108 per cent of the second. So over almost the whole journey from a published matrix to the best one, the surface is a slope and not a bowl — and a table of eigenvalues taken there describes a neighbourhood the optimum is nowhere near. XYZ scaling is the exception, at 1.08 of the distance, because its slope is the steepest in the table.

The slope arrives before the bowl

The adaptation transforms colour management actually uses are not optima of anything. At every one of them the objective has a slope, and the slope is the larger term over almost the whole distance to the best matrix — so a table of curvatures taken there describes a bowl nobody meets on the way anywhere.

applied · Delivery 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

Named alongside it

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

Chromatic adaptationCondition numberDegrees of freedomHessianAnisotropyBasisDeclared inputQuadratic formLevel setToleranceCAT16Luther condition

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