Colour Science & Perception
what the eye settles, and what it cannot
Concept

White LED — where it appears

A blue emitter behind a phosphor, whose spectrum is a spike and a hump with a trough between them where cyan would be. The trough is why such a lamp renders some colours poorly, and its lack of any emission below the pump wavelength is why nothing fluoresces under it.

Named by 45 essays across 7 fields — each of them below, with the objects they name alongside it.

A 100 hertz drive, and whether anybody sees it. 3 cycles of a 100 hertz drive at 100 per cent modulation. The number beside each is how far above the threshold for seen flicker its loudest harmonic sits: above one and a stationary observer sees the flutter, below it and only something moving does. Fusion is at 60 Hz at 100 cd/m², and moves 12.5 Hz for every decade of light.

A lamp has a waveform

A lamp modulating a thousand times a second is a hundred times past the frequency at which flicker fuses, and it is plainly visible — as a dotted trail, during any glance across the room. The reason is not a new measurement; it is the spatial contrast sensitivity function, arriving from an unfamiliar direction.

light · Light Colour temperature, and the number nobody quotes beside it. The Planckian locus in the 1960 UCS diagram — the only diagram on which correlated colour temperature is well defined — with four sources and the perpendicular from each to its nearest point. The temperature is where the foot of the perpendicular lands; Duv is how long the perpendicular is. halophosphate sits 0.0246 off the locus at 4513 K, which is a visible green cast that its colour temperature does not mention.

White is a region

A lamp is not sold as a chromaticity. It is sold as 4000 K, and what that means is that its chromaticity fell inside a quadrangle — which two lamps can occupy at opposite corners, eleven ΔE00 apart. In a room with either of them, the same twelve surfaces differ by one unit.

light · Light Several sources on one scale. planck, led, narrowband, each normalised to its own peak and drawn on shared axes with the colour each produces beside it. Every one of these is sold as white light and every one is ordinarily described that way; what they have in common is a chromaticity, and very little else.

A screen is a poor lamp

A display's white is a light. Shone on a surface it renders colour worse than a fluorescent tube — and the wider the gamut, the worse it gets, because the narrow primaries that buy a large triangle are exactly the ones that leave holes in the spectrum.

light · Light The colour of a beam, against the angle it leaves at. A conformal converter — an even layer laid straight onto the die — makes light leaving at θ cross 1/cos θ times as much of it, so it is more completely converted and the beam is warmer at its edge. From the axis to 75° the correlated colour temperature falls by 556 K and the whole difference is ΔE00 16.4. The flat line is the same emitters under a dome, whose path length is the same in every direction by construction: no angular colour at all, exactly, which is what a remote converter is sold for.

A lamp has a direction

A white LED is a blue die under a converter, and light leaving at an angle has travelled further through the converter than light leaving straight up. So the mixture is different in every direction, the beam is bluer in the middle than at its edge, and the number on the box is one direction's worth of a device that has no single colour.

light · Light Dimmed to a fiftieth, three ways. Switching a lamp on and off faster than anyone can see scales the spectrum and changes nothing else, so its colour temperature is a horizontal line and its chromaticity moves by 7.3e-14 ΔE00 — exactly, to floating point. Reducing the drive current moves the die's peak and cools the junction, which moves the mixture: ΔE00 3.5 across the range. A filament has no spectrum of its own to move; it is a blackbody at whatever temperature the power leaves it at, and it falls 1672 K. Three methods, one instruction, three colours.

Three ways to dim a lamp

"Dimmed to ten per cent" names three different colours. Switching the lamp on and off faster than anybody can see changes the chromaticity by nothing at all — exactly, to floating point. Reducing the current moves it a few units. Reducing the power to a filament moves it eighteen hundred kelvin down the Planckian locus, and stays on the locus exactly the whole way.

light · Light A lamp switched on, and what it is still doing minutes later. The junction warms from ambient to 80 °C with a time constant of 150 seconds, and three quoted slopes act as it does: the die's peak moves, the die loses efficiency, and the converter loses quantum yield. The output falls 25 per cent and the colour moves ΔE00 2.6. Nine tenths of the way takes 405 seconds. The vertical mark is the eye's own slow adaptation constant, 60 seconds, for scale.

A lamp switched on is not the lamp measured

A luminaire takes about seven minutes to reach nine tenths of its working temperature, loses a quarter of its output on the way, and moves ΔE00 2.6 while it does. That is slower than every clock in the eye — so for the first minutes after a switch is thrown, both ends of the measurement are moving, and every appearance claim here has assumed one of them was still.

light · Light A surface in a room, from the moment the light goes on. How far a blue surface is from the colour it will settle at, second by second, for an observer who walked in from the snow as the lamp was switched on. It starts ΔE00 27 away, is still 6.1 away after a minute — the point at which the eye is conventionally said to have adapted — and does not fall under a unit until 295 seconds.

