A tolerance in the wrong coordinates

A specification writes a primary’s tolerance as a region on the chromaticity diagram, and nobody manufactures a chromaticity.

The claim #

A tolerance stated in chromaticity is stated in coordinates nobody controls, and translating it into the two numbers somebody does control changes its shape, changes which direction is cheap, and discards most of it as unreachable.

• What a maker sets is a peak wavelength and a bandwidth, for a quantum dot, a phosphor or an LED. Two numbers, both in nanometres.
• The map from those two to a chromaticity is strongly anisotropic. Its two singular values differ by a factor of 4.1 on the green primary, 13.8 on the blue and 11,505 on the red.
• On the red primary the two knobs do nearly the same thing, because the spectral locus is almost straight there — so a tolerance region has a direction along which no single-peak emitter can move at all.
• Most of a chromaticity region is unreachable. Two per cent of a grid of two and a half thousand candidate emitters lands inside the green primary’s region; the rest are colours no single-peak source produces.
• And what survives translates into ±3.5 nanometres of peak wavelength with a great deal of latitude in width — a specification a production line can check, which the region it came from was not.

Two coordinate systems and no dictionary #

A primary is chosen for four things, and every one of those requirements is a statement about a chromaticity: how well a gain in the resulting basis adapts, how much area three points enclose, which surfaces fall inside them, whether a light of that colour exists. Chromaticity is the right coordinate system for all four, and the regions in that essay are drawn in it.

A person making the emitter has different numbers. A quantum dot’s emission is set by its size; a phosphor’s by its host and its activator; an LED’s by its band gap. In every case the two things that can be dialled are where the peak sits and how broad it is, and the chromaticity is a consequence.

So a specification and a process are written in different variables, with a nonlinear map between them, and the map is where the trouble is.

How anisotropic the map is #

Differentiating the emitter’s chromaticity with respect to its peak and its width gives a 2×2, and its singular values say how much chromaticity a nanometre buys in the best and worst directions.

• Green, at 530 nanometres and 30 wide: 7.6 × 10⁻³ and 1.9 × 10⁻³ per nanometre. A condition number of 4.1.
• Blue, at 460 and 25: 2.5 × 10⁻³ and 1.8 × 10⁻⁴. A condition number of 13.8.
• Red, at 630 and 30: 2.6 × 10⁻³ and 2.2 × 10⁻⁷. A condition number of 11,505.

The same argument runs on a camera’s dyes, which are designed in the same kind of plane and specified in a different one.

The red primary’s number is the finding. Above about 610 nanometres the spectral locus is very nearly a straight line: every monochromatic wavelength from there to the long end sits on almost the same line in the diagram, so a narrowband source anywhere in that range has a chromaticity constrained to it. Moving the peak slides along the line; broadening the band slides back along the same line towards the white. The two knobs push in nearly the same direction, and the perpendicular direction is empty.

That is a fact about the observer’s colour-matching functions rather than about any emitter. It is the same fact that makes a red primary’s chromaticity insensitive to its exact wavelength and it is why display reds are specified loosely in wavelength and tightly in almost nothing else.

The green primary is the well-behaved one at 4.1, and even there the two knobs are not interchangeable: a nanometre of peak is worth four times a nanometre of width, in the best direction. A specification that gave both the same tolerance would be four times too tight on one of them and four times too loose on the other, which is the ordinary consequence of an anisotropic map and is invisible from the chromaticity side.

The blue primary sits between at 13.8. Its locus runs steeply and curves, so peak and width push in genuinely different directions — but the short-wavelength end of the diagram is compressed, so a nanometre buys less chromaticity there than anywhere else, and the absolute latitude in nanometres is correspondingly wider.

What is reachable, and what is not #

A tolerance region is two-dimensional. The set of chromaticities a single-peak emitter can produce is also two-dimensional — a surface parameterised by peak and width — so the two might have been expected to overlap generously. They do not.

Taking a grid of sixty-one peaks and forty-one widths around the green primary, forty-nine of two thousand five hundred and one candidates land inside the region: two per cent.

The reason is that the reachable surface is a thin band through the region rather than a patch covering it. A Gaussian emitter’s chromaticity is pinned by two constraints at once — it must lie inside the locus, and it must lie on the curve traced by the family — and a two-parameter family in two dimensions covers an area only when its Jacobian is well conditioned. Where the Jacobian is nearly singular, as on the red primary, the family degenerates towards a curve and covers almost nothing.

So most of what a colorimetric tolerance permits is not a manufacturing latitude at all. It is a region of the diagram that happens to satisfy the requirements and that nobody can build with one emitter.

