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The thread: Computed, not quoted — page 17

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.

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.

Four ways to fill in a clipped highlight: a glossy surface with a reflection of the lamp, under tungsten. Twenty-four chart surfaces under tungsten, as a glossy surface with a reflection of the lamp, taken up a ramp until their raw channels reach the sensor's ceiling. Each line is the mean colour difference, at equal lightness, between the true colour and what one response to the clipped reading makes of it: clipping to white, carrying the clipped values through, filling the clipped channel from the surface's own ratio, and filling it from the surface's colour plus the lamp's. At 0.4, where most surfaces have one channel clipped, the four leave 5.87, 4.37, 5.56, 0.00; at 2, 2.90, 26.97, 8.18, 8.18.

What a camera does

Filling in a highlight is a claim about the surface

A converter that rebuilds a clipped channel has to say what the highlight was. A matt surface over-exposed keeps its own colour, and filling the lost channel from that colour is exact. A glossy highlight is the surface's colour plus a reflection of the lamp, and the same fill leaves it 7.6 colour differences too colourful — while a fill that solves for surface and lamp is exact. Neither works once two channels are clipped, and a tungsten lamp keeps a highlight in the one-channel band more than twice as long as daylight does.

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.

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.

What light is

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.

A gloss finish's loss of colour, read by the light and by a viewer in the room. Four changes against the matt room as the coloured walls are made glossier, from roughness 0.8 to 0.15: the chroma of the room's reflected light as a colorimeter reads it; the mean chroma of the six faces as CIECAM16 sees them adapted to the lamp and adapted to the room's own average light; and how far the faces sit from that average in the model's uniform space. At roughness 0.2 the light loses 8.3 per cent, the faces 8.6 per cent to the lamp-adapted viewer and 13.9 to the room-adapted one, and the spread 11.0 per cent against 11.2 read against the lamp.

What a scene does

A gloss room looks less colourful than it measures

A gloss finish takes 8.3 per cent of the chroma out of a green room's reflected light, and a viewer adapted to the room should discount a loss that affects everything alike. The appearance model says the opposite. Adaptation removes the colour the whole room shares, leaves the colour that differs from face to face, and the finish takes as large a share of that as of anything — so to a viewer standing in the room the faces lose 13.9 per cent of their chroma, not 8.6.

The corner of a patch of skin against its own shadow, magnified four times by Lanczos, three lobes. A map of the neighbourhood of one corner of a square patch — the patch fills the lower right, the field the rest — once it has been magnified four times by Lanczos, three lobes. Each cell is shaded by how much lighter (warm) or darker (cool) the stored-value result is than the light's, the deepest shade 18.2 units of lightness. Within three source pixels of the corner the stored-value result is up to 14.5 colour differences lighter, 14.5 outside the patch against 3.1 inside it, and up to 7.3 darker.

What it takes to deliver it

The corner of a resized patch is lighter than its edges

A resize taken on stored values is darker than the resize of the light wherever its weights are positive, and lighter beside an edge where a kernel's negative lobes fall. In two dimensions the kernel is a product, and just outside a patch's corner a Lanczos magnification of skin against its shadow comes out 14.5 colour differences lighter — more than along either edge. A reduction to a quarter is mostly an average and errs lighter by at most two. And an unsharp mask, whose negative weights are its whole purpose, is lighter on the stored values at every amount on every pair, and never darker.

A soft proof exact for one observer, and three proofs tuned for readers. For each display, the median over printed patches of the 95th percentile reader's mismatch between screen and print, for four ways of choosing the display's three drive levels: exact for the reference observer; least squares over a population of a hundred; tuned on that population's 95th percentile; and tuned on the two hundred readers it is scored on, which no workflow could do. On a wide-gamut LCD the four give 4.57, 4.99, 4.74, 4.39, and the three tuned proofs cost the reference observer 0.70, 0.82, 0.69. On an OLED panel the four give 5.12, 5.72, 4.99, 4.86, and the three tuned proofs cost the reference observer 1.35, 0.95, 0.67. On a laser projector the four give 7.54, 7.00, 6.81, 6.50, and the three tuned proofs cost the reference observer 1.88, 1.23, 1.14.