The room settles after the eye does

Four clocks run in a room and only three of them are in the observer. The slowest is the lamp, which takes four hundred seconds to reach nine tenths of its colour change — so a minute after the light goes on, when the eye is conventionally said to have settled, most of what is left is the lamp, and three quarters of that could not be adapted away by an observer of any speed.

light · Light A colour rendering index, computed for everybody instead of for one observer. Each bar spans what 120 eyes make of one lamp: the same reflectances, the same reference illuminant, the same arithmetic, different colour-matching functions. The mark is the standard observer's own answer, which is the number printed on the box. One lamp's spread is 3.8 points wide while the whole range the standard observer puts these lamps in is 1.8 — so a ranking quoted to a tenth of a point is a statement about one set of tables. Two of the marks fall outside the population's range entirely.

The index is one observer's opinion

A colour rendering index is printed to a tenth of a point and computed for a single set of colour-matching functions that no standard names as a choice. Scored by two hundred eyes instead, one lamp's spread is wider than the whole range the standard observer puts five lamps in — and two lamps the standard separates by four tenths of a point come out the other way round for every member of the population.

light · Light The same flicker, written across the frame. A rolling shutter exposes each row of the sensor at a different moment, so a lamp that flickers above fusion is recorded as bands. There are 1.7 of them here — the readout time times the lamp's frequency, which is a camera setting and not a property of the light — spanning 2.37 stops. Held at matched luminance the lightest band and the darkest are still ΔE00 5.40 apart in colour, because the two drive currents are two spectra and the shutter caught one of each.

The shutter samples the lamp

A lamp switched between two drive currents at a hundred hertz is one steady colour to a person and two spectra to a camera. A thousandth-of-a-second exposure catches whichever phase the shutter opened at — two and a third stops of exposure and five units of colour, decided by nothing but timing — and a rolling shutter writes the difference across the frame as bands.

imaging · Capture A viewing booth is a lamp, and a lamp has an angle. The same sample at five positions across a booth's plane, with the lamp 55 centimetres above it. The curve is how far each position is from the middle, in the units the judgement is made in. It reaches ΔE00 2.40 against a tolerance of 1, while the illuminance uniformity — the quantity the standard actually bounds — never falls below 73 per cent and stays comfortably inside it. The flat trace is the same booth with a source whose converter is the same thickness in every direction: ΔE00 0e+0, exactly, by construction.

The booth is a luminaire

A standard viewing booth is specified by its light's chromaticity, its rendering and the uniformity of its illuminance across the sample plane. Illuminance is a photometric integral, so two positions can be inside the uniformity tolerance and lit by two different spectra — and a sample at the far corner of a compliant booth is ΔE00 2.4 from one in the middle, against a tolerance of one.

applied · Delivery The same census, sorted by where the change of light came from. Each row is a change of illumination. The pale bar is how far it moves an ordinary surface for an observer who does not adapt; the solid bar at its left end is what is left after the observer has applied the one gain adaptation gives them, which is the ratio of the two whites in the CAT16 basis and is not fitted to anything. Sorted by where the change came from. The two kinds of light that existed before electricity sit at the top and leave the smallest share of themselves behind; the discharge lamps are worse, and the worst of them is d65 to a triphosphor tube at 33 per cent.

Which lamp changes are free

The changes of light that existed before electricity commute with one another to a couple of parts in a thousand, so one set of axes handles all of them. The lights the lighting industry invented do not, and the worst pair in the census is seventy-six times further from commuting than the best.

light · Light Every departure under every light. Six departures across six lights, each cell the difference between two observers in ΔE₀₀, drawn as a bar whose length is the number. The rows are not multiples of one another: the lens is worst under tungsten and the pigment peaks are worst under a three-emitter LED, because a departure is a pairing and which light is being paired with decides it. The laser projector's row is empty, and that is not a fact about lasers — on this collection's five-nanometre grid a three-line spectrum is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about exactly.

The lens is worst under tungsten

An ageing lens costs 4.38 ΔE₀₀ under a tungsten lamp and 2.20 under a fluorescent tube. The pigment peaks cost 2.84 under a three-emitter LED and 1.87 under the same tube. Six departures across six lights do not form a single ordering, which is what a pairing looks like and what a dominant term would not.

eye · Cones The three cone absorptances at two settings of the macular pigment. Solid and dashed are the same construction at the two ends of two standard deviations of the reported spread. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 31.5 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

The macular is a band, not a filter

The lens absorbs everything below 500 nanometres with a long tail; the macular pigment absorbs forty nanometres either side of 460 and nothing else. The two have similar sizes and completely different distributions, and the reason is that one is broad and one is narrow — which is what decides whether a sample is affected at all.

eye · Cones Every departure under every light. Six departures across six lights, each cell the difference between two observers in ΔE₀₀, drawn as a bar whose length is the number. The rows are not multiples of one another: the lens is worst under tungsten and the pigment peaks are worst under a three-emitter LED, because a departure is a pairing and which light is being paired with decides it. The laser projector's row is empty, and that is not a fact about lasers — on this collection's five-nanometre grid a three-line spectrum is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about exactly.

