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The thread: The instrument is the reader — page 8

This is the one subject where the page is displayed on the apparatus under discussion, and the reader's own eye is the measuring device. Several figures here are experiments rather than illustrations.

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 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.

a fluorescent tube, mercury lines on a phosphor bed, with a 5-nanometre grid marked on it. The light drawn at a fifth of a nanometre, with the 5-nanometre tabulation points marked beneath. 38 of the 53 points carry more than a twentieth of the peak. What a summation over those points computes is not an approximation to the area under this curve; on a spectrum with features narrower than the spacing it is a different quantity, and the difference depends on where the points fall rather than on how many there are.

What light is

The index is a choice too

Every colour in this collection is a sum over eighty-one numbers running from 380 to 780 nanometres in steps of five. That index is not a property of the eye, the light or the sample — it is a tabulation, and it holds three separable decisions that behave completely differently from one another.

What an instrument's slit width does to a tabulated colour. The horizontal axis is the full width of a triangular slit, from zero — perfect point sampling — to twenty nanometres; the vertical is the distance from the true colour, logarithmic. For a smooth light the lines are flat: a slit narrower than any feature changes nothing. For a line spectrum they fall off a cliff at the left. Point-sampling a mercury line at five nanometres costs 1.26 ΔE₀₀ and integrating the same spectrum through a five-nanometre slit costs 0.014. A spectrometer does not sample a spectrum; it integrates one, and the blur everybody would remove if they could is what makes a five-nanometre table safe.

What light is

The slit is what makes it legal

Point-sampling a mercury line at five nanometres costs a colour difference of one unit, and a laser projector thirty-seven. Integrating the same spectrum through the five-nanometre slit every spectrometer already has costs 0.014 and 0.19. The blur anybody would remove if they could is what makes a coarse table honest.

How much the answer moves when the 5-nanometre grid is slid through one cell. Each bar is the spread of one light's colour across five grid origins, all at the same 5-nanometre step, in ΔE₀₀. A smooth light barely moves, and what movement it has is the end cells rather than the sampling. The fluorescent tube moves by 3.18 units and the laser projector by 35.0, because their emission lines are narrower than the step and whether a sample lands on one is a coincidence of arithmetic. This is the measurement that separates a quadrature error from an aliasing error, and no average over origins can substitute for it.

What light is

Where the grid starts

Holding the step at five nanometres and sliding the grid's origin through one cell moves a fluorescent tube's computed colour by 3.18 ΔE₀₀ and a laser projector's by 35.0. Refining the step does not fix it and averaging over origins hides it. It is the one tabulation fault with no smooth error to cancel against.

The two tabulation choices over forty-two surfaces, under a 6500 K thermal radiator. Each column is one choice, measured over a family of forty-two analytic reflectances rather than on a single example: an absorption band of stated centre, width and depth. The four marks are the smallest, the median, the ninety-fifth percentile and the largest cost in ΔE₀₀, logarithmically. Under a smooth light the range is worth 9.1 times the step at the median, so a collection wanting one repair should widen its range rather than refine its step — and under a fluorescent tube the ranking reverses outright.

What light is

Which end to buy

Refining a five-nanometre grid to one buys a daylight calculation 0.05 ΔE₀₀ and widening its range buys 0.54. Under a fluorescent tube the same two purchases are worth 0.83 and 0.0001. The ranking reverses completely, and what decides it is one length compared against one other length.

The rectangle sum against the trapezoid sum, under a 6500 K thermal radiator. Colorimetry's summation is the rectangle rule at the tabulated points. The trapezoid rule differs from it by exactly one thing — half a cell at each end of the range — and the gap between these two lines is therefore that term and nothing else. At five nanometres it is a factor of 14.3, which means the number everybody calls a sampling error is mostly a truncation error wearing the step's clothes. On a light whose lines are narrower than the step the two rules agree to three decimal places, because there the error really is the sampling.

What light is

The endpoint term has a name

The five-nanometre error on a smooth light falls linearly with the step, which is not what a sampling error does. It is the half-cell at each end of a truncated range, it is first order where the sampling is second, and halving two weights removes fourteen fifteenths of it for nothing.

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.

Where the model breaks

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.

What a tabulation step costs, by light, on paper. The horizontal axis is the tabulation step in nanometres, from one to twenty; the vertical is how far the resulting colour is from the same integral taken at a tenth of a nanometre over the same range, in ΔE₀₀, on a logarithmic scale. Each line is one light. The three with no feature narrower than the step fall smoothly and stay below a tenth of a unit at five nanometres, which is the grid used throughout. The fluorescent tube and the laser projector do not fall at all: their lines are narrower than any step drawn here, so the answer depends on where the samples land rather than on how many there are. The sample is held at paper throughout.

