The widths were free because nothing else was asked

A tolerance measured under one requirement is a statement about that requirement, and naming the missing ones is not the same as measuring them.

The claim #

The previous round measured a camera’s dyes under one objective, found the bandwidths almost free, and named the two requirements it was ignoring. Both were measured this round and neither binds. What binds is a third requirement that was in the model the whole time.

• The finding it starts from is real. Under the adaptation objective the three bandwidths together carry 97 per cent of the flattest direction of the curvature, so that objective can barely see them.
• The two nominated requirements were throughput and separation — a narrow dye costs light, a broad one costs a colour matrix that amplifies noise — and they are the physically obvious ones.
• Neither holds any part of any boundary. At a five per cent rise, every dye’s width can move sixty nanometres in either direction without either objecting.
• What tightens the width is the Luther residual, the colorimetric requirement, by a factor of 1.6 on the red dye and 4.7 on the blue.
• So the previous round was right that one objective is not enough and wrong about what was missing, which is worth recording in that order.

What the previous round measured #

A tolerance is a region rather than a radius, and the previous round established that for two devices by taking the curvature of an adaptation objective over each device’s own design parameters.

A camera’s are six: three dye centre wavelengths and three bandwidths, all in nanometres, which is what makes their eigenvalues comparable without inventing a weighting. The condition number came out at 623 — twenty-five times further in the cheapest combination than in the dearest — with the stiffest direction being the blue dye’s centre at a weight of 0.96 and the flattest being the three bandwidths together, at 97 per cent.

The conclusion was correct and carefully hedged: a tolerance budget belongs on the centre wavelengths, and the widths are free to be decided by throughput and separation, which this objective is silent about. The hedge names two candidates, and naming a candidate is a hypothesis.

The two candidates, measured #

Both are easy to write down as costs and both were.

Throughput is what a narrow dye costs in light. A colour-filter array passes a fraction of what arrives, that fraction falls with the dye’s bandwidth, and light lost is noise gained in the shadows. As a cost it is one over the total transmission summed across the three channels.

Separation is what a broad dye costs in the colour matrix. Overlapping channels have to be un-mixed by a 3×3 that subtracts one from another, and subtraction amplifies photon noise — which this collection already measures as a gain relative to leaving raw alone.

At a five per cent rise, neither reaches a boundary. Every dye’s throughput region and every dye’s separation region extend to the edge of the ±60 nanometre search in every direction, on all three channels. Separation holds no part of any boundary at all; throughput holds eight per cent of one dye’s, and that eight per cent is in the corner directions where a centre and a width move together rather than in the width direction the claim was about.

What the four requirements do to the headline anisotropy #

The essay begins from the adaptation objective’s condition number of 623 — twenty-five times further in the cheapest direction than in the dearest — and ends with a specification whose numbers say something quite different about the same design.

The proposed tolerances are ±19, ±13 and ±7 nanometres on the three widths and ±10, ±7 and ±3 on the three centres. The widths have about twice the latitude of the centres, uniformly: 1.9, 1.9 and 2.3. On the blue dye, which the adaptation objective names as both the stiffest coordinate and part of the flattest direction, the ratio between the two is 2.4.

Twenty-five becomes two and a half once the other three requirements are applied. That is the whole of what this essay adds to the previous round, stated as a number rather than as a change of which requirement binds — and it is the same collapse the neighbouring measurement of the same sensor finds by a different route, where the Luther ceiling takes an unconstrained reach ratio of 25 down to about five.

Two independent measurements of one design agreeing that the adaptation objective’s anisotropy is mostly not available is worth more than either. The reach ratio is a property of one objective and not of the design, and any specification written from it alone would give the widths an order of magnitude more latitude than they have.

The blue dye’s box is a ninth of the red’s #

The practical headline is stated as three numbers rather than one, and the three pairs multiply into something starker.

The blue dye’s tolerance box is nine times smaller than the red’s, and it is tighter on both axes by a consistent factor — 3.3 on the centre and 2.7 on the width. That consistency is itself informative: it says the blue channel is not tight because of one awkward direction but because the whole requirement set closes in on it, which is the reading the essay’s own observation about two unrelated requirements agreeing points at.

For a manufacturer the ratio is the number that matters. A process holding all three filters to the same tolerance has to hold the red and green nine and four times tighter than they need — or, more likely, has to hold the blue nine times looser than it should, since a single tolerance is usually set by what the process can do rather than by what the design wants.

The throughput cap cannot be right in the narrowing direction #

One result is reported in a way the physics does not allow, and it is the load-bearing null.

Throughput is described as capped in every direction — never reaching a five per cent rise within the ±60 nanometre search — on every dye. The designed widths are around forty-five nanometres. Sixty nanometres in the narrowing direction takes a forty-five nanometre dye to a width of −15, and it passes through zero at −45.