The booth and the eye disagree

A viewing booth is specified to simulate daylight, and daylight is the light under which observers disagree least. So a booth is the condition in which a specification is least likely to be tested against the thing that breaks it, and the shop it will be judged in roughly doubles the term.

applied · Delivery The share of each light outside 380–780 nanometres, as power and as visual product. Two measurements of the same truncation. The upper bar is the fraction of the light's radiant power that lies outside the range; the lower is the fraction of the product of light, sample and observer — which is what a colour is made of. A thermal radiator puts a fifth of its power outside and about a thousandth of its colour, because the observer is zero where most of that power is. The gap between the two bars is the observer's own tails doing their job, and reading the upper number as though it were the lower is how a range gets argued about without being measured.

A lamp is two audits at once

A lamp's ultraviolet content decides what its tabulation costs and its blue content decides what its observer costs, and the two run in opposite directions with colour temperature. So no lamp is good for both audits and no lamp is bad for both, and a single figure of merit for either is a figure of merit for one property of a spectrum.

light · Light A satin finish pulls each room towards the lamp's white — a 3000 K radiator. Six rooms lit by a 3000 K radiator, on the ab plane of an instrument referenced to daylight, whose own white is the cross at the centre. Each open circle is a matt room and the arrow runs to the same room with satin walls. The filled diamond is the lamp's own white on that plane. Every arrow points within 9.2 degrees of the diamond and covers between 8.1 and 11.5 per cent of the distance to it. Whether the daylight instrument then reads more chroma or less depends only on whether the room was nearer the cross than the diamond is.

A finish adds colour only to a daylight meter

Measured against daylight's white, a satin finish under six lamps and six wall colours takes anything from −6 to 42 per cent of a room's colour, and one room reads as more colourful glossy than matt. Measured against the white of the lamp each room is actually lit by, the same thirty-six rooms lose between 7.7 and 12.9 per cent and none gains. The whole spread was the lamp's own colour, and the finish does one simple thing to every room: it pulls the room's colour a tenth of the way towards the lamp's white.

scene · Scene How many directions a slope limit costs, under four lamps. For each transition width and each lamp, the share of the solid's directions that lose more than one per cent of their reach — each lamp's solid against its own ideal. Daylight, a tungsten lamp and a phosphor LED run close together. The three-emitter LED, whose power sits in lines at 455, 530, 625 nanometres, costs 62 directions at forty nanometres where daylight costs 183, and by eighty — about the spacing of its lines — it costs 214 against 218.

Three lines spare a slow pigment

A pigment that cannot switch faster than forty nanometres loses more than a per cent of its reach in 183 of the object-colour solid's 305 directions under daylight. Under an LED whose light sits in three narrow lines, it loses that much in 62. Between the lines almost nothing is measured, so a slow reflectance can do its changing there — until its transitions are as wide as the lines are far apart, at which point the lamp stops helping and the count jumps to daylight's.

limits · Limits What the rods cost a match, by where their signal enters and under which lamp. For five lights, the median colour difference over forty-two surfaces between the reference observer and the same observer with a rod signal a tenth of each cone's peak added — into all three cone channels, into the long- and middle-wavelength channels only, or into the short-wavelength channel only. Under daylight the first two are 1.03 and 0.90: whether the rods reach the S pathway hardly matters. Under a phosphor white LED they are 1.24 and 0.45, a factor of 2.75, and the S-only route alone costs 0.96.

The rods' route is priced by the lamp

A rod signal in a dim room disturbs a colour match, and how much depends on which of the cone pathways it reaches — a weight the physiology leaves uncertain, especially for the blue–yellow pathway. Under daylight the uncertainty is nearly free: a rod signal that skips the S pathway costs 0.90 at the median surface against 1.03 for one that enters all three. Under a phosphor white LED it is worth a factor of 2.75, 0.45 against 1.24. What decides it is one number per lamp: how large the rod signal is compared with each cone class's own catch of the light.

eye · Cones What a fourth emitter costs, by where it is put. A three-emitter LED with one more emitter added at each position from 470 to 610 nanometres, and for each lamp the number of the object-colour solid's directions that lose more than a per cent of their reach to a 40-nanometre transition limit. The three-emitter lamp itself costs 62 of 312. A fourth emitter in the middle of the blue-to-green gap, at 490 nanometres, costs 172; one at 540, beside the green emitter, costs 65. The two dips sit on the existing lines and the two peaks sit between them.

