The matches carry their own lens

Age can size the rod signal, if the lens is known found an experiment that decides how large the rod signal in the colour pathways is: asymmetric matches between daylight and a phosphor LED, on observers of different ages, with the fitted weight’s trend across ages as the result. It also found what could ruin it. A lens ten years older than assumed moves the match as far as 0.38 of the rod signal’s S weight at fifty-five, and a tenth of an optical density of extra macular pigment as far as 0.42 — confounds the size of the drift the experiment is looking for. Measured on each observer, they would have to be known to two or three years’ worth of lens density and a few hundredths of macular density.

That comparison was of sizes. It asked how far each cause moves the match, averaged over a family of surfaces, and found them alike. It did not ask how. Each cause moves each surface’s match by a different amount and in a different direction, and a fit that takes all three as unknowns can tell them apart to exactly the extent their patterns differ. The essay ended on that question, and on a second lamp pair chosen to make the patterns differ more.

One session fits all three #

From one forty-two-setting session of daylight against the LED, with the S weight, the lens and the macula all fitted together, the lens comes out to within 1.2 years and the macula to within 0.016 of an optical density — the lens better than the 2.6 years a densitometer was asked for. The price is the weight’s precision, which falls from ±0.026 to ±0.058, and it is the macula’s price more than the lens’s. Adding a second session under the LED against a three-emitter lamp brings the weight back to within a sixth of its precision with both known, ±0.028, and the lens to within 0.56 years.

• The patterns differ. Across the surfaces, the weight’s pattern and the lens’s have a cosine of 0.58, and the weight’s and the macula’s 0.51; a fit separates what is not parallel.
• Some pairs of lamps separate them for free. Tungsten against the LED has the weight’s and the lens’s patterns nearly orthogonal, and fitting both costs the weight under five per cent.
• A densitometer is not needed, in the model, once two sessions are run. It would still be the check that the model’s lens is the eye’s.
• The separation survives a fourth thing eyes vary in. Fitting the S cones’ peak wavelength as well costs the two-session design another third of the weight’s precision, and fixes the peak to 0.34 nanometres.

Three causes, three patterns #

The observer is fifty-five, with the rod signal entering the S channel at weight one half. Each of the collection’s forty-two test surfaces — dips in reflectance at seven centres, three widths and two depths — is matched across the two lamps, and the match is recorded as the CIELAB difference between the surface under one lamp and under the other. Three things can move those forty-two matches: a change in the rod signal’s weight, an older or younger lens, and more or less macular pigment.

The three are the same size, which is why each looked like a confound of the others. Half the weight, ten years of lens and a tenth of macular density each move the b* of some surfaces’ matches by more than a unit. But they do not move the same surfaces. Across the first dozen surfaces — dips in the blue and violet — the weight and the lens rise together, both lifting the matches’ yellow-blue coordinate. Across the next dozen, dips in the green, the weight pushes the matches down and the lens barely moves them. The macula follows the weight for the dips at 430 nanometres, leaves it for those at 470, and joins it again in the green, because it absorbs over a narrower band than the lens and sits in front of the cones but not the rods.

This is the whole basis of the result. A fit sees forty-two surfaces in three coordinates, 126 numbers, and asks which combination of the three causes accounts for them. Where two causes move every number in proportion, no fit can separate them; where they move different numbers, it can, and the precision it achieves is set by how much of each pattern is not shared.

What freeing each costs, pair by pair #

Daylight against the LED is the best pair when everything else is known, fixing the weight to ±0.026 from one session. The reference lamp must not move found it the best pair for the weight, and this confirms it. Freeing the lens costs it only a fifth, to ±0.032, because the weight’s and the lens’s patterns share a cosine of 0.58 and leave most of each other free. Freeing the macula as well more than doubles it, to ±0.058: the macula’s pattern overlaps both the weight’s and, oppositely, the lens’s, and three partly alike patterns share more between them than any two.

Tungsten against the LED behaves differently. With everything known it fixes the weight less well, ±0.043. With the lens and the macula both fitted, ±0.044 — nearly nothing lost. Its weight pattern is almost orthogonal to its lens pattern, a cosine of 0.04, and only slightly aligned with its macula. So it is a worse pair for the weight and a much better one for the weight in an eye whose lens and macula are unknown.