There is a second reading of the same fact that is worth having, because it sounds like bad news and is not. A degenerate map is a map with a direction along which nothing can go wrong. On the red primary, the direction perpendicular to the locus is one no emitter can move in — so a manufacturer cannot drift that way either, and a tolerance that extends in that direction is extending into a region no process visits.

So the two per cent figure is not a measure of how much of a tolerance is wasted. It is a measure of how much of the chromaticity plane’s freedom is irrelevant to a maker, and a specification could shrink to the reachable band without excluding anything anybody could build.

What the tolerance becomes #

Pulled back into the coordinates somebody sets, the green primary’s region is:

• peak wavelength from 528 to 535 nanometres — a span of 7, so ±3.5 about the centre;
• bandwidth from 25 to 45 nanometres — a span of 20, so ±10.

That is a specification a production line can check. A peak wavelength is measured with a spectrometer in seconds; a chromaticity requires an integration against a standard observer and a decision about what white to compare with.

It is also asymmetric in an informative way: the width has nearly three times the latitude of the peak, in nanometres, on the same emitter. A maker asked to hold both to the same tolerance would be over-specifying the width by a factor of three, and a maker asked to hold only the chromaticity would have no idea which of the two to control.

The condition number is not the latitude ratio #

The essay makes the anisotropy do double duty: it is quoted as a property of the map — a nanometre of peak is worth four times a nanometre of width — and then used to say what a specification would get wrong, four times too tight on one of them and four times too loose on the other. The pull-back it computes two sections later disagrees.

The green primary’s tolerance comes out at ±3.5 nanometres of peak and ±10 of width, a latitude ratio of 2.86. Its condition number is 4.0. Those are not the same number and they measure different things: the condition number is the map’s anisotropy, and the latitude ratio is the map’s anisotropy combined with the region’s own shape, which is itself elongated and not aligned with the map’s singular vectors.

Using the condition number as a latitude ratio would over-specify the width by 40 per cent — demanding ±14 nanometres of latitude where ±10 is what the region supports. That is the wrong direction for a specification to be wrong in, and it is the direction the essay’s sentence points.

The general form is worth stating because it is the essay’s own lesson one level down. A pull-back of a region through an anisotropic map has two sources of anisotropy and they do not multiply cleanly; the only way to get the latitudes is to carry the region through, which is exactly what the essay does and then describes using the wrong one of the two numbers it computed.

What the same arithmetic predicts for the red primary #

The reachability figure is computed for green alone — 49 of 2,501 candidates, or two per cent — and the essay says the red primary’s family degenerates towards a curve and covers almost nothing without putting a number on it. The condition numbers supply one.

If the reachable band’s width in chromaticity scales inversely with the map’s condition number, then the same region on the blue primary would admit about 0.56 per cent of a grid this size — fourteen of 2,501 candidates — and on the red primary 0.0007 per cent, which is 0.02 candidates.

On any grid of that size the red primary’s region would contain no reachable emitter at all. Not a narrow band but an empty set, and the pull-back would be undefined rather than tight.

That is a prediction rather than a measurement and it is the one worth running, because it changes what the essay recommends. The proposed specification — a peak and a width in nanometres — is constructible for green and, on this estimate, is not constructible for red from the region as given. For red the honest specification would have to run the other way: name the emitter first and let the chromaticity follow, since the locus’s straightness means almost any peak from 620 to 700 lands in much the same place.

Which is, as the essay observes in passing, exactly what display reds are specified like in practice. The industry’s asymmetry between how tightly a red and a green are specified in wavelength is the condition number of this map, arrived at by everybody who has ever had to hold one.

A bounding box is not the region #

One caution about the specification the essay proposes, which follows from its own picture of the reachable set.

The accepted candidates form a diagonal band through the peak–width plane — that is what a two-parameter family with an anisotropic Jacobian produces, and it is what the hero figure shows. The pull-back is then quoted as two independent ranges, 528 to 535 and 25 to 45, which is the band’s bounding box.

A box strictly contains a diagonal band, so a part at a corner of the box — peak 528 with a width of 45, or peak 535 with a width of 25 — is very likely outside the region the box was derived from. A specification written as two independent ranges therefore accepts parts the colorimetric tolerance rejects, which is the opposite of the essay’s own claim that the nanometre specification is strictly inside the chromaticity one.

The repair is small and standard: quote the two ranges with a coupling — a permitted width that depends on the peak, or equivalently a linear combination held within a bound. That is one more number on a drawing and it is the difference between a specification that is conservative and one that is merely convenient.

What a maker would have to be told #

Setting out the specification this analysis implies is short, and the contrast with the one it replaces is the point.

Replaced: the green primary shall lie within the region bounded by the following twelve chromaticity pairs. Checkable with a spectroradiometer, an integration and a standard observer; not checkable at a deposition station.

