Which index to buy an instrument for

An audit that ends in a list of four things a model gets wrong is not much use to anybody who has to measure something on Tuesday. The useful question is which of the four to spend money on, and the answer is not the ranking by size.

The claim #

The two departures that cost most are removed without buying anything, and the two that need hardware are the two that cost least.

• The wavelength index costs 6.98 ΔE₀₀ on a coated printing paper, and is removed by specifying the lamp — which the graphic arts standard already does, and calls M1.
• The range costs 6.70 ΔE₀₀ on the same paper, and is removed by widening a table from 380 nanometres down to 300, which is a published table and free.
• The place index costs 1.96 on a pigmented plastic, and needs a second aperture — a port and a second reading.
• The direction index costs 1.00 on an eggshell paint, and needs a goniospectrophotometer, which is the most expensive instrument on the list by an order of magnitude.
• And the two rankings are almost exactly reversed, which is a fact about how measurement standards have developed rather than about physics.

The four, and what each would take #

The wavelength index. The sample returns light at wavelengths it did not receive, so its response is a matrix and its diagonal is what an instrument reports. Measuring the whole matrix needs a bispectral fluorimeter — a monochromator on the way in and another on the way out — which exists, costs what a laboratory instrument costs, and is used in a handful of national metrology institutes.

But the whole matrix is not needed for a colour. What is needed is the sample’s response under one stated light, and that is obtained by stating the light. ISO 13655’s M1 condition does exactly that: a D50-like source with its ultraviolet content specified, so that two instruments excite the same fluorescence. The measurement is unchanged; only the lamp is specified.

The range. The site’s own grid stops at 380 nanometres because the visible band does. The CIE’s daylight basis functions are published from 300, for the express reason that the ultraviolet of daylight is what excites a brightener. Extending the grid is a table lookup and a longer array; the cost is arithmetic, and the benefit on a brightened stock is 6.70 ΔE₀₀.

The place index. A second reading at a second aperture, and the difference between them. Bench instruments have two ports already; hand-held ones do not, and the data format has no field for a second number even where the hardware exists.

The direction index. A goniospectrophotometer measures the response at a grid of incoming and outgoing angles. It is slow, it is expensive, and its output is not a colour but a function of four variables — so the instrument’s cost is only the beginning of the cost.

Why the two rankings are reversed #

The reversal is not a coincidence and it is not luck. It follows from the pairing form: a departure is the product of something the sample does and something the light does, and removing a departure means setting either factor to zero, while measuring one means characterising both.

The two cheap ones are cheap because their light-side factor is under the measurer’s control. A lamp’s ultraviolet content is a property of the instrument, so it can be specified. A grid’s lower bound is a property of the arithmetic, so it can be changed. In both cases the field factor can be set to a known value by fiat.

The two expensive ones are expensive because their light-side factor belongs to the situation. An aperture is under control, but the sample’s kernel is not, and the reading depends on the ratio. A room’s directional structure is not under anybody’s control at all — that is what a room is.

So the practical rule falls out of the algebra: spend on the departures whose light-side factor the measurer owns, and measure the ones whose light-side factor owns the measurer. That is the reverse of the intuition, which is to spend where the numbers are largest.

What a second reading is worth #

The place index is the one where the arithmetic of buying is clearest, because the instrument needed is small and the benefit can be stated exactly.

Two readings at two apertures give a difference, and the difference is a measurement of the sample’s own transport rather than of its colour. On coated paper the two readings at four and eight millimetres differ by about a quarter of a ΔE₀₀; on a pigmented plastic by 1.00; on marble by 5.90. So the difference is simultaneously the diagnosis — is this sample translucent enough to matter — and the magnitude — by how much.

That is an unusually good property for a measurement to have. Most diagnostics tell a laboratory that something is wrong and leave the size to a second procedure. This one is a single subtraction, uses hardware many instruments already contain, and produces a number in the unit the specification is already written in.

What stops it is not the hardware and not the arithmetic. It is that a tolerance cannot cross a condition: two laboratories reporting one number each, from two different ports, are not in disagreement about the sample, and until a specification has somewhere to record which port was used there is no way to say so.

Where the money actually goes #

A ranking by instrument price is not a ranking by cost, because the instrument is usually the cheapest part.

A goniospectrophotometer’s real cost is that its output is not a colour. It returns a function of four variables, and turning that into something a specification can hold means choosing a summary — a set of angles, a fitted model, an average — which is another convention, and one that would have to be agreed before the instrument helped anybody. The automotive industry has been through exactly this for effect pigments and settled on a small set of stated angles, which is a summary rather than a solution.

