A second lamp pair finds a lens band's centre

A dimmer session hides a lens band in the rods looked for the way a colour-matching experiment on older observers could tell a band of extra absorption in the lens from the macular pigment, and found it was not a dimmer session: more rod signal gave the band more of the fitted rod weight to hide in. What separated the band from the macula was a fourth fitted number, the band’s amplitude at 450 nm, which recovered a tenth of a density to within a hundredth in a single daylight session.

A parameter for a band at 450 nm describes a band at 450 nm. Given a band at 440 it found two thirds of it; given one at 410 or 420 it found almost nothing, and the fit read that band as five or six years of lens age instead. The yellow pigments that accumulate in an ageing lens absorb at centres that differ from one eye to the next, so a fit that is to find them needs the centre as well as the size — a fifth number, and one the matches depend on nonlinearly. The essay predicted that one session cannot place a band’s centre below 430 nm to better than about fifteen nanometres, because there a band and an older lens change the matches almost identically, while two sessions with a lamp whose blue line sits near 440 place it within five. If so, the lamp an experiment on older observers most needs is one with a narrow line where the lens’s pigments and the macula overlap.

One session trades a low band for age; a second pair places every centre #

The prediction is right about where the difficulty is and about how to cure it, and wrong about which lamp does. From one daylight-against-LED session, a band’s centre is placed to within six nanometres anywhere from 435 to 460 nm, and only to 8.5 to 11.5 nanometres from 410 to 425, where the fitted centre trades with the lens age at correlations up to 0.89. There the band’s size is barely measured either: a tenth of a density at 410 nm comes with a standard error of 0.09. A second session, the LED against the three-emitter lamp, places every centre from 410 to 460 within five and a half nanometres and every band’s size to a third of itself or better. A second session against the fluorescent tube, whose mercury line sits at 436 nm, places a band at 440 better than any other design, to 2.8 nanometres, and a band at 415 worse than the three-emitter session.

• Below 430 nm, moving a band is ageing a lens to a single session; above it, the macula’s flank gives the centre something to be measured against.
• Freeing the centre costs the low bands their size. A band at 410 nm is found only as a correlated blur of centre, amplitude and age.
• The cure is a second pair of lamps, not a special line. A pair that differs across the whole blue decorrelates the centre from age at every wavelength.
• A line helps where it sits. The tube’s 436 nm line is the best tool for a band at 440 and not a general one.

The figure is the census. The one-session curve rises to its worst between 415 and 420 nm and falls to three and a half near 440; each two-session design lowers it most where it was worst. The three-emitter design is the flattest, under five and a half nanometres everywhere; the two tube designs dip lowest near their line and rise above the three-emitter design below 425.

Five numbers from the same matches #

The observer, the band and the matches are the earlier essays’, and the lens is the one the eye stops at the lens described taking the short end of the spectrum: a sixty-year-old whose lens carries, on top of the model’s age curve, a band of absorption a tenth of an optical density high and fifteen nanometres wide; cones through the lens and the macular pigment and rods through the lens alone, the rod signal a tenth of each cone’s peak and half-weighted in the short-wave channel. A session is a set of asymmetric matches between two lamps across the same family of surfaces, each setting with its stated noise.

What is new is the fit’s fifth parameter. The five are the rod weight, the lens age, the macular density, the band’s amplitude and the band’s centre. Because the matches depend on the centre nonlinearly, the question asked of them is local: at the true observer, how precisely could the five be recovered — the standard errors from the matches’ sensitivity to each parameter, taken as central differences at the truth, and the correlations between them, which say which errors can be traded for which. At the band at 450 nm with its centre held fixed, the other four standard errors are the earlier essay’s to within seven per cent, the difference being that those derivatives were taken at the observer without the band.

The census runs the centre from 410 to 460 nm in five-nanometre steps, under four designs: the one daylight-against-LED session; that session with the LED against the three-emitter lamp, the pair earlier fits found best for the rod weight; and two designs adding the fluorescent tube, against the LED and against daylight.

