Colour Science & Perception
what the eye settles, and what it cannot
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The thread: Three numbers — page 6

An infinite-dimensional spectrum is projected onto three cone responses, and everything colour science can do — and every way it fails — follows from that single collapse.
The angle between two departures, which nothing in the audit records. Every pair of the six departures on every surface — 2520 pairs — binned by the angle between their two deviations in the local metric. The distribution reaches both ends: 440 pairs sit under thirty degrees and point almost the same way, and 615 sit above a hundred and fifty and point almost opposite. On 1095 of the 2520 the two together cost less than the larger of them alone. A table of magnitudes cannot say which case it is in. Matching and measuring

A size is not a direction

An audit that reports magnitudes cannot say what two of them cost together. Over two and a half thousand pairs of observer departures the angle between them in the local metric runs from one degree to a hundred and seventy-nine, and on forty-three per cent of them the two together cost less than the larger of the two alone. Adding the angle predicts the composition to one and a quarter per cent; Pythagoras is out by twenty-eight.

 The straight piece under the cube root, and where it stops. CIELAB's lightness against relative luminance, over the bottom 5.0 per cent of the range. Below Y = 0.008856 it is a straight line of slope 7.787; above it, a cube root. The two meet at L* 8 in value and in slope, exactly — the CIE's two constants are chosen to make that true. The dashed curve is the pure cube root, which reaches negative lightness before it reaches zero luminance and has an infinite slope there. The break is marked, and the axis runs to Y = 0.050. Matching and measuring

The straight piece under the cube root

CIELAB's lightness is described everywhere as a cube root and below a stated luminance it is a straight line, spliced on with two constants chosen so the join is exact in value and in slope. Every black a delivery chain reaches is inside that straight piece — a press black at L* 2.4, a projected black at 1.1 — where the compression does not compress, the price of a deviation is flat to two parts in a thousand, and the second derivative the composition of two departures needs does not exist.

 What one tolerance accepts, around four colours. The surface in tristimulus values that ΔE₀₀ 1.0 draws around four colours, each outline scaled to its own size so the shapes can be compared. The volumes they enclose differ by a factor of 1.0e+5 across the sRGB cube, and the longest axis of one shell is between 3.3 and 29.9 times its shortest. A tolerance is written as one number and is a different set of samples at every colour it is applied to. Matching and measuring

What one number accepts

A delivery tolerance is written as a single colour difference and it acts on three tristimulus values, so what it actually accepts is a closed surface. Measured over sixty-four colours in the sRGB cube, the volume inside that surface varies by a factor of a hundred thousand and its longest axis is between three and thirty times its shortest. The same contract, applied to a dark colour and a light one, is two different requirements.

 What a code lattice costs, and where. Twelve thousand colours quantised to 8 bits per channel through the sRGB transfer function and read back, with lightness across the bottom and the colour difference the rounding cost up the side. The mean is 0.191 and the worst case is 1.15, a factor of 6.0. The bars are band means, and they rise: the encoding spends its codes in the shadows, so the top of the ramp is where the lattice is coarsest against a metric that does not compress as hard. Matching and measuring

A lattice has no derivative

Every departure priced here was priced by perturbing something and reading the answer, which requires the thing being perturbed to have a derivative. A file written on a code lattice does not have one — its output is flat almost everywhere and jumps on a set of measure zero — so quantisation can be bounded and never propagated. The bound is 1.15 colour differences at eight bits per channel against a mean of 0.19, and it is worst where the encoding spends fewest codes.

 Two primaries mixed, and the line a reader assumes they take. The additive mixture of two sRGB primaries, walked in twenty steps, plotted in the a and b of CIELAB. The filled points are where the light actually goes, which is exactly straight in tristimulus values because that is Grassmann's second law. The open points are the straight line between the two readings. They part company by 26.7 ΔE₀₀ at their furthest, at 60 per cent of the way along, and the half-and-half mixture misses the midpoint by 21.1. Matching and measuring

The mixture line bows

Grassmann's second law says an additive mixture is exactly linear in tristimulus values, and tested here it is exact to floating point. Nothing downstream of the three numbers preserves it. The physical half-and-half mixture of two colours sits a median of 5.5 colour differences from the midpoint of their two readings and up to thirty; on a green and a blue display primary it is twenty-one, which is a quarter of the distance between them.

 Three exchanges, two of which move the colour. The documented pipeline is a white balance, a colour matrix, a tone curve and a clip. Each bar is what happens when two neighbours change places, over 30 surfaces the modelled sensor captures: the filled bar is the mean and the tick is the worst patch. Exchanging the balance and the matrix costs 9.2 colour differences at the mean and 13.2 at the worst. Exchanging the curve and the clip costs exactly nothing, and that is a theorem rather than a small number: a monotone curve onto the unit interval commutes with clamping to it. What a camera does

The order is not in the documentation

A raw converter performs a white balance, a colour matrix, a tone curve and a clip, and every account of the process lists them in that order without saying the order decides anything. Exchanging the first two moves the picture by nine colour differences at the mean and thirteen at the worst patch. The twenty-four arrangements collapse into five outcomes running out to sixty-five, and nothing a converter ships says which of them it is.

