Identifiability — where it appears
Named by 59 essays across 9 fields — each of them below, with the objects they name alongside it.
The matches do not name the cones
Colour matching is the whole empirical basis of colorimetry, and it fixes the observer's three curves only up to a nonsingular 3×3 — nine numbers that no match, in any quantity, to any precision, can see. One particular choice of those nine is used throughout here, and it was made for a different purpose.
A confusion point is a missing pigment
The nine numbers colour matching leaves free are fixed by three points on a chromaticity diagram, each of them the place where everything one class of dichromat cannot tell apart converges. Two of the three lie outside the diagram entirely, which is not a defect — a direction in tristimulus space need not correspond to a light.
The diagram has no area
A chromaticity diagram is a projective picture of a three-dimensional space, and the freedom colour matching leaves in the observer acts on it as a projective map. Straight lines and mixture ratios survive that; area, distance and angle do not — so half of what the diagram is used to say is a statement about the paper.
Two thirds is not a property of the eye
It has been said from the beginning here that about two thirds of the chromaticity diagram cannot be shown on a screen. The figure is right, on the diagram it was measured on, and across twelve published diagrams the same triangle covers anything from 8.5 to 38.4 per cent of the same locus. Counting stimuli instead gives an answer that does not move.
No diagram makes them circles
Every chromaticity diagram is a projective picture of the same measurement, so how badly MacAdam's ellipses fail to be circles can be minimised over the whole family of them. The best plane there is still leaves the average ellipse twice as long as it is wide — which makes the residual a fact about the eye rather than about anybody's choice of primaries.
A difference needs a basis too
A linear change of coordinates leaves every colour match exactly where it was. A cube root does not commute with one — so a lightness–chroma space, and every colour difference computed in it, is a property of the basis that happened to be in place before the nonlinearity. CIELAB's basis was chosen in 1931 for reasons that had nothing to do with difference.
An image does not determine the light
A photograph of a scene under one illuminant gives three numbers per surface and asks for the illuminant plus three numbers per surface. The count closes only if reflectances lie in a two-dimensional model, and no number of surfaces helps — at three dimensions the alternative scenes can be written down, and they reproduce every sensor response exactly.
The chart decides the profile
A camera's colour matrix is nine numbers fitted to a set of patches somebody chose, and the number that comes with it is the error on those patches. On a chart with no chromatic range that number is 0.19 ΔE00 and the matrix delivers 1.65 — and a second matrix, indistinguishable on the chart, delivers 2.03.
A matrix is fitted under one light
A camera's colour matrix is nine numbers determined by a chart photographed under a particular illuminant, and the error it quotes is the error under that illuminant. Fitted under a tungsten lamp and used under a cold sky it delivers eight times as much — and the two-matrix scheme every real profile uses turns out not to be a compromise at all.
Three numbers cannot see a line
An instrument that returns three filtered readings of a spectrum determines a three-dimensional projection of it and is exactly blind to the other seventy-eight. On daylight that costs almost nothing; on a fluorescent tube, three quarters of the lamp lies in the part no reading reaches, and adding filters recovers it slowly.
The colour is right first
A reconstruction of a lamp from twelve filtered readings gets its colour right to half a unit while two thirds of its spectrum is still unmeasured. That combination is not a partial success — it is the exact condition under which a spectral prediction made from the reconstruction will be confidently wrong.
A fit can be exact and empty
Every fitted object here reports one number, the residual on the data it was fitted to, and every one of them has two more that nobody publishes — how many of its parameters the data actually determine, and how large the family of equally good answers is. The third column is where the failures live.
Which of these is a convention
Nine ordinary sentences from this collection, put through one test — do they mean the same thing after the observer's three curves are replaced by a nonsingular combination of themselves? Five survive and four do not, and none of the four is wrong, because each is a statement about a set of coordinates being read as a statement about an eye.
A profile is a fit between its nodes
A printer profile is a table, exact at every patch that was printed and interpolated everywhere else — and everywhere else is where every job lives. Going from a hundred and thirty-five patches to three and a half thousand is twenty-seven times the work for eight times the accuracy, and the exponents say that is as good as it gets.
