Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Matching is not appearance — page 4

CIE XYZ predicts when two lights will match under identical viewing conditions. It was never a model of how anything looks, and most of the confusion in applied colour comes from using it as one.
Every basis against both objectives at once. A scatter with the mean adaptation residual across the illumination census on the horizontal axis and the mean axis ratio of MacAdam's ellipses in a lightness–chroma space on the vertical. Lower is better on both. The two winners sit at the two ends of an empty diagonal: the basis that adapts best leaves 7.70 on the vertical and the basis that discriminates best leaves 1.79 on the horizontal, each worse on the other objective than every published transform. The basis built from the dichromat confusion points is at (1.65, 2.60) — best at neither and within a factor of two of both floors, which no other entry in the picture manages. What the brain does

No basis is good at both

The same nine numbers decide how well a von Kries gain reproduces a change of light and how nearly a lightness–chroma space makes the discrimination ellipses circles. Minimise either one and the other collapses. The basis built from the receptors is best at neither and is the only entry in the table respectable at both.

 How far from circles every basis leaves the ellipses. Eight bases ranked on the mean ratio of the long to the short axis of MacAdam's twenty-five discrimination ellipses, measured in a lightness–chroma space built on that basis. The range runs from 1.61 for best for discrimination to 7.70 for best for adaptation. The ordering is not the ordering on the other objective and is nearly its reverse. What the brain does

One matrix doing two jobs

CIECAM16 adapts in CAT16 and then applies its response compression in the same axes, so a single matrix decides both how well the model handles a change of light and how uniform the space it produces is. The two jobs have different best answers, and the matrix was chosen against only one of them.

 Five answers to how far the ellipses are from circles. Five mean axis ratios on the same twenty-five measured ellipses, measured the same way in every row: the boundary points carried through, the longest radius over the shortest, averaged. What differs is which class of map is allowed. The first two rows are chromaticity diagrams, which divide by a sum; CIE xy as printed leaves 2.95 and the best diagram there is leaves 2.02. The last three are lightness–chroma spaces, which divide by a white point; CIELAB as specified leaves 3.44, the best space with no compression leaves 2.33, and the best space with a cube root in it leaves 1.61. Neither family contains the other, and only the last one gets below two. Difference and uniformity

A compression goes below the floor

Elsewhere this collection minimised the anisotropy of MacAdam's ellipses over every chromaticity diagram there is, found 2.02, and called the residual a property of the eye. It is a property of the eye seen through a projective picture. A cube root after the right basis reaches 1.61 on the same twenty-five ellipses.

 The floor as a function of the exponent, and the fixed basis beside it. Two curves against the compression exponent on a logarithmic axis from 1 to 10. The lower curve is the best mean ellipse axis ratio any basis can reach with that exponent applied after it, and it falls from 2.33 at no compression to 1.66 at a square root and 1.61 at a cube root, then hardly moves — 1.57 at a tenth root. The upper curve is CIELAB's own basis at the same exponents and gets steadily worse, from 3.57 to 3.77. Almost everything a compression buys arrives with the first step away from linearity, and after that the exponent is choosing between 1.66 and 1.61 while the basis is choosing between 1.61 and 3.44. Difference and uniformity

The exponent was never the argument

A century of colour science has argued about whether the eye's response is a cube root, a square root or a logarithm. Minimise the anisotropy of MacAdam's ellipses over every basis, at each of eight exponents, and the floor moves by under three per cent between a cube root and a tenth root — while the basis moves it by a factor of two.

 Every change of light in the census, under three bases. Fourteen changes of illumination, each drawn three times: the residual left by a gain in the basis built from the dichromat confusion points, in CAT16, and in the basis that minimises the average. The ordering between the three is the same on nearly every row, and — the part an average hides — the worst row is the same row for every basis nobody fitted, which is two bounces off the same wall. The difficulty belongs to the change rather than to the choice of axes — except for the fitted winner, whose worst row is D65 to a triphosphor tube instead. What a fit buys is not a uniform improvement; it is the abandonment of the one change everybody else is beaten by. What light is

Which changes of light pay for it

A fitted adaptation basis beats the receptors by 0.68 units on average, and an average is a poor description of what it does. On six of fourteen changes of light it is worse, and the whole of its advantage comes from four — a tungsten lamp and three coloured walls.