The second reading at a second aperture has the opposite property. Its output is a number in the same unit as the first, so it fits into every existing tolerance without any new agreement at all. The cost is a field in a data format.

And the two spectral departures cost nothing at the instrument and something at the standard: getting an industry to agree on a lamp took fifty-five years, which is the real price of the cheapest item on the list.

What a specification would have to say #

Four departures give four fields, and the industry has written down one and a half of them.

The one is the measurement condition — M0, M1, M2, M3 — which fixes the wavelength index by fixing the lamp. It is in the graphic arts standards and it is complied with, because press and proof would otherwise not match on brightened stock.

The half is the geometry — 45°/0° or d/8°, specular included or excluded — which addresses a piece of the direction index. It fixes what the instrument does and says nothing about the room the result will be looked at in, which is the larger half.

The two that are missing entirely are the aperture and the grid. An aperture is usually recorded as a hardware note rather than as a condition, and no tolerance anywhere is stated relative to one. A grid’s lower bound is a property of a computation nobody reports at all.

Both absences follow the same pattern: a field appears in a specification when two parties have disagreed over it in a way that cost money. Brightened paper cost money, so M1 exists. Gloss cost money, so geometry designations exist. Translucent plastics cost money in an industry that mostly wrote its own standards, which is why the aperture advice lives in ASTM documents about plastics rather than in the colour standards proper.

What was computed, and how #

The four ordinary cases are each a stated sample under a stated light, computed in the same unit and with the same colour arithmetic.

For the two spectral rows the sample is a coated printing paper with the brightener loading a coated printing paper has, and the light is D50 or D65 on the wide grid. The wavelength row compares the sample measured with the lamp’s ultraviolet present and filtered; the range row compares the same lamp integrated from 300 nanometres and from 380.

For the place row the sample is a pigmented plastic sheet through a four-millimetre radius against no aperture. For the direction row it is a pigmented surface with an ordinary sheen at a window against the same surface under an overcast sky.

The one thing the ladder is not is a comparison of the departures themselves, and the chart says so in its own caption. Change the plastic for marble and the third row goes from 1.96 to 12.65; change the eggshell for a varnish and the fourth doubles; change the paper for an unbrightened one and the first two go to nothing at all. Each bar can be moved by a factor of ten by choosing a different example, and what survives the objection is that all four are above the tolerance a specification is written in.

Twice the difference is the departure #

The second aperture’s double duty — simultaneously the diagnosis and the magnitude — is better than that, because the two are related by a constant.

The difference between a four-millimetre and an eight-millimetre reading is 45 per cent of the departure on coated paper, 51 on a pigmented plastic and 47 on marble — three materials spanning a factor of twenty-four in the departure and agreeing to six points in the share.

So the rule is arithmetic rather than a judgement. Double the difference between the two readings and the answer is the departure, to within ten per cent: 0.48 against 0.53, 2.00 against 1.96, 11.80 against 12.65.

That is a stronger claim than the essay makes, and it changes what the second port buys. It is not a diagnostic that tells a laboratory to go and do something else; it is the correction. A reading at four millimetres plus a reading at eight gives the sample’s own colour, to a tenth of the departure, without any knowledge of its kernel, its scattering coefficient or its diffusion length — none of which anybody publishes for any material.

The constant is what makes it work, and it is not a coincidence: the departure falls as about one over the aperture, so a doubling removes half of it, and the half removed is what the subtraction measures. One over the aperture is the whole content of the correction.

The reversal, as a correlation #

Almost exactly reversed is a phrase and the four rows support a number.

Rank the departures by size: wavelength, range, place, direction. Rank them by what removing each costs, most expensive first: a goniospectrophotometer, a second port, a specified lamp, an afternoon of arithmetic — direction, place, wavelength, range.

The Spearman correlation between the two is −0.80. Not −1.00, and the single inversion is the one worth knowing: the largest departure is not the cheapest to remove. The range is cheaper still, being a table lookup nobody has to be persuaded of, while specifying a lamp required an industry to agree.

So the honest form of the claim is that size and removal price are strongly anti-correlated and not inversely ordered, and the exception sits exactly where the essay’s own economics predicts it: the convention that needed agreement is the one that slipped a place.

What the conventions have already bought #

The four bars total 16.64 ΔE₀₀ at the stated examples, and their shares say what has been paid for and what has not.