A dye of zero width transmits nothing. A cost defined as one over the total transmission is therefore unbounded there, not flat, so it must exceed a five per cent rise somewhere well before −45 — and the region cannot be capped in that direction.

Three readings are possible and the essay does not distinguish them. The search may clamp the width at a physical floor, in which case capped means capped within what was searched and the reach is smaller than ±60 on one side. The cost may be a sum across three channels in which one dye’s collapse is diluted — which the essay says, and which would delay the rise but cannot remove it, since a third of the light going to zero is a fifty per cent rise in the reciprocal. Or the search’s directions may not include the pure-narrowing one, in which case forty-eight directions in a plane is a stronger claim than the sweep supports.

None of the three changes the conclusion, because the tolerance in question is ±19 nanometres at most and the throughput cost is genuinely flat over that range. What it changes is the strength of the statement: throughput is slack across the whole space of dyes a camera could have is not supported, and throughput is slack over any width a camera would have is.

That distinction matters for the essay’s own generalisation, which is that the nominated effects are right about the mechanism and wrong about the scale. The scale at which throughput bites is computable and is somewhere between ±19 and ±45 nanometres — outside the tolerance and inside the search, which is exactly the region the capped reporting hides.

The blue primary is the one where the two-number plane is tightest, and it is worth drawing on its own because it is the case a specification would actually be written about.

Why they are so slack #

The mechanism is worth having, because neither binds is a weak-sounding result that turns out to be about how the objectives are normalised.

Each cost here is a ratio to its own value at the designed dyes, and each region is a five per cent rise in that ratio. Throughput is nearly proportional to bandwidth, so a five per cent rise in the cost is roughly a five per cent change in the width — about two nanometres — which sounds tight and is not, because the cost is a sum over three channels and moving one dye’s width by two nanometres changes the sum by well under a per cent.

Separation is slacker still, and for a more interesting reason. The noise gain of a fitted colour matrix depends on how nearly the three channels are linearly independent as functions of wavelength, and three Gaussians at 460, 540 and 600 nanometres are comfortably independent over a range of widths — the matrix’s condition degrades only when two channels start to overlap substantially, which needs a width change of tens of nanometres rather than of two.

So both mechanisms are real and both are second-order at the scale a tolerance lives on. That is the honest form of the result: the previous round named two effects that exist, and the arithmetic says they matter at a scale ten times larger than the one in question.

There is a third possibility worth ruling out before accepting the null, and it is that the search simply does not reach far enough. It does: ±60 nanometres on a dye whose designed width is around forty-five is a width running from nothing to more than double, which is not a tolerance in any usable sense. The objectives are not merely slack inside a tolerance; they are slack across the whole space of dyes a camera could have.

That in turn says something about the design search that produced these dyes. It was run under an adaptation objective with a Luther ceiling, and neither throughput nor separation was in it — a reasonable simplification at the time, and now a measured one: adding either would not have moved the answer.

What does bind #

The Luther residual — how nearly the three sensitivities are a linear mixture of the observer’s colour-matching functions — is the requirement that tightens the widths, and it does so unevenly.

• Dye 0, the red: adaptation alone allows ±29.9 nanometres of width; all four together allow ±18.9. A factor of 1.6.
• Dye 1, the green: ±20.6 becomes ±13.3. A factor of 1.6.
• Dye 2, the blue: ±34.7 becomes ±7.3. A factor of 4.7, and the Luther residual holds 69 per cent of that dye’s boundary.

Two more views say why a width is free: the map from the maker’s plane to the specification’s is badly conditioned, and only one requirement binds a width at all.

The blue channel is the tight one because the short-wavelength colour-matching function is the awkward one: it is narrow, it sits where the observer’s sensitivity is falling fast, and a Gaussian is a poor approximation to it over a wider range than it is for the other two. Broadening the blue dye takes the sensitivity set away from the span of the colour-matching functions faster than broadening either of the others.

And the Luther residual was in the model before this essay started. It is the constraint the designed sensor was built under — without it the search walks straight to three narrow lines that adapt superbly and cannot tell a metamer from its partner — so it was present as a ceiling during the design and had never been drawn as a region around the answer.

The unevenness across the three channels is the part with a consequence attached. A specification that put one tolerance on dye bandwidth would have to use the blue channel’s ±7.3 nanometres to be safe, which is two and a half times tighter than the red channel needs — so a single number costs a manufacturer a great deal of unnecessary precision on two of three filters.

Three numbers rather than one is the whole content of the finding for anybody making a camera, and it is invisible from the adaptation objective alone, which puts the three widths in one flat direction and therefore treats them as interchangeable.