The cones an appearance model uses
CIECAM16 adapts in three axes whose rows are labelled L, M and S, and they were fitted to corresponding-colour experiments rather than measured on receptors. Run the dichromat construction backwards on them and they commit to a deuteranope confusion point 1.45 away in chromaticity from the measured one — which is a test the axes were never asked to pass.
The three numbers a gain cannot see
Colour matching leaves nine numbers free. Three dichromat confusion points fix six of them and three choices of unit fix the rest — and it turns out that a von Kries gain is exactly blind to those last three. So the dichromat data do not merely constrain an adaptation basis. They determine it, with nothing left over to fit.
The best axes are not receptors
If the axes that make a von Kries gain work were nearly the cone fundamentals, the published adaptation transforms would be cone-like and their departures would be slack in a fit. They are not. Minimise the residual over all nine free numbers and the winner sits further from the measured dichromat confusion points than any of them.
A compression goes below the floor
Elsewhere this collection minimised the anisotropy of MacAdam's ellipses over every chromaticity diagram there is, found 2.02, and called the residual a property of the eye. It is a property of the eye seen through a projective picture. A cube root after the right basis reaches 1.61 on the same twenty-five ellipses.
The exponent was never the argument
A century of colour science has argued about whether the eye's response is a cube root, a square root or a logarithm. Minimise the anisotropy of MacAdam's ellipses over every basis, at each of eight exponents, and the floor moves by under three per cent between a cube root and a tenth root — while the basis moves it by a factor of two.
A constraint costs what it points at
Three primary chromaticities remove three of the nine numbers in an adaptation basis and cost one per cent. Three dichromat confusion points remove six and cost seventy. Counting what a constraint removes predicts neither, because an optimum is a long bowl and what matters is which way the constraint points.
The rank is the invariance
A von Kries gain cannot see the scale of a row of its basis. That is an identity, proved in a line, and it can be measured instead — as the rank of a second-derivative matrix. Both objectives this collection minimises over the observer's nine free numbers have a Hessian of rank exactly six, and the three directions they cannot see are the three scalings, to a hundredth of a degree.
A constraint is a direction and a distance
Four restrictions on the same nine numbers cost nothing, nothing, two per cent and seventy. How many parameters each removes predicts none of it. What does is the quadratic form evaluated along the displacement — and showing that it does means walking in towards the optimum rather than arguing at the edge, because at the edge the prediction is out by a factor of three.
A template cannot place a point
This collection's model of an eye is good to a few tenths of a per cent at predicting what a cone catches, which is far more than enough to place a spectrum. Asked where that eye's confusion points are, it puts the protanope's at (0.99, 0.20) against a measured (0.75, 0.25) and the deuteranope's anywhere from (1.1, −0.5) to (−18, 11) depending on which stimuli the fit was made over.
The price is also the person
The receptor construction costs seventy per cent above the unconstrained floor, which is a number usually quoted as a property of the construction. Propagated across a population of eyes it runs from a quarter above the floor to a hundred and thirty per cent above it, and the population's own spread is wider than the entire gap between the published transforms the seventy per cent was being compared against.
A corner is corrected by one row
A grey-card gain map makes the corner of a frame exactly right on grey and leaves coloured patches 1.86 colour differences wrong at twenty-five degrees. A three-by-three matrix fitted at that position halves it — and six of its nine numbers do nothing, because the moved filter edge is in one channel. The three that matter rebuild the lost red from green and blue, they carry to another lamp better than a gain map in eleven cases of twelve, and in the twelfth, a row fitted under tungsten and used in daylight, they leave the grey 9.2 off.
Two matrices do not reach a white LED
A camera profile's two matrices, blended by the scene's colour temperature, are as good as a matrix fitted anywhere along daylight. Under a white LED or a fluorescent tube the same blend leaves colours twice as far off as a matrix fitted under that lamp, and no weight inside the profile's range repairs it. What decides it is not how far the lamp sits from the Planckian locus — a triphosphor tube sits nearly on it and fares worst — but whether its spectrum has lines in it, which a white balance reading cannot see.
