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

Fluorescent — where it appears

A discharge lamp whose mercury lines stand on a phosphor continuum, so its spectrum carries structure at a scale no daylight has. The mercury lines are about a nanometre wide, so their width in any plot on a five-nanometre grid is a property of the sampling rather than of the mercury.

Named by 15 essays across 3 fields — each of them below, with the objects they name alongside it.

triphosphor fluorescent — three narrow phosphors plus the mercury lines, and the white it produces. The spectral power distribution of a triphosphor source, normalised to its own peak, and the colour a perfect white reflector takes under it: chromaticity (0.3379, 0.3389), correlated colour temperature 5258 K at Duv -0.0035. The white looks ordinary. The spectrum producing it does not.

A lamp is not a blackbody

A fluorescent tube puts a third of its light into four mercury lines. A white LED is a blue spike with a hole beside it. Both are sold by a colour temperature, and a colour temperature says nothing about either.

light · Light What separates the separations, and under which light. 11 separations of a single colour, differing only in how much of it is carried by black rather than by the three chromatic inks. Under D50 they agree to ΔE00 = 0.00 — the solver was asked for that and delivered it. Under illuminant A they spread to 6.75. They are metamers of one another, and the whole family is invisible to any instrument reading a single illuminant.

The lamp in the shop decides

Two packages printed to the same specification, verified under the same standard illuminant and passed, can differ by nearly seven colour differences on a shelf under a tungsten lamp. Nothing went wrong at either press. The specification named one light, the separations that satisfy it are metamers of one another, and the shop chose the second light.

applied · Delivery 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 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 An instrument, as the only thing it really is. The 3 filters a bank of that size puts across the visible range, each drawn against wavelength. Everything the instrument can report about a spectrum is 3 numbers — the integral of the light against each of these — so the set of spectra it cannot tell apart is everything orthogonal to all 3 of them, which is 78 dimensions of the 81 this site works in. Three of these is a colorimeter in spirit; the eye is three of them too.

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.

light · Light A 2-nanometre notch, and what two five-nanometre grids record of it. The reflectance of a sample with a 2-nanometre notch at 552.3 nm, drawn finely from 535 to 570 nm. The dark ticks are the samples of a five-nanometre grid starting at 380 nm and the pale ticks those of the same grid started half a step later. The true minimum is 0.06; the first grid's deepest sample reads 0.70 and the second's 0.08. The sample has put a line into a calculation whose light has none.

The line can be in the sample

The rule for which tabulation defect to fix compares the light's narrowest feature with the grid's step, and it named its own failure case — a sample with structure narrower than the step. Given one, a two-nanometre notch under a thermal source with no feature at all costs 2.1 colour differences at five nanometres and moves 2.0 when the grid slides, which is what a fluorescent tube costs on a smooth pigment. Under that tube a notch twelve nanometres wide, more than twice the step, moves 8.1 when it sits on the mercury line.

light · Light Two matrices blended by colour temperature, under fourteen lamps. For each lamp, with the neutral held exact as a converter holds it: the mean colour difference over twelve test surfaces with a matrix fitted under that lamp (the short bar) and with the tungsten and daylight matrices blended at the weight its correlated colour temperature gives (the long bar). Smooth lamps on or near the locus sit within 6 per cent of their own matrix. The lamps with lines or narrow bands in them sit a median of 2.2 times theirs, from 1.38 for a broadband tube to 5.2 for a three-emitter source.

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.

imaging · Capture A 12-nanometre notch at 546.1 nm under a fluorescent tube, tabulated five ways. The cost against a tenth-nanometre reference, on a five-nanometre grid, of a notched sample under a fluorescent tube, mercury lines on a phosphor bed, when the two factors of the colour are tabulated as points, when the lamp alone is measured through a five-nanometre slit, when the sample alone is, when each is measured through its own slit, and when the light the sample reflects is measured through one slit. The costs are 2.535 for both sampled at points, 0.379 for the lamp through a slit, 2.724 for the sample through a slit, 0.727 for both through their own slits, 0.019 for the product through one slit. The tick on the two-slit bar is the same two tables summed at a tenth of a nanometre, 0.749: what the separate slits leave is not the grid's.

Two slits are not one slit

A spectrometer's slit is what makes a coarse table honest, for a lamp and for a notched sample alike. But a colour is a sum over the product of the two, and a notch measured through one slit and a lamp measured through another are not the product measured through a slit. Under a smooth light the difference is nothing. Under a fluorescent tube a notch on the mercury line comes out 0.73 colour differences off from two slits — worse than no slit at all for some notches — and 0.02 off from one slit on the reflected light.

light · Light The confusion matrix, and which corner the common lamps are in. Twelve fixtures sorted two ways. Down the page is what their spectra are; across is what flicker says. The two corners on the diagonal are 7 fixtures the classifier gets right. The 3 missed are structured lamps that do not flicker — a white LED and a warm LED on constant drivers, and a three-emitter fixture — and those are the lamps most modern interiors are lit by. The 2 false alarms are smooth lamps that do flicker: a halogen lamp on mains and a tinted radiator, both of which a photograph of a room is quite likely to contain.