What it takes to deliver it

A proof cannot be tuned for readers who disagree

A soft proof matched exactly for the standard observer is five colour differences wrong for one reader in twenty. Giving up that exactness to tune the display's three drives for a population instead moves the ninety-fifth percentile reader by 4 to 14 per cent even when the tuning is done on the very readers it is scored against — because what readers see is mostly each other's disagreement, and three drives act on every reader at once.

Matching and measuring

A press is charged for the direction it barely moves

A press run varies almost entirely along the direction a colour tolerance forgives. Split along the tolerance's own axes under four stated mixes of press variation, its tightest axis holds about one per cent of the sheet-to-sheet variation and pays a quarter to a third of the price — and on a blue overprint four fifths, for the balance between two inking units rather than the level of either.

The angle between the two filters against how completely the eye has adapted. The angle, in the local metric, between what an older lens does to a reading and what a denser macular pigment does, on 120 smooth reflectances, as the degree of adaptation runs from nought to one. The median angle is 8 degrees unadapted and 156 at complete adaptation, and almost all of the turn happens in the last tenth: it passes a right angle at a degree of 0.928. The marks are the degrees CIECAM16 gives five rooms — an overcast sky 1.00, an office 0.94, a lit living room 0.86, a dim room 0.75, a cinema 0.66 — so only the outdoor one is at the end of the dial.

Difference and uniformity

Two filters cancel only in a bright enough room

An older lens and a denser macular pigment cancel each other once an eye has adapted — and that result belongs to the end of a dial nobody stands at. Read at the degree of adaptation CIECAM16 gives an ordinary room, the two barely cancel; in a living room they add, and in a cinema they cost six times what they cost under the sky. The room has to be about as bright as an office before the cancelling begins at all.

The same pairs, held at one colour difference, read in a unit that knows the room. 23 pairs of reflectances built to sit at exactly ΔE₀₀ 1.000 under D65, read in CAM16-UCS as the adapting luminance runs from a third of a candela a square metre to ten thousand, in an average surround. ΔE₀₀ has no argument for the room, so in that formula every pair stays at 1.000 all the way across — the flat line. In the model's unit the same pairs rise from a median of 0.76 to 1.30, and they do not rise together: at the bright end they run from 1.11 to 1.65.

Difference and uniformity

A tolerance has no light level

Twenty-three pairs built at exactly one colour difference stay at exactly one in every room, because the formula has no argument for the room. Read in the unit that does have one, the same pairs are 0.72 in a cinema, 1.04 in an office and 1.30 in direct sun — and inside any one room they spread by half again, so no single conversion between the two units exists at all.

What is left of one colour difference when the two colours alternate. 23 pairs built at exactly ΔE₀₀ 1.000, alternated at a rate, with each part of the difference scaled by its own temporal channel and the formula then applied unchanged. At rest every pair is the flat line at one. By 7 hertz the median is 1.11 and the pairs run from 0.57 to 3.04 — a factor of 5.3 between pairs the formula calls identical. By sixty hertz the largest of them is 0.15.

Difference and uniformity

A difference has no rate

A colour difference formula answers for two patches that are both there and stay. Alternate the same two colours and the difference is not scaled but taken apart: the colour half is gone by fifteen hertz and the lightness half is four times louder at eight, so twenty-three pairs the formula calls identical run over a factor of six at the rate the eye is best at, and are worth nothing at all above sixty.

The straight line between two colours, and the formula's own shortest path. Five gradients seen from above, in the a and b plane of CIELAB: the straight line between the two colours in grey, and the shortest path under ΔE₀₀'s own local metric in colour. The lightness coordinate bows too and is not drawn. red to green saves 3.6 per cent and leaves the straight line by 13; blue to yellow saves 7.5 per cent and leaves the straight line by 21; cyan to magenta saves 3.9 per cent and leaves the straight line by 10; black to white saves 0.0 per cent and leaves the straight line by 0; red to blue saves 0.3 per cent and leaves the straight line by 4 CIELAB units at the widest. Black to white is the flat case: its shortest path is the straight one.

