Colour Science & Perception
what the eye settles, and what it cannot
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The thread: The instrument is the reader — page 12

This is the one subject where the page is displayed on the apparatus under discussion, and the reader's own eye is the measuring device. Several figures here are experiments rather than illustrations.
What each pair of lamps pays to fit the lens and the macula as well. For each pair of lamps, the standard error of the rod signal's S weight fitted from one session — with the observer's lens and macula known (top bar), with the lens fitted too (middle) and with both fitted (bottom). Daylight against the LED is the best pair with everything known, ±0.026, and pays most of its advantage for the macula; tungsten against the LED pays almost nothing for either. What the eye does

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.

 What the classifier misses, against what the camera gets wrong. Every lamp as the pair of channels reads it against the camera's mean colour error under it, with its two-matrix profile blended at the lamp's colour temperature. The band is the range of the six smooth census lamps, 1.14 to 1.62. Left of the line the classifier calls a lamp smooth; no lamp there is above the band. The violet-pumped LEDs it misses leave the camera at least as accurate as a smooth lamp does, two of them more so. What a camera does

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.

 The camera's error, and the two-channel estimate of it. Each of the fifty-four lamps as the camera's true mean colour error against the estimate from the 450 and 500 nm channels, fitted with that lamp left out. The held-out error is ±0.14 ΔE00 against ±0.16 from the 450 nm channel alone and ±0.30 from the mean. The dashed lines at 1.5 mark the repair decision: points right of the vertical and below the horizontal are lamps that need a repair and are not given one. What a camera does

Two channels estimate what one can only sort

A narrow sensor channel was fitted to call lamps smooth or structured, and the label turned out to be a proxy for the thing a camera cares about: its own colour error under the lamp. Scored against that error directly, the best channel is still the one at 450 nm — the band where camera and observer differ most comes close and no closer. Read as an estimate rather than a verdict, a second channel at 500 nm, useless as a vote, earns its place: together they predict the camera's error to within 0.14 ΔE00 over fifty-four lamps, and deciding when to repair from that estimate raises a quarter of the false alarms the label raises.

 Where a sharpened serif letter's error is seen, in black type and reversed, 10 point. The serifed letter E at 10 point, sharpened on stored values rather than on light, as the eye's three spatial channels leave the difference at forty centimetres, in black type on paper and in paper-coloured type on black, on one scale. Dashed circles mark the serifs' corners: their tips and their brackets against the stems. In black type the error lies along the strokes' dark side and the serifs' corners hold little of it; reversed, it gathers round the serif tips, which hold 11 per cent of the letter's seen error against 3 in black type. What it takes to deliver it

A serif moves the error and adds little

When a scanned page is sharpened on its stored values, the error the eye sees lies on the dark side of every edge, so black sans-serif type does not gather it at its corners. The worry was that a serif changes that: every stroke end gains a bar and every bar an inside angle, the configuration where large letters did gather error. It does not. Black serif type's brackets hold three quarters of their share of the error and its serif tips half. Reversed, the serif tips hold two and a half times theirs. Across both, serifs lengthen the outline by a tenth and the error by a few per cent — they move it, and in black type away from the corners.

 Each end colour's move into the eroded press, over the margin, in CIELAB and in ΔE₀₀. The 9 end colours of the census gradients, each moved to the nearest colour of the press eroded by a margin, for margins of one, two and four. Squares: the margin in CIELAB units and the move in CIELAB. Circles: the margin in ΔE₀₀ and the move in ΔE₀₀. The dashed line at one is a flat face and the one at 1.73 a cube's corner. In CIELAB every end colour moves further than a cube's corner at two units, the solid yellow ×4.6, ×3.9, ×3.8. In ΔE₀₀ six of them move between 1.06 and 1.34 times the margin, and only the solid yellow, at ×1.91, ×2.04, ×2.10, is still sharper than a cube's corner. Matching and measuring

A tolerance's own unit rounds the corners

Shrinking a coated press by a margin in CIELAB units made its corners pay several times the margin — the solid yellow nearly four. A print buyer states a margin in ΔE₀₀, and in that unit the corners pay about half as much: six of nine saturated end colours move within a third of the margin, and only the solid yellow is still sharper than a cube's corner. The prediction that the dark corners would stay sharp was wrong; they round too. What the tolerance's unit costs instead is volume — two ΔE₀₀ keep two thirds of the press where two CIELAB units kept four fifths.

