A template cannot place a point

A model can be accurate about everything it was built for and useless about the one quantity somebody wants out of it, and the reason is usually geometry rather than accuracy.

The claim #

A pigment template good to a fraction of a per cent on what a cone catches cannot place a confusion point, because a copunctal point is where two nearly parallel planes meet. The variation between one pair of eyes and another can still be propagated — by moving the measured points rather than by deriving them.

• The fit is excellent. Over reflectances under three illuminants, a 3×3 predicts the reference eye’s cone catches from standard tristimulus values with relative residuals of 5.4 × 10⁻⁴, 1.0 × 10⁻³ and 8.4 × 10⁻³.
• The points it implies are not. Protanope at (0.99, 0.20) against a measured (0.7465, 0.2535); deuteranope at (−18.1, 11.2), which is not on any diagram.
• And they move with the fitting set. Refitted over daylight alone the deuteranope’s point is at (−5.2, 3.3); over the monochromatic lights, at (1.14, −0.45). A swing of 22.5 in chromaticity.
• The construction runs well in the other direction. Perturbing the three points by a thousandth moves the adaptation residual by one per cent; perturbing the basis by a thousandth moves the implied deutan point by between 0.4 and 9 thousandths.
• So the template supplies the difference and the measurement supplies the position. G = M(member)·M(reference)⁻¹ applied to the basis the published points give, which returns the published points exactly on the reference member.

What a confusion point is, geometrically #

A dichromat is missing one cone class. Two stimuli they cannot tell apart agree in the two classes they have, so the set of confusions through any stimulus is a line, and every one of those lines passes through one point: the chromaticity of the direction the missing receptor would have responded along. Three classes give three points, and three points are six numbers, which is exactly what colour matching leaves undetermined about an observer’s curves once the three row scales are accounted for.

Written as arithmetic, the protanope’s point is the chromaticity of the cross product of the M and S rows of the cone matrix. That is where the trouble is. The M and S rows, as directions in tristimulus space, are not far apart in the sense that matters: their cross product is a direction that is nearly determined by small differences between them, and small differences between them are precisely what a template gets slightly wrong.

A copunctal point is a quotient of small differences. The deuteranope’s is quoted at (1.4, −0.4), a long way outside the diagram, which is the visible symptom: a point at infinity is a direction stated as a position, and stating a direction as a position is what makes it ill-conditioned.

What the template does well #

The model of an eye here is five measurements: the lens’s optical density entered as an age, the macular pigment’s peak density, the cone outer segment’s axial density, and the peak wavelength of each of the three pigments. Each is a nomogram or a filter with a reported spread, and together they produce three spectral sensitivities.

Asked to predict what those sensitivities catch from a set of stimuli, expressed through the standard observer’s tristimulus values, the model is very good. Over a hundred and twenty smooth reflectances under three illuminants, the best 3×3 leaves relative residuals of 5.4 × 10⁻⁴ on the long-wave cone, 1.0 × 10⁻³ on the middle one and 8.4 × 10⁻³ on the short-wave one.

That is more than enough accuracy for everything this collection normally asks of it: how much two members of a population disagree about a match, how far a display’s primaries move a population’s acceptance, how the median eye compares with the standard. Those are all questions about catches, and catches are what the fit is good at.

What it does badly, and by how much #

Asked for the confusion points, the same fit returns the protanope’s at (0.991, 0.205) — 0.25 away in chromaticity from the measured (0.7465, 0.2535), which is a quarter of the way across the diagram — and the deuteranope’s at (−18.1, 11.2).

Refit the same template over a different stimulus set and the numbers change completely:

The deuteranope’s point moves 22.5 units of chromaticity between the first and the third. The diagram is 0.8 wide.

A quantity that moves by more than the picture it is drawn on when the fitting set changes is not being measured. The three fits differ from each other in the cone matrix by fractions of a per cent — all three are excellent predictors of what the cones catch — and their implied points are in different parts of the plane.

The monochromatic fit is the closest to the published values and it is the worst fit: its residuals are 8.1 × 10⁻², 9.1 × 10⁻² and 4.9 × 10⁻¹, two orders of magnitude above the others, because it weights the ends of the spectrum where a template is least reliable. Its being closest is a coincidence and treating it as a vindication would be exactly the wrong lesson.

The construction runs well the other way #

The ill-conditioning is one-directional, and that turns out to be the useful fact.