 Three numbers for one set of ellipses, and which of them is which. Two curves and a horizontal line, against the size the ellipses are drawn at. The line is the analytic axis ratio — the ratio of the singular values of the map's own derivative, which is what "does this space make discrimination contours circles" means. The upper curve is a very finely sampled ring, which sits 0.6 per cent above the line at full size and converges onto it as the ellipse shrinks, because the gap between them is the second-order distortion of the map across a real ellipse rather than an error. The lower curve is the forty-eight-point sample used for this until now: it does not converge onto anything, because its error is set by the sample and not by the size. Difference and uniformity

Three numbers for one ellipse

How far a colour space is from making a discrimination contour circular has three different answers — what a coarse sample of the boundary reports, what a converged sample of a contour of stated size reports, and what the map's own derivative says. They differ by up to a factor of two, they mean different things, and only one of them is what the question is asking.

 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. What the brain does

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.

 The worst case is wherever the box stops. Four horizontal tracks, one per parameter of a painted wall. Each track spans the range an ordinary paint is allowed to occupy, with a second, wider range drawn behind it, and two markers show where the search for the worst change of light came to rest under each. Under the narrower box the answer sits on the wall in centre and width; under the wider one, in centre, width, base. The residual rises monotonically towards a narrower notch at a shorter wavelength on a darker wall, so there is no interior maximum to find. The worst change of light is 21.3 ΔE00 under one box and 28.4 under the other, and the census's own worst row is 3.37. What a scene does

The worst case is where the box stops

The worst change of light this collection quotes is two bounces off a green wall, and it is the worst of fourteen changes somebody wrote down. Searching the family those fourteen were drawn from reaches six times further — and does not stop, because the residual rises monotonically towards a narrower notch on a darker wall. There is no worst case in this family, and the number anybody quotes for one is a number about their own constraint.

 Best on the average, undefined at the edge. Two rows of bars sharing one set of labels. On the left, each adaptation basis's mean residual over the fourteen changes of light the census lists — Bradford is the shortest bar at 1.14 ΔE00 and is what colour management uses. On the right, the same bases against the worst change the same family of painted rooms can produce. Three of the five have no bar there at all, marked instead with the gain that replaced it: under a deep narrow notch their reading of the white passes through zero, so the diagonal is a division by nothing and the model stops being defined rather than merely doing badly. Bradford's middle gain reaches -1.0e+19. CAT16, which exists because CAT02 was withdrawn for going negative in practice, is one of the two that survives. What it takes to deliver it

Best on the average, undefined at the edge

Bradford has the lowest mean residual of any adaptation transform over the census of illumination changes, which is why colour management uses it. Inside the family that census was drawn from, its middle row's reading of the white passes through zero — so the gain is a division by nothing, and the model stops being defined rather than merely doing badly. CAT16, which exists because its predecessor did this, does not.

 What the confusion points charge, across a population. A histogram of 200 members of a population of eyes, each scored by what the adaptation basis their own confusion points determine leaves after the gain. It runs from 1.22 to 2.24 ΔE00 with a median of 1.78. Vertical marks show the unconstrained floor at 0.97, the published transforms, and the single observer this site quotes at 1.65. The distribution straddles Hunt–Pointer–Estévez and reaches below CAT16: 18 per cent of members are better served by their own receptors than by a matrix built to make a gain behave, and 2 per cent than by the current recommendation. Where the model breaks

The price is also the person

The receptor construction costs seventy per cent above the unconstrained floor, which is a number usually quoted as a property of the construction. Propagated across a population of eyes it runs from a quarter above the floor to a hundred and thirty per cent above it, and the population's own spread is wider than the entire gap between the published transforms the seventy per cent was being compared against.

 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.

 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.

 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.

 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.

 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

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.

 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. What the brain does

A model judged in another model's unit

An appearance shift is a change in what an observer would report, not a change in a stimulus, so measuring one with a matching difference means first asking what stimulus a settled observer would need to be shown to give the same report. That step is not bookkeeping — it is the whole distinction the field rests on, and it costs a factor of 1.7 across the menu.

 Every figure a surround dial would apply to, weighed. The eleven generators in the appearance family, by the size of the drawing each emits — which is what a dial multiplies, because every frame carries the figure's whole body. The dashed rules are the thresholds the dial policy here is written in: under 3 KB gets thirteen stops, under 6 gets nine, under 10 gets seven, and above about 12 KB nothing gets a dial at all. The median body here is 2.25 KB and 4 of the eleven are inside the most generous tier, against a policy written when the distribution had a median of 4.0 KB. Two phases deferred a surround dial on the grounds that appearance swatches are not small. The exception is unique-hues at 14.1 KB, which is over the ceiling that says it should have no dial — and it has one, on chroma, declared in the same file as the ceiling. What it takes to deliver it

A dial has a price

Two rounds declined to give the appearance model's surround a continuous control, both times because the frames would be too heavy, and neither time was the weight taken. It is 2.25 kilobytes at the median — the light end of these figures rather than the heavy end — and one figure is already breaking the rule the deferral was made under.