Two rooms with one lightness scale
CIECAM16's surround and its background both reach lightness, and they reach it through one product. So the rooms fall into classes: a television in a lit living room against a mid grey returns exactly the lightness a print on a desk against a background of 2.8 does, for every sample, to the last bit of a double. It returns 0.76 of its chroma and 0.81 of its brightness. Two of the model's four viewing-condition parameters are one parameter, and only colour tells them apart.
Flicker sorts lamps the wrong way
A camera cannot see a spectral line in its own white, so the essay on the two-matrix profile named the classifiers a device might have instead, and the first of them was flicker. Flicker is measurable, and it measures the wrong thing. It sorts lamps by how their power is delivered while the matrix needs them sorted by how their spectrum is shaped, and the two are independent: a white LED on a constant driver is perfectly steady and strongly structured, and it is what most indoor photographs are lit by.
The converter can choose except where it matters
A converter rebuilding a clipped highlight has to say whether the surface was matt and over-exposed or glossy and carrying a reflection, and the evidence is whether its raw chromaticity slides towards the lamp's on the way up. Read through the sensor's own noise, the slide is clear on most of the chart from the few tens of pixels a specular highlight holds — and on the surfaces where it takes half a frame, confusing the two models costs more than the median. A ratio cannot see a common scale, and an exposure supplies one.
Four places to clamp are two pipelines
The essay on clipped noise ended on a procedure: a black frame and a dim grey card at a high amplification would place each raw converter's clamp. A procedure is a claim that a measurement identifies something, and this one identifies less than it looks. A clamp at zero commutes with the white balance and with the tone curve and not with the colour matrix, so the four positions are two pipelines — and the measurement separates them, on a card at half a per cent of white rather than on the black frame.
Two lamps decide what one lamp could not
Under one lamp a handful of surfaces cannot tell a glossy highlight from a matt over-exposure without most of a frame, because there the two differ by a scale a ratio cannot see. In a room lit evenly by a 3000 K lamp and daylight, every surface on the chart is decided from ten pixels. A body can sit at one lamp's white, not at two, and the rescue holds only while the second lamp carries a fifth of the light and its white sits far enough from the first.
Two sensors disagree about deep red, not lines
A phone's ambient-light sensor and its camera read the same lamp differently, and the difference sorts fourteen lamps into smooth and structured without a single mistake — where flicker made five. It is not reading their lines. Almost all of it comes from the sensor's clear channel collecting deep red the camera's infrared cut throws away, so daylight with its far red trimmed is called structured and a white LED with a far-red emitter is called smooth.
A narrow channel has to be read on its own
An ambient sensor with a clear channel tells structured lamps from smooth ones by reading deep red, and one without it reads structure but breaks at one per cent of calibration error. A narrow fourth channel between the camera's peaks was proposed to have both. Pooled into the sensor's disagreement with the camera, it reads structure and breaks at one and a quarter per cent — no better than the design it was meant to rescue. Read on its own, as its departure from what the camera predicts for it, a channel at 450 nm holds to four per cent, the clear channel's figure, and far red does not move it.
Age can size the rod signal, if the lens is known
A rod signal leaking into the colour pathways has no natural size in the model: read as a fraction of each cone's own peak it shrinks as the lens yellows, read as a fixed amount it grows relative to the cones, and the two readings differ by a factor of two over a lifetime. An asymmetric match on observers of different ages can tell them apart, because they predict opposite trends: one says a weight fitted at seventy-five will look sixty per cent larger than at thirty-two, the other says it will not move. A handful of observers per age would show it. But a lens mistaken by ten years looks like forty per cent of the weight, and a tenth of an optical density of macular pigment as much again, so the experiment is only as good as what it knows of each eye's lens and macula.
