Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Computed, not quoted — page 16

Every swatch begins as a spectral power distribution and is carried through the colour-matching functions as it is drawn. None is a hex code recalled from a table.
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. What a camera does

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.

 Where a stated appearance has no stimulus under it. The chroma the inverse can still return a light for, at lightness 50, all the way round the hue circle. Below the curve a stated appearance corresponds to a stimulus; above it the formula still returns three numbers and one of them is negative. Over a lattice of 10488 appearances spanning the whole space a specification is written in, 10.8 per cent are of that kind, and only 44 per cent are colours a display could show. What the brain does

An appearance is not always a stimulus

CIECAM16's inverse is a closed form that returns three numbers for every lightness, chroma and hue it is handed. Whether those three numbers are a light it does not ask, and over a lattice spanning the space a specification is written in, 10.8 per cent of them are not — one tristimulus value negative, or a relative luminance above the white's. Only 43.8 per cent are colours a display could show at all.

 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. What the brain does

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.

 Where averaging the model's answers is and is not averaging its argument. Each row takes a spread of situations, averages the model's predictions across them, and compares that against the model's prediction for the average situation. The bar is the gap as a share of the spread itself. Over the differences between observers it is 1.4 per cent — the model is very nearly linear there. Over the range of adapting luminance one room covers in a day it is 59 per cent, and from indoors to outdoors 74. What the brain does

Where the model's curve does not matter

This round has been about what a nonlinearity does to an average, and CIECAM16 is the most nonlinear thing in the collection. Over the spread a population of observers produces, the average of its predictions is its prediction for the average to within 1.4 per cent — the nonlinearity is there and the excursion is too small to reach it. Over the range of adapting luminance one room covers in a day, the same gap is 59 per cent of the spread, and from indoors to outdoors 74.

 The audit's six departures, read twice. Each departure priced in the matching unit it was published in and in the appearance unit an appearance prediction would be judged by. The model does not scale them by one factor: it amplifies the smallest by 1.58 and the largest by 1.03, so the range between them narrows from 3.34 to 2.17. The mean ranking is unchanged and 14 of the 42 surfaces reorder. What the brain does

The audit, read as appearances

The previous round priced six observer departures in a matching unit and left them there. Read in the appearance unit an appearance prediction would actually be judged by, they are not the same six numbers scaled by a constant — the model amplifies the smallest by 1.58 and the largest by 1.03, so the range between the largest and smallest term narrows from 3.34 to 2.17. The published ranking survives in the mean and changes on fourteen of the forty-two surfaces it was averaged over.

 What a stated lightness pins down, and where. A lightness quoted to 0.05 of a unit, inverted, and the luminance it fixes. Read as a fraction of the colour's own luminance the requirement is 0.90 per cent at J 10 and 0.097 at J 95, a factor of 9.3. Read in absolute luminance it is the other way round, by a factor of 6.5. Both readings are true and they answer different questions. What the brain does

A stated lightness is two requirements

A specification quotes an appearance to a stated precision — a lightness to a tenth of a unit, say — and the same precision everywhere. Inverted, a twentieth of a unit of lightness fixes the luminance to nine tenths of one per cent at the bottom of the scale and to a tenth of one per cent at the top, a factor of 9.3. Read in absolute luminance it is the other way round by a factor of 6.5, and both readings are true.

 The budget's three numbers, and the units they are in. The published three-stage budget's own figures, with each one's unit named, beside the same stage re-measured in a single unit over the same colours. Two of the three are colour differences between stimuli and the third is a distance between appearances, and the budget adds them. The fourth row is a stage the budget has no entry for: the colours the separation cannot reach even after the mapping has moved them, which comes to 1.45. What it takes to deliver it

The budget adds two units

This collection publishes a three-stage error budget for a colour-management chain and prints its sum. Two of the three stages are colour differences between stimuli and the third is a distance between appearances, and the conversion between those is not a constant. Re-measured in one unit over the same colours the chain has four stages rather than three, its end-to-end error is 5.05 against a sum of 7.66, and the profile's contribution where a job actually lands is 3.2 times its published average.