Perturbing the basis by a thousandth of its own scale, in random directions, moves the implied deuteranope point by between 4 × 10⁻⁴ and 9 × 10⁻³ across the eight matrices this collection tabulates — an amplification of between 0.4 and 9. That is moderate, and it means the numbers this collection quotes for how far each published transform sits from the measured points are sound.

Perturbing the points by a thousandth moves the adaptation residual of the resulting basis by 0.018 ΔE00, which is one per cent of it; by a hundredth, by 0.21; by five hundredths, by 6.85, which is more than four times the residual itself.

Two qualifications on that, and both are in the numbers already quoted.

The asymmetry is a large-perturbation property, not a small one. At a thousandth, the points-to-basis direction amplifies by about ten — a tenth of a per cent on a point moves the residual by one per cent of itself — and the basis-to-points direction amplifies by about six, since nine thousandths on a point whose magnitude is 1.456 is six tenths of a per cent. Those are the same order. What separates the two directions is what happens next: the basis-to-points amplification stays where it is, and the points-to-basis response turns over. Its exponent is 1.07 between a thousandth and a hundredth — linear — and 2.17 between a hundredth and five hundredths.

So the sentence to carry is not that one direction is conditioned and the other is not. It is that one direction is linear and the other is not, and a linear amplification of ten is a thing that can be reasoned about while a quadratic one is not.

And the population’s own spread sits at the turn. Extrapolating the second exponent, a perturbation of the confusion points by 0.027 makes the resulting basis’s adaptation residual as large as the residual itself — the point at which the construction has stopped being a perturbation of anything. The protanope’s cloud across two hundred members has a root-mean-square radius of 0.025.

Those two numbers are the same to within a tenth, and the coincidence is worth stating carefully because it is easy to over-read. They are not the same operation: the transfer applies a fitted 3×3 to the whole basis, and the perturbation study moves the three points directly, so a cloud radius of 0.025 does not mean the population is applying perturbations of 0.025 in the sense the sensitivity study measures. What it does mean is that the population’s variation and the construction’s breakdown are at the same scale, with no order of magnitude between them.

That is the honest caution to attach to everything downstream. The transfer is well founded — it returns the measurement exactly when asked for no change, which is the check that matters — and the spread it produces is not comfortably inside the region where the construction behaves linearly. It is at the edge of it. A population twice as wide as the one modelled here would be outside, and the reported spreads on the five variates are literature figures with their own uncertainty.

Which sharpens what the essay’s own robustness assertion is doing. The population median cost moves by 1.2 per cent between two broadband fitting sets and by 12 per cent when the monochromatic set is used, and that twelve per cent is reported rather than asserted away. The 0.025-against-0.027 comparison says why twelve per cent is the right order to expect: a modelling error concentrated in the two variates that are filters moves the basis, the basis moves the points, and the points are near the scale at which the response stops being linear.

So the map from points to basis is well behaved for small perturbations and violently sensitive for larger ones, and the map from basis to points is mild. The construction is a good way to build a basis from measured points and a bad way to report a basis as points, which is a sentence about conditioning rather than about either object.

Doubling the population is the check that the cloud’s width is a property of the eyes rather than of how many were drawn.

The transfer #

What is wanted is the spread of the receptor construction across a population — what happens to an argument that rests on three quoted numbers when the people those numbers came from are replaced by other people. The template cannot supply the points. It can supply the difference between one member’s cones and another’s, and that difference is a linear map with none of the ill-conditioning in it.

Fit each member’s cone matrix M(p) over the same stimulus set as the reference member’s M(ref), and take G = M(p)·M(ref)⁻¹. That is the 3×3 carrying one pair of eyes to another. Apply it to the basis the published points give:

B(member) = G · koenigMatrix().

On the reference member G is the identity to 2 × 10⁻¹⁷ and the construction returns the published confusion points to 2 × 10⁻¹⁶, which is the check that it is a perturbation of a measurement rather than a second model. Everything the template’s absolute error does is common to M(p) and M(ref) and cancels in the ratio; what survives is the part that differs between the two, which is what the five variates model and what the whole question is about.

What was computed, and how #

The cone matrix is an ordinary least-squares fit of each member’s three catches against the standard observer’s three tristimulus values, over a stimulus set that is named rather than assumed. Three sets are available and the default is smooth reflectances under three illuminants, on the grounds that it is the closest of the three to what a matching experiment shows anybody.