A fourth emitter spends the gap it fills

A three-emitter LED lets a pigment that takes forty nanometres to switch reach most of its ideal solid, because the lamp is dark where the pigment is slow. Adding a fourth emitter to render better takes that darkness back — but only if it is put in the middle of a gap. At 490 nanometres it costs 172 of the solid's 312 directions against the three-emitter lamp's 62, and renders two points worse. At 540 it costs three directions and renders two and a half points better.

limits · Limits The rescue is spent on light between the lines, not on width. The three emitters of a narrow-band LED broadened together, from their nominal widths up to six times them, plotted against the light left in the darkest of the lamp's two gaps as a share of its peak. Up is the share of the object-colour solid's directions that lose more than a per cent of their reach, at three transition limits, with each limit's cost under daylight marked at the right. At forty nanometres half of the rescue is gone by a floor of 7.4 per cent — emitters only 1.30 times their nominal width — and all of it by about a fifth. At twenty nanometres and at eighty there is little to lose either way.

The gap has to be dark, not the line narrow

A lamp whose light sits in three narrow emitters lets a blunt pigment reach most of its ideal solid, and the reason was given as the spacing of the lines. Broadening those emitters without moving them says otherwise. At 1.3 times their nominal width the lamp still looks like a line spectrum, its closest spacing has not changed at all, and half the rescue is gone — because the darkest point of the narrow gap has risen from one per cent of the lamp's peak to seven.

limits · Limits Which two lamps to stand a mesopic match between. Every pair of the five lights, by how far a match made under one and set under the other moves as the rod signal's weight into the S channel goes from nothing to equal — the median over forty-two surfaces, which is the signal an experiment has to resolve. The best pair is daylight against phosphor LED at 0.58 ΔE₀₀; the worst is tungsten against fluorescent tube at 0.09, a factor of 6. The count at the right is how many settings it takes to resolve the weight to a tenth at half a colour difference of scatter per setting.

The reference lamp must not move

To measure an uncertain weight, use the condition in which the answer depends on it most. That is right about half of an asymmetric colour match and exactly wrong about the other half: a match measures a difference of two displacements, and a reference field that also moves with the weight cancels the signal the test field carries. Daylight is the least sensitive of five lamps and belongs in every one of the three best pairs — 75 settings against a phosphor LED, 2,804 against the pair of lamps the principle as stated would have chosen.

eye · Cones Two ways to ask how much rod signal an older eye has. The rod signal's catch of each lamp divided by the S cones' own catch, against the observer's age, for five lamps — drawn twice. The upper curves take the rod signal at a fixed absolute size, and every one of them roughly doubles from twenty to seventy-five: an older lens cuts the blue before the S cones see it and the rods, peaking further into the green, lose much less. The lower curves take the rod signal at a tenth of each cone's own peak absorptance, which is the model's own definition, and they barely move at all. Nothing about the retina differs between the two; only the normalisation does.

A rod signal has no natural size

An older lens absorbs where the S cones are sensitive, so it should make the uncertain rod-to-S-cone weight cheaper. Measured, the rod signal's catch of a phosphor LED as a share of the S cones' own catch more than doubles from twenty to seventy-five — and the share the model actually uses falls by a fifth. The two differ by the S cone's peak absorptance, which the lens takes 60 per cent of, and which entered the model as a normalisation rather than as a claim.

eye · Cones How far a mesopic match moves as the S weight opens: two rooms against one field. For every pair of the five lamps, the median over forty-two surfaces of how far a match moves as the rod signal's weight into the S channel goes from nothing to equal: pale for the two-room match, each half adapted to its own lamp; dark for a bipartite field whose two halves share one adaptation. The field's signal is larger for every pair, by ×1.5 to ×7.7.

One field keeps what two rooms divide out

An asymmetric match can measure how strongly the rods feed the blue-yellow pathway, but set with the observer adapted to each lamp in turn it needs seventy-five settings on the best pair of lamps and hours of waiting between them. Putting the two lamps on the two halves of one field was proposed as the quick version, at the cost of a weaker signal. The signal is not weaker. Under one shared adaptation it is three times stronger for daylight against a white LED, and the best pair needs six settings. The adaptation that makes the slow version slow is also what was dividing the rods' contribution out of each half.

eye · Cones 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.

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.

imaging · Capture 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.

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.

imaging · Capture 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.

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.

imaging · Capture 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.