 The same blend, taken on the stored values and on the light. Six pairs blended at 50 per cent, once by averaging the values as they are stored and once by averaging the light they stand for. Every resize, every antialiased edge and every transparency composite in an ordinary pipeline does the first. The two land 15.8 colour differences apart at the mean and 19.3 on a red against a green, and the stored-value blend is the darker on all six, by up to 23 units of lightness. What it takes to deliver it

An average on the stored values

A resize, an antialiased edge, a transparency composite and a chroma subsample are all averages, and in almost every pipeline they are taken on the numbers as stored. The numbers as stored are encoded, the encoding is a compression, and a half-and-half blend of black and white taken that way lands 18.7 colour differences from the half-and-half blend of the light — 22.7 units of lightness darker, on every pair, always in the same direction.

 The same colour, delivered from two documents. A ramp of chroma, mapped into a press's gamut under the perceptual intent from two source solids: the whole of sRGB, and the same solid with its chroma limited, which is what a document containing only muted colours amounts to. Under a colorimetric intent the two answers are identical to the floating-point floor. Under the perceptual one they are 0.69 apart at the mean and 2.64 at the worst, on colours the destination could hold either way. What it takes to deliver it

An intent is not a function of the colour

A rendering intent is chosen per job and applied per pixel, which suggests it is a function of the colour. The colorimetric ones are, exactly — the same colour from two documents comes out identical to the floating-point floor. The perceptual one is not — it compresses the whole source solid into the destination, so the same colour delivered from a wide document and from a muted one lands 2.6 colour differences apart, on colours the destination could hold either way.

 Out to the device and back, once and again. Each intent applied 3 times in succession to the same ramp. The upper bar is what the first pass moves and the lower is what the second moves. relative-colorimetric is idempotent — the second pass moves nothing, to the floating-point floor — and the other two are not: perceptual moves another 2.73 and saturation moves another 3.09. A file converted twice is not a file converted once. What it takes to deliver it

A second conversion is not a repeat

A colour converted to a device and back is described as lossy where the device cannot hold it and exact where it can, which suggests that a colour surviving one round trip survives any number. One of the four intents is idempotent to the floating-point floor. The other two are not — a perceptual conversion moves a ramp 3.70 colour differences on its first pass and another 2.73 on its second, and after three passes it is still moving by 2.42.

 Which way each tolerance is tightest, colour by colour. Sixty-four colours on a lattice through the sRGB cube, with lightness across the bottom. For each, the tolerance ΔE₀₀ 1 is pulled back into tristimulus values and its shortest and longest axes are found. The filled points are the angle between each colour's shortest axis and the direction common to all of them: a median of 15 degrees, ninety per cent within 32. The open points are the same for the longest axis, at 23. The common short axis is 3.1 degrees from X up and Y down, which is the direction of a*. Matching and measuring

A tolerance has a grain

A colour tolerance pulled back into tristimulus values is a long thin shape, and across the whole sRGB cube its shortest axis points the same way, fifteen degrees off at the median — the direction in which X rises as Y falls, which is the direction of a*. So what one colour difference allows depends on which way a delivery drifts. An instrument's X filter may age 1.4 per cent before the tolerance is used up, its Z filter 5.1, and the light on the sample 9.3.

 What a small neutral deviation costs as the grey darkens, in two lightness scales. The price of a fixed fraction of deviation on a neutral — how much lightness it is worth per unit of luminance — from L 40 down to L 0.25, both axes logarithmic. CIELAB's L is a straight line below L 8 and its price there is exactly flat. The appearance model's J′ has no straight piece: its price keeps rising, by a factor of 5.0 across the same range, as the luminance to the power -0.442 — the derived exponent is -0.441 in an average surround. Matching and measuring

The appearance model has no straight piece

CIELAB's lightness is a straight line below L* 8, so a deviation near black has a fixed price and a black has a floor under it. CIECAM16 has no such piece. Its lightness near zero goes as the luminance to a power set by the room, the price of a deviation rises without limit as the black deepens — as Y to the power −0.44 in a lit room and −0.58 in a dark one — and on a projected black a lightness unit in the model allows under half the luminance change a unit of L* allows.

 Four mixtures of display primaries, bowing by different amounts in two units. The half-and-half mixture of each pair of sRGB primaries, measured from the midpoint of the two readings as a share of the pair's own separation. The upper bar is ΔE₀₀ and the lower is the appearance model's J′a′b′. In ΔE₀₀ green and blue bows most, at 25 per cent; in the model it is red and blue, at 20, and blue and yellow falls from 22 to 12. The right-hand column gives the model's distance with its power correction, which is smaller than the Euclidean one for every pair here. Matching and measuring

Which mixture bows most depends on the ruler

The physical half-and-half mixture of two lights misses the midpoint of their two readings in any unit a person is shown, and the share it misses by is about the same in ΔE₀₀ and in the appearance model's own space — 12.8 and 14.6 per cent at the median. Which mixtures miss most is not. The rank correlation between the two units is 0.57, a display's blue and yellow is the worst pair in one and the mildest in the other, and the smaller number usually quoted for the appearance model is its power correction rather than a straighter line.