What light is

A grid is not a resolution

Ten essays into this collection there is one sentence about wavelength sampling, and it is that five nanometres is enough. Ten measurements later there are three decisions, three mechanisms, three repairs and two rankings, and the word resolution names none of them.

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.

What the eye does

The third factor is a construction

Every figure in this collection names its observer, which was the whole point. None of them says what an observer is made of. Three curves are not a measurement of the eye; they are a projection of one, with a field size, an age, a macular density, three peak wavelengths and a luminance constraint inside them.

What choosing a space to divide the white out in is worth. Three pairs of routes to the same colour, over forty-two surfaces: dividing the white out in tristimulus values, in a published cone space, and in the observer's own cones. The first two agree to 0.59 ΔE₀₀ at the median. Either of them differs from the observer's own cones by more than fifteen. That is why the two exact conditions in this round are exact only in the eye's own coordinates: the identity belongs to the receptors, and every published arithmetic works in a basis somebody else chose.

What the eye does

The identity is in the eye's own coordinates

Two conditions in this round are exact — a gain on each cone is not a different observer, and three curves for one space are one observer. Imposed in a published cone space rather than the eye's own they leave 13.0 and 8.11 ΔE₀₀ standing. The identities belong to the physiology and every arithmetic in use works somewhere else.

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.

Where the model breaks

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.

The three cone absorptances at two settings of the field size. Solid and dashed are the same construction at the two ends of the CIE's own second observer. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 16.0 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

What the eye does

A field size is two changes

The CIE publishes two standard observers and the difference between them is usually described as a field size. What actually differs is a macular pigment the light no longer passes through and a cone outer segment the light no longer travels the length of — two changes, in two places, with different signs and different sample dependence.

The three cone absorptances at two settings of the macular pigment. Solid and dashed are the same construction at the two ends of two standard deviations of the reported spread. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 31.5 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

What the eye does

The macular is a band, not a filter

The lens absorbs everything below 500 nanometres with a long tail; the macular pigment absorbs forty nanometres either side of 460 and nothing else. The two have similar sizes and completely different distributions, and the reason is that one is broad and one is narrow — which is what decides whether a sample is affected at all.

$What the Lambertian assumption costs a room, against how rough its walls are. The horizontal axis is the roughness of the two coloured walls; the right-hand end is nearly matt, which is what a radiosity calculation assumes. One line is the distance in ΔE₀₀ between the floor's colour and what radiosity gives for the same room — 4.89 at an eggshell finish, falling to 0.97 at the matt end. The other is the chroma of the bounce, which falls as the walls get glossier: what an interface returns is a Fresnel reflection and carries no pigment, so the fraction of the return that goes into the lobe is a fraction that arrives at the floor white. The lobe is taken out of the body term rather than added beside it, which is what a real finish does.$ What a scene does

The solver had no slot for gloss

Every scene result in this collection is computed by radiosity, and radiosity is not an approximation that could be made more accurate. Its unknown is one number per surface, and a surface that returns light differently in different directions does not have one. A missing slot cannot be wrong by a small amount.

The directional solver reduces to the radiosity solver exactly. A solver with a new unknown in it is worth nothing until it reproduces the one it replaces. Setting every wall's bidirectional distribution to ρ/π collapses all thirty ordered-pair radiances onto their patch's radiosity divided by π, and the answer agrees with this collection's existing radiosity solution to 9.8e-16 relative — the floating-point floor. That is the check that makes every other number in this family a statement about lobes rather than about a new piece of arithmetic, and it is the reason the reduction is drawn rather than mentioned.

What a scene does

Thirty unknowns instead of six

A directional transport solver is worth nothing until it reproduces the one it replaces. Setting every wall's bidirectional distribution to ρ over π collapses thirty ordered-pair radiances onto six radiosities and reproduces this collection's existing answer to 9.8 × 10⁻¹⁶ relative — which is the only reason anything else it says can be believed.

Where the viewer stands, at a wall roughness of 0.2. The chroma of the light the floor sends towards each of the five other faces of the room, at one roughness. The spread is 2.00 ΔE₀₀ between the extremes. A radiosity solution assigns one radiosity to the floor and therefore cannot have a spread at all — the whole width of this chart is a quantity the method has no slot for, rather than one it approximates badly. The two side walls see the most because they are where the coloured light comes from, and the direction the lobe favours is the direction it came from.