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.

imaging · Capture Six ways of tabulating one notch on one mercury line. A 12-nanometre notch centred on a fluorescent tube's 546.1 nm line, its colour computed on a five-nanometre grid six ways, on a logarithmic scale. Point sampling costs 2.54 colour differences and two separate slits 0.727. Sharpening both blurred tables with the published three-term correction takes it to 0.188 — a real improvement, four times better — and one slit on the light the sample actually reflects gives 0.019. The correction recovers the part of the damage that is a blur, and the part that is left is not a blur.

A linear repair for a bilinear loss

The Stearns correction sharpens a table blurred by a triangular slit, and the obvious question was whether applying it to a lamp's table and a sample's restores the product the two of them are wrong about. It restores four fifths of the damage and cannot touch the rest: a three-term filter is linear, the covariance two separately blurred tables discard is bilinear in the two factors, and no linear operator applied to each factor separately produces a bilinear term. What is left is ten times the one-slit answer, at every position of the notch.

light · Light One instrument, one slit, two requirements. The slit's width swept, with three measurements on a logarithmic scale: a smooth sample under the line lamp, where only the lamp's structure is at stake; the notched sample under a smooth lamp, where only the notch is; and the real case, both at once. The lamp wants a slit of 5 nanometres and the sample wants 1, and each wants what it wants for the same reason: a slit should spread a feature the grid cannot resolve and leave one it can. The real case is best at 3 nanometres — which is neither requirement's answer — and costs 0.247 there, an order of magnitude more than either requirement alone.

One slit, two requirements

A line lamp wants a wide slit, because a wide slit spreads a line where a coarse grid can see it. A notched sample wants a narrow one, because a wide slit fills the notch the grid could have resolved. An instrument has one slit. Measured on the two requirements separately the best widths are five nanometres and one; measured on the two together the best is three, which is neither — and it costs twelve times what the lamp alone would cost and thirty-four times what the sample alone would.

light · Light How far each lamp's sensor reading is from what the camera predicts, with RGB + clear. Fourteen lamps, six smooth and eight structured, each scored by how far an ambient-light sensor with red, green, blue and clear channels reads from what the camera's white predicts through a map fitted on smooth radiators and daylights. The smooth lamps score at most 0.057 and the structured at least 0.133; the dashed line is the threshold at the gap's geometric middle, 0.087. The lights the map was fitted on score at most 0.0113. Every lamp falls on its own side of the line.

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.

imaging · Capture Every pair of slits, over 68 notches. The colour error, in ΔE₀₀ from the truth, for every pair of slit widths — the lamp's table blurred through the width down the side, the sample's through the width across — on 68 notches, the mean over all of them under a fluorescent tube. Circle area follows the error. The best pair is 5 nm on the lamp and 1 nm on the sample, at 0.26; the best single slit, on the diagonal, is 5 nm at 0.36. One slit on the reflected light, at 5 nm, averages 0.016.

A second slit buys a quarter

A line lamp wants a five-nanometre slit and a notched sample a one-nanometre slit, so an instrument with a slit for each should do much better than one with a single compromise. Over sixty-eight notches under a fluorescent tube, it does better by 28 per cent. One slit on the reflected light does fifteen times better than the best pair, and an oracle choosing the best pair for every notch is still six times worse. The error was never that the factors were flattened; it was that they were flattened separately.

light · Light Three bounds against the error they bound, over 68 notches under a fluorescent tube. Each notch placed across by its actual colour error from blurring the lamp and the sample separately, and up by a bound on that error, both on logarithmic scales; the dashed diagonal is where a bound equals the error, and a valid bound sits above it. Cauchy–Schwarz with the true window variances is above the diagonal on every notch, a median 14.7 times the error. Estimated from the blurred tables it falls below on 6 of 68, as low as 0.45 of the error. The Bhatia–Davis bound from the tables and declared ranges is above on every notch and a median 196 times the error.

The tables cannot bound what they discarded

A colour computed from a lamp's blurred table and a sample's blurred table is wrong by the covariance the two blurs threw away, and Cauchy–Schwarz bounds a covariance by two variances. With the true variances the bound always holds and sits fifteen times above the error. With variances read from the tables it fails on six of sixty-eight notches under a fluorescent tube and twenty-two under a laser projector — on the line, where the error is largest. A blurred table does not carry the width of a line, and the covariance depends on it.

light · Light The calibration error a narrow fourth channel survives, by where it is placed. For a red, green and blue ambient sensor with one more channel 10 nm wide, centred from 450 to 640 nm: the first calibration error at which some lamp of fourteen is misclassified. Dashed: the channel pooled into the whole residual, as the ambient sensor's disagreement was read; its best is ±1.50 per cent. Solid: the channel read on its own, as its departure from what the camera's white predicts for it; its best is ±4.0 per cent, at 450 nm. Crosses mark centres where the channel read alone does not separate the lamps at all. The two horizontal lines are the red, green and blue design (±1 per cent) and the design with a clear channel (±4 per cent).

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.

imaging · Capture

Named alongside it

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

Spectral structureBandpassLED emissionSpectrophotometryWavelength gridIdentifiabilityInstrumentCalibrationColour matrixCorrelated colour temperatureMeasurement conditionTransmittance

All concepts