A hairline spills its error onto the paper

A serif moves the error and adds little asked what a serif does to the error of sharpening scanned type on its stored values rather than on light. It drew four capitals with serifs a third of a stem thick and found that, in black type at text sizes, the serifs’ tips hold about half their share of the seen error and their brackets three quarters. The reason was the rule that sharpened type errs on its dark side: the difference the eye keeps lies on the darker side of each edge, and black type’s dark side is a stroke too narrow to wrap a corner.

That essay closed on a case the rule might not survive. A hairline serif, a tenth of a stem thick, has almost no dark side at all. Display faces use hairlines — a didone’s serifs are barely there — and they are set from 18 to 48 point, where a letter is large enough for the eye to see a corner as a corner. The prediction was that the sharpening halo would spill round a hairline’s tip onto the paper, as it spills round reversed type’s corners into the ground, and that hairline tips in black type would begin to gather error above about 24 point and gather more as the size grew.

The tips never gather; the junctions do #

Pooled over four sub-pixel positions, black type’s hairline serif tips hold 0.58 of their share of the seen error at 12 point and 0.35 to 0.43 of it from 18 to 48 point — less than slab serif tips at every display size, and not rising with size. The corners that gather in black display type are the junctions of bar and stem, which climb from ×1.05 of their share at 12 point to ×1.64 at 48 without serifs and within a tenth of that with either kind. Reversed, hairline tips hold 2.2 to 3.6 times their share.

• The prediction fails. Hairline tips in black type hold less than their share at every size, and less than slab tips from 18 point up.
• The error that would gather at a hairline spills onto the paper, where the eye’s lightness scale is shallow and counts little of it.
• Black display type gathers at its junctions, the inside angles whose dark side is a whole stem — and serifs, of either weight, change that by under a tenth.
• Where the letter lands against the pixels matters more than its size: four positions a quarter of a pixel apart move a hairline tip’s share by up to a factor of 3.7.

Three sets of letters on one grid #

The earlier census drew its capitals — E, H, L and T — on a grid of seventeen by twenty-one cells, with three cells to a stem and serifs one cell thick. A hairline a tenth of a stem cannot be drawn on that grid. So the four letters are drawn here with ten cells to a stem, on a grid fifty-six cells wide and seventy tall, in three versions: without serifs; with slab serifs three cells thick, the earlier essay’s third of a stem; and with hairline serifs one cell thick. Both kinds of serif reach three cells past the stroke on each side, horizontal at a stem’s free end and vertical at a bar’s.

The finer grid is checked against the coarser one before anything is read from it. Drawn with ten cells to a stem, the slab-serifed letters at twelve point give black type’s serif tips the same share, ×0.46, that the three-cell letters gave, and a corner concentration within seven hundredths of theirs. The difference is the reach: three cells of ten is a little less than one of three.

Each letter is set at 12, 18, 24, 30 and 48 point on a 300-dot page — cap heights from 35 to 140 pixels — sharpened by the same unsharp mask on stored values and on light, and the difference filtered by the collection’s model of the eye at forty centimetres, the distance the eye counts a corner’s error, not its peak used for print. Every corner of the outline is labelled by what made it: a stroke’s own end, a junction of bar and stem, a serif’s tip or a serif’s bracket. Each part’s share of the seen error is then set against its share of the outline near corners, and a ratio of one is a fair share.

What the halo does round a hairline #

In black type the error lies along the ink. The stems carry it on their inner side, the three arms carry it along their edges, and the two inside angles where the middle arm meets the stem are the brightest places on the letter. The hairline serifs hanging from the arms’ ends are thin lines of ink with thin lines of error beside them, and their tips hold very little. Reversed, the picture turns inside out: the error lies in the dark ground, it wraps every outer corner, and the hairline tips — outer corners with ground on three sides — are among the places it gathers most.

What the proposal expected was a third picture: black type whose hairline tips looked like reversed type’s, because a hairline’s dark side is too thin to hold the halo. Part of that is right. The halo does spill round a hairline onto the paper, because a line one or two pixels wide cannot contain an unsharp mask of radius 1.5 pixels. What the proposal missed is where on the lightness scale the spilled error lands. The straight piece under the cube root priced lightness near black: a difference in light costs far more against ink than against paper, because the lightness scale is steep in the dark and shallow in the light. The error a hairline spills goes to the paper side, and there a given difference in light is a small difference in lightness. Reversed type’s spill goes into a dark ground, which is why its corners gather and black type’s do not.

So a hairline does not turn black type into reversed type. It removes the little dark side a serif had, and with it most of the error the serif’s tip could have held.

Where the scan lands the letter #

Before the size can be read, the pixel grid has to be averaged out. Placed with its outline at one fixed offset from the pixel grid — the way the earlier essays placed their letters — black type’s hairline tips hold 0.49 of their share at 12 point, 0.28 at 18, 0.88 at 24, 0.32 at 30 and 0.69 at 48: a zigzag no mechanism predicts. The zigzag is the grid. At 24 point a cell is exactly one pixel, so every edge of an unshifted letter falls on a pixel boundary and is rendered as a hard step; at 18 point a cell is three quarters of a pixel and most edges fall inside pixels as partial coverage. A hard step and a partly covered pixel sharpen differently, and a hairline one pixel wide is all edge.