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.

imaging · Capture What each design buys each family of lamps. The camera's colour error under each lamp, estimated with the lamp left out, and each family's held-out error under four designs: the ratio of the 500 and 450 nm channels alone; one line on the ratio and a 640 nm channel for every lamp; two stages with each lamp's family given; and two stages with the family found by the nearest other lamp. The census's halophosphate tube is missed by 0.405 under the ratio and 0.011 under either two-stage design; the LEDs go from ±0.127 to ±0.111 with the family given and ±0.155 with it found.

Two stages catch the tube and bill the lamps they misplace

A narrow-channel estimate of a camera's colour error missed the halophosphate tube because the census held nothing like it, and a line that learned the tube's family from four siblings cost every other lamp. The proposed cure was two stages: recognise a lamp's family, then apply that family's own line. Given each lamp's family, it works as predicted and better — the tube missed by 0.011 ΔE₀₀, the fifty-four lamps estimated to ±0.133 against the ratio's ±0.141. Found by the nearest other lamp in the sensor's departures, the family is right for the tube and wrong for five lamps, and those five, missed by ±0.41, take the whole census back to ±0.162. The gain is real; the classifier is where it is lost.

imaging · Capture How many neighbours a lamp's family is read from, and what the estimate costs. The two-stage estimate's held-out error over the census's fifty-four lamps when a lamp's family is read from its one to six nearest neighbours, and used only if all of them agree (upper bars) or most of them do (lower), with the halophosphate tube's miss on each bar; otherwise the census-wide ratio line. The ratio alone is ±0.141 and the families given ±0.133. Blue bars catch the tube; orange ones lose it.

A classifier that abstains gives up the tube

Found by the nearest other lamp, a lamp's family cost a two-stage estimate of a camera's colour error more than it bought, because five lamps were placed confidently in the wrong family. The proposed repair was a classifier that abstains — falls back on the census-wide line — unless a lamp's three nearest neighbours agree. It abstains on the wrong lamps. The misplaced ones have three agreeing neighbours, all white LEDs; the lamp it gives up is the halophosphate tube the two stages were built to catch, whose third neighbour is a cyan LED, and its miss returns from 0.011 to 0.445. Two agreeing neighbours look better than knowing every family. Chosen without looking at the lamp being estimated, the rule is worth exactly what the ratio alone is worth.

imaging · Capture Three family lines through the departures, and whom they claim. Every lamp of the census and the halophosphate tube's four siblings by its 450 and 640 nm departures, coloured by family, with the principal line of each family that has three or more members — the radiators, the white LEDs and the halophosphate tubes — solid over the stretch its members span and dashed beyond. Circled are the 29 lamps whose nearest line is another family's. The radiators' short line runs along the LEDs' long one and, extended, through the violet-pumped LEDs.

A family's line runs through its neighbours

The halophosphate tubes differ from one another by one number — where their main phosphor band sits — and so lie along a line in the three departures an ambient-light sensor reads. The proposal was to classify a lamp by the family whose line passes nearest, which would keep the tube a vote of neighbours gives up. It keeps the tube, under every version that does not rescale the distances. It also places eighteen white LEDs on the radiators' line, because that line — fitted to six radiators close together — runs along the LEDs' own and, extended, straight through the violet-pumped ones; and the radiators' estimate, extrapolated to them, misses the triphosphor tube by 7.6. Clamped to the stretch each family spans, the lines still cost ±0.285; given to the halophosphate family alone, its line claims a cyan LED and costs ±0.179. The one line classifier that comes in under the ratio line's ±0.141 does so by two thousandths, having placed the tube with the LEDs.

imaging · Capture What treating radiators and LEDs as one family does to the estimate. Held-out error over the census's fifty-four lamps for the ratio line alone, for the two-stage estimate with the separate families found by the nearest lamp and given, and with radiators and LEDs merged into one family of smooth sources, found by the nearest lamp, with the halophosphate tube's miss. Merged, the estimate comes to ±0.127 — below the ratio's ±0.141 and below the separate families even when every lamp's family is given — and keeps the tube.

Radiators and LEDs are one family to the sensor

An ambient-light sensor's three narrow departures see the collection's six smooth radiators and forty-four white LEDs along one line: the radiators' own short line lies on the LEDs' long one. Estimating a camera's colour error family by family, the census had treated them as two, and every classifier that tried to tell them apart paid for it. Merged into one family of smooth broadband sources, found by the nearest other lamp as before, the two-stage estimate comes to ±0.127 over fifty-four lamps held out — below the ratio line's ±0.141, and below the ±0.133 the separate families managed when every lamp's family was simply given. The halophosphate tube is still caught. The radiators, which the prediction said the merge would cost a little, are estimated better inside it than inside their own family, because a three-coefficient line fitted over five radiators was worse for them than the census-wide ratio line.

imaging · Capture How far the merged design misses each lamp it never saw. For fourteen lamps outside the census — radiators and daylights, tinted and filtered lamps, white LEDs varied in their phosphors, and a three-band LED — the camera's colour error less the estimate the merged design makes from the ambient-light sensor's departures, fitted on the whole census. The dotted lines are the census's own held-out error, ±0.13. The radiators, daylights and tinted lamps fall inside or on it; 4 lamps fall outside, every one of them an LED.