 The length of one grey ramp, cut into more and more steps. A neutral ramp from L 1 to L 100 cut into between one and ten thousand equal steps, each step measured and the steps added, in four units, both axes logarithmic. ΔE*ab and the model's Euclidean J′a′b′ give the same length at every step count, 99 and 96. ΔE₀₀ settles at 74.6 once the steps are small. The power-corrected ΔE′ does not settle: 25 in one step, 137 in a hundred, 755 in ten thousand, growing as the number of steps to the power 0.37. Difference and uniformity

A distance raised to a power has no length

CAM16-UCS's colour difference is its Euclidean distance raised to the power 0.63 and multiplied by 1.41. That is still a metric — the triangle inequality holds on every one of four thousand random triples — and it has no length. A grey ramp from black to white measures 25 units in one step, 137 in a hundred and 755 in ten thousand, growing as the number of steps to the power 0.37, and halving the size of a step triples the number of steps that fit.

 The angle between the lens and the macular pigment, before and after adaptation. On each of 120 surfaces the angle, in the local metric, between what an older lens does to the reading and what a denser macular pigment does, binned in ten-degree steps. Read without adaptation, where both filters yellow the observer's white along with everything else, the two point nearly the same way: a median of 8 degrees. Read after each observer has adapted to its own white, the median is 156, and the two together cost less than the larger alone on 115 of the 120. Difference and uniformity

Two yellow filters cancel on a slope

An older lens and a denser macular pigment both take blue out of the light, and read before adaptation they move a colour in nearly the same direction, eight degrees apart. Once each eye has adapted to its own white they point a median 156 degrees apart on smooth reflectances and together cost less than the lens alone. On surfaces with a narrow absorption band they still sit 26 degrees apart and add. What decides it is the width of the surface's own features.

 Which appearances a surface can have, lightness by lightness. The same lattice of lightness, chroma and hue a specification is written in, 10488 points, inverted under daylight with the observer adapted to it. Each row is one lightness, split into three shares: appearances a reflecting surface can have, appearances that are a light but that no surface can return, and appearances with no light under them at all. Over the whole lattice the first is 66 per cent, the second 23 and the third 11. At J 90 a surface can have 38 per cent of the row. What the brain does

A third of the appearance box is no surface

An appearance specification is written as a lightness, a chroma and a hue, and the model's inverse turns any such triple into three numbers. A tenth of the space turns into something that is not a light at all. A further quarter turns into a light no reflecting surface can return, because a surface cannot give back more than all the light at any wavelength. So a third of the space a paint, a print or a dye is specified in cannot be made from paint, print or dye, and at lightness 90 nearly two thirds cannot.

 How far the answer for the average is from the average answer, over eight spreads. For each spread of inputs, the distance between the mean of the model's answers and its answer for the mean input, as a share of the spread of the answers. Over a population of observers it is 1.4 per cent. Over surfaces it is 21 on smooth natural reflectances, 27 on a banded family with lightness in it, and 9 on a set of pale surfaces. Over the light one room sees in a day it is 59. What the brain does

The average surface does not look average

Over a population of observers the appearance model is so nearly linear that the mean of its answers is its answer for the mean, to 1.4 per cent of the spread. Over the surfaces in a scene it is not. On 240 smooth reflectances the gap is 21 per cent of the spread, and the mean surface looks 4.2 units lighter than the surfaces look on average. The grey that matches the average light is a 47 per cent reflectance; the grey that matches the average look is a 43 per cent one.

 One infrared-cut filter, crossed at four angles. The transmittance of the same interference filter for light crossing it at the centre of a frame and at three steeper angles, from 560 to 760 nm. Its half-transmission edge is at 665 nm straight on and moves to 658, 646 and 630 nm, following the edge wavelength times the square root of one minus the squared sine of the angle over the square of the stack's effective index, 1.8. The deepest reds are what it takes away, and a lamp decides how much light there was in them. What a camera does

The corner of the frame has another filter

A camera's infrared-cut filter is an interference stack, and light crossing it at an angle sees its edge moved towards the blue — from 665 nm at the centre of a frame to 646 at twenty-five degrees. The corner's red channel loses a tenth of its response under daylight and a sixth under tungsten. A grey-card gain map makes the corner exactly right under the lamp it was made under; under a tungsten lamp the same grey is 2.6 colour differences wrong, under a white LED 4.2, and even under its own lamp the coloured patches are not corrected, because a moved edge is not a gain.

