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

Every essay — page 21

Page 21 of 40, continuing through the fields in the same order.

What light is What the eye does Matching and measuring Difference and uniformity What the brain does What a scene does What a camera does Where the model breaks What it takes to deliver it

SeriesObserversNamed objectsRefutationsSearch

Where the model breaks

Seventeen observers in 1931, an error in the blue that was never fully repaired, colour vision deficiency, and the display this page is being read on.

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.

7 figures 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.

A choice with no magnitude

An audit can multiply a width by 1.25 and report an elasticity. It cannot multiply CIEDE2000 by anything. Auditing a structural choice needs a different instrument, and building one shows that six published numbers in this collection each carry a factor of about two of unit-choice — after the change of scale has been taken out.

9 figures 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.

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.

7 figures 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.

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.

7 figures 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.

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.

8 figures 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.

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.

6 figures Which of the collection's published quantities a departure can be pushed through. The six quantities the previous round recomputed under six different colour-difference units, and whether the same treatment works for a departure. Two do: the adaptation census and the metameric pair both take reflectances and a light, which is what a departure acts on. Four do not, and the reasons are different in each case rather than a single obstacle. A unit is a function applied to the answers, so it can be swapped at the end of any computation; a departure changes the object at the start, so it has to be accepted by every stage in between. That is the practical difference between auditing a convention and auditing a structure.

A departure is not a unit

The previous round audited six published quantities by swapping the unit they were quoted in — a function applied at the end of each computation. Nothing of that shape works here. A departure changes the object at the start, so every stage in between has to accept it, and only two of the same six quantities can take one at all. The fourth cannot even be expressed in the interface.

7 figures 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.

The audit that changed the object

Three rounds audited this collection's numbers, its sets and its conventions, and each one ended by naming the same thing it could not reach — that a surface is a reflectance. This round reached it, found that four departures share one algebraic form, had two of its predictions refused, and discovered that its own hemisphere quadrature had been passing a convergence check by luck.

7 figures How far this collection's analytic observer is from the tabulated one. The construction every figure in this family uses is three pigment absorptances through one fitted 3×3, and this is the residual of that fit against the CIE's 1931 functions: root-mean-square error as a percentage of each curve's peak, and the colour difference it produces over forty-two surfaces. The short-wavelength function is the worst at 16.4 per cent, which is where a pigment template is weakest and where the ocular media are doing most of the work. The median colour difference is 1.42 ΔE₀₀, so this is an observer of the right shape rather than a copy of the table — and every departure in this family should be read beside that number rather than against zero.

The reference had to be built

A five-nanometre error cannot be measured with five-nanometre data. Interpolating the tables and integrating finely measures the interpolator, not the grid — so the audit of this collection's index had to be run against an observer made of formulae, and the price of that is a residual of 1.42 ΔE₀₀ that every number in the section is read beside.

6 figures What this collection's grid does to its own observer audit. Two bars per light: the mean departure of the observer computed on this collection's five-nanometre grid, and the same computation on a quarter-nanometre one. For five of the six lights the two agree to two decimal places, which is what a well-sampled spectrum looks like. For the laser projector the coarse answer is exactly zero and the fine one is 2.34 — the largest in the table. On a five-nanometre grid a three-line spectrum with lines at 465, 532 and 638 nanometres is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about to the last bit. The two departures do not compound here; the first conceals the second.

The grid hid the observer

On this collection's five-nanometre grid a three-laser projector's observer disagreement is exactly zero. On a quarter-nanometre grid it is 2.34 ΔE₀₀ and the largest in the table. The two audits of this round meet here, and the first one does not compound with the second — it removes it.

5 figures The arguments a standard observer does not have. Seven choices inside a set of colour-matching functions, each with the shape it takes and what it is worth in ΔE₀₀ on a red pigment under a 6500 K radiator. Six are measurements: a field size, an age, a macular density, a cone optical density, three peak wavelengths and a rod contribution. The seventh is not — a change of basis is a change of curves and not a change of observer, and its entry is exactly zero because the space an experiment measures is what an observer is. Printing that zero beside the others is the clearest statement of what the other six are measurements of.

An observer is a contract

Six departures, all above a delivery tolerance, none dominant, and no reader matching the standard on any of them. That would be a demolition if the standard observer were a description of an eye. It is not — it is an agreement about what a number means, and an agreement is not falsified by being unlike anybody.

6 figures Where a point-sampled patch stops resolving a lobe. The horizontal axis is the wall's roughness, logarithmic; the vertical is how far the answer moves when the cone quadrature is refined from eight directions to sixteen, also logarithmic. A patch in a cube subtends a cone of angular radius 25.8° at the face opposite, and a microfacet lobe of roughness a is about a radians wide, so a lobe below about 0.15 is narrower than the quadrature that samples it. The movement at 0.05 is 21.2 ΔE₀₀ and at 0.3 it is 0.033. This is where the method stops, not where the paint does: matt, eggshell and satin finishes are inside it and a high-gloss varnish is not.

Where a patch stops being a point

A patch in a cube subtends a cone of 25.8° at the face opposite. A microfacet lobe of roughness 0.05 is about three degrees wide. Point-sampling the second inside the first returned a chroma of thirty-four thousand and a negative lightness, and the boundary between working and not working is measured rather than declared.

5 figures