Nine black levels sharpened the press's corners

A margin costs a press its corners kept only the colours of a coated press that lie at least a stated distance inside it, and asked how far that pushes the colours on its boundary. A colour on a flat face moves the margin. A colour at a corner moves further, because it has to get clear of two or three faces at once, and the census read each corner’s move as a measure of its sharpness. Every one of the nine end colours in the census moved further than a cube’s corner does — 1.73 times the margin — and the solid yellow moved nearly four times. A tolerance’s own unit rounds the corners restated the margin in ΔE₀₀ and found most of them rounding, and the yellow stays sharp because it leans explained why the yellow did not. That essay’s account failed at the other eight corners, and its closing section proposed that they were not cones at all.

They are not, and the ink recipes say so without measuring anything. The solid yellow is yellow ink at full strength and nothing else; the green is cyan and yellow at full strength; the red is magenta and yellow at full strength. Every ink in those three recipes sits at nought or at full, so none can move in both directions, and each colour is a vertex of the press. The other six have one ink part-way — yellow at three quarters over full cyan, say, or black at a half under full cyan and yellow — and that ink can move either way without leaving the press’s boundary. Each of those six sits on an edge, and an edge has one number that decides how far a margin pushes it: the angle between its two faces.

The press model gives that angle exactly, from the way each ink’s coverage moves the colour. The prediction was that the edges’ moves would follow it. They did not: the angles said the six edges should move between 1.06 and 1.82 times the margin, and the census had them at 1.73 to 2.30 at a margin of four units, a quarter to two thirds too far. Something about the edges was sharper than their own faces allowed.

The corners were the sampling’s, not the press’s #

The press the census measured was built from three-ink solids at nine levels of black, one every eighth of full black, and nothing filled the space between two levels. Walked at thirty-three levels instead, everything else the same, every one of the nine corners moves within seven per cent of what a perfect corner of its own tangent shape would, at margins of four and eight CIELAB units — and walked at sixty-five, the moves change by less than an eighth of the margin. At nine levels the six edges had moved 24 to 63 per cent further than their shapes allow, and the three vertices 24 to 37 per cent.

• The dihedral model was right. An edge’s move is set by the angle between its faces; the census was measuring a press that did not have those faces.
• The missing space is between black levels. Each black level is a separate solid, and the faces black helps to make were scalloped into a row of ridges.
• The sharpest corner is still the solid yellow, at 3.2 times the margin rather than 3.8, and three corners — both half-black end colours and the yellow with a quarter of cyan — are rounder than a cube’s.
• Three earlier results change in their numbers, and one of them in a sentence: not every corner of the press is sharper than a cube’s.

In each row the upper bar is the move a perfect corner of the end colour’s own shape makes, the middle bar is what the finely sampled press gives, and the lower bar is the census. The upper two agree in every row. The lower one stands out in every row, and by most in the two half-black end colours, whose edges run along black itself.

A corner’s shape, from its recipe #

The tangent shape of a corner is the set of directions its recipe can move in. At the solid yellow the press can add cyan, add magenta or add black, and it can take yellow away; each of those four moves the colour in a direction the press model’s derivatives give, and those four directions span a pointed cone with four faces. A perfect corner of that shape, shrunk inward by a margin, has its tip at the point that sits the margin’s distance inside every face. For the solid yellow that point is 3.02 margins from the tip. For the green it is 2.11 and for the red 2.30.

At an edge one of the directions has an opposite: the free ink can rise or fall, so the colour can slide along the edge both ways. Seen along the edge, the other directions fan out in the cross-section, and the outermost two bound a wedge. The six wedges open at 67 to 141 degrees: the two edges where yellow runs part-way under full cyan are the sharpest, at 67 and 70 degrees; the yellow with three quarters of cyan opens at 71 and with a quarter at 92; the two half-black edges, which run along black, open at 117 and 141. A wedge of dihedral angle θ moves its edge 1/sin(θ/2) times the margin, so the prediction runs from 1.82 at the sharpest edge to 1.06 at the flattest — nearly a face.