New LEDs miss the line their cousins drew

An ambient-light sensor's three departures can estimate a camera's colour error under a lamp, and merging radiators and white LEDs into one smooth family brought that estimate to ±0.127 over fifty-four lamps, each left out in turn. Every one of those lamps had relatives in the pool. Fourteen lamps the census never saw test what the relatives were worth. The prediction was that new smooth lamps would land on the line and tinted incandescents fall off it by four tenths or more. The radiators and daylights land on it, within ±0.13 each. The tinted lamps do too — the pink-glass incandescent, placed among the LEDs by its nearest neighbour, misses by 0.12. The lamps that fall off are the new white LEDs: ±0.20, half again the census's error, two of them by a quarter. Their misses follow the sensor's 640 nm channel at a slope of −0.72, explaining four fifths of their scatter, because the line learned what that channel means from LEDs varied only in their blue pump. Adding the fourteen to the pool does not help.

imaging · Capture A phosphor grid's camera error is a V about the locus. Forty-eight white LEDs varied in pump, converter share and red phosphor (circles, shaded by red phosphor) and the census's forty-four (squares), placed by how far each white sits from the Planckian locus and by the camera's mean colour error under it. Across the grid the error rises away from the locus on both sides — correlated 0.68 with the distance on the green side and -0.64 on the pink — and is lowest near it. Every one of the census's LEDs sits on the green side, from 0.007 to 0.034.

LEDs varied in their phosphors fall off every line

The merged smooth family estimates a camera's colour error from an ambient-light sensor's departures to ±0.127, and five new white LEDs, varied in their phosphors where the census's forty-four vary their pump, missed by ±0.196 because the family's line had learned almost nothing about the sensor's 640 nm channel. Rebuilt on forty-eight LEDs varied in pump, converter and red phosphor, the line learns it: its 640 nm weight goes from −0.13 to −0.74, most of the way to the −0.85 the new LEDs implied. The prediction was that the new LEDs would then land within ±0.13. They miss by ±0.229, worse than before, crossing to the other side of the line, and the census's own held-out error more than doubles, to ±0.294. No pool of LEDs restores it, and no line in the sensor's departures reads the grid's own LEDs better than ±0.29. Across the grid the camera's error is a V about the Planckian locus, rising away from it on the green side and on the pink — and every one of the census's LEDs sat on the green side.

imaging · Capture Building on the locus does not make a line. Lines in the sensor's departures fitted to LEDs alone, each LED left out and estimated from the others: the free grid's forty-eight, whose whites fell either side of the locus, and the thirty built on it, all together and one pump at a time. Removing the V about the locus brings the merged row from ±0.291 to ±0.282 and the three departures from ±0.298 to ±0.248; only the LEDs pumped at 452 nm, taken alone, fall on a line.

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.

imaging · Capture A channel at its own pump reads the long pump better and the short one hardly at all. For the ten locus-built LEDs at each pump: the spread of the camera's colour error, and how far a line in one blue departure — at 440, 450 or 465 nm — with the 500 and 640 nm departures misses it, each LED left out. The 465 nm LEDs fall from ±0.264 with the 450 nm departure to ±0.135 with their own; the 440 nm LEDs from ±0.171 to ±0.149; the prediction was ±0.08.

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.

imaging · Capture Two LEDs of one colour, one with a red phosphor. Two white LEDs pumped at 452 nm, both on the Planckian locus at 4,000 K: one with no red phosphor and its converter peaking at 579 nm, one with the full red phosphor and its converter pushed to 529 nm, drawn at equal total power. The red phosphor adds light past 620 nm and the shorter converter takes it away from 570 to 610, so a channel anywhere in the red sees the two changes together.

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.

imaging · Capture Each added channel buys less, and none buys the last light. For a modelled ambient sensor of three to twelve Gaussian channels spread evenly from 445 to 650 nm, its matrix fitted in least squares to the room's forty-seven lights with each light held out of its own fit: how many lights tint the moon picture beyond one ΔEITP unit at a tenth of a candela. 3: 33, 4: 18, 5: 6, 6: 2, 7: 1, 8: 1, 10: 2, 12: 3; channels as wide as their spacing or half as wide again give nearly the same counts.

