A tint at the edge of a page

Two essays on this site take a colour difference and make it smaller. Spreading it as a pattern costs most of it, because the eye’s own filter removes fine detail. Moving it into the periphery costs most of it again, because thresholds rise away from the centre of gaze.

The previous phase named the obvious next question and did not answer it: what happens when both act at once, which is the ordinary situation of a printed page. Its guess, written down in its closing note, was that the two would multiply.

They do not multiply. At five degrees off axis the product prediction is fifty-three times too generous, and by twenty degrees it is out by a factor of a hundred and forty-five million.

That is a large enough error to be worth being precise about what was guessed and what is measured. The guess was not careless: two independent reductions of the same quantity, each measured on its own, ordinarily do compose by multiplication, and both of the earlier essays were careful. What went wrong is that the two reductions are not independent and are not even two — they are one anatomical fact read on two different axes, and one of those axes is inside the function the other one scales.

The claim #

Eccentricity does two things to a fine pattern and only one of them is a threshold.

• It raises the threshold, by the E2 rule, which is the effect the peripheral essay measured: a difference presented ten degrees out is worth about a fifth of what it is worth at fixation.
• And it lowers the cutoff frequency, because the whole spatial scale is magnified. A screen sits near the top of the frequency range, where the filter is falling steeply, so a modest shift along the frequency axis removes far more of the screen than the threshold elevation does.
• The second dominates completely. The tint is 8.1 times threshold where the reader is looking, is still visible at two degrees, and is gone by five — while the product prediction has it visible across the whole page.

Why the two effects are not independent #

The threshold elevation and the frequency shift are not two mechanisms; they are two consequences of one.

The periphery’s receptive fields are larger. That is the whole of it. A larger field means less sensitivity to any given contrast, which is the threshold elevation, and it means a lower spatial cutoff, which is the frequency shift. The E2 rule — that the scale factor is 1 + e/E2 for a stated constant per channel — is normally quoted as a statement about thresholds, and it is equally a statement about frequencies, because the two are the same magnification read on two axes.

Applying only the threshold half is therefore not a conservative approximation. It is a model in which the periphery has larger receptive fields for the purpose of losing sensitivity and normal-sized ones for the purpose of resolving detail, which is not a model of anything.

And the reason it produces such an enormous error is that a halftone is at the wrong end of the curve. A contrast sensitivity function falls steeply past its peak. A pattern near the peak, at three or four cycles per degree, barely cares: a scale factor of three moves it into a region where the curve is flat. A pattern at twenty-six cycles per degree, already halfway down the high-frequency slope, is moved by the same factor of three to seventy-eight, which is past the cutoff altogether.

What was computed, and how #

The substitution that makes it exact. A screen of ruling r, seen through a filter whose cutoff has been divided by s, is the same measurement as a screen of ruling r·s seen at the fovea. So the peripheral filter is not a second model with its own landmarks: it is the existing filter called with a different argument, which is the only way to keep the two halves of the join from drifting apart.

The reading is harmonic rather than rasterised, and that is not a detail. screenVisibility builds the screen as a field of samples and takes its transform, which means it refuses a ruling at or past the grid’s Nyquist frequency — correctly, since rasterising such a screen draws its alias. A real screen scaled by an eccentricity factor of nine is past the grid immediately. screenLoudness takes the harmonics of the lattice analytically and rotates the harmonic vectors, so it has no grid to alias against and no upper limit on the ruling.

Both readings agree exactly at the fovea, where neither effect acts, and assertTheTwoReductionsDoNotMultiply requires that as well as the divergence — a version that disagreed at zero degrees would be reporting a bug rather than a finding.

The threshold half contributes nothing to the shortfall #

The claim above says the frequency shift dominates. It does more than dominate: the threshold elevation cancels out of the shortfall exactly, and contributes nothing at all to it.

The arithmetic is one line. The product prediction is the foveal loudness divided by the scale, $8.10/s$. The correct answer is the foveal loudness at the magnified frequency, also divided by the scale, $L(26s)/s$ — because the substitution that makes the peripheral filter exact leaves the threshold division in place. So the ratio between the two is

$$\frac{8.10/s}{L(26s)/s} \;=\; \frac{L(26)}{L(26s)}$$

and the scale factor is gone. Reconstructing the shortfall column that way reproduces every entry to four figures: 4.412, 52.94, 5,226 and 1.5 × 10⁸.

So the whole of the discrepancy is a property of the sensitivity function’s slope, evaluated at two frequencies, and none of it is a property of the E2 rule’s threshold half. That half is in both predictions, identically, and divides out.

