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

Sharpening — where it appears

Recombining three broad, overlapping channels into narrower and more independent ones, so that a change of light acts more nearly as three separate gains. Every fitted adaptation transform does it, and it moves the axes away from the receptors rather than towards them.

Named by 13 essays across 4 fields — each of them below, with the objects they name alongside it.

How cone-like a basis is, against how well it adapts. Each basis placed by how far its own implied deuteranope confusion point falls from the measured one (horizontal) and by how much an adapted observer is left with in it (vertical). The construction from the confusion points sits at zero on the horizontal by definition and near the top on the vertical. Nothing near the left of the picture is near the bottom: the closer a basis is to the receptors, the more a von Kries gain leaves behind. The unconstrained winner sits at 1.63 on the horizontal, further from the measurement than any published transform except CAT02 and Bradford.

The best axes are not receptors

If the axes that make a von Kries gain work were nearly the cone fundamentals, the published adaptation transforms would be cone-like and their departures would be slack in a fit. They are not. Minimise the residual over all nine free numbers and the winner sits further from the measured dichromat confusion points than any of them.

eye · Cones A display's primaries, scored as the adaptation basis they are. Four primary sets ranked by the mean ΔE00 an adapted observer is left with when the white point moves — which for a display is a gain on R, G and B, and so a von Kries adaptation in the inverse of its own primary matrix. sRGB leaves 2.36, as much as scaling XYZ directly and therefore as much as having no cone basis at all. Rec. 2020 leaves 1.09, better than every published adaptation transform fitted to corresponding-colour data. Nobody chose that: it is what wanting a wider gamut does to a primary's spectral selectivity.

The gamut race chose the basis

Twenty years of arguing about how much of the diagram a display should cover has produced primaries whose inverse is a better adaptation basis than any transform ever fitted to corresponding-colour data. On the invariant count of what those displays can actually show, the same twenty years produced nothing at all.

matching · Gamut The dyes a camera has, and the dyes an adaptation basis would want. Three sensor sensitivities drawn twice: faintly, the silicon-and-filter-array set this collection models, and boldly, three Gaussian dyes chosen to make the inverse of their own response matrix a good basis for a white-balance gain. The designed dyes sit at 610, 542, 449 nm with widths of 35, 26, 30 nm — narrower and further apart than the real ones, which is what sharpening looks like when a search rather than a committee does it. They leave 0.97 ΔE00 against the real sensor's 1.62, and they are held within 0.28 of the Luther condition so that the result is still a camera.

A sensor designed for its inverse

A camera's white balance is a gain in a basis made from its own dyes and the light in the room. Choose the dyes for that basis instead of for cost and quantum efficiency, hold the sensor within a stated distance of the Luther condition, and the design reaches the best adaptation figure any basis achieves — and then the room moves it.

imaging · Capture What choosing a space to divide the white out in is worth. Three pairs of routes to the same colour, over forty-two surfaces: dividing the white out in tristimulus values, in a published cone space, and in the observer's own cones. The first two agree to 0.59 ΔE₀₀ at the median. Either of them differs from the observer's own cones by more than fifteen. That is why the two exact conditions in this round are exact only in the eye's own coordinates: the identity belongs to the receptors, and every published arithmetic works in a basis somebody else chose.

The identity is in the eye's own coordinates

Two conditions in this round are exact — a gain on each cone is not a different observer, and three curves for one space are one observer. Imposed in a published cone space rather than the eye's own they leave 13.0 and 8.11 ΔE₀₀ standing. The identities belong to the physiology and every arithmetic in use works somewhere else.

eye · Cones 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.

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.

applied · Delivery 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.

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.

applied · Delivery Three stored-value defects, before the eye and after it, a print at 40 cm. Each defect's largest colour difference between the version computed on stored values and the version computed on light, read twice: as a pixel carries it, and after both images have been through the visual system's three channels at 83 pixels a degree. The two rankings are not the same. The unsharp mask has the largest error per pixel — 16.62 — and keeps only 38 per cent of it; the corner has the smallest at 7.14 and comes out at 9.87. The filter is not a blur applied to a difference: it is applied to each image, and the difference is taken after.

The eye keeps the lightness errors

Four essays have priced what a resize taken on stored values costs, in colour differences between pixels. A reader does not see a pixel. Put the two versions of each image through the visual system's own three channels and the ranking reverses: the unsharp mask, the largest error per pixel at 16.6, is seen at 6.3 on a printed page, while the corner's 7.1 is seen at 9.9. What decides it is not the size of an error but how much of it is colour.

applied · Delivery Where a sharpened E errs, per pixel and as seen: black type on paper, 10 point. The letter E at 10 point on a 300-dot-an-inch page, black type on paper, sharpened by an unsharp mask on stored values and on light; the two maps show the colour difference between the results, pixel by pixel on the left and after the eye's three spatial channels at 40 centimetres on the right, on one scale. The thin line is the letter's outline and the dashed circles mark two pixels round each corner — 8 outer corners and stroke ends, 4 inside angles. The circled zones are 31% of the outline and hold 28% of the per-pixel error and 25% of the seen error.