Matching and measuring

The straight line is not the shortest gradient

A colour difference formula says what a small step costs at every colour, and that is enough to ask which path between two colours is shortest. It is not the straight line: between red and green the shortest path bows thirteen CIELAB units away, saves four per cent — and stays inside sRGB on every step where the straight line leaves it. The formula's own answer for the two endpoints, meanwhile, is neither length.

The metamerism index, computed three ways, as the reference match loosens. The special metamerism index of 6 metameric pairs under an incandescent test light, against how well each pair matches under the reference light. Uncorrected, the index absorbs the reference mismatch and rises from 2.86 to 3.76. Corrected multiplicatively it rises to 3.31 and additively to 3.87. All three are the same number when the pair matches exactly, which is the only case the definition covers.

Matching and measuring

The metamerism index has two corrections

The index for a metameric pair is defined for a pair that matches exactly under the reference light, and no real pair does. The standard's remedy is to correct the sample first, and it names two corrections — scale the tristimulus values, or add the difference. On six pairs matched to one colour difference, the two answers differ by a tenth to four tenths of an index unit; at two, by a whole one.

What a slope limit costs the object-colour solid, direction by direction. For each transition width, how much of its ideal reach the solid keeps: the median direction, the tenth percentile, and the worst. At twenty nanometres the median keeps 0.997 and the tenth percentile 0.985; at eighty they keep 0.946 and 0.766, and at 160 0.826 and 0.417. The worst direction falls from 1.00 at five nanometres to 0.20 at 160, with directions reaching under five units beyond black set aside. The cost is in a corner only at widths sharper than an ordinary pigment's.

Where the model breaks

The limits assume a pigment that switches instantly

The hardest boundary in colorimetry is reached by reflectances that jump between nought and one at a wavelength, and no material does that. Constrain the jump to take twenty nanometres — a sharp dye — and the median direction of the object-colour solid loses under half a per cent of its reach. Constrain it to eighty, an ordinary pigment, and the median loses five per cent, the tenth percentile nearly a quarter, and seven directions in ten lose more than one. The cost is in a corner only for chemistry sharper than paint.

How much of a display's gamut each observer names differently. Each of 60 observers names every colour of the displayable gamut that the panel can make — 823 of them — and the histogram is how much of that gamut each observer names differently from the standard observer. On an OLED panel the median observer renames 8.1 per cent and the furthest 13.1 per cent. Nobody agrees with the standard observer about all of it.

What the brain does

A name moves with the reader

Three earlier essays have moved a colour's name by changing the distance function, the room and the space. All three held the observer fixed. Handed the same light from the same display, sixty observers rename between 3.5 and 13.1 per cent of the gamut against the standard observer, a quarter of its colours have a dissenter in twenty, and the narrower the display's primaries the worse it gets.

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.

What the brain does

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.

What the choice of adopted white is worth, in a room lit by two lights. A room lit half by daylight and half by an incandescent lamp. The light on the surfaces is fixed; what varies is the white the model is told the observer has adapted to, running from the lamp on the left to the window on the right. The median of twelve surfaces moves 32.6 CAM16-UCS units if the lamp is adopted and 24.5 if the window is, against the room's own mixture in the middle. The model offers no way to choose, and its degree of adaptation is 0.94 at every point of the dial.

What the brain does

A room with two lights has no white

An appearance model takes one adapting white. A desk beside a window has two, and the mixture falling on the paper is not the same thing as the white the person reading it has adapted to. Mixing the lights is arithmetic. Choosing the white is not, and the choice is worth sixty units of appearance — most of which is a cast, and not all of which is.