Seven channels end the trade and lose the tube

A three-channel ambient sensor fitted to a room's forty-seven lights trades its error on LEDs against its error on lamps, and a fourth channel moved that trade without closing it. The prediction for more channels was that six, spread from 445 to 650 nm, bring every lamp, mixture and LED under a ΔEITP unit with the lamps within 0.0025 in the CIE 1960 plane, and that nothing short of eight reads the halophosphate tube. Six channels leave the 2,200 K lamp beyond a unit and the lamps at 0.0040; seven bring every lamp, mixture and LED in, with the LEDs and lamps both under 0.001, and the trade is gone — both families improve together. The halophosphate tube is read from six. But the triphosphor tube, held out of its own fit, is read only by three and four broad channels and lost by every finer sensor, its tint growing to thirty-three units at twelve channels. A sensor learns the lights it was fitted on, and a finer one extrapolates further from a light with no relative in the census.

difference · Metric Given relatives, the tube is read by every fine sensor. The census's triphosphor tube, held out of its own fit, read by sensors of three to twelve channels fitted to the census alone, to the census with three triphosphor siblings at 3,000 to 4,000 K, and with four at 3,000 to 5,000 K. Alone it is beyond a unit from five channels and reaches 33 at twelve; with siblings it is within a unit at every count from five, 0.04 at seven.

Relatives, not twins, teach a sensor a tube

An ambient sensor of seven evenly spread channels, fitted to a room's forty-seven lights, reads every lamp, mixture and LED within a ΔEITP unit with each held out of its own fit, and loses the triphosphor tube, which nothing else in the census resembles — thirty-three units out at twelve channels. The prediction was that four triphosphor siblings, the same phosphors mixed for 3,000 to 5,000 K, would bring the tube within a unit at seven and eight channels and stop its rise past eight. They do both: the tube reads at 0.04 with seven channels and 0.06 with twelve, and seven or eight channels then read all fifty-one lights. A twin is not what does it. Three siblings, every one warmer than the tube and the nearest sixty mireds from it, read it within a unit at every count from five. What the siblings cost is paid at the finest sensors: at twelve channels the Planckian lamps' worst error rises fivefold, while from five to eight it moves by under a quarter.

difference · Metric Off the locus, the 600 nm line reads the red as well as on it. For white LEDs built on the Planckian locus and at a Duv of −0.003 and +0.003, the red-phosphor strength read from the 600 nm departure, the mireds and their product by the line learned on the thirty locus LEDs, less the true strength. On the locus, each left out, it misreads by ±0.120; below the locus by ±0.110 and above by ±0.117, reading too little below and too much above.

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.

imaging · Capture Only a curve per temperature brings the bin back. The camera's colour error under eighty-nine white LEDs — thirty on the Planckian locus, fifty-nine at a Duv of ±0.003 — read through the red phosphor estimated from the 600 nm line and an error line within each pump, each LED left out, as the Duv enters the lines in turn: not at all, as one term, as a slope in mireds, as a slope for each temperature, and as a slope and a curve for each. Beside them the line learned on the locus alone. With a slope and curve per temperature: ±0.073 on the locus, ±0.056 below, ±0.048 above.

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.

imaging · Capture The curve holds inside the bin and fails outside it. Four error lines learned on eighty-nine white LEDs at a Duv of nought and ±0.003, each read at those three values with every LED left out and at ±0.0015 and ±0.006 on 119 LEDs none of them was fitted on. The slope and curve per temperature reads ±0.045 and ±0.043 inside the bin and ±0.153 and ±0.161 at twice its edge, where a slope per temperature alone reads ±0.093 and ±0.110.

The bin's curve holds between its points and breaks beyond them

A camera's colour error under a white LED was read from an ambient sensor across a manufacturing bin by a line with a slope and a curve in Duv for each colour temperature, fitted at three Duv values — on the locus and 0.003 either side — and reading the LEDs beside the locus within ±0.056 and ±0.048 ΔE₀₀. A curve through three points passes through them whatever happens between them. The prediction was that 119 new LEDs built at ±0.0015 are read within ±0.06, and those at ±0.006 worse than ±0.10. Both halves hold: ±0.045 and ±0.043 inside, ±0.153 and ±0.161 outside. But the error's own mean course stays on its parabola within 0.02 out to ±0.006. The line's bend is not the error's: it is the gap the line leaves between the locus LEDs and the bin's edges, and a gap spent as a curvature quadruples at twice the distance.

imaging · Capture With the level placed, the line bends as the error does. At each temperature, the curvature in Duv of the camera error's mean course through nought and ±0.003, beside the curvature each slope-and-curve line implies, averaged over the three pumps. At 2,700 K the course bends by +7,916, the line without a level by +2,803 and the line with one by +8,198; at 4,000 K the course bends by +2,196, the line without a level by +14,591 and the line with one by +2,195; at 6,500 K the course bends by +363, the line without a level by −13,349 and the line with one by +445.