At the other end, tungsten against the fluorescent tube cannot separate the weight from the macula at all: their patterns have a cosine of −0.84, and fitting everything raises the weight’s standard error from ±0.155 to ±0.347. A pair chosen for its sensitivity to the weight alone could as easily be this one.

The cosines are the reason in every row. Pairs whose weight pattern is nearly orthogonal to both others lose almost nothing by fitting them; pairs with a large cosine to either lose in proportion. The figure is the design table for anyone choosing lamps for this experiment: rank by the weight’s standard error with the confounds fitted, not with them known, and the pairs come in a different order.

Why the macula costs more than the lens #

The fit pays more for the macula than for the lens, and the reason is in where each sits. The rods are a fourth curve introduced the rod signal as a curve peaking near 500 nanometres added to all three cone channels. The lens absorbs below about 450 and sits in front of everything — cones and rods alike — so an older lens dims the rods’ short-wavelength tail as well as the S cones’, and its pattern in the matches is partly its own. The macular pigment absorbs between 400 and 520, which is the rising half of the rod curve, and it sits in front of the cones but not the rods. More macular pigment therefore lowers the S cones’ signal while leaving the rod signal alone, which is almost exactly what a larger rod weight in the S channel does from the other direction: it changes the proportion of rod to cone in the S signal. That is why the weight’s and the macula’s patterns have a cosine of 0.51 under daylight against the LED, and why the fit spends most of its information separating those two.

The field size is a lever on this. A field size is two changes found that the difference between the two standard observers is mostly macular pigment the light no longer passes through as the field widens, and a matching field of ten degrees averages over retina where the macula is thin. A larger field would shrink the macula’s pattern in the matches and with it the price of fitting it — at the cost of more rods in the field, which is a larger rod signal to measure and a larger weight to fit. Whether that trade favours the larger field is a computation this census does not make; it is the obvious next design variable after the choice of lamps.

The two confounds are also not independent of each other. Their patterns share a cosine of −0.25 under daylight against the LED: more lens and more macula move some surfaces’ matches in opposite directions. One match names the observer showed a single Rayleigh match fixing two parameters of a colour-vision deficiency because the two move its solution in different directions; the same geometry is at work here with three.

The lens, from the matches #

Eight of the ten pairs fix the lens more precisely from one session than the densitometer was required to, and the best — daylight against the three-emitter lamp — to ±0.6 years. The two that do not are the LED against the tube and tungsten against the tube, where the weight’s and the macula’s patterns are so alike that the fit spends its information disentangling them.

This inverts the conclusion of the essay before. The lens does not have to be measured separately if the matches are fitted for it, and the matches measure it better than the requirement a separate measurement would have to meet. The requirement was derived by asking how large a lens error would look if it went unmodelled. Modelled, it looks like itself.

Two sessions, fitting everything #

The best combination is daylight against the LED with the LED against the three-emitter lamp: the weight to ±0.028 with lens and macula fitted, against ±0.024 with both known — within a sixth. The lens to ±0.56 years and the macula to ±0.007. Every one of the ten best combinations includes daylight against the LED, which is the pair most sensitive to the weight, and adds a pair whose confound patterns differ from its own.

That is the answer to the closing question of the age essay, and it came out better than the question hoped. It asked for a second pair responding to the lens “many times more than to the S weight.” The best second pair does not do that — its lens and weight sensitivities are similar — but its patterns lie in different directions from the first pair’s, and two partly overlapping views from different angles pin down three unknowns better than one view of each.

A fourth parameter #

Real observers differ in more than lens and macula. The peaks move the flanks priced the spread in the cones’ peak wavelengths, and a shifted S cone moves the matches too. Fitted as a fourth unknown, it costs the single daylight-against-LED session another two fifths — the weight to ±0.080 — and the two-session design a third, to ±0.037, with the S peak itself fixed to ±0.34 nanometres. The two-session design pays for each addition modestly; the one-session design pays for each in full. That is the practical case for the second session: not that it measures the weight better, but that it keeps measuring it well as more of the observer is left free.