What a scene does

The floor is a different colour from the door

With a lobe on the walls the floor sends chroma 18.76 towards the front of the room and 20.94 towards the side walls, a spread of two colour differences. A radiosity solution assigns the floor one number, so the whole of that spread is a quantity the method has no slot for rather than one it estimates badly.

Two ways of putting a lobe on a wall, and the sign they disagree about. The chroma of the floor's return against the wall's roughness, computed twice. In one the interface's return is taken out of the body term — light reflected at the boundary never reaches the pigment, which is what a real finish does. In the other it is added beside the body term, which is what a microfacet model does if nobody couples the two. The first says a gloss wall makes the room less coloured and the second says more, and the gap at the glossiest end is 2.87 units of chroma. Neither is a numerical error; the difference is a modelling decision that is usually made by omission.

What a scene does

Two ways to put a lobe on a wall

Take the interface's return out of the body term and a gloss wall makes the room less colourful. Add it beside the body term and the same wall makes the room more colourful. Same solver, same room, one line of energy accounting, and the two answers differ by nearly three units of chroma at the glossy end.

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 the model breaks

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.

Every scene in this collection was matt

The green wall, the corner, the bounce series and the metamer separation are all computed on Lambertian surfaces, because the solver that produced them requires it. Each would move by between one and five colour differences on an ordinary satin finish, and none of those essays says what finish it means.

Each departure over forty-two surfaces rather than one. The same six departures measured over a family of forty-two analytic reflectances — an absorption band of stated centre, width and depth — with the smallest, the median, the ninety-fifth percentile and the largest marked. Every one of them spans more than a factor of three, and the ranking between them is not stable across the family: what decides a departure's size is which sample it is asked about, because a departure is a pairing and the sample is one of the two factors. Quoting any single number for what an observer's age is worth is quoting a choice of example.

Difference and uniformity

A tolerance with an observer in it

A delivery tolerance is written in ΔE₀₀ against the 1931 observer, and six departures of that observer combine to about three of the same units on an ordinary saturated sample. A one-unit tolerance is being asked to contain a three-unit uncertainty that nothing in its budget mentions.

The conditions under which an observer's departure is exactly zero. A departure of the observer is the pairing of something belonging to the observer with something belonging to the stimulus, so emptying either factor empties the product. The axis is logarithmic in what is left when the condition is imposed. Six rows empty the stimulus's factor — a perfectly neutral sample is the same colour for every observer, at any age and any field size — and two empty the observer's, since a gain on each cone and a change of basis are both absorbed exactly. All eight are identities rather than small numbers. The last two are the same two conditions imposed in a published cone space rather than in the observer's own, and they are worth eight and thirteen units: the identity is about the eye, and the arithmetic everybody uses is in somebody else's coordinates.

What it takes to deliver it

The instrument is one observer exactly

A spectrophotometer has no observer uncertainty. It computes through the 1931 tables to the last bit, reproducibly, forever — which is its whole value and is also why it cannot report the three colour differences of observer spread between itself and whoever is looking at the sample.

What a camera does

The chart was measured by an observer too

A camera profile is fitted so that the camera's numbers reproduce the chart's measured tristimulus values. Those values were computed through the 1931 observer, so the fit inherits every departure in this round — and the fit's own residual, at 0.19 ΔE₀₀ on the chart, is fifteen times smaller than the term it cannot see.

What the brain does

The census under another observer

This collection's largest computed result is an adaptation census — fourteen changes of light judged over a hundred and twenty-five constructed surfaces. Every number in it was computed through one observer, and the observer's own departures are between one and two and a half units on the same surfaces, which is the size of the effects the census reports.

Every departure under every light. Six departures across six lights, each cell the difference between two observers in ΔE₀₀, drawn as a bar whose length is the number. The rows are not multiples of one another: the lens is worst under tungsten and the pigment peaks are worst under a three-emitter LED, because a departure is a pairing and which light is being paired with decides it. The laser projector's row is empty, and that is not a fact about lasers — on this collection's five-nanometre grid a three-line spectrum is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about exactly.

What it takes to deliver it

The booth and the eye disagree

A viewing booth is specified to simulate daylight, and daylight is the light under which observers disagree least. So a booth is the condition in which a specification is least likely to be tested against the thing that breaks it, and the shop it will be judged in roughly doubles the term.

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