Why a low band looks like an older lens #

The figure shows what the fit is trying to tell apart. An older lens adds density that rises steeply towards the shortest wavelengths. A band at 420 nm adds a hump; the same band moved a few nanometres to the red takes density from its short side and adds it on its long side, so its change is a dip followed by a rise. At 420 nm, most of what a lamp pair can see of that dip-and-rise falls where the ageing lens’s own change is also steep — the dip at 405 to 415 against the age term’s steep rise there — and a fit can buy most of a moved band with a slightly different age. At 450 the moved band’s change sits on the macular pigment’s short-wave flank instead, where ageing the lens changes little, and the two are distinct.

The macula is the other neighbour. Its absorption is a band of its own, forty nanometres either side of 460 — the macular is a band, not a filter is the essay about how narrow it is — and a lens band near 450 overlaps it almost entirely. With the band’s amplitude free, the earlier essay found that overlap to be the difficulty at 450; with its centre free as well, the centre gets something to push against there. A lens band moved from 450 towards 460 slides under the macula, and one moved towards 440 slides out from under it, and those two changes do not look like more or less macular pigment. The macula is what makes a band near 450 easy to place, and its absence below 430 is what makes a low band hard.

A lens of the wrong shape lands in the macula found the same neighbourhood from the other side: a lens edge a few nanometres off, which is a change of the whole short-wave slope, was read as macula. Every departure of the lens from the model’s shape is read as whichever standard parameter produces the most similar change, and below 430 that parameter is age.

The correlation between the fitted centre and the fitted age says the same in numbers: 0.83 to 0.89 for a band from 410 to 420 nm in one session, falling to 0.41 at 450 and 0.15 at 460. A correlation of 0.89 means the matches cannot distinguish a band moved by its standard error from a lens aged by its own; the centre’s standard error is large because an error in it costs almost nothing once the age absorbs it.

The prediction put that trade below 430 and put the cost at fifteen nanometres. The trade is where predicted, and the cost is a little smaller — eight and a half to eleven and a half.

Freeing the centre costs a low band its size #

The fourth parameter recovered a band at 450 nm to within a hundredth when its centre was known. With its centre free, one session measures a band at 450 to ±0.026 and one at 440 to ±0.028 — still clearly present — but a band at 410 nm to ±0.090 and one at 415 to ±0.070, most of its own size. The fit can no longer say whether a low band is there, because a band it does not have to place can be traded against age and against its own centre at once.

This is the ordinary cost of an extra parameter, paid where the parameter is weakest. It is also the practical reason the earlier essay’s fixed-centre parameter worked as well as it did at 450: a fixed centre is a strong assumption, and near 450 it happened to be both true and easy.

A second pair decorrelates the centre from age #

Adding the LED against the three-emitter lamp cuts the centre–age correlation at 410 nm from 0.83 to 0.41 and places the centre there to 4.8 nanometres. Across the whole range it holds the centre’s standard error between 3.2 and 5.5 and the band’s size to ±0.034 or better. The three-emitter lamp’s blue is a single emitter near 455 nm and the LED’s a pump near 452 with a broad phosphor beside it, so the difference between those two lamps across the band’s range has a different shape from the difference between daylight and the LED. Two differences of different shapes are what it takes to separate a band’s position from an age: one session gives one difference, and at the short end the band and the age move it together.

The lens is worst under tungsten found an ageing lens costing twice as much under one lamp as under another, and no single ordering of lamps across the eye’s departures; here, likewise, what places the lens’s band is not any one lamp but the difference between two pairs of them.

The tube’s line helps where it sits #

The prediction’s instrument was a lamp with a narrow line near 440 nm, and the fluorescent tube modelled throughout has one at 436. A second session against the tube places a band at 440 nm to 2.8 nanometres, the best of any design, and a band at 435 to 3.3. A line at 436 makes the tube’s light at the band’s centre change sharply as the band moves across it, and a match against the tube feels that directly.

Below 430 the line is on the band’s long-wave side, and it helps less than the three-emitter pair does: at 415 nm the tube design places the centre to 6.5 nanometres against the three-emitter design’s 5.2, and at 410, 5.5 against 4.8. The tube also carries mercury’s line at 405 nm, which ought to help a band at 410; it does, a little, but a line sees a band only at one wavelength, and a broad emitter’s flank sees it along the whole flank.