These are statements about the press model, not about any sampling of it. They were the numbers the census should have met. A margin costs a press its corners had read its moves as cones of one angle each, which is what a move can be turned into without a shape to hand; its own section on where the method stops warned that a real corner “is a ridge or a wedge or a tip of something irregular”. The recipes say which, for every one of the nine.

Thirty-three levels settle it #

The press was rebuilt four times with nothing changed but the number of black levels. Its cells stay one CIELAB unit on a side, its chromatic inks are still walked at thirteen steps each and filled between them, and its ink limit is still 320 per cent. At nine levels every end colour overshoots its shape by a fifth or more; at seventeen about half of it has gone; at thirty-three all nine sit within seven per cent; at sixty-five they move by less than an eighth of the margin more. The worst remaining gap at thirty-three levels is the half-black green, six per cent beyond its wedge at four units and five at eight, and it is still settling at sixty-five, where it moves 1.15 times the margin against its wedge’s 1.17.

The chromatic sampling does not matter in the same way, and it was checked rather than assumed. Walking the three chromatic inks at twenty-five coverages instead of thirteen, at thirty-three levels of black, changes no end colour’s move by more than 0.05 of the margin at eight units, and none by more than 0.13 at four — the half-black yellow, from 1.07 to 1.20, which is one cell’s worth of a four-unit move and the kind of wobble the grid gives at any sampling. Going from thirty-three black levels to nine moves the green by 0.76 at four units and the half-black yellow by 2.77 at one. The chromatic lattice was filled between its nodes — a cell of three inks’ coverages is a solid, and the estimator fills every CIELAB cell inside it — and black was not.

Why the black levels made ridges #

The fourth ink is not for colour is about what black adds to a press: depth at the bottom and a neutral that holds its hue, and at every lightness above about forty, no extra chroma at all. The press’s volume estimator takes that seriously in the way it builds a four-ink press. A four-ink press’s colours are the union of three-ink presses, one at each amount of black. That is exactly true when black runs continuously. Sampled at nine amounts, the union is nine overlapping solids, each a slightly darker and duller copy of the one before, and wherever the press’s boundary is made by black changing, the boundary is a row of the nine solids’ edges with notches between them.

The two half-black end colours sit on exactly such a boundary. They are yellow and cyan at full strength with black at a half, or yellow alone with black at a half, and the edge they lie on is the line along which black runs from a quarter to three quarters. At nine levels, black at a half is one of the sampled levels: the end colour sits at the tip of one solid’s ridge, and the neighbouring solids at three eighths and five eighths bulge past it on either side but leave a notch around it.

A notch is a narrow wedge. The cross-section through the half-black yellow on the coarse press subtends 61 degrees three units out and does not reach its faces’ 141 degrees until eight; on the finely sampled press it subtends 147 degrees at three units and 148 at eight. A margin of a unit or two sees only the notch, and pushes the end colour in as though it sat at the tip of a 60-degree spike. That is why the half-black yellow moved 4.1 times the margin at one unit on the census’s press and 1.73 at four: the move was falling towards the edge’s true value as the margin grew past the notch’s depth, which a tolerance’s own unit rounds the corners read as the dark corners rounding with scale. On the finely sampled press it moves 1.33, 1.35, 1.07 and 1.08 times the margin at one, two, four and eight units — a nearly flat face at every scale.

The half-black green shows the same notch less deeply: 58 degrees at three units on the coarse press against 146 on the fine, and both near the wedge’s 117 by twelve units. The light edges are not along black, but black is still one of the directions their recipes can move in — every light end colour prints with no black, and adding a little is one of the ways off its edge — so their cross-sections were notched on one side, and they overshoot by a quarter to a third on the coarse press for the same reason. The three vertices have black as one of their four faces, and their tips were narrowed by the same row of notches: the solid yellow’s cross-section is sharpest near its tip, which is where a small margin reads it.

What a margin does to one corner as it grows #

The solid yellow is the corner the three earlier essays were most interested in, and the figure is its whole history on one axis. On the census’s press it moves 4.6 times the margin at one unit, 3.9 at two, 3.8 at four and 3.2 at eight. At seventeen levels the small margins are unchanged — the notch at the tip is still there — and the larger ones have fallen. At thirty-three and sixty-five levels the curves lie together, at 3.4 at two units, 3.2 at four and 3.0 at eight, and the tangent cone says 3.02.