The assertion in the build has two halves and the second is the one that rules out the obvious repair. The point read straight off the template must be at least 0.2 from the measured one — the observation. And it must move by at least a whole unit of chromaticity between two fitting sets — which says the problem cannot be fixed by choosing better stimuli, because the quantity is not stable under that choice.

Robustness of the transfer is asserted separately: the population median cost is 1.780 fitted over three illuminants and 1.802 over one, a difference of 1.2 per cent, and 1.573 over the monochromatic set, a difference of 12 per cent. The first is required to be small; the second is required to be large, and is reported rather than asserted away, because it is the honest uncertainty on everything downstream.

Where the three confusion points actually sit is the picture behind all of this, and two of the three are outside the diagram proper.

What the cloud looks like once the transfer is applied #

The point of the whole construction is a distribution, and it is worth reporting its shape here rather than only its consequences.

Across two hundred members the protanope’s point sits in a cloud of root-mean-square radius 0.025 in chromaticity, centred 0.013 from the quoted value — inside its own spread, which is the reassurance that the transfer has not moved the anchor. The tritanope’s radius is 0.073 and its centre sits 0.038 from the quoted point. The deuteranope’s radius is 0.345, which is enormous as a chromaticity and is the ill-conditioning arriving from the other side: the point is far off the diagram, so a small change of direction is a large change of position.

Distances between the three are therefore not comparable in chromaticity, and every comparison in the essays downstream is made in units of each point’s own spread instead. That is not a cosmetic choice. Ranked by raw distance the deuteranope’s point dominates everything and the ranking is a ranking of how far off the paper each point is.

Which of the model’s own numbers is doing the displacing is a question the same machinery can answer, and the answer turns out not to be the pigments at all.

Where the model stops #

This is not a published study of between-observer variation in confusion points. It is what this collection’s own pigment model implies, propagated — and whose eyes those are is a question the model answers only as a distribution. A real dichromat also differs from a normal observer in ways no template of this kind carries, and every essay that uses the resulting cloud says so.

The transfer is a linear map and a filter is not. The lens and the macular pigment multiply a sensitivity curve pointwise, and a pointwise product of a curve and a filter is not in the span of three other curves. G fits it well over broadband stimuli and poorly over monochromatic ones, which is exactly the twelve per cent quoted above — the modelling error is concentrated in the two variates that are filters.

And nothing here says the template’s changes are the right size. It says what follows if they are. The reported spreads for lens density, macular density, axial density and the three peaks are from the literature, and the propagation is only as good as they are.

The same population read as a distribution rather than as a cloud says where the published transforms fall inside it.

The generalisation #

A model’s accuracy is a statement about a quantity, and a different quantity computed from the same model can be arbitrarily worse. Half a per cent on a cone catch and a quarter of the diagram on a confusion point are the same model, on the same day.

The bridge between the two is the condition number of whatever function turns one into the other, and it is worth computing before trusting a derived quantity. Here the function is a cross product of two nearly parallel rows followed by a projective division — two ill-conditioned steps in a row — and both are visible on inspection.

The second half is the repair, and it generalises further than the diagnosis. Where a model is trustworthy about differences and not about absolutes, use it for differences. Anchor on the measurement, propagate with the model, and check that the composition returns the measurement exactly when the model is asked for no change at all. That last check is worth insisting on: it is cheap, it is exact rather than approximate, and it is the difference between a perturbation of somebody’s data and a second model wearing its labels.

Who found it, and when #

König’s construction of cone fundamentals from dichromat confusion data is of the 1880s and 1890s, and Smith and Pokorny’s fundamentals — which this collection checks its algebra against and recovers to four decimals — are from 1975. The construction has always run from measured points to a matrix, which is the well-conditioned direction, and the literature reports the points as measurements rather than deriving them from a pigment model.

Where pigment models are used to predict confusion points, the usual practice is to report the confusion lines or the dichromatic confusion loci rather than the copunctal point itself, and that is not a stylistic preference. A line is a robust object and its intersection with a distant line is not, and a discipline that draws lines instead of points has quietly avoided the whole problem.

Where the ladder goes next #

With the transfer in hand the question the quoted numbers could not answer becomes answerable: what does the receptor construction cost, across a population of eyes rather than for the one that was measured? The answer moves more than the table it is being compared in, which makes a number this collection has quoted as a property of a construction into a number about a person.