The matches carry their own lens
A lens mistaken by ten years moves a colour match as far as half the rod signal being measured, and a tenth of an optical density of macular pigment as far again, so an experiment on the rod signal looked as if it needed a densitometer for every observer. It does not. Across a family of surfaces the three move the matches in different patterns, and one session of forty-two settings fits all three together — the lens to within a year and a quarter, better than a densitometer was asked for. The price is the weight's precision, and a second pair of lamps nearly removes it.
The lamps two channels miss are smooth to the camera
A narrow channel at 450 nm tells a phone which lamps are structured, and a white LED pumped in the violet might hide from it; a channel at 500 nm does the same job, and an LED with a cyan emitter should blind it. Reading both and calling a lamp structured if either departs was supposed to cover both families. It covers the cyan family only because the 450 nm channel already did, and half the violet-pumped LEDs escape both channels. Every one that escapes leaves the camera's colours as accurate as a smooth lamp does. What the classifier misses is what the camera does not need to be warned about.
A lens of the wrong shape lands in the macula
One session of colour matches can fit an observer's rod signal, lens and macular pigment together — but it fits a lens of the model's shape, one exponential in wavelength scaled by age, and a real lens need not be that shape. Give a sixty-year-old a lens with the right density at 400 nm and its edge ten nanometres off, and the fit puts the error mostly in the macula, moves the rod signal's weight by six tenths of its standard error — six times the predicted tenth — and leaves a misfit the session cannot see. A second lamp pair makes it worse: 3.5 standard errors. Fitting the edge as a fourth number removes the bias, costs the weight three to fourteen per cent, and finds the edge.
A band in the lens reads as age or as macula
A lens whose absorption edge is steeper or shallower than the model's biases a matching session's fit, and fitting the edge's position repairs it. An ageing lens also grows bands: yellow pigments that add a shoulder of absorption just past 400 nanometres. The edge parameter takes none of one. A tenth of an optical density centred at 410 to 430 nm is read as four to six years of extra lens age; centred at 440 to 460, as three to ten hundredths of macular pigment. Either way the rod signal's fitted weight moves, by up to 1.4 of its standard errors in one session and 1.8 in two, and the session's misfit stays within its noise. The lens model has to carry the band, or the experiment has to state what it does not know.
A dimmer session hides a lens band in the rods
A band of extra absorption in an older lens is read by a colour-matching session as lens age or as macular pigment, and it pulls the fitted rod weight with it. The rods see the lens and not the macula, so a dimmer session, where they carry a third of the signal instead of a tenth, was predicted to tell the band from the macula. It does the opposite. With the band at 450 nm left out of the fit, the rod weight's shift grows from 1.7 of its standard errors to 2.5, the misfit a session could notice shrinks by a third, and the macula and the band trade more tightly. The rods see the band — and the rod weight, which the fit is free to move, is exactly what absorbs it. What separates the band from the macula is fitting its amplitude as a fourth number, which works in daylight already.
A second lamp pair finds a lens band's centre
A band of extra absorption in an older lens is found by a colour-matching fit only if the fit has a parameter for it, and a band parameter at a fixed wavelength finds only a band near that wavelength. Freeing the band's centre as a fifth number asks whether the matches can place it. From one daylight-against-LED session they place a band above 430 nm to within six nanometres and one below it only to eight to twelve, where moving the band and ageing the lens change the matches together — and there they barely see the band at all. A second session, the LED against a three-emitter lamp, places every centre within five and a half nanometres. The fluorescent tube's line at 436 nm helps where it sits and nowhere else.
A lost lens band goes to age or the macula
Two colour-matching sessions determine the centre of a band of extra absorption in an older lens to about five nanometres, if a fit can find it. Fitted in full from eight starts across 400 to 470 nm on twenty simulated experiments, two sessions find a band at 415, 430 or 445 nm from every start in fifteen to eighteen experiments of twenty, against eighteen for a fit started at the truth. The prediction was that a low band would be lost from high starts to the macula's flank. The start barely matters with two sessions, and a lost band does go somewhere — but a band at 415 or 430 goes to lens age, three to five years of it, and only a band at 445 goes to the macula. Every loss is a worse fit than the session's best, so eight starts find the band as often as the truth does. With one session the start decides.