 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. Where the model breaks

Three choices reached

Two rounds ago this collection named three things it rested on and could not audit — a unit, a diagonal and a template. All three are now reached, and the interesting part is not the three answers but that four of the round's own predictions were refused by the arithmetic and one of its measurements was wrong in a way only a cross-check caught.

 The six arguments a surface's response has, and the one this model keeps. A surface's response to light is a function of six arguments: the wavelength, direction and place the light arrives with, and the wavelength, direction and place it leaves with. The model every colour here is computed from keeps one number per wavelength, which means it takes the diagonal of the first pair, integrates the second away, and assumes the third pair equal. Each departure drawn here restores one of them. The fourth departure is not on the diagram: the wavelength grid is the range of the index that was kept rather than an index that was dropped, which is why it is the cheapest of the four to fix and was still not fixed. Where the model breaks

The model has six arguments

Every colour computed here is an integral of a reflectance against a light against three curves, and a reflectance is one number per wavelength. A real surface's response is a function of six arguments — the wavelength, direction and place light arrives with, and the three it leaves with — so the model keeps one of them, takes a diagonal, integrates two away and assumes two more equal.

 Eight conditions under which the model equation is exact, and how exact each one is. Each of the four departures vanishes if either of its two factors is empty, which is eight conditions. The axis is logarithmic in the residual that is left when the condition is imposed. Three of the eight are identities: the fluorophore's loading is zero so the emitted term is an empty sum, and a Lambertian surface or a uniform field makes the pairing's second argument identically zero. The other five are limits — a Gaussian excitation band has no edge, an opaque sample still has a kernel a few microns wide, a four-metre aperture is still finite, and the observer is small rather than absent at 380 nanometres. Each limit is drawn with the sequence its residual falls along as the condition is pushed, because a small number is not evidence of a limit and a falling sequence is. Where the model breaks

Either factor being zero

Four different departures from the colour integral turn out to have one algebraic form — each is an inner product of something the sample does that the model has no slot for with something the light does that the model assumed away. Either factor being zero makes the departure exactly zero, and the sizes of the two factors decide neither how large it is nor which way it goes.

 An aperture and a gloss lobe, apart and together. Six materials, each measured through a four-millimetre radius and each given a gloss lobe, alone and at the same time. The pale bar is what the two cost added together as if they were independent; the dark one is what they cost when both are present. Every material comes out below the sum, by between 0.8 and 3.6 ΔE₀₀. The two departures partly cancel: the aperture removes light that went into the material and came back out too far away, and the interface returns light that never went in at all. Measuring either one alone therefore overstates what both together do, which is the opposite of the way interacting errors are usually assumed to behave. Difference and uniformity

Two departures that partly cancel

A glossy translucent sample has two of this round's four departures at once, and the expectation was that they would compound. They do the opposite. An aperture takes light away that went into the material and came back too far out; an interface returns light that never went in at all — so measuring either alone overstates what both together do, on every material tested.

 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 the eye does

Two blurs before the eye's own

A translucent object arrives at the eye already blurred, by a kernel that is a property of the material rather than of the optics. On every material in this collection's table that blur is coarser at ordinary reading distance than the finest detail the eye can resolve — so the softness of marble or skin or wax is not a failure of vision, it is the object.

 Two slabs with one reflectance, and two colours through an aperture. Two constructed media whose bulk reflectance agrees at every wavelength to fifteen figures, and whose diffusion lengths differ by a factor of four. The upper curve is that shared reflectance — both slabs lie on it exactly. The two patches on the right are what a 4 millimetre radius returns from each, and they are 6.3 ΔE₀₀ apart. This is a metamerism with no observer in it: the two samples are the same colour to anybody under any light, and the instrument separates them because it is measuring a kernel through a hole rather than measuring a reflectance. What the brain does

The appearance model has no slot for it

Two slabs with identical reflectance and different kernels come out of CIECAM16 with identical lightness, chroma and hue — to six decimal places, because the model's input is a stimulus and their stimuli are the same. Translucency is a recognised appearance attribute with no correlate in any appearance model, and the reason is structural rather than an oversight.

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