What the experiment now looks like #

Two sessions per observer, forty-two settings each, under daylight against the LED and the LED against a three-emitter lamp, fitted for the S weight, the lens, the macula and the S peak together. Per observer, the weight comes out to about ±0.04 and the lens to under a year. The age essay’s drift — a fitted weight 0.41 larger at seventy-five than at twenty if the rod signal has a fixed size — is then ten standard errors per observer, and the design it described, with a handful of observers at each end of the age range, becomes a design that does not depend on any auxiliary measurement.

A densitometer becomes a check rather than a requirement. The fit returns a lens density for each observer; a densitometric measurement of the same lens either agrees with it, which confirms that the model’s lens is the eye’s, or does not, which says the model is missing something the matches are absorbing into the lens. The observer has no age is the collection’s reminder that the age model is an average; this turns the average into an individual measurement that the colour experiment makes for free.

How the fit was computed #

For each pair of the collection’s five non-laser lamps — tungsten, daylight, the phosphor LED, the three-emitter LED and the fluorescent tube — and each of the forty-two surfaces, the match is the CIELAB of the surface under the first lamp minus under the second, for the collection’s observer at lens age fifty-five with the rod signal at a tenth of each cone’s peak, entering L and M at weight one and S at weight one half. Its sensitivity to each parameter is a finite difference: 0.1 of weight, five years of lens, 0.05 of macular density and one nanometre of S-cone peak. Each setting’s error is taken as 0.5 ΔE split evenly over the three CIELAB axes, one setting per surface. The Fisher information is the sensitivities’ inner products over that error, and a parameter’s standard error is the square root of the corresponding diagonal of its inverse, with the other parameters in the fit free and those outside it fixed.

These standard errors are smaller than the age essay’s, and on purpose. That essay followed the reference lamp must not move in dividing a setting’s precision by the median match signal over the surfaces, a summary that ranks lamp pairs; a fit uses every surface’s full shift, and the surfaces that move most carry most of the information. The rankings agree; the absolute numbers are a fit’s.

What this leaves out #

The model is taken as right. A fit separates causes by the patterns the model says they have. If the eye’s lens absorbs with a different spectral shape from the model’s, the fit will attribute the difference to whichever parameter best absorbs it, and could attribute part of it to the weight. That is what the densitometric check is for.

The standard errors are local. They are the curvature of the fit at one observer, fifty-five with weight one half; the patterns change with age and weight, and the design’s performance would need checking at the ends of the age range before being relied on there.

And the settings are independent. Real settings drift within a session, and an observer’s error across surfaces is correlated. Correlated errors would make some directions of the fit worse than computed here, most likely the ones where the patterns are alike.

Still open: whether the fitted lens agrees with the model’s shape #

The fit returns a lens age for each observer, meaning a lens density with the model’s spectral shape. A real lens’s absorption is not only denser with age but differently shaped, and the model’s age function is an average of both.

The calculation is this fit on synthetic observers whose lenses have a different shape — the same density at 400 nanometres as a sixty-year-old’s, but with the absorption edge moved by ten nanometres — and the question is how much of the shape error goes into the fitted weight. The prediction is that it goes mostly into the fitted lens age and the macula, because both absorb short wavelengths, and that less than a tenth of the weight’s standard error goes astray. If more does, the densitometer’s check is needed after all, and the right instrument is one that measures the lens’s shape as well as its density.

Sizes are not patterns #

The habit is about asking whether two effects of equal size are also effects of the same shape.

The age essay compared three causes by how far each moves a match and found them equal, and from that concluded that each confounds the others. The conclusion followed from the comparison but not from the physics, because a match is not one number. It is a set of forty-two colour shifts, and two causes that move the set by the same total can move it along different directions. Asked that way, the same numbers said the reverse: the causes are separable, and the experiment measures its own confounds.

The failure mode is to compare magnitudes where a fit compares directions. A confound is a confound only to the extent that its pattern lies along the effect’s. How far it moves things is a statement about how badly it matters if ignored, not about whether it can be separated.