Sharpened type errs on its dark side

A sharpened square hides its halo along its edges and shows it at its corners, so a sharpened page of type was predicted to show its error at the corners of its letters, several times out of proportion to their length. Black type does the opposite: its outer corners and stroke ends hold between two thirds and nine tenths of their share. As the eye leaves it, the error lies on the dark side of every edge, and a corner holds as much of it as it has dark ground around it — a quarter of a disc where a black corner points out into paper, three quarters where black wraps round a pale one.

applied · Delivery Where a sharpened serif letter's error is seen, in black type and reversed, 10 point. The serifed letter E at 10 point, sharpened on stored values rather than on light, as the eye's three spatial channels leave the difference at forty centimetres, in black type on paper and in paper-coloured type on black, on one scale. Dashed circles mark the serifs' corners: their tips and their brackets against the stems. In black type the error lies along the strokes' dark side and the serifs' corners hold little of it; reversed, it gathers round the serif tips, which hold 11 per cent of the letter's seen error against 3 in black type.

A serif moves the error and adds little

When a scanned page is sharpened on its stored values, the error the eye sees lies on the dark side of every edge, so black sans-serif type does not gather it at its corners. The worry was that a serif changes that: every stroke end gains a bar and every bar an inside angle, the configuration where large letters did gather error. It does not. Black serif type's brackets hold three quarters of their share of the error and its serif tips half. Reversed, the serif tips hold two and a half times theirs. Across both, serifs lengthen the outline by a tenth and the error by a few per cent — they move it, and in black type away from the corners.

applied · Delivery Where a sharpened E with hairline serifs is seen to err, 48 point. The capital E with hairline serifs, a tenth of a stem thick, set at 48 point on a 300-dot page, sharpened on stored values rather than on light, as the eye's three spatial channels leave the difference at forty centimetres: black type on the left, reversed on the right, on one scale. Circles mark the serif tips and the junctions where the middle arm meets the stem. In black type the error follows the ink and gathers in the junctions' inside angles, and the serif tips hold 2 per cent of it; reversed, it spreads into the dark ground and the tips hold 6 per cent.

A hairline spills its error onto the paper

Sharpening a scanned page on its stored values errs on the dark side of every edge, which is why black type's serifs hold less than their share of the error at text sizes. A hairline serif, a tenth of a stem, was expected to change that at display sizes: its dark side is so thin that the halo would wrap its corners. It does not. From 12 to 48 point, black type's hairline tips hold a third to three fifths of their share, less than slab serifs' tips, and the share falls as the letter grows. What gathers at display sizes is the junction of bar and stem, with or without serifs — and where the scan lands the letter against the pixels moves a tip's share more than its size does.

applied · Delivery Three weights of one letter, placed by the stem's width in pixels. The four capitals E, H, L and T at three weights — stems six, ten and fourteen cells thick on the same spacing — set in black at 12, 18, 24, 30 and 48 point on a 300-dot page, sharpened on stored values rather than on light and read through the eye at forty centimetres, pooled over four sub-pixel positions: how much of the seen error the junctions of bar and stem hold, over their share of the outline, against the stem's width in pixels.

A junction gathers by the width of its stem

Black type sharpened on its stored values gathers its error at the junctions of bar and stem, and gathers more as the letter grows. The mechanism offered for that was the stem growing past the eye's pool, which predicts that size is only standing in for the stem's width in pixels. Drawn at three weights and five sizes, the junctions' share ranks with the stem's width at 0.97 and with the point size at 0.85, and where two weights have stems within six tenths of a pixel of each other their junctions agree to within a tenth. The width is nearly the whole story. The part it leaves over is small and one-sided: at the same stem width, the heavier letter gathers a little more in five pairs of six.

applied · Delivery What closing the counters adds, stem width by stem width. At each stem width, how much larger a share of the seen error the closed spacing's junctions and stroke ends hold than the open spacing's. The junctions gain on average 0.043 and the stroke ends 0.011; width by width the junctions' gain runs from -0.046 to 0.115.

A closed counter adds a little to a junction

Sharpened on its stored values, black type's junctions — the inside angles where a bar meets a stem — gather error by the stem's width in pixels, and at equal widths a heavier letter gathers slightly more. A heavier letter differs in two ways: it is smaller around its stem, and its counters are narrower. Holding the stem and moving the spacing separates them. At equal stem widths, closing the counters adds about four hundredths of their share to the junctions and one hundredth to the stroke ends, and twice as much to E and H, whose bars face each other across a counter, as to L and T. The prediction was the counter, and the counter it is — by an amount no larger than a sub-pixel position moves it. Drawing the letters apart found the stem census's light H with its crossbar floating.

applied · Delivery Condensing and shortening each carry half of what closing the counters added. At equal stem widths, the mean gain in the junctions' share of the seen error from the extended letter to the condensed one at one height, from the tall letter to the short one at one width, and from the open spacing to the closed one, which changed both together. The first two are +0.023 and +0.023; together +0.047, against +0.043 for both at once. Each half is within one and a half standard errors of nought.

Width and height each carry half the counter's gain

Closing a sharpened letter's spacing at a fixed stem width raised its junctions' share of the error the eye sees by about four hundredths, and the spacing moved the letter's width and height together. A type family's condensed and extended faces separate them: the counters across the letter narrow while its height and stem stay put. The prediction was that the condensed letters' junctions would carry the whole four hundredths. They carry half — 0.023 — with the stroke ends moving by four thousandths. Shortening the letter at one width carries the other half, also 0.023, and the two add to within four thousandths of what closing both did. Each half is found where its own counter lies — condensing gains most in the H, shortening in the E — and the T, which has no counter across it, gains nothing when condensed. Neither half, on its own, is larger than one and a half times the scatter a sub-pixel position produces.

applied · Delivery

Named alongside it

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

Image differenceSpatial frequencyContrast sensitivityViewing distanceLightnessLuminanceBasisChromatic adaptationThe von Kries transformColour managementGammaInterpolation

All concepts