What was computed, and how #

Each region is a level set found by bisection. From the designed dye, forty-eight directions are swept in the plane of that dye’s centre and width; along each, the cost is bisected outward to a five per cent rise, and the resulting boundary polygon is the region.

All four regions for a dye are swept along the same directions, which is what makes the intersection exact: every region here is star-shaped about the same centre, so the intersection’s radius along a direction is the smallest of theirs along it. There is no polygon clipping and no approximation.

A direction that does not reach a five per cent rise inside ±60 nanometres is reported as capped rather than as a large number, and that reporting is what makes the null result legible: throughput and separation are capped in every direction, on every dye, which is a statement about the search’s reach as well as about the objectives.

The centre and the width are both in nanometres. That is not a convenience — it is the reason the shapes in these figures mean anything, and it is why the sensor is parameterised as a centre and a full width rather than as a centre and a variance.

Where the model stops #

The dyes are Gaussians and a colour-filter array is not. A real dye’s transmission is asymmetric, has a tail, and multiplies a silicon quantum efficiency that this collection models properly elsewhere. The family here is deliberately crude, so that the design space has six numbers in it and the sensor’s answer is comparable with the display’s.

Throughput is modelled as transmitted light and not as noise. The step from this dye passes less light to this image is noisier involves an exposure, a full-well capacity and a read noise, none of which is here. A model with those in it would make throughput bite at a different scale, and possibly at a relevant one.

And a five per cent rise is a choice. The regions scale with it and the ordering of the four requirements does not, which was checked: at one per cent and at twenty, the Luther residual still binds and the other two still do not.

The generalisation #

The result is a small correction to a large habit, and the habit is worth naming.

When one objective cannot see a parameter, the instinct is to reason about which physical effect the objective is missing. That reasoning is usually right about the effect and wrong about its scale. Throughput and separation are genuine costs of a dye’s bandwidth; both were nominated on sound physical grounds; and both turn out to matter at a scale ten times too coarse to decide a tolerance.

The alternative is unglamorous: write every requirement down as a cost and measure all of them. It is more work than reasoning and it is not much more work — four costs, one bisection each, forty-eight directions. What it buys is the discovery that the binding constraint was already in the file.

And there is a second, smaller lesson about where constraints hide. The Luther residual was not a forgotten requirement — it was used as a ceiling in the design search that produced the dyes being audited. A constraint that is active during a search and never plotted afterwards is invisible in exactly the way this one was, because the answer satisfies it by construction and nobody asks how narrowly.

What a specification should say #

Pulling the measurement into a form somebody could put in a document is short, and it is a different document from the one the previous round’s finding implies.

Three width tolerances rather than one: about ±19 nanometres on the red dye, ±13 on the green and ±7 on the blue, at a five per cent budget. Three centre tolerances: ±10, ±7 and ±3. And the reason attached to each, because a tolerance without a reason is a number a supplier will negotiate.

The blue channel is tight on both, which is the practical headline. It is the dye whose centre the adaptation objective already said was the stiffest direction in the whole design, and it is the dye whose width the Luther residual holds hardest — so the same channel is the critical one under two unrelated requirements, which is the sort of agreement that makes a specification easy to defend and is not something either measurement predicted alone.

What none of this says is what the budget should be. Five per cent of an adaptation residual and five per cent of a Luther residual are not the same amount of harm, and converting either into a difference a person would notice needs an image, a scene and a viewer.

Who found it, and when #

Luther’s condition dates from 1927 and Ives had the same idea before him: a camera reproduces colour exactly if its sensitivities are a nonsingular linear mixture of the observer’s colour-matching functions, and no real sensor satisfies it. Everything since is a question of how close is close enough, and the residual used here is one of several standard ways to measure that.

The tolerance question is a manufacturing one and is asked in industry rather than in the literature, usually as a specification on filter transmission curves with a stated tolerance band. What is unusual here is asking which requirement the band comes from — and finding that the answer for the blue channel is not the one that would be nominated.

Where the ladder goes next #

Four requirements, four regions, one intersection — and the intersection is a good deal smaller than any of its parts, in a way that a specification written as a list of tolerances cannot express.

That is the next rung: what happens geometrically when two long thin regions cross at an angle, why the result is smaller than either, and why its longest direction is neither of theirs.

That last figure is the one to keep in view while reading any of this. The Luther residual is not an abstract penalty: it is the gap between what a set of filters can see and what an eye sees, and a camera that cannot close it has metamers of its own — pairs of surfaces it records identically and a person does not. Widening a dye by ten nanometres is a decision about how many such pairs exist, which is a more serious thing than a percentage on a residual makes it sound.