A level for each temperature lets the curve be the error's

An error line with a slope and a curve in Duv for each colour temperature read white LEDs inside a manufacturing bin within ±0.045 ΔE₀₀ and LEDs at twice the bin's edge only within ±0.16, though the camera's own error stays on its parabola there. Its curve was borrowed: the line's shared terms could not place each temperature's level, and the square of the Duv absorbed the gap. The prediction was that one more number a pump, the square of the mireds, brings the locus LEDs within ±0.05 and those at ±0.006 within ±0.08. It brings the locus to ±0.025 — three times better than the locus-only line that started this run of essays — and the outer LEDs to ±0.049 and ±0.058. The curvature the line now implies at each temperature is the error's own within a fifth; what it leaves is the warm whites' scatter at the bin's edges.

imaging · Capture The red phosphor bends nothing and tilts the warm whites. The 2,700 K LEDs' residual at each Duv under the eleven-term line, with the square of the Duv times the estimated red phosphor added, and with the Duv times the estimated red added instead. The bend term leaves ±0.006 at ±0.068 and ±0.088, against ±0.069 and ±0.087; the slope term takes them to ±0.046 and ±0.067.

The red phosphor tilts the warm whites and bends nothing

An error line with a level, a slope and a curve in Duv for each colour temperature reads white LEDs at 4,000 and 6,500 K within ±0.04 ΔE₀₀ across and beyond a manufacturing bin, but warm whites at twice the bin's edge only within ±0.087. The prediction was that the red phosphor sets a warm white's bend, so that one term more a pump — the square of the Duv times the estimated red — brings them within ±0.05. It moves them by a thousandth. The red phosphor enters the slope instead: the Duv times the red takes them to ±0.046 below the locus and ±0.067 above. And the rest points upstream. The first stage reads a warm white's red phosphor squeezed toward the middle of its range, the warm misses follow that misreading times the Duv at r = 0.79, and given the red each LED was built with the warm whites come within ±0.05 — though the same substitution doubles the error at 4,000 K.

imaging · Capture The squeeze belongs to the temperature, not the Duv. For the 2,700 K LEDs at a Duv of ±0.003 and ±0.006, the first stage's mean estimate of the red phosphor against the red each was built with, by three first stages; the dashed line is a perfect reading. With one Duv term, no red and a full red are read at 0.13 and 0.80; with the Duv times the departure and the mireds at 0.14 and 0.80; with a level and a departure slope for each temperature at 0.02 and 1.01.

The line had already undone the squeeze

An ambient sensor's first stage reads a 2,700 K white LED's red phosphor squeezed about a third toward the middle of its range, and the camera-error line's warm misses outside the bin follow that misreading times the Duv. The prediction was that giving the first stage the Duv times its 600 nm departure and times the mireds would undo the squeeze. It changes nothing: the squeeze belongs to the temperature, and a level and a departure slope for each temperature read the warm red four times better, within ±0.035. But the eleven-term error line does not move — no warm LED by more than 0.006 — because at one temperature a squeeze is a rescaling its red and red-times-mireds terms had already undone. Only the red-slope line, whose one coefficient a pump spans three temperatures, gains: to ±0.034 below the locus, within 0.002 of the red known. And the cold whites pay for it.

imaging · Capture Split between two estimates, the line keeps both ends. The warm whites at ±0.006 and the cold whites at −0.003, read by the red-slope line on the squeezed red, on the un-squeezed red, and split — the eleven terms on the squeezed estimate, the red slope on the un-squeezed. The split line reads the warm whites at ±0.034 and ±0.057 and the cold at ±0.025; the squeezed line reads them at ±0.046, ±0.067 and ±0.021, the un-squeezed at ±0.034, ±0.057 and ±0.040.

Two estimates of one red each read their own end

Two first stages read a white LED's red phosphor from an ambient sensor: one squeezes the warm whites' red a third toward the middle, the other reads it four times better, and a camera-error line with a red slope prefers the first for cold whites and the second for warm. The prediction was that a line handed both — the eleven-term line on the squeezed estimate, the red slope on the faithful one — keeps each end's gain. It does, within 0.004 at the cold end. But the second estimate is not needed: the squeezed red alone, its red slope multiplied by the mireds, reads every LED within a ten-thousandth of the split line. And the two estimates' difference is worth a term of its own. Added to the line, it reads the cold whites within ±0.016 at every Duv and on the locus within a thousandth, and moves the warm whites and the middle by nothing.

imaging · Capture

Named alongside it

The objects these essays reach for when they reach for this one.

Held-out validationCamera rawSpectral power distributionIlluminantCorrelated colour temperatureModelling assumptionLeast-squaresSpecificationOverfittingQuality controlFluorescentPrediction

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