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

Tone curve — where it appears

The mapping from input level to output level in a reproduction, which decides how a scene's range is fitted into a medium's. It is chosen rather than derived, and the choice is about which end of the range to compress.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

Everything between the photons and the picture, and what each stage decides. The 8 stages of a camera pipeline. Only the second is physics; every one after it is a decision somebody made, and the reason two cameras pointed at the same scene disagree is that they made different ones.

Raw is not a picture

A raw file is three integrals per site and a list of decisions nobody has taken yet. Every one of those decisions has a defensible answer and none of them has a correct one, which is why two converters open the same file and disagree.

imaging · Capture A clipped channel turns the hue of what is left. CIELAB hue shift against exposure for one saturated stimulus, measured against the same stimulus rendered without clipping. Nothing moves until the first channel reaches the ceiling at 0.25 stops; after that the recorded hue rotates by as much as 67 degrees, with nothing in the scene having changed colour.

A blown highlight turns

An exposure that clips nothing changes no hue at all. Once one channel reaches the ceiling the recorded hue rotates by as much as sixty-seven degrees, with nothing in the scene having changed colour — and every response a converter can make to that is an invention.

imaging · Capture Everything between the photons and the picture, and what each stage decides. The 8 stages of a camera pipeline. Only the second is physics; every one after it is a decision somebody made, and the reason two cameras pointed at the same scene disagree is that they made different ones.

A photograph is not a measurement

A photograph is a measurement made by an instrument whose kernel nobody published, under an illuminant nobody recorded, corrected by a matrix fitted to somebody else's surfaces, with two thirds of every pixel invented. It supports relative claims well and absolute ones badly, and it is used for the second.

imaging · Capture Two mechanisms, one ramp, and the patch that separates them. A cyan ramp printed by a press with 10 per cent mechanical gain and a Yule–Nielsen exponent of 2.4. Fitting a mechanical gain at n = 1.0 gives 23 per cent and fitting one at n = 2.4 gives 10 per cent; both reproduce the ramp to under 0.66 of a lightness unit, and the curves lie on top of one another. The two-ink overprint below was in neither fit, and there they are ΔE00 = 3.20 apart.

A dot is larger than it was asked to be

Two quite different things make a printed midtone darker than its coverage says — ink spreading under pressure, and light scattering sideways inside the paper. Both bend the tone curve the same way and neither touches its ends, so the measurement every press is characterised by cannot say which happened. A patch that nobody measures can.

applied · Delivery What a dither mask is worth, read as components, in two dimensions. Five luminance ramps, each quantised to 8 bits with and without a high-passed mask of the same power. The bars are the most visible single sinusoidal component of the error, as a multiple of the contrast that component needs to be seen: above the line at one it is visible. The mask lowers it by 20–22×, on every ramp — which the one-dimensional model on this site says it does not, and that disagreement is the finding.

Every threshold was measured with a grating

An earlier essay here claimed that the model cannot explain why dither works, and named two missing pieces. One of them was real and worth thirteen times the guess; the other was not needed. The piece nobody named was the detector — and reading the same model two ways changes the answer by a factor of fifty.

limits · Limits A mid-grey's lightness across a continuum of rooms. The lightness a mid-grey is predicted to have, plotted along the continuous surround parameter running from an average room to a dark one. The three rooms the standard tabulates are marked on it: average at the left, dark at the right, and dim 61% of the way between them rather than halfway. The whole span is 9.71 units of lightness and the step from average to dim is 5.64 of it — 58% — so choosing one of the three rows is a decision worth most of the range.

The surround is three rows of a table

An appearance model takes the room as three constants, and the standard tabulates three rooms. Every appearance figure in this collection is drawn at one of them. The parameter they are three points of is continuous, and the middle row is not in the middle.

brain · Appearance A black level slightly wrong, through the balance, under daylight. A grey ramp from half a per cent to seventy-two per cent reflectance, with a pedestal error of a tenth, three tenths and one per cent of white left in all three raw channels before the white balance, against the same ramp with none. The horizontal axis is the grey's reflectance, logarithmic. At a three-tenths error a two per cent grey is 2.4 colour differences off, most of it chroma, and a seventy-two per cent grey 0.21. An equal offset in the raw channels is not equal after three different gains.