The small margins do not settle as cleanly as the large ones, and the reason is the grid rather than the black. A one-unit margin on a one-unit grid is a single layer of cells; which cell centre counts as “inside by one” depends on where the tip falls within its own cell, and the cell-centre method the census used moves a tip by up to half a cell in any direction whatever the margin. At four and eight units that half-cell is an eighth or a sixteenth of the margin and the move per unit is stable; at one it is half, and the numbers are the grid’s. Every claim here is stated at four and eight. A grid is not a resolution made the same separation for wavelength sampling — a step is several decisions, and which one matters depends on the quantity asked of it — and the press’s one-unit cells are a grid in the same sense: fine enough for a volume, coarse for a one-unit margin.

In the tolerance’s own unit #

A print buyer states a margin in ΔE₀₀, and the census was repeated on the finely sampled press in that unit: every printable cell’s depth measured as the smallest ΔE₀₀ to the press’s boundary, the press eroded by one, two and four ΔE₀₀, and each end colour moved to the nearest cell that survives, the move measured in the same unit. No end colour moves further on the fine press than on the coarse one, and eight of the nine now move between 1.02 and 1.25 times the margin at every one of the three margins. On the census’s press, six of nine had been in 1.06 to 1.34.

The solid yellow does not change: 1.91, 1.95 and 1.91 times the margin at one, two and four ΔE₀₀, against 1.91, 2.04 and 2.10. It is still the only corner sharper than a cube’s, and the sentence in the earlier essay that said so stands. Its CIELAB tongue, re-measured from the fine press, leans 26.5 degrees towards lightness rather than 23, and the account the yellow stays sharp because it leans gave for it — a cone about that lean, rescaled by ΔE₀₀’s own lengths at the tip — falls ten degrees short of the margin’s opening at two ΔE₀₀ and eight at four, where it had fallen ten and six short.

The half-black yellow is the corner whose story changes. The earlier essay found it rounding with scale, from 1.86 times a one-ΔE₀₀ margin to 1.32 at four, and read that as the dark corners rounding too. On the fine press it moves 1.06, 1.03 and 1.02 times the margin: there was no dark corner to round. The rounding was the notch being outgrown.

What a ΔE₀₀ margin costs in volume barely moves. Two ΔE₀₀ keep 69.5 per cent of the fine press’s cells against 67.9 of the coarse press’s, and two CIELAB units 82.3 against 80.4. The press itself is 3.3 per cent larger at thirty-three levels than at nine — 454,366 one-unit cells against 439,844 — which is the volume the notches had taken, and at sixty-five it is 3.7 per cent larger.

What the three earlier essays now say #

Each of the three carries a note pointing here, and none of their arguments falls.

A margin costs a press its corners found that the press has no pinch, that projected routes on it neither trap nor tear, and that the cost of a margin lies at the corners. All three stand. Its corner numbers were the coarse press’s: the solid yellow’s 3.7 times the margin is 3.2 on the fine press, the red and green’s three times is 2.4 and 2.2, and “every end colour moves further than a cube’s corner” is true of six of the nine. Its section on the grid had said that a finer grid would sharpen the corners a little, since a coarse grid rounds a spike’s tip. A finer black sampling does the opposite, and by more.

A tolerance’s own unit rounds the corners found that a ΔE₀₀ margin rounds the corners and leaves the solid yellow sharp. Both stand, and the rounding is more complete than it reported. Its finding that the dark corners round too, against its own prediction, was the black sampling.

The yellow stays sharp because it leans found the yellow’s tongue leaning 23 degrees and a cone about that lean accounting for the yellow’s opening within ten degrees. On the fine press the lean is 26.5 degrees and the account is ten and eight degrees short at two and four ΔE₀₀. Its other finding — that the account fails at the other eight corners — also stands, and this essay gives the reason: six of them are wedges, and a cone rescaled by ΔE₀₀ is the wrong shape to rescale.

The routes of a press makes the cheap route a search and a projection has no reason to detour ran on the same coarse press. Their findings are about which starting paths get trapped and whether projection removes the trap; the notches add small concavities between black levels that a penalised path could catch on, and whether any trap there belonged to a notch rather than to the press has not been checked.