A macula three nanometres short is a lens band
Two colour-matching sessions can fit a band of extra absorption in an older lens, its size and its centre, and the test proposed for real older observers was to count those with a band more than twice its standard error. That test assumes a significant band can only be a band. The observer model's macula peaks at 460 nm; a real one is two carotenoids whose mix moves its peak by a few nanometres. Simulated with its macula peaking three nanometres short and no band at all, an observer at sixty is fitted with a lens band of 0.06 at 438 nm, three standard errors, and noisy sessions find it in nineteen of twenty experiments. Six nanometres short, the phantom is larger than the band the earlier essays searched for. The confusion runs both ways: a real band fitted with the macula's peak free reads as a macula two to three nanometres short. Neither fit alone can tell them apart. Comparing the two can, with two sessions — never preferring the macula for a real band, and preferring it in sixteen to nineteen experiments of twenty for a short macula.
A second session keeps a band and the macula apart
A lens band and a macula peaking a few nanometres short look alike to a colour-matching fit that has only one of them, and an older eye may have both. Fitted with all six numbers — rod weight, lens age, macular density, the band's amplitude and centre, and the macula's peak — nine simulated observers at sixty, each with a band at 415, 430 or 445 nm and a macula moved three nanometres either way or not at all, test whether the matches can carry both. The prediction was that the band at 445 and a short macula would trade almost completely, a correlation beyond 0.95 and the band's standard error tripled. That is what one session does: correlations of 0.91 to 0.95 at 430 and 445 nm, and a band found in at most twelve experiments of twenty. Two sessions do not. Their correlations stay between 0.42 and 0.83, the band is found in twelve to twenty experiments for every observer, and a moved macula in twelve to nineteen — while an unmoved one is found moved in at most three. The price of the extra number is paid by the band nearest the macula, whose standard error rises by three quarters; the rod signal pays almost nothing.
A line on the macula's flank buys less than a repeated session
Fitted with six numbers, two sessions of colour matches keep a lens band and a moved macula apart, but the band at 445 nm pays for it: correlated with the macula's shift at 0.82, its standard error 1.8 times its value when the macula is not fitted. A moved macula changes its long flank past 470 nm, where no lens band reaches, so a third session — the LED against a lamp with a narrow line at 487 nm — was predicted to bring the correlation under 0.6 and the band's error within a quarter of its band-only value. The correlation falls to 0.80 and the error to 1.5 times. A third session that simply repeats the second does better on both errors. The flank line pins how much macular pigment there is, not where it peaks, and it ties the band's height to its centre. Moved to 465 nm, onto the macula's peak, the same session brings the band within about a fifth of its band-only error — but no position of the line brings the correlation below 0.78. The band at 445 nm and the macula's peak are told apart by their short flanks, and a third session adds matches rather than a new kind of evidence.
Knowing the macula's density frees two sessions, not the line on its peak
Fitted with six numbers, an older observer's colour matches cannot fully split a lens band at 445 nm from a macula whose peak has moved: their correlation stays above 0.78 under every third session tried. The proposal was to measure the macular pigment's density from outside the matches, leaving five numbers, and the prediction was that a line on the macula's peak would then split the pair below 0.6 while two plain sessions stayed near 0.8. It runs the other way. With the density known exactly, two sessions fall from 0.83 to 0.62 and the line at 465 nm only from 0.86 to 0.74. The density was the road the band and the peak traded along — correlated with each at about −0.75 in two sessions' fit — and the designs that had already narrowed that road gain least from closing it. A realistic outside measurement closes almost none of it: the matches already know the density to 0.027, so a flicker measurement good to 0.05 moves the correlation by three hundredths. And far from the macula, at 415 nm, a known density binds the band to the peak more tightly than before.
