Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Computed, not quoted — page 10

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 far a fitted transform is from anybody's eyes. Five groups of three bars: for each published adaptation transform, the distance from the population's own cloud to the confusion point that transform is committed to, measured in the population's standard deviations on that point. On the protanope's point every one of them is between 2.2 and 14.4 out, and on the deuteranope's between 3.7 and 11.8. On the tritanope's, 5 of the five are within three standard deviations — indistinguishable from a member of the population. The claim that these matrices are not cone responses is safe, and the evidence for it is two points out of three. Matching and measuring

Two points out of three

Every published adaptation transform implies three dichromat confusion points, whether or not it was fitted to any. Measured in the population's own standard deviations they are two to fourteen out on the protanope's point and four to twelve on the deuteranope's — and between half a standard deviation and two and a half on the tritanope's, which is inside the population. The claim that these matrices are not cone responses is safe. The evidence for it is two points.

 A population of receptor bases, in the plane the published ones live in. The two axes this collection scores an adaptation basis on — the residual across the illumination census on the horizontal, ellipse anisotropy on the vertical, lower better on both — with 200 extra points on it. Each is the basis a member of the population's own confusion points determine. The cloud is not a point: it runs from 1.22 to 2.24 ΔE00 horizontally, which is wider than the whole spread of the published transforms marked on it. The observer this site quotes sits inside the cloud and near one edge of it, and the sentence "the receptor basis costs seventy per cent" is a sentence about that one point rather than about the construction. What the brain does

A trade between matrices, not people

Across the space of possible bases, adapting well and discriminating well pull in opposite directions — the two optima sit at the ends of an empty diagonal. Across a population of actual observers the same two costs move weakly together, at a correlation of +0.29. The trade-off is a property of the set of matrices somebody could choose, not of the eyes anybody has. One measurement inside the population does trade, and it is the macular pigment.

 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. What a camera does

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.

 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. What it takes to deliver it

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.

 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. What the eye does

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.

 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. What the eye does

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.

 How wrong the ellipses would have to be for a pair to change places. One bar per adjacent pair in the uniformity table: the relative error on each ellipse's own axes at which that pair changes places in one draw in twenty. No error on the data is quoted anywhere — the question is inverted, so what is reported is how large an error would have to be, and a reader with an opinion about MacAdam's experiment can compare it with their own number. The nearest pair goes at 0.171; 2 of the 7 pairs do not reverse under any error this search covers. Difference and uniformity

How wrong would the data have to be

Twenty-five ellipses measured on one observer in 1942 are the ruler every colour space here is judged against, and they have never been given an error. Rather than invent one, the question is turned round, and asks how large an error would have to be before the ranking changed.

 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. Difference and uniformity

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.

 A mid-grey's lightness across a continuum of rooms. The lightness a mid-grey is predicted to have, plotted along the continuous surround parameter running from an average room to a dark one. The three rooms the standard tabulates are marked on it: average at the left, dark at the right, and dim 61% of the way between them rather than halfway. The whole span is 9.71 units of lightness and the step from average to dim is 5.64 of it — 58% — so choosing one of the three rows is a decision worth most of the range. What the brain does

The surround is three rows of a table

An appearance model takes the room as three constants, and the standard tabulates three rooms. Every appearance figure in this collection is drawn at one of them. The parameter they are three points of is continuous, and the middle row is not in the middle.

 Every adaptation number here assumes a complete adaptation. Three curves and their mean: the colour difference an adapted observer is left with after a change of light, against the degree of adaptation from zero — no adaptation at all — to one. Every adaptation figure in this collection is computed at one, the right-hand end. The appearance model's own formula puts the degree at 0.941 for an average surround at a hundred candelas, marked, where the residual is 2.21 ΔE00 rather than 1.27 — larger by a factor of 1.74. The left-hand end is exactly the unadapted change, which is not an approximation but an identity, and is what says the curve interpolates between the two things it claims to. What the brain does

A discount nobody measured

Every adaptation number in this collection assumes an observer who adapts completely. The appearance model's own formula says they do not — it puts the degree at 0.94 in an ordinary room — and the difference is not a rounding. It is a factor of 1.7 on the residual every one of those figures reports.

 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. Where the model breaks

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.

 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. Where the model breaks

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.

 The family does have a worst case, at a band no pigment can cut. The worst change of light a painted wall can produce, at each band width, with the wall's centre wavelength, depth and base optimised at every point. The horizontal axis is logarithmic in the width. The curve rises as the band narrows, turns over at about 6.02 nanometres, and falls again — a band that narrow returns too little light to move the white much. The previous round's search reported no worst case because its box stopped at ten nanometres, marked, which is on the wrong side of the turn. The peak is 28.54 ΔE00 against 28.38 at that floor, which is 0.6 per cent higher: wrong in principle, right in practice to a fraction of a per cent. What light is

A notch a pigment cannot cut

The worst change of light a painted room can produce has no maximum inside the box the search was given, which the previous round reported as a family with no worst case. Bounded by what a molecule can actually do, it has one — at a band six nanometres wide, narrower than any pigment and narrower than the box.