Shifting the same letters by a quarter, a half and three quarters of a pixel along both axes shows how large the effect is. At 24 point the four positions put the hairline tips at ×0.88, ×0.28, ×0.24 and ×0.28 — a factor of 3.7 from one position to the next — and at 12 point one position gives ×1.33, the only value in the whole census where black type’s tips hold more than their share, while the other three give under half. At 30 point, where a cell is a pixel and a quarter, the four positions differ by a factor of only 1.3; at the other four sizes, by 1.8 or more.

No scan chooses its position. A page lands on the sensor wherever it lands, and a letter repeated along a line of text lands at every offset in turn. So every number in the rest of this essay is pooled over the four positions, and pooled the tips’ share varies across sizes by a factor of only 1.65 — less than half of what one position does at one size. The earlier essays’ letters sat at one offset. At text sizes that did not mislead them: at 12 point this grid’s slab tips, pooled over the four positions, hold ×0.57 of their share against the three-cell letters’ ×0.46 at one position, and both are well under one.

Hairline tips against slab tips #

In black type both kinds of tip stay below their share at every size, and the hairlines stay lower: 0.58 against 0.57 at 12 point, then 0.40 against 0.51 at 18, 0.35 against 0.47 at 24, 0.35 against 0.49 at 30 and 0.43 against 0.51 at 48. Neither curve rises through the display range. The small upturn at 48 point is the tip’s own geometry: at 140 pixels of cap height a hairline is two pixels thick and begins to have a dark side of its own, but it is still well under half its share.

Reversed, both kinds of tip gather, and the hairlines more. At 12 point hairline tips hold 3.6 times their share against slab tips’ 2.4; the two converge as the size grows, to 2.2 against 1.8 at 30 point and to within two hundredths of each other at 48. A hairline tip reversed out of black is a thin light line ending in a dark ground, and the ground wraps it on three sides at every size. Its share falls with size for the reason the eye counts a corner’s error, not its peak gave for all corners: the eye pools over a fixed patch of the page, and on a larger letter that patch covers less of the corner and more of the edge.

The two polarities differ by a factor of five to eight at the hairline tip, and that is the practical point. White hairline type on a dark ground is where a display face shows sharpening on stored values most, and it shows at the serif tips. In black type the same serifs are nearly invisible to the error.

Which corners hold it at display sizes #

At 30 point in black type only the junctions hold more than their share, ×1.54. The stroke ends hold ×0.61, the hairline brackets ×0.42 and the hairline tips ×0.35. The brackets are inside angles, like the junctions, but their dark side is the hairline — one pixel and a quarter at this size — and there is not enough ink round them for error to wrap into. A junction’s dark side is a full stem, twelve and a half pixels at 30 point, and the inside angle holds three quarters of a disc of it.

Reversed, every relation inverts. Stroke ends hold ×2.31 and hairline tips ×2.20, while the junctions give up almost all of theirs, ×0.36, and the brackets ×0.49. In reversed type the dark side is the ground, which wraps every outer corner and never fills an inside angle; the letter’s own corners, which are all it has to give, give accordingly.

The pattern is the one the earlier essays found at text sizes, and the hairline does not disturb it. What changes with size is how much each kind of corner holds, and that is the next figure.

The junctions climb with the letter #

The junctions’ share rises at every step, in all three sets of letters: from ×1.05 at 12 point to ×1.27, ×1.36, ×1.44 and ×1.64 at 48 without serifs, and from ×1.13 to ×1.73 with hairlines. The stroke ends move the other way, from ×0.76 to ×0.52 without serifs. Adding serifs of either kind moves the junctions by under a tenth at every size, and moves the stroke ends by about the same.

The mechanism is the one sharpened type errs on its dark side found between its 16- and 24-point letters, now followed further. An inside angle gathers error in proportion to how much dark lies inside the eye’s pool around it. At text sizes a stem is a few pixels wide and the pool reaches past it into paper; the angle holds little more than an edge. As the letter grows the stem grows past the pool, the angle fills with ink on three sides, and its share climbs. At display sizes black type does have corners that gather, and they are its junctions, not its serifs.

This is also why the prediction’s reasoning, though wrong about the hairline, was right about the size. It expected the corners of black type to begin collecting error once the letter was large enough; they do, but only where the dark side is wide enough to fill a corner, and a hairline never is.

What hairlines cost #

Both kinds of serif lengthen the four letters’ outline by nine per cent. The hairlines raise the total seen error by 0.7 to 3.8 per cent across sizes and polarities, the slabs by 1.6 to 6.4. A hairline is almost all edge, so it adds edge in full; but its edge is the thinnest dark side a letter has, and sharpening’s error on it is correspondingly small. An average on the stored values and a resize with a negative weight in it found that this kind of error is made where a mask’s negative weight reaches across an edge; a hairline gives the negative weight almost nothing dark to reach into.