A black level is multiplied by the balance

Every raw value carries a pedestal that is subtracted before anything else, and a white balance is then a different gain in each channel. Subtracting a constant and multiplying by one commute only when the constant is zero or the gains are equal. So a pedestal left three thousandths of white too high becomes 2.4 colour differences in a two per cent grey, most of it chroma, in the colour the lamp starves — and pushing that shadow four stops in editing makes it 8.9. An offset in proportion to the lamp's own white is the exception, and it is exactly grey.

imaging · Capture The error between a profile's nodes is a bias, not a scatter. Two quantities against the lattice size: the mean colour difference between the nodes, and the mean signed lightness error. If the interpolation erred in both directions the second would be near zero while the first was not. They lie on each other — 0.136 against 0.133 at a nine-step lattice — so the whole of what a profile does between its patches is to lighten. It errs light because a press's response is convex in ink coverage: the first drop of ink removes more light than the last, and a straight line between two points on a convex curve lies above it. At three steps 400 of 400 samples err light.

A profile interpolates light

A profile is exact at its patches and wrong between them, and how wrong has been measured twice. Which way it is wrong has not. At a nine-step lattice the mean signed lightness error between the nodes is +0.136 against a mean colour difference of 0.133 — the error is not a scatter but a bias, and it lightens. The repair costs nothing measured and is forbidden by how a profile is checked: let the table be wrong at its own patches.

applied · Delivery The four places a clamp could sit are two pipelines. Every pair of clamp positions, with the largest difference their delivered values reach over a grid of raw inputs that includes negative ones. Two of the six are exactly zero: a clamp at zero commutes with the white balance, which is a positive scale applied channel by channel, and with the tone curve, which is monotone and fixes zero. It does not commute with the colour matrix, which is the only step that mixes the channels — so the four positions collapse to two, before the matrix and after it, and no measurement of any scene can say more than which side a converter is on.

Four places to clamp are two pipelines

The essay on clipped noise ended on a procedure: a black frame and a dim grey card at a high amplification would place each raw converter's clamp. A procedure is a claim that a measurement identifies something, and this one identifies less than it looks. A clamp at zero commutes with the white balance and with the tone curve and not with the colour matrix, so the four positions are two pipelines — and the measurement separates them, on a card at half a per cent of white rather than on the black frame.

imaging · Capture Per channel, a tenth of green lands where a half was graded. Each mixture's CIECAM16 hue quadrature on the master and through the per-channel curve, between unique green at 200 and unique blue at 300. On the master a tenth of green sits at 312.9 and a half at 277.5; per channel on the 100 cd/m² display the tenth lands at 260.9, past the master's half.

An appearance model sees the per-channel sky turn twice as far

BT.2390's tone curve, applied to each channel of a 1,000 cd/m² master shown on a 100 cd/m² display, raises a sky-blue's green share from a tenth to 69 per cent of its blue and turns its ICtCp hue 23°, where the same curve applied to the largest channel with all three scaled keeps the hue within 3°. ICtCp's hue is not an appearance, and the prediction was that CIECAM16, read under the display's own white, still shows the per-channel turn as more than twenty hue-quadrature units and more than ten ΔEITP, the scaled curve under three of each. The appearance model sees it as fifty-two units, more than half the way from unique blue to unique green, and its lightness more than doubles. The scaled curve keeps hue and lightness to a quarter of a unit. And ΔEITP between master and display ranks the two curves backwards: in absolute light the per-channel sky is the nearer one.

matching · Gamut A blue's intensity sits below where the curve compresses. BT.2390's curve from the 1,000 cd/m² master to displays of 100 and 300 cd/m², with the level a pure blue, a tenth-green and a half-green sky present to it when the curve is applied to ICtCp intensity, and the level their blue channel presents when it is applied per channel or to the largest channel. The skies' intensity corresponds to 79 to 387 cd/m², against knees of 27.9 and 161.4; their blue channel sits at 1,000, the top of the curve.