How the census was repeated #

The press is the site’s coated CMYK press at a 320 per cent total-ink limit, built by the volume estimator the census used: at each sampled black level the three chromatic inks are walked on a thirteen-step lattice, every lattice cell’s eight corners are printed and taken to CIELAB under D50, and every one-unit CIELAB cell inside the resulting solid is marked printable; the four-ink press is the union over black levels. Black was walked at 9, 17, 33 and 65 levels from nought to full. Each cell’s depth is its Euclidean distance to the nearest unprintable cell, less half a cell, from an exact distance transform; each end colour’s move at a margin is its distance to the nearest printable cell centre at least that deep. At nine levels these are the published census’s moves to the last digit, which is checked.

Each end colour’s recipe comes from the ring of two-ink overprints the gradient census drew its ends from. The directions its recipe can move in are central and one-sided differences of the press model at a hundredth of coverage. An edge’s dihedral angle is the angle between the outermost two of those directions projected across the edge; a vertex’s cone is found by taking every pair of directions whose plane has all the others on one side as a face, and its eroded tip by solving for the nearest point at unit depth over every set of up to three active faces. The cross-section’s opening is the share of a circle about the end colour, perpendicular to the edge, whose points fall in printable cells, sampled at 1,440 points. The ΔE₀₀ depths are the earlier census’s exact local search, repeated on the fine press.

What this leaves out #

The tangent shape is the press model’s, at the end colour. A corner whose faces curve would move differently from its tangent shape at large margins; at four and eight units these do not, but at sixteen or thirty they might, and nothing here says.

Thirty-three levels is converged at four and eight units, not at one. The half-black green is still settling at sixty-five. The one-unit numbers are the grid’s, on any sampling.

Other results rest on the same press. Every volume quoted for the press in these essays — its size against sRGB’s in neither gamut contains the other, the share black adds, the share a fifth ink buys — was estimated at nine black levels. The press is 3.3 per cent larger at thirty-three, and a share that compares two presses built the same way is likely to move less than that, but none of those has been recomputed.

The ring is one ring. Nine end colours from the census’s ten crossing gradients. A corner elsewhere on the press — a cyan-magenta blue, a three-ink dark — has not been read.

Still open: how ΔE₀₀ rounds a wedge #

The yellow’s account treated a corner as a circular cone and rescaled it by ΔE₀₀’s three lengths at the tip. Six of the corners are wedges, and a wedge rescaled by an anisotropic metric is still a wedge, with a new dihedral angle that depends on how the metric stretches the plane across the edge. The calculation is each edge’s two faces carried into ΔE₀₀’s local coordinates at the end colour, the new dihedral angle read from them, and the predicted move set against the fine press’s ΔE₀₀ census. The prediction is that it accounts for all six edges within a tenth of the margin, because the fine press’s CIELAB moves meet their wedges within seven per cent and ΔE₀₀ is smooth over a few units. If it does, the margin census is explained end to end — vertices by cones, edges by wedges, and each in either unit by the press model’s own derivatives. If the two half-black edges miss, the reason is likely to be ΔE₀₀’s lightness weight changing across a few units at L* 26 and 49, where the metric is least uniform.

A measurement can be of the instrument #

The habit is about which part of a calculation a surprising number belongs to.

The census’s corner numbers were surprising in a direction that looked like a finding: sharper than a cube, sharper than any smooth solid, and in the edges sharper than their own faces allowed. The explanations offered over three essays were all about the press — its tongues, its lean, the way ink coverage trades near the limit. None was about the way the press had been built, because the volume estimator had been checked against published volumes and had passed. It passed because a volume is an integral, and a row of shallow notches along a few edges changes an integral by three per cent. A corner is a local quantity, and the same notches changed it by half.

The test is to build the thing being measured a second way and see which numbers move. The room census met the same test when the tall room gains once its walls are divided re-solved it on finer faces and found a published sign reversed. Here the chromatic lattice could be nearly doubled and the corners stayed within a cell’s wobble; black could not be refined at all without them moving by half.