 Where a sharpened E with hairline serifs is seen to err, 48 point. The capital E with hairline serifs, a tenth of a stem thick, set at 48 point on a 300-dot page, sharpened on stored values rather than on light, as the eye's three spatial channels leave the difference at forty centimetres: black type on the left, reversed on the right, on one scale. Circles mark the serif tips and the junctions where the middle arm meets the stem. In black type the error follows the ink and gathers in the junctions' inside angles, and the serif tips hold 2 per cent of it; reversed, it spreads into the dark ground and the tips hold 6 per cent. What it takes to deliver it

A hairline spills its error onto the paper

Sharpening a scanned page on its stored values errs on the dark side of every edge, which is why black type's serifs hold less than their share of the error at text sizes. A hairline serif, a tenth of a stem, was expected to change that at display sizes: its dark side is so thin that the halo would wrap its corners. It does not. From 12 to 48 point, black type's hairline tips hold a third to three fifths of their share, less than slab serifs' tips, and the share falls as the letter grows. What gathers at display sizes is the junction of bar and stem, with or without serifs — and where the scan lands the letter against the pixels moves a tip's share more than its size does.

 A local-dimming panel's backlight behind a small highlight on black. The backlight of each of a panel's 32 by 18 dimming zones behind a small highlight on black, on a logarithmic scale from one per cent to full, after a diffuser that spreads each zone's light with a standard deviation of 1 zone. The zones driven by the picture are the darkest or the brightest; the diffuser's halo fills 3.5 per cent of the screen with levels in between, and each zone's floor follows its backlight. Difference and uniformity

Two floors trade the screen for the halo

A local-dimming panel in a dark room has no single black: its floor varies twenty-six times across a scene, and one declared floor misreads ΔEITP's grey steps by a factor of 2.6. The proposal was two floors, one for dimmed zones and one for lit ones, on the reasoning that a picture's zones cluster at the two ends. They do cluster, and two floors still do not work. Placed for the worst zone they halve the error and misread almost the whole screen a little; placed on the two clusters they read most of the screen exactly and misread the halo between as badly as one floor. How many floors a panel needs is set by its diffuser, not by the picture — and the only declaration that works everywhere is the backlight map.

 How far each unit moves a colour's balance, at the most saturated colour BT.2020 allows. For every fifteen degrees of hue at lightness 25, 50 and 75, the most saturated base colour whose one-unit lightness and chroma pairs stay inside a BT.2020 display, and the factor by which each unit's ratio of the two pairs changes from a 1.5 to a 10,000 candela display. CAM16-UCS moves every one by ×0.76 to ×0.89. ΔEITP moves them from ×0.54, a yellow-green at hue 90, to ×1.64, a red at hue 30 — with the red-purples beside it and the dark blues well below. Difference and uniformity

At the gamut's edge the reds move as far as the violets

As a display brightens, ΔEITP shifts a saturated colour's balance between lightness and chroma by an amount set by its three quantised signals, and CAM16-UCS shifts every colour alike. The largest shifts found so far were a yellow-green at ×0.63 and a dark blue at ×1.62, both at the census's edge — and the blue turns out to be a colour no BT.2020 display can show. Walked out to the display's real boundary, the yellow-greens go lower, to ×0.54, and the dark blues never pass ×1.35, because BT.2020's blue edge is at low chroma. The top of the range goes to the reds and red-purples at ×1.6: at the edge the L-minus-M term grows to two thirds of the S term. Pairing the two extremes still needs a third of the observers the violet experiment needs.

 Where a lens edge 10 nanometres off goes in a one-session fit. A sixty-year-old observer whose lens edge sits 10 nanometres longer (upper bars) or shorter (lower bars) than the model's, fitted by the model from one daylight-against-LED session: the shift in each fitted parameter, in that parameter's standard errors. The macula takes the most, 2.4 of its standard errors; the weight moves by 0.59 of its own, 0.035 in absolute terms, against a prediction of under a tenth for an edge ten nanometres off. What the eye does

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.

 Where one session puts a band in the lens, by the band's wavelength. A band of a tenth of an optical density in a sixty-year-old's lens, centred from 410 to 460 nm, fitted from one daylight-against-LED session by the model with weight, lens age, macula and the lens edge free: each parameter's shift in its own standard errors. Near 410 to 430 nm the band is read as lens; from 440 it is read as macula; the edge takes little of it anywhere; and the rod weight moves at every centre, by 0.37 to 1.39 of its standard errors. What the eye does

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.

 What straightening the 570 nm channel buys an estimate of the camera's error. The root-mean-square and worst held-out errors of linear estimates of the camera's error, over fifty-four lamps with the three-emitter source kept in: from the 570 nm channel as read and through three transforms, from 450 nm alone, and from 450 nm with a log-transformed 570 nm channel or with 500 nm. Every transform rescues 570 nm from worse than the mean; none reaches 450 nm; and beside 450 nm it adds nothing, where 500 nm does. What a camera does

A straightened channel repeats the one beside it

A phone's narrow sensor channel at 570 nm ranks a camera's colour error under a lamp almost as well as the one at 450, and estimates it worse than guessing the mean, because one lamp reads ten times higher than any other. Read through a logarithm, the channel is rescued: its estimate improves from ±0.57 to ±0.23. It still does not reach the 450 nm channel's ±0.16, and put beside that channel it adds nothing at all, where a 500 nm channel cuts the error to ±0.14. Straightened, 570 nm sorts the lamps as 450 nm already does; and the two lamps it reads highest have the most different errors, which no transform of one reading can tell apart.