A curve on intensity cannot see a blue's brightness

ITU-R BT.2390's tone curve applied to each channel of a bright master turns a tone-mapped sky's appearance fifty-two hue-quadrature units towards green on a 100 cd/m² display; applied to the largest channel with all three scaled, it keeps hue and lightness to a quarter of a unit. BT.2390 also describes applying the curve to ICtCp's intensity and scaling its chroma, which keeps ICtCp's hue by construction, and the prediction was that it keeps the appearance's hue within three units and its lightness within two. It keeps the hue, within 2.3 units. It cannot keep anything on the display: a saturated blue's intensity is the PQ level of 79 cd/m², under the 300 cd/m² display's knee, so the curve barely touches it and every sky asks for two to seven times the blue the display has. Clipped channel by channel, the sky turns as far as the per-channel curve turned it; scaled back into the gamut, it lands where the scaled curve put it. The step back into the gamut decides the appearance, not the curve.

matching · Gamut Holding ICtCp's hue and lightening the chroma still turns the sky. Each mixture mapped by BT.2390 on ICtCp intensity on the 100 cd/m² display and brought into its gamut three ways — clipped channel by channel, scaled until the largest drive is full, and by its ICtCp chroma alone at fixed intensity and hue — read as the change in CIECAM16 hue quadrature from the master's. The chroma-only return moves −16.8, −14.4, −12.6, −17.0, −9.2 units at no green to half green; scaled into gamut, −1.3, −1.6, −1.8, −1.6, −2.2.

Lowering only the chroma turns the sky white and its hue with it

A tone curve applied to ICtCp's intensity keeps a high-dynamic-range sky's hue, but a saturated blue's intensity lies under the curve's knee, so the colour it asks for lies outside every display's gamut. The prediction was that bringing it back by lowering its chroma alone — intensity and ICtCp hue held — keeps the CIECAM16 hue within three units, lifts the lightness by ten and halves the chroma: a paler blue of the right hue. The ICtCp hue and intensity are kept to a hundredth. Everything else fails. The chroma must fall to a fifth or less on a 100 cd/m² display, the sky rises from a lightness of 22 to 76 and higher, and its hue quadrature moves eight to seventeen units toward green — a pure blue's by 17. Holding ICtCp's hue while lowering chroma does not hold the hue a viewer sees, and the intensity the curve chose was never a blue's.

matching · Gamut Holding the true hue saves one or two units of seventeen. Each mixture's CAM16-UCS distance from the master on the 100 cd/m² display: mapped on intensity and returned by ICtCp chroma, returned along CIECAM16's hue at the same lightness, scaled into the gamut, and the master's drives shown unmapped. The CIECAM16 return lies 16.7, 17.2, 17.3, 16.8, 15.7 from the master, against 18.9, 18.7, 18.5, 17.6, 16.8 for ICtCp's; the master's own drives 4.14, 4.10, 4.07, 4.02, 4.05.

The sky a dim display can show best is the master's own

A tone curve applied to ICtCp's intensity maps a saturated high-dynamic-range sky to a colour no display can show, and bringing it back by lowering only its ICtCp chroma turns it pale and toward green. The prediction was that a return along CIECAM16's own hue line, at the lightness the curve chose, holds the hue by construction, needs a deeper chroma cut, and still lies at least twice as far from the master as simply scaling the drives. It holds the hue to a hundredth, keeps more chroma than ICtCp's return rather than less, and lies two and a half to four times as far as scaling — because holding the true hue removes only one or two of seventeen units of difference. The rest is the lightness the curve held. And the nearest a dimmer display can come to the master's appearance turns out to be the master's drives, shown unmapped.

matching · Gamut

Named alongside it

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

Camera rawDynamic rangeTransfer functionColour appearance modelColour matrixHue quadratureWhite balanceClippingGamut mappingQuantisationSpectral sensitivityDeclared input

All concepts