 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. What a camera does

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.

 A soft proof exact for one observer, as two hundred others see it. Each display is driven to match each of thirty printed patches exactly for the reference observer, so for that observer screen and print are the same colour to fourteen decimal places. The bars are what two hundred observers drawn from the population make of the same pairs: the median observer's difference, median over the patches, and the ninety-fifth percentile observer's: 1.8 and 4.8 on the wide-gamut LCD, 2.0 and 5.4 on the OLED, 3.1 and 7.5 on the laser projector. The narrower a display's primaries, the larger both become. What it takes to deliver it

A soft proof is exact for one reader

A display can be driven to match a printed patch exactly for the standard observer — three equations, three unknowns, agreement to fourteen decimal places. Two hundred observers drawn from a realistic population see the same screen and print a median of 2.1 colour differences apart on an OLED panel and 5.4 apart at the ninety-fifth percentile. On a laser projector the ninety-fifth percentile is 7.5. The patch that fails worst is unprinted paper, and in the chain's own unit the ninety-fifth percentile reader's stage is larger than every one of the four stages a delivery chain is budgeted for.

 How far the chain falls short of its sum, and how much of that is the unit. For each rendering intent, three ratios of the chain's end-to-end error to the sum of its four stages. The lowest bar is the published one, in the power-corrected unit. The middle bar is what that unit reports for a chain whose stages point the same way and add exactly — the exponent on its own. The top bar is the same chain measured in the model's own Euclidean space. Under the colorimetric intent the published ratio is 0.66, the exponent alone gives 0.74 and the chain in a space that can add 0.84: 77 per cent of the shortfall is the unit. What it takes to deliver it

A chain measured in a unit that cannot add

A delivery chain's four stages, measured in the power-corrected appearance difference, sum to 7.66 while the chain end to end measures 5.05, and the shortfall was read as the stages partly cancelling. A chain whose four stages lay in a straight line and added exactly would still read 0.74 of its sum in that unit, because a distance raised to the power 0.63 cannot add. Measured in the model's own Euclidean space the same chain reaches 0.84 of its sum. Three quarters of the published shortfall was the exponent.

 One edge magnified four times, three ways: text on a page. A step between the two colours of text on a page, magnified four times by linear interpolation, by bicubic and by a three-lobe Lanczos kernel, once on the stored values and once on the light. The curves are the stored-value result's lightness minus the light's, sample by sample across the edge; below the line the stored-value resize is darker. Linear interpolation is darker everywhere it differs. The two kernels with negative lobes swing above the line beside the edge, where their negative weights fall — bicubic by up to 1.8 colour differences and Lanczos by 3.7. What it takes to deliver it

A resize with a negative weight in it

A blend taken on stored values is always darker than the blend of the light, because the encoding is concave and a convex combination of a concave function's values lies below the function of the combination. A bicubic or Lanczos resize is not a convex combination. Beside an edge its negative weights make the stored-value result lighter than the light's own — by up to 6.1 colour differences with bicubic and 12.2 with Lanczos on skin against its shadow — and on edges between full-scale values the clip hides the overshoot and the old guarantee appears to hold.

 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. What light is

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.

 The room's reflected light and its colour, against how glossy the walls are. Four changes against the matt room as the coloured walls are made glossier, from roughness 0.8 on the left to 0.15 on the right. The room's reflected light rises by up to 19 per cent and its chroma falls by up to 9.1 per cent. The walls' own outgoing chroma falls fastest, by 14.6 per cent, and the floor's follows the room's. A lobe does not move colour from one face to another: the room as a whole has less of it. What a scene does

A gloss finish takes colour out of the whole room

A gloss wall makes the floor's return less colourful seen from the front of a room and more colourful seen from the coloured walls, which leaves open whether the lobe removes colour or only moves it. A ledger of every flux between the room's faces answers it. At an eggshell finish the room's reflected light gains 16 per cent in quantity and loses 8.3 per cent of its chroma, and the loss is nearly the same for blue, green and orange walls while the walls' own losses range from 13 to 25 per cent. Only the painted walls receive light as colourful as before.

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