The one convention that exists has already removed 42 per cent of the total, for no recurring cost to anybody. The convention that could exist would remove another 40, and the two items needing hardware are 18 per cent between them.

That is the economic argument made concrete rather than asserted. Four fifths of the damage is removable by agreement and one fifth by purchase, and the agreement half was reached once, sixty years after the problem appeared, and has cost nothing since.

It also prices the outstanding item precisely. Widening the grid is worth 40 per cent of everything this round measures, costs an afternoon, and needs nobody’s consent — and it is the one item on the list this collection has not done. Of the four departures, the cheapest to fix and the second largest is the one still open, which is exactly the pattern the essay’s own economics says should not survive and which survives because nobody outside the computation can see it.

Where the model stops #

The cost side of this comparison is a computation and the price side is a judgement, and the two do not have the same standing.

Instrument prices change, and a departure that needs a laboratory today may need a phone tomorrow — the history of colour measurement is largely a history of expensive instruments becoming cheap ones. Directional measurement is the most likely of the four to move, since multi-angle instruments already exist for effect pigments and are ordinary in the automotive industry.

The comparison also assumes each departure is worth removing at all, which depends entirely on the sample. A laboratory measuring opaque, matte, unbrightened material has all four factors near zero and needs none of this. The ranking is for somebody who has established that a departure is present, which is a prior question and is answered by measuring either of its two factors.

The generalisation #

There is a decision rule here worth separating from the colour, and it applies wherever a model is known to be incomplete.

Rank the corrections by what it costs to eliminate them, not by how large they are. A large error whose cause is under control is cheap; a small error whose cause belongs to the environment is expensive, and the expense is recurring, because every new situation needs it measured again.

The second half of the rule is the one that is easy to forget. An error removed by a convention — a specified lamp, a stated grid — stays removed for everybody thereafter at no further cost. An error removed by a measurement has to be measured every time. So the value of a convention is the cost of the measurement it replaces, multiplied by the number of measurements nobody will now have to make, and that number is usually enormous.

That is the whole economic argument for measurement standards, and the four rows here are an unusually clean instance of it: the two departures with conventions attached are the two largest, and they are largest partly because having a convention made them safe to leave large.

That is the departure a second port would remove, and it is the cheapest of the four to remove. The reason the four cannot then be added into one budget is in the figure below, and it is not conservatism.

What the four are departures of, and how much of the collection each one reaches, are the two facts a purchasing decision actually turns on.

The order the four should be attacked in #

Putting the two rankings together gives an order of work, and it is worth writing out because it is not the order anybody would guess.

First, state the lamp. It is the largest departure on brightened material, the standard exists, and compliance costs nothing beyond buying the right instrument next time. Anybody measuring paper who is not on M1 is leaving 7 ΔE₀₀ on the table.

Second, widen the grid. It is nearly as large, it costs an afternoon of arithmetic, and it is the only item on the list that no laboratory has to be persuaded of, because it is invisible to everybody outside the computation. This collection has not done it yet either, which is the round’s own outstanding item.

Third, take a second aperture reading on anything that might be translucent, which is more materials than the phrase suggests: plastics, textiles, paint films on light substrates, food, teeth, skin, stone, and paper thin enough to see through.

Fourth, and only if the object is glossy and the viewing matters, worry about the room. It is the smallest of the four on ordinary material, it is the most expensive to measure, and the honest answer for most work is to control the viewing rather than to characterise it — which is what a viewing booth is for.

Who found it, and when #

The economics of measurement conditions is not usually written down, but it can be read off the order in which standards appeared.

Geometry designations came first, in the 1930s and 1940s, because gloss is visible and disputes about it are old. The M-conditions came in 2009 with ISO 13655’s revision, sixty years after brighteners entered papermaking and about fifteen after they became universal — which is roughly how long it took for proofs and press sheets to disagree often enough to force the issue.

The aperture has no such history because the industries it bites hardest — plastics, textiles, food — settled it internally with advice rather than with a condition. And the grid has none at all, because it is a property of a computation and computations do not have disputes.

Where the ladder goes next #

The obvious missing instrument is the cheap one: a hand-held spectrophotometer with two ports and a data format with room for both readings. Everything needed for it exists, and what does not exist is a specification that would ask for it.

The obvious missing convention is a stated field for viewing. A specification names the illuminant and the observer, and this collection has argued at length that a viewing condition is an argument rather than a footnote. What it does not name is the field’s directional structure, and the pairing says that is exactly the term that decides whether a gloss sample looks right in the room it ends up in.