The earlier essay’s conclusion — serifs move the error more than they add to it — holds with hairlines and holds more strongly. What they move, in black type, is error out of the serif and back along the stems.

What this means for sharpening display type #

A sharpening test for black display type should look at the junctions. A capital E, H or F at 36 point or more, sharpened on stored values, shows its error in the inside angles where bars meet stems, at one and a half to one and three quarters of their share, and not at its serifs whatever their weight. A test pattern of unserifed capitals at the display size measures what a didone at that size will show.

A test for reversed display type needs hairlines. White hairline serifs on a dark ground hold two to three and a half times their share of the seen error, more than slab serifs do up to 30 point, and they are where a reversed headline in a didone will show a sharpening made on stored values.

A test at either polarity has to average over sub-pixel position, or a single rendering of the pattern will report a hairline tip anywhere in a factor of three of its average. A line of repeated letters does this for free; a single letter centred on a pixel does not.

And the cure is the one every essay on this error has given. Sharpening in light rather than on stored values removes the difference; a halftone is a luminance object is why the lightness error is the one that shows, and why the remedy is in how light is handled rather than in the letterform.

How the letters were set #

The letters are unions of cells on a fifty-six-by-seventy grid with ten cells to a stem and a three-cell margin each side for serifs. Hairline serifs are one cell thick and slab serifs three, and both reach three cells beyond the stroke. Each letter is rendered at a cap height of 0.7 of the point size on a 300-dot page by exact area coverage, at offsets of 0, 0.25, 0.5 and 0.75 of a pixel along both axes, mixed in light, encoded as sRGB and sharpened by an unsharp mask of radius 1.5 pixels and amount one, on the stored values and on light. The colour difference between the two is filtered by the collection’s three spatial channels at forty centimetres and read as CIEDE2000 per pixel. Letters up to 30 point sit in a 128-pixel field and the 48-point letters in a 256-pixel one, filtered as periodic: its border is plain paper for more than fifty pixels, and against a mirrored border the periodic one moves a part’s share by under one per cent at two thirds of the cost. Each pixel’s error goes to the nearest point of the outline; each part’s share of the error within two pixels of its corners is divided by those corners’ share of the outline’s length, and the four letters and four offsets are pooled by adding lengths and errors before dividing.

What this leaves out #

The serifs are unbracketed. A real didone’s hairline meets its stem through a curved fillet, which removes the sharp inside angle a bracket here has. The brackets already hold under half their share in both polarities, and a fillet can only make that smaller.

The letters are rectilinear. Rounds — the bowls of a didone’s O and C, with their thick-and-thin stress — carry the hairlines of a display face at least as much as its serifs do, and a hairline curve has no corner at all; its error is an edge’s. What this census says about hairline thickness should apply to them, and what it says about corners should not.

The eye is the collection’s model of it, contrast-sensitivity functions measured at threshold and applied to errors above it, with the caveat every threshold was measured with a grating attaches. The finding rests on two properties of that model — pooling over a few pixels at print distance and a lightness scale steep in the dark — and a different curve would move the ratios without changing which side of a hairline is dark.

And the page is printed and read at forty centimetres. A display face is also used on posters read from metres away and on screens; at those distances the pool covers more of the letter and the corner’s share approaches the edge’s.

Still open: whether a stem’s width in pixels alone sets a junction’s share #

The junctions climb because their dark side, the stem, grows past the eye’s pool as the letter grows. If that is the whole mechanism, then size is standing in for something simpler: the stem’s width in pixels at the viewing distance. A bold face at 18 point and a light face at 36 might have stems of the same width, and should then have junctions holding the same share.

The calculation is this census with the stem’s weight varied independently of the size — letters with six, ten and fourteen cells to a stem on grids scaled to keep the letter’s proportions — and each junction’s share plotted against its stem’s width in pixels rather than against the point size. The prediction is that the three weights fall on one curve, rising from about one at a stem of three pixels to about ×1.7 at twenty, and that a bold face at text size therefore gathers at its junctions as a regular face does at display size. If they do not fall on one curve, the letter’s counters — the paper enclosed by its strokes — play a part the stem alone does not, and the test pattern needs a weight as well as a size.

A trend read at one sampling is a trend in the sampling #

The habit here is about averaging over what the measurement did not choose before reading what it did.

The first census of hairline tips found them gathering at 24 point and not at 18 or 36, which would have supported half the prediction and suggested a curious resonance with size. It was the pixel grid: at 24 point the unshifted letters’ edges fell on pixel boundaries. Four positions a quarter of a pixel apart moved one number by a factor of 3.7, and pooled over them the resonance disappeared and the tips sat below their share at every size.

The failure mode is to vary one parameter across a range and read the curve, when a second parameter nobody varied changes along with it. Point size on a fixed grid changes the pixel phase of every edge; the phase was the signal and the size was the label on it. A letter on a real page takes every phase in turn, and a measurement meant to say what that page will show has to take them too.