One grey scale, three backgrounds. CIECAM16's lightness against the luminance factor of a neutral sample, with only the background changed. A grey reflecting 19% reads 46.7 on a near-black background and 32.8 on a near-white one. The three curves are not three shapes: each is the same curve raised to a different power, because the background reaches lightness only through the exponent z, which runs 1.621 to 2.374 across the three.

What the brain does

A dark background moves every difference and no match

CIECAM16's background is one number, and it reaches lightness as one exponent. That is enough to change what a grey looks like and not enough to change which of two greys is lighter — so a match survives the background exactly, a corresponding colour is invariant to it, and a tolerance is not. The effect the background is usually invoked to explain is absent from the model entirely.

What the brain does

The cancellation is exact and cheap to lose

CIECAM16 has no crispening, and adding one was expected to be expensive: the background's exactness in a corresponding colour comes from its being a common exponent, and a function of the sample's own level is not one. It is expensive in kind and not in size. A term that raises a straddling pair's lightness difference by half moves a corresponding colour by five thousandths of a tristimulus unit — a thousandth of what stating the background differently at the two ends already costs.

The pairs that change places in chroma are a wedge with two straight edges. Every pair of samples, plotted by the log ratio of the two samples' background-free responses across and the log ratio of their chroma at a background of 2 up. A pair changes places between that background and one of 80 exactly when its chroma ratio and its response ratio point in opposite directions and the chroma ratio is the smaller — which is the wedge between the horizontal axis and a line of slope -0.2598, half the change in the lightness exponent. 754 of 14028 pairs are inside it, and the condition names every one of them and nothing else: 0 disagreements between the line and the model. The pale dots are one pair in eleven of those outside; the filled ones are every pair inside.

What the brain does

The reversals have a straight edge

Twenty-one of 276 pairs change places in chroma between a dark background and a light one, and the reason given was that chroma is a product of a term carrying the exponent and a term that does not. That is true and it is not a description of which pairs. Chroma at one background is a single common factor times a power of lightness at another, so a pair reverses exactly when its chroma ratio and its lightness ratio point opposite ways and the first is the smaller — a wedge with two straight edges, which names every reversal and nothing else.

One lightness scale, drawn as a contour across the rooms. Three surrounds up the page and the background's luminance factor across it, on a square-root scale so that the model's own exponent base is linear in the axis. Each curve joins the rooms whose lightness exponent is the same, and every room on one curve returns the same lightness for every sample. The marked curve is the one through a television in a lit living room against a mid grey: it also passes through a print on a desk against a background of 2.8 and a projection in a dark room against 47.0. A curve that leaves the plot has no member in that surround, because each surround multiplies a base that runs only from 1.48 to 2.48.

What the brain does

Two rooms with one lightness scale

CIECAM16's surround and its background both reach lightness, and they reach it through one product. So the rooms fall into classes: a television in a lit living room against a mid grey returns exactly the lightness a print on a desk against a background of 2.8 does, for every sample, to the last bit of a double. It returns 0.76 of its chroma and 0.81 of its brightness. Two of the model's four viewing-condition parameters are one parameter, and only colour tells them apart.

Every pair of departures, before adaptation and after it. The fifteen pairs of the six audited observer departures. Each row runs from the angle between that pair's two deviations with no adaptation to the angle with complete adaptation; an angle past ninety degrees is a pair pointing apart, which is where a pair can cost less together than the larger of the two costs alone. 3 pairs gain that behaviour as the eye adapts, 3 keep it, 5 lose it and 4 never have it. The pair followed here — the lens against the macular pigment — is in the smallest group that is not empty, and every result quoted from it generalises in the wrong direction.

Difference and uniformity

Adaptation turns more pairs off than on

One pair of observer departures was followed across the degree of adaptation and found to cancel only in a bright enough room. The same calculation takes any two, and run over all fifteen pairs it says something the single pair does not: adaptation is a rotation rather than a mechanism for making departures oppose each other. Five pairs lose their cancellation as the eye adapts, three gain it, three keep it and four never have it — and the pair everybody quotes is one of the three it turns on.

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