LEDs built on the locus still fall off the line
A camera's colour error under a lamp can be estimated from three narrow channels of a small sensor, read as departures from a smooth lamp of the same colour, and the estimate held for the census's white LEDs, which varied only their blue pump. Rebuilt on LEDs varied freely in pump, converter and red phosphor, it failed: the error was a V about the Planckian locus, and no line in the departures read it. A maker does not let the white fall where it falls, so the proposal was to build the LEDs as a maker does — the converter tuned for each pump and red phosphor until the white sits on the locus — and the prediction was that the V would go and the line return, the census's held-out error to within ±0.15. Thirty of thirty-six such LEDs can be built. On the locus the V is gone and the line is not: the LEDs alone are missed by ±0.25 to ±0.28, the census's lamps by ±0.27. Taken one pump at a time, the departures read the LEDs pumped at 452 nm to ±0.05 and those at 440 and 465 nm no better than ±0.17. Within one pump the LEDs are readable, and across pumps they are not.
A density a tenth wrong costs the band more than fitting it
An older observer's colour matches carry a lens band and a macula that can trade against each other, and fixing the macular pigment's density from an outside measurement breaks much of that trade when the density is right. A flicker measurement and a colour match average the pigment over different fields, so the fixed density can be wrong by a tenth or more, and the prediction was that a tenth too high takes about a third of a standard error off a band at 445 nm and moves the macula's peak by under half a nanometre. It takes 1.4 standard errors off the band and moves the peak 1.25 nm short. At 415 nm the same error pushes the band the other way, up by 2.4 standard errors, and runs its fitted centre 5.6 nm shorter and the lens's age three years younger. Bias and error together, a density fixed a tenth wrong gives the band at 445 nm a larger error than leaving the density free; the break-even is about eight per cent. Given as a measurement with its own standard error of 0.03 instead of fixed, the same wrong density biases the band by under half its error.
A wrong density shifts every experiment alike near the macula
Read from the expected matches, a macular density fixed a tenth too high lowers an older observer's lens band at 445 nm by 1.4 of its standard errors and raises a band at 415 nm by 2.4. Real experiments carry noise, and the prediction was that their fits scatter about those biases with the fixed fit's own error, so that a tenth's mistake puts the band beyond twice its error in about a third of experiments at 445 nm and two thirds at 415. Twenty simulated experiments per case, each fitted with the density free and held at 0.9, 1 and 1.1 of its truth, give five of twenty at 445 nm and sixteen at 415 under two sessions — eight and eighteen with a line on the macula's peak. Near the macula the bias is a rigid shift: every experiment moves by the first-order amount to within a tenth of a standard error. Far from it the bias is lopsided — up by 0.058 for a density too high, down by 0.032 for one too low — and a density too low erases the band altogether in one experiment of twenty.
A channel at each pump cannot see the recipe
Thirty white LEDs built onto the Planckian locus defeat a line in a small sensor's three departures, except within one pump: the 452 nm LEDs are read to ±0.047 by the 450 nm channel, the others no better than ±0.15. The proposal was a channel at each pump — 440 and 465 nm beside the 450 — and the prediction that each pump's LEDs would be read from their own channel within ±0.08. The 465 nm LEDs' own channel halves their miss, to ±0.135; the 440 nm LEDs' moves it from ±0.171 to ±0.149. No line in any three of the five departures reads either pump better than ±0.1. Yet the camera's error is simple: a line in what each LED is made of — its red phosphor, its colour temperature and their product — reads every pump within ±0.07, the 440 nm LEDs to ±0.043. The error is easy; the channels cannot see the recipe that sets it. Knowing each LED's pump halves the pooled miss, to ±0.119, and two blue channels can tell only one pump of three from the others.
The converter gives the red phosphor away
A camera's error under white LEDs built onto the Planckian locus is simple in what each LED is made of — its red phosphor and its colour temperature — and a small sensor's narrow departures cannot read the red phosphor. The proposal was a channel at 610 nm, where a garnet converter falls and a red phosphor rises, and the prediction that it reads the red-phosphor strength within a tenth of its range. Within a pump it reads nothing: ±0.36 to ±0.43 at 440 and 452 nm against a spread of ±0.36, and no red channel from 600 to 680 nm does better than ±0.22. The reason is the locus. To keep the white's colour, each step of red phosphor pushes the converter's peak shorter, by 29 to 55 nanometres from none to the most, and in every red channel the two changes cancel. What survives is the converter: the 600 nm departure reads its peak to ±5 nm. Read across all thirty LEDs at once, the converter gives the red phosphor away — ±0.12 — and the error read through that estimated recipe comes within ±0.08 at every pump, the 440 nm LEDs at ±0.078, half their own channel's miss.