So a line is a precise tool for a band at its own wavelength, and a second pair of broad lamps is a general one. An experiment that expects bands anywhere between 410 and 450 is better served by the second.

What an experiment on older observers should do #

Fit the band’s centre if the band’s position is unknown, and expect to measure it only where the matches can: within six nanometres above 430 from one session, within eleven or twelve below.

Add a second lamp pair rather than a special lamp. The LED against the three-emitter lamp, already the best pair found for the rod weight, places every band centre from 410 to 460 within five and a half nanometres and every band’s size to a third of itself. A lamp with a narrow line is worth adding only if the band is expected at that line.

Read a fitted lens age with a low band in mind. The observer has no age made the point that a lens yellows in one direction for everyone, so an observer without an age is not an average over people; a fitted age is likewise not the eye’s age but the age of the model’s lens shape that best explains the matches. Below 430 nm a band and an older lens are nearly the same change to one session’s matches; a fitted age from one session is partly a band, and the matches carry their own lens holds only for a lens of the model’s shape.

How the census was computed #

The observer is built as in a band in the lens reads as age or as macula: cone absorptances at the population’s median peaks and standard density, the lens density of the model’s age function at sixty plus a Gaussian band of density 0.1 and standard deviation 15 nm, the standard macular pigment, and the rod signal through the lens added at a tenth of each channel’s peak, half-weighted in the short-wave channel. A session’s matches are the CIELAB differences between two lamps’ renderings of the same family of surfaces, with noise σ per setting. Sensitivities are central differences at the true observer with steps of 0.1 in rod weight, five years, 0.05 of macular density, 0.02 of band density and two nanometres of band centre; the standard errors and correlations come from the inverse of the Fisher matrix these build, summed over a design’s sessions. The lamps are the earlier essays’: a 6500 K radiator, a phosphor LED with its pump at 452 nm, a three-emitter lamp with emitters at 455, 528 and 625, and a fluorescent tube with mercury lines at 405, 436, 546 and 578 on three phosphor bands.

What this leaves out #

The standard errors are local. They say how precisely the five could be recovered near the truth; a fit started far from the truth might find a different minimum, especially for the centre, whose effect on the matches is not linear. The census says what the matches contain, not that a fit will always extract it.

The band is Gaussian and fifteen nanometres wide. A real lens’s extra absorption is broader and asymmetric, and a wider band would trade with age over a wider range of centres.

The observer is one sixty-year-old. The trade between a band and an age depends on the lens density the band sits on, and an eighty-year-old’s steeper lens would move the region of difficulty.

Still open: whether a fit started far away finds the centre #

The standard errors say the matches from two sessions determine a band’s centre to about five nanometres. A fit has to find it, and a fit over a nonlinear parameter can settle in the wrong place.

The calculation is the full nonlinear fit, started from centres spread across 400 to 470 nm, on simulated sessions with noise, for bands at 415, 430 and 445 nm, and the question how often the fit returns to within twice the standard error of the truth. The prediction is that from two sessions it finds a band at 430 or 445 from any start, and a band at 415 from starts below 440 but not above, where the fit reaches the macula’s flank first and settles there, trading the band for extra macular pigment — the confusion the earlier essay found at 450 running the other way. If so, a practical fit should start its band below the macula and let it move up.

A parameter’s precision is a question about its neighbours #

The habit is about what makes a parameter hard to measure.

A band’s centre low in the blue is hard to place, and the obvious reading is that the matches carry little information there. They carry a good deal — the band is a tenth of a density, well above the noise — but the information arrives in a shape the lens age can also produce. The band’s centre is imprecise not because the matches are blind to it but because a neighbouring parameter can explain the same change. The remedy followed from that reading, not the other: a second set of matches in which the two parameters produce different changes.

The failure mode is to treat a large standard error as missing data and look for more light at the difficult wavelength. The test is to look at the correlations first, because they name the parameter doing the hiding.