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

Hue quadrature — where it appears

A scale that spaces the hues by their appearance rather than by an angle in a space, anchored on four unique hues. It exists because equal angular steps in an opponent space are not equal hue steps, and the discrepancy is largest in the blues.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

Three lightness scales, seventy years apart. Munsell value, CIELAB's L* and CIECAM16's J against luminance, all rescaled to run 0 to 100. Each is somebody's answer to how evenly spaced lightness steps map onto light. They put the midpoint of the scale at 19.8%, 18.4% and 28.0% of the white's luminance respectively — close enough to be three measurements of one thing, far enough apart to be three measurements rather than one restated twice.

Colour by catalogue

A colour order system arranges physical samples on a regular lattice so a person can find one and name it. The space it samples is not regular, and everything interesting about such systems is what happens at that mismatch.

matching · Gamut The four unique hues, and the axes they are said to define. A constant-lightness, constant-chroma ring in CIECAM16, with the four unique hue anchors marked and CIELAB's a and b axes drawn through the same circle. If a* really were the red-green axis the anchors would fall on the crosshairs. Unique red sits 25° off, and the four are not 90° apart in any case. Hatched sectors are hues this display cannot reach at this chroma.

Why there are four unique hues

Observers agree that four hues are elementary and that no colour is reddish green. Nothing in the three receptors predicts either fact, and the axes of every standard colour space miss the four by tens of degrees.

brain · Appearance A hue circle through a per-channel curve. 28 colours on a circle of constant lightness 55 and chroma 38, each put through the tone curve one channel at a time and read back. The curve is a function of a single number and has no idea what hue is, and it rotates the circle by up to 4.4 degrees — largest at hue 260 — while raising chroma by a factor of 1.27 and lightness by about 1 units.

A contrast control is three controls

A tone curve is a function of one number at a time and knows nothing about hue. Applied to each channel separately it rotates a hue circle by up to seventeen degrees, raises chroma by a factor of 1.27 at ordinary strength, and lifts lightness — so a photographer who moves a contrast slider has moved three things and the interface names one of them. All three scale with the curve's strength, monotonically, and the hue rotation depends on which hue it is.

imaging · Capture The same highlight, clipped in two places. A ramp running from inside the sensor's range to 1.6 times over it, clipped at the sensor and clipped after the matrix. Below the ceiling the two are identical to the floating-point floor. Above it they part, reaching 21.0 colour differences and 78 degrees of hue. Clipping late keeps a highlight neutral and clipping early keeps its hue, and converters do both.

Two converters and one highlight

The clip is the only step in a raw pipeline that destroys information rather than moving it, and it is the step whose position varies most between converters. Below the sensor's ceiling its position changes nothing at all, exactly. Above it, clipping at the sensor and clipping after the matrix land twenty-one colour differences and seventy-eight degrees of hue apart, and which hues are affected is a property of the camera's own dyes.

imaging · Capture The model's hue scale, and the four numbers it is built from. Hue quadrature against hue angle. The four anchors are the unique hues, each with its own weight, and between them the scale is a hyperbolic interpolation rather than a straight line. The quadrant from blue back to red spans 143 degrees of hue angle for its hundred units of quadrature, against 70 for red to yellow.

The hue scale has four corners

CIECAM16 reports hue twice — as an angle in its own opponent plane, and as a quadrature interpolated through four unique hues with four fitted weights. The second is the one hue tolerances and hue-preserving mappings are written in, and it is piecewise — its slope jumps at each of the four anchors, by a factor of 1.74 at green and by 0.41 at blue, and it runs 4.1 times faster near yellow than near blue.

brain · Appearance How far CIECAM16 moves a monochromatic hue when the light is brightened thirtyfold. A monochromatic stimulus at a relative luminance of 2 and of 60, read through CIECAM16 in one room, and the difference between its two hue angles. The model was never fitted to this effect and has it anyway: the shift is positive at the short end, negative through the greens, positive again in the yellows and reds, and crosses zero at 459, 495, 502, 570 nanometres. The marks are the invariant wavelengths the literature reports — 474, 506, 571.

The model has a hue shift it was never given

A monochromatic light changes hue as it is brightened, except at three wavelengths that do not move — an effect measured since the nineteenth century and not among the things CIECAM16 was fitted to. The model has it anyway: brightening a stimulus thirtyfold moves its hue angle, and the places where the movement crosses zero land at 459, 495, 502 and 570 nanometres against the reported 474, 506 and 571. The sizes are another matter.

brain · Appearance 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.

Transfer functionColour appearance modelDynamic rangeTone curveCIECAM16CIELABColour appearanceGamut mappingLightnessNamingSpecificationStructural choice

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