One session has a second valley
Every essay on older observers since the second session has read its fitted lens bands and macula shifts from one census: nine simulated observers, three designs, twenty noisy experiments each, every fit a damped descent from two starts. Two of forty experiments refitted in the last essay found a better fit than the census had. The prediction was that the census misses one fit in twenty, almost all where the band is small, and that the misses change no count it reported by more than a few hundredths. Refitted from three more starts, 66 of its 540 experiments have a better fit — 45 of 180 under one session, 9 and 12 under two and three. Sixty of the 66 better fits move the macula's peak more than two nanometres further short, most shrinking the band, and 31 lie lower by under one unit of χ²: a second valley, nearly level with the first. Two sessions' counts stand to within one or two experiments. One session's do not: with the best fits an unmoved macula is reported moved in eight experiments of twenty, where the census said one.
Beside the locus the converter still gives the red away
On the Planckian locus, a white LED's red phosphor pushes its converter to a shorter peak, so an ambient sensor's 600 nm departure, which reads the converter, reads the red phosphor too — within ±0.12 of a spread of ±0.36 — and through it the camera's colour error within ±0.074 ΔE₀₀. Real LEDs are binned a few thousandths off the locus, and the prediction was that a Duv of 0.003 moves the converter about as far as a fifth of the red phosphor does, so that the line learned on the locus misreads off-locus LEDs by a fifth of the range and their error falls back to ±0.12. The converter moves under two nanometres. The line reads the off-locus LEDs' red within ±0.11 and ±0.12, as well as it reads the locus's own held out — with a bias of about a twentieth that follows the Duv's sign. The error reads a third to a half worse, ±0.096 and ±0.107, and most of that is the camera's error itself moving off the locus with nothing in the recipe changed.
Across a bin, the camera's error bends with the temperature
White LEDs a Duv of 0.003 off the Planckian locus have their red phosphor read by an ambient sensor as well as LEDs on it, and their camera colour error read a third to a half worse, because the error itself moves with the Duv. The prediction was that one Duv term in the error's line, learned over all eighty-nine LEDs, brings those beside the locus back to the locus's ±0.074 ΔE₀₀ and costs the locus LEDs nothing. One term buys little: ±0.109 below the locus and ±0.085 above. The error turns across the locus one way at 2,700 K and the other at 4,000 and 6,500 K, and bends most where it is warmest, so the Duv has to enter as a slope and a curve for each temperature — ten numbers a pump, not five — before both sides come back, to ±0.056 and ±0.048. That line then reads the locus LEDs at ±0.073, worse than the ±0.062 a line with no Duv in it manages once it is learned across the bin.
A bound on the macula trades false moves for hidden ones
Fitted thoroughly, one session of an older observer's colour matches finds a second valley in a quarter of its experiments — the macula's peak moved ten or more nanometres from the model's, where no real macula sits — and reports a moved macula where there is none in up to eight experiments of twenty. The prediction was that holding the macula's shift within six nanometres either way, the range real pigment mixtures span, restores every observer's one-session counts to within two of the census's and leaves about one fit in four pressed against the bound. The false moves vanish: an unmoved macula is reported moved in one, two and one experiments of twenty for the three bands, against eight, six and two unbounded. Seven of nine observers keep their count of moved maculas within two of the census's. But only one fit in six ends on the bound, and beside the band at 445 nm the bound hides real moves: a macula truly three nanometres short is found in three experiments of twenty, where the census found seven. Two sessions barely notice the bound.