 The winner survives the census's own construction; the middle of it does not. One row per perturbation of a constant the adaptation census is built from — the imaginary wall's centre wavelength, its width, its depth, its base, the macular filter's density and the two lens ages — each moved by an amount plausible for that quantity in its own units, up and down, and then all of them together. Each row shows where the five published transforms rank under it. Bradford holds the first column in all 14 rows. The second and third columns, which the table as built separates by six parts in a thousand, change places in 2 of them — so that ordering was never a fact about the transforms. What light is

The census is a construction too

Five of the fourteen changes of light this collection scores adaptation transforms against are not measurements of anything — they are a wall somebody invented, at a wavelength somebody chose. Moving those constants by amounts plausible in their own units moves the mean residual by two fifths and never changes which transform wins.

 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. What a camera does

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.

 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. What a camera does

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.

 A room applies its wall a different number of times at each wavelength. The mean number of bounces the surviving light has made, wavelength by wavelength, in a closed room whose walls are the green paint the adaptation census uses. It runs from 0.33 in the band the wall absorbs to 5.67 in the band it reflects — a factor of 17.00 — because the light that survives many bounces is the light the wall was reflecting all along. The census has one bounce and two bounces as separate rows and a search treats the count as a free integer; a room has neither, and what it has is bounded by the walls reflecting less than everything. What a scene does

A room bounds its own bounces

The adaptation census has one bounce and two bounces as separate rows, and a search over the family treats the count as a free integer it always takes to the largest value offered. A room offers no integer at all — it applies a geometric mixture of every number of bounces, and that mixture is bounded by the walls reflecting less than everything.

 The three worst walls, drawn as the reflectances they are. Three reflectance curves, one per bound: the wall each search settled on. All three are dark over most of the spectrum with a single band near the short-wavelength end — the arithmetic bound's is 10 nanometres wide, the physical one's 40, and a paint somebody sells the same. None of them is a saturated colour: their excitation purities are 0.18, 0.52, 0.52 against a ceiling of 0.6, which is why the purity constraint never bites. What breaks an adapted observer is a wall that takes most of the light away, not one that is a strong colour. What a scene does

The darkest wall anybody sells

Asked which property of a paint decides the worst change of light a room can produce, anybody would answer how saturated it is allowed to be. A ceiling on saturation never comes near binding, because the worst wall is dark rather than colourful — and the constraint that does bind is one nobody would nominate.

 What a display's red primary is allowed to be, under four requirements at once. A close view of the chromaticity plane around one designed primary, 0.101 units across. Four outlines: the set of positions the primary can take before each of four requirements gets one per cent worse — how well a gain in the display's own basis undoes a change of light, how much of the diagram the three primaries enclose, how many real surfaces fall inside them, and whether a light of that colour exists at all. The shaded region is where all four hold. It is 5% of the smallest outline's area, because the outlines are long and thin and cross at an angle rather than nesting. adaptation holds 42% of its boundary, gamut holds 10% of its boundary, realisable holds 48% of its boundary. Matching and measuring

A primary is chosen for four things

A display's primaries have to adapt well, cover the diagram, hold the surfaces anybody photographs, and be colours a light can actually have. Drawing all four tolerance regions around one primary shows that no single requirement decides where it can go, and that one of the four never decides anything.

 The same tolerance, in the two numbers somebody actually sets. The plane a maker of a single-peak emitter works in: peak wavelength across, full width at half maximum up. Each marker is a candidate emitter whose chromaticity falls inside the colorimetric tolerance drawn for this display's green primary. They occupy a narrow band — peaks from 528 to 535 nanometres, a span of 7, against widths from 25 to 45 — so a tolerance stated as a region in chromaticity becomes ±3.5 nanometres of peak and a great deal of latitude in width. 2.0% of the 2501 candidates land inside at all: most of a region drawn in chromaticity is a colour no single-peak emitter makes. Matching and measuring

A tolerance in the wrong coordinates

A display primary's tolerance is written as a region in chromaticity, because that is where the colorimetry lives. Nobody has a knob for chromaticity. What a maker of an emitter sets is a peak wavelength and a bandwidth, and the map between the two is so anisotropic that on the red primary its condition number is over eleven thousand.

 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. Matching and measuring

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.

 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. What it takes to deliver it

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.

 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. What it takes to deliver it

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.

 Where a delivered colour's error actually comes from. Three bars and two totals. The profile's interpolation between its nodes contributes 0.13 ΔE00; the rendering intent, at the colorimetric setting, moves nothing that was already printable and contributes 0.00; and looking at the result in a dim room rather than the booth it was proofed in contributes 4.30, which is 97 per cent of the total. The two totals are the three added — 4.43 — and combined in quadrature — 4.30. The gap between those two rules is 0.127, about the size of the entire profile stage, and no standard says which rule to use. What it takes to deliver it

A delivery tolerance is three tolerances

A colour reaches a reader through a profile, a gamut mapping and a room, and each has a number somebody chose. Each has an essay of its own here, and none of those essays had put the three costs in one column. Put there, the stage nobody controls carries ninety-seven per cent of the total.

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