 Which way each narrow channel departs from what the camera predicts. Each lamp's two narrow channels, their departures from the camera's prediction kept signed. 37 of the fifty-four lamps, every structured one among them, lie in the lower right: more light than predicted at 450 nm, less at 500 — a peak and a trough. 8 lie in the upper left, the opposite pattern, most of them violet-pumped LEDs whose pump misses 450 nm. In both the departure of the two readings' ratio is the sum of the two distances. The 7 in the upper right are the most heavily cyan-filled LEDs, whose fill puts light at 500 nm, and two daylights near the origin. What a camera does

One ratio reads a peak and a trough

Two narrow sensor channels, at 450 and 500 nm, estimate a camera's colour error under a lamp to ±0.14 ΔE₀₀; the proposal was to read their ratio as one number instead. Taken as one channel's departure over the other's, the ratio is worthless — no better than the mean, because for smooth lamps both departures are nearly nothing. Taken as the departure of the two readings' ratio from what the camera predicts for it, one number estimates the error to ±0.141, as well as the two channels as two. The reason is a sign: under every structured lamp the channels depart in opposite directions, a peak at 450 and a trough at 500, and the ratio adds them; where a cyan fill lifts 500, it subtracts them, as the error falls. And no five per cent calibration error moves any of these estimates by a hundredth.

 A correction measured under one tube, carried to seven other lamps. For each lamp, the mean error over sixty-eight notch filters of the separable table (upper bar), of the separable table corrected by the ratio of product to separable colour measured under the census tube (middle), and of the product measured under the lamp itself (lower). Under the census tube the correction is the product exactly; under tubes with other line widths or a moved phosphor bed it removes nine tenths of the error; under a tube with its lines re-mixed, 63 per cent; under the laser and the LEDs it makes things worse. What light is

A correction travels with its lines

Keeping a sample's reflectance and a lamp's spectrum as separate tables costs fifteen times the error of measuring their product under a fluorescent tube. The middle course proposed was to measure the product once under a representative tube, store the ratio of product to separable colour with the sample, and apply it under any tube. Under tubes whose lines are narrower or wider, or whose phosphors sit elsewhere, it removes nine tenths of the error. Under a tube with the same lines in a different mix it removes under two thirds, and less the further the mix moves. Under an LED or a laser it makes things worse — a white LED five times worse than no correction. A class of lamps, for this purpose, is a mix of lines.

 How well the table's entries around the 546 nm line fit each width of line. For the tube's green mercury line, 1.2 nm wide, the seven table entries around it fitted by a line of each width from 0.2 to 12 nm through the same slit on a sloping background, centre, height and background free: the best fit's miss. Through a one-nanometre slit the miss has a sharp minimum at the true width and only widths near it pass half a per cent; through two, a broader dip; through three and five the curve is flat from the narrowest width tried up to about two, and any of them passes. What light is

A narrow table declares its own lines

A bound on a colour table's bandpass error needs one number from the lamp's maker: its narrowest feature, 1.2 nm for a fluorescent tube. Through a slit about as narrow as that feature, the table's entries around each line carry its width. Measured to half a per cent, a one-nanometre table's entries admit only line widths within four per cent of the truth, and a bound built on the narrowest of them is within two per cent of the bound on the maker's number. Through two nanometres the table still declares a usable width. Through three it cannot rule out a line five times narrower, and the bound it supports is the global one. The failure is as sharp as predicted, and never unsafe.

 Directions each pigment model loses, under three broadband lamps and a lamp of lines. For each lamp, how many directions of the object-colour solid a pigment model cannot reach within one per cent of the ideal: a slope limit, absorption bands no narrower than forty nanometres, and bands whose minimum width is fixed in energy. The order is the same under every lamp. The broadband lamps cost every model far more than the three narrow emitters: under daylight 89, 73 and 70 against 31, 23 and 20. Where the model breaks

The lamp outweighs the pigment model

Three models of what a real colorant can do — a slope limit, absorption bands forty nanometres wide, and bands whose width is fixed in energy — were ranked under lamps of narrow emitters, slope dearest and energy cheapest. Under daylight, tungsten and a white LED the ranking holds, every time. What changes is everything else: each model loses two and a half to three and a half times as many directions of the object-colour solid under a broadband lamp as under three narrow emitters, and the difference between the models is smaller than the difference between the lamps. Under tungsten, whose power rises smoothly to the red, the three models are nearly one.

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