A soft prior on the macula frees the lens band
One session of an older observer's colour matches, fitted thoroughly, can drift into a valley where the macula's peak sits ten or more nanometres from any real eye's, and a hard bound at ±6 nm removes those false moves at the cost of hiding some real ones — beside a lens band at 445 nm, a macula truly three nanometres short is found in three experiments of twenty where the census found seven. The prediction was that a Gaussian prior four nanometres wide, which lets a fit go further when the matches insist, keeps the false moves away and restores those real ones. It keeps the false moves away — none, one and one for the three bands. It does not restore the real moves: it finds them as seldom as the bound or more seldom, two of twenty beside 445 nm, because it pulls every fitted shift towards nought, to a mean of 1.2 to 1.9 nanometres for moves of three. What it buys instead is the lens band. With the macula held near the standard, the band beside it is found in ten to twelve experiments of twenty, where the census found three to seven.
A wider prior gives the moves back by spending the band
One session of an older observer's colour matches, fitted under a Gaussian prior four nanometres wide on the macula's peak shift, finds the lens band beside the macula in ten to twelve experiments of twenty and real three-nanometre moves of the macula as seldom as a hard bound does. The prediction was that a prior six nanometres wide keeps the false moves at two or fewer, finds the real moves within two of the bound's counts and keeps two thirds of the band's gain, and that at eight the far valley reopens at 415 nm. Three parts hold. The fourth fails: six nanometres keeps 55 per cent of the band's gain. Read across the widths the prior traces one trade — 14 real moves and 44 bands at four nanometres, 22 and 34 at six, 25 and 25 at eight, against the unconstrained fit's 27 and 22 — and the hard bound, at 19 and 20, is worse than six nanometres on both.
A second session leaves the prior one job
Under one session of an older observer's colour matches, a prior on the macular pigment's peak shift trades the lens band against real moves of the macula, six nanometres wide at the knee of the trade. Two sessions already find the band in twelve to twenty experiments of twenty and real three-nanometre moves in twelve to eighteen, and still report an unmoved macula moved in two to three. The prediction was that the six-nanometre prior, applied to two sessions, removes those false moves and costs nothing else. It removes none of them — two, two and none remain, about the rate at which a test at twice the standard error calls a move by chance. It costs nothing at five observers of six. Beside 415 nm, with the macula moved short, it finds four fewer moves: three of the best fits there were in the far valley, eight to twelve nanometres short, and counted as moves because their sign was right. With two sessions the far valley is the prior's only remaining job.
A third session finds the edge band and counts bands misplaced
Two sessions of colour matches find an older observer's lens band in eighteen to twenty experiments of twenty at every place but one: a band at 415 nm, beside the lens's own short-wave edge, found in twelve. The prediction was that a third session — a phosphor LED set against a lamp with a cyan line — finds it in at least seventeen, and that the experiments losing it under two sessions fit the lens three or more years old. Three sessions find it in eighteen, and every loss under two did fit the lens older, but by under three years in five of eight. The count hid something else. It credits a band wherever a fit puts it, and up to twelve of 180 fits put it sixty to eighty nanometres from its place, on the macula's long flank. With three sessions the six-nanometre prior no longer closes that valley.
A band held near the lens closes the valley the prior could not
Three sessions of colour matches find an older observer's lens band beside the lens's own edge as predicted, but twelve of 180 fits place the band sixty to eighty nanometres away on the macula's long flank, each with the macula fitted far short — a valley the six-nanometre prior on the macula's shift no longer closes. The prediction was that holding the band's centre to 400–460 nm, where a lens band can be, empties that valley, finds the 415 nm band with the macula short in at least eighteen experiments, and raises the moves found beside 430 nm from nine to twelve. The first two hold: no fit is left in the valley, and the band is found in nineteen. The third does not: 430 nm stays at ten. And the valley turns out to have been nearly level with the truth — holding the band costs ten of the twelve fits under four units of the fit's chi-squared, and every one's macula comes home.
Named alongside it
The objects these essays reach for when they reach for this one.
Macular pigmentLens yellowingModelling assumptionStandard errorCamera rawLeast-squaresDegrees of freedomCalibrationCone fundamentalsHeld-out validationWhite balanceBasis