Instead: the green emitter’s peak shall be 531.5 ± 3.5 nanometres and its full width at half maximum between 25 and 45. Checkable with a spectrometer in seconds, at the point in the process where the number is actually set.

The two are not equivalent, and the difference runs one way: the second is strictly inside the first, because it is the first intersected with what an emitter can do. A part meeting the second meets the first; a part meeting the first may be unbuildable, or may be built with a spectrum that is not a single peak at all and whose behaviour under a different observer is another question entirely.

That last clause is not a quibble. A chromaticity tolerance is satisfied by any spectrum landing in the region, and two spectra landing in the same place are metamers for the standard observer and not for anybody else. A specification in nanometres pins the spectrum; a specification in chromaticity pins only its integral.

What was computed, and how #

The emitter is a Gaussian in wavelength with a stated peak and full width at half maximum, integrated against the 1931 observer to a chromaticity. The Jacobian is taken by central differences in nanometres on both parameters, which is the unit both live in — so its two columns are comparable and its singular values are a statement rather than a choice of units.

The singular values come from a closed form for a 2×2 rather than from an iterative routine, which matters because the entries are of order 10⁻⁴ and one of the singular values on the red primary is 10⁻⁷. Handing a general-purpose decomposition a matrix like that is asking it to compute a small singular value out of a squared condition number, which is exactly the failure this collection has documented before.

Membership in the region is tested by comparing a candidate’s radius along its own direction with the region’s boundary radius in the nearest of forty-eight sampled directions. That is nearest-angle rather than interpolated, which is adequate because the grid of candidates is coarser than the boundary’s sampling, and it is the reason the reported share is a share of a grid rather than an area.

Where the model stops #

A single Gaussian is a caricature of an emitter. A real quantum dot has an asymmetric line shape with a tail to the red; a phosphor has structure from its host lattice; an LED has a shoulder from its own band tail. Each of those is a third parameter and a third parameter changes the reachability arithmetic entirely — a three-parameter family covers an area in two dimensions with room to spare.

And most displays do not use one emitter per primary. A white LED behind colour filters, which is how most panels work, has a spectrum that is a product of a broad source and a filter, and its two knobs are the filter’s centre and width rather than the emitter’s. The map is different, the anisotropy is different, and only the shape of the argument transfers.

And the observer is in the map twice. The chromaticity of an emitter is an integral against the colour-matching functions, so both the region and the reachable set are computed for one observer; twenty-four observers give twenty-four triangles, and a nanometre tolerance derived for the average is not the tolerance a particular person’s eye implies. Nothing here propagates that, and it would widen the peak tolerance rather than narrowing it, since a spread of observers spreads the region.

The pull-back also inherits everything the region inherited. The region is a level set of four requirements at a one per cent budget, and every caveat about that budget, those requirements, and the display they are computed for applies unchanged to the nanometres it becomes.

The generalisation #

Two things generalise and both are about the gap between a specification and a process.

A tolerance should be stated in the coordinates somebody controls, and translating it is not a change of units. It is the same lesson a difference measured in one basis and read in another teaches about colour differences, arriving in a factory rather than in a metric. The map is nonlinear and anisotropic, so a symmetric region in one coordinate system is an asymmetric sliver in the other, and the ratio of the two axes changes by an order of magnitude between primaries of the same display.

And the reachable set matters as much as the region. A specification describes a set of acceptable outcomes; a process describes a set of achievable ones; and what matters is the intersection. When the achievable set is a low-dimensional surface — which it is whenever a device has fewer knobs than the space it is specified in has dimensions — most of a specification’s permissiveness is fictional.

The second point has a name in other fields — the difference between the feasible set and the design space — and its usual consequence is that a specification written by one team and met by another describes something neither of them has drawn.

Who found it, and when #

That the spectral locus is nearly straight above 610 nanometres is elementary and is why the CIE’s own diagrams show the long-wave end as a line: all wavelengths from about 620 to 700 have essentially the same chromaticity, and a display red anywhere in that range is colorimetrically almost the same red. Every textbook mentions it and few draw the consequence for a tolerance.

The manufacturing translation is done constantly in industry and rarely published, because it is a supplier’s problem and the answers are commercially specific. What is unusual here is doing it explicitly from a colorimetric region rather than from a target chromaticity, and reporting the condition number of the map as the quantity that decides how badly the two coordinate systems disagree.

Where the ladder goes next #

Three essays have now taken a tolerance region seriously as a shape: what it is under one requirement, what happens when several cross, and what it becomes in the coordinates somebody sets. All three are about a device, where the parameters are manufactured.

The same geometry applies to a set of numbers nobody manufactures — the nine coefficients a colour match leaves free — and there it has a stranger property: three of the nine directions are exactly flat everywhere, and they stop being the objective’s own principal directions the moment the point stops being an optimum.