 Where the mosaic is filled in, along one row through an edge. A Bayer row across a step from 0.9 to 0.08, in units of the sensor's own ceiling, reconstructed in linear light and reconstructed after the tone curve, with the second undone so the two are compared at the same point in the chain. Away from the edge they agree to 8.3e-14, because a constant interpolates to itself under any curve. At the edge they differ by 5.78 colour differences. Interpolating encoded values pulls an edge towards its dark side. What a camera does

One step has no choice

Four of a raw converter's operations can be arranged twenty-four ways. The reconstruction cannot be arranged at all — a colour matrix needs three numbers and a mosaic site has one, so filling in the mosaic is forced to the front by arithmetic rather than by convention. What is not forced is whether it happens in linear light or after the curve, and that decision costs 5.8 colour differences at an ordinary edge and nothing at all four sites away from it.

 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. What a camera does

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.

 A stop taken in raw, and the same lightness reached afterwards. Each row is a stop of exposure applied to the raw values, against a gain applied after the whole pipeline and solved so that an eighteen per cent grey comes out at the same lightness. The two are then the same brightness by construction and differ by 4.2 colour differences at the mean and 11.1 at the worst patch. A stop is a scalar in front of the curve and is not a scalar behind it. What a camera does

A stop is not a stop afterwards

Doubling the light is exactly a factor of two in raw values and in tristimulus values, which is the one thing about exposure everybody is sure of. A stop taken after the tone curve is a factor of something else, and matching the two on an eighteen per cent grey leaves the rest of the frame between three and four colour differences apart at the mean and up to eleven at the worst patch. The gain that matches one stop is 2.47 rather than 2.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

 Noise clipped at zero, averaged over a shadow, under tungsten. A grey ramp from black to ten per cent reflectance under tungsten, captured at three illustrative noise levels, with every negative raw reading set to zero before the readings are averaged over an area. Each line is the colour difference between that average and the noiseless grey. At high gain a half per cent grey is 1.04 off and a black frame 0.79; at very high gain the worst is 2.95, at 1.0 per cent. The same readings averaged before any clip come back exactly, at every level. The tint is gone once every channel sits several deviations above zero. What a camera does

Clipped noise does not average away

Noise on a raw reading is as likely to fall below the true value as above it, which is why averaging an area removes it. A converter that sets negative readings to zero keeps the upper half and throws the lower away, and the mean of what is left is the signal plus a pedestal. With no black level error anywhere, a half per cent grey under a tungsten lamp comes out 1.04 colour differences off at high gain, 8.98 after a four-stop push — and a blur that removes every trace of the noise leaves the tint where it was.

 Three corrections for the corner of a frame, each made under daylight. The mean colour difference over twenty-four coloured patches between the centre of a frame and its corner, against the angle light arrives at, after three corrections each fitted under daylight, D65 and used under it: a grey-card gain map, a correction confined to the red channel's row, and a full three-by-three matrix. All three leave the grey exact. At 25° the gain map leaves 1.86, the red row 0.93 and the matrix 0.90; at 35°, 3.81, 1.95 and 1.81. Six more numbers buy almost nothing, because the moved edge is in one channel. What a camera does

A corner is corrected by one row

A grey-card gain map makes the corner of a frame exactly right on grey and leaves coloured patches 1.86 colour differences wrong at twenty-five degrees. A three-by-three matrix fitted at that position halves it — and six of its nine numbers do nothing, because the moved filter edge is in one channel. The three that matter rebuild the lost red from green and blue, they carry to another lamp better than a gain map in eleven cases of twelve, and in the twelfth, a row fitted under tungsten and used in daylight, they leave the grey 9.2 off.

 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. What a camera does

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.

 What a five-nanometre grid costs a steep-sided notch, against its width, under a 6500 K source. The cost of a five-nanometre grid starting at 380 nm, against a reference at two hundredths of a nanometre, for a sample with a flat-bottomed notch at 552.3 nm under a 6500 K thermal radiator, against the notch's width from 2 to 30 nm, for edges rising in 0.4, 2.2, 6.6 nm. With the steepest edges the cost is 0.02 at 10 nm, 1.99 at 12.5 and 0.03 at 20: it rises and falls with the step as its period and does not die away as the notch widens. With the softest edges it stays under 0.10 at every width. What light is

The cost of a steep notch repeats every step

A Gaussian notch is safe on a five-nanometre grid once it is a couple of steps wide. A flat-bottomed notch with steep sides never is. Its cost on the grid rises and falls with its width, with the step as its period — 0.02 colour differences at ten nanometres, 1.99 at twelve and a half, 0.03 at twenty — and it does not die away as the notch widens. What sets its size is how fast the edges rise, and an interference filter's edges rise in under a nanometre.

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