That is a cleaner statement of the finding than two effects that do not multiply, because there are not two effects being combined. There is one prediction that moved the frequency and one that did not, and the difference between them is the curve. The guess was not wrong about how two factors compose. It was wrong about there being a second factor at all — it applied the magnification to the threshold and forgot to apply it to the argument, which is a single omission rather than a composition error.

The consequence is that the shortfall can be read straight off the sensitivity curve without any peripheral machinery. Taking the log-log slope of the foveal loudness between the frequencies the magnification visits gives −2.5, −4.9, −9.0 and −17.5, steepening as roughly the 1.1 power of frequency — which is what differentiating an exponential cutoff of the form $e^{-(f/w)^{1.2}}$ produces, and is the arithmetic behind the essay’s own observation that a screen is at the wrong end of the curve.

And it makes the result portable. Since the shortfall is $L(f)/L(fs)$, it depends on where the pattern sits on the curve and on the magnification, and on nothing else — not on the channel’s E2 constant except through $s$, not on the coverage, not on the threshold. The essay’s second worked example, a twelve cycle-per-degree screen at half coverage, has a smaller shortfall for exactly that reason and would have the same one at any coverage.

Which also sharpens the general rule at the end. That rule says to ask what each correction is a correction to before multiplying. The sharper version this identity supplies is: a magnification acts on an argument, and applying it anywhere else is not a partial correction — it is a different model. The product prediction is not the right answer scaled down; it is the answer for an observer whose receptive fields grew for the purpose of losing sensitivity and stayed the same size for the purpose of resolving detail, which is the sentence the essay uses and which the cancellation makes exact rather than rhetorical.

What it means for a printed page #

A page held at ordinary reading distance is about thirty degrees wide. The fovea covers under two.

So a halftone screen is visible on the word being read and on nothing else. Not faintly visible; not visible if one looks for it. At five degrees — which is two or three centimetres away on the same line — the model says the screen is at a twentieth of threshold.

Three consequences, and each of them is something the trade already knows and explains differently.

A tint looks smoother than it measures. A densitometer reading a flat tint reports the same dot structure everywhere on the sheet. A person reports smooth paper with a texture wherever they happen to be looking, and the texture follows their gaze, which is exactly the report and exactly what nobody believes when they hear it.

Screen quality matters at the point of inspection and nowhere else. A press operator with a loupe is at the fovea by construction, and everything they can see about the screen is invisible to the reader at the same distance. That is not a criticism of the loupe; it is why quality control needs a criterion rather than an opinion.

And a moiré is a different object from a screen. A beat between two screens is at a low spatial frequency by construction — that is what a beat is — so it does not fall off the peripheral cutoff the way its parent screens do. A page whose screens are individually invisible everywhere except at fixation can carry a moiré that is visible across the whole sheet, which is why moiré is the failure printers fear and dot visibility is not.

A heavy tint at a shallow screen angle is the case a printer would call safe, and the three channels disagree about it.

Where the model stops #

The E2 constants are quoted and they differ per channel. The achromatic constant is 2.5 degrees, red–green 1.2 and blue–yellow 4.0, so the three channels shrink at three different rates — which is why a colour difference in the periphery turns as well as fading, and which this essay’s single-channel computation does not exercise. A chromatic screen would fall off faster than the luminance one measured here.

There is no eccentricity-dependent shape change, only a scale. Real peripheral sensitivity is not a scaled copy of foveal sensitivity — the low-frequency end behaves differently, and the periphery is relatively better at motion and worse at pattern in ways a single magnification factor cannot express.

The coverage and ruling are one choice. A screen at twenty-six cycles per degree and thirty-five per cent coverage is an ordinary book’s screen at an ordinary reading distance, and the numbers scale with both: a coarser screen survives further out and a finer one disappears sooner, monotonically, with no interesting structure in between. What does not change with the choice is the shape of the divergence between the two readings, because that comes from the slope of the sensitivity function rather than from where on it any particular screen sits.

And nothing here has an eye that moves. A reader’s gaze crosses the page several times a second, so every part of the sheet is at the fovea within a second or two of being looked at. What the model describes is an instant, and the experience of a printed page is an integral over a scanpath that this site has no machinery for. That is the same gap the drift essay leaves from the other direction: eye movements are in this site as a velocity and never as a trajectory.

The generalisation #

The sentence worth carrying is: two effects that each scale a quantity do not compose by multiplication unless they act on the same axis.

That is a general statement and it is the third time this site has been caught by it. The dither measurement changed by a factor of twenty when the detector changed rather than the filter. The plane model found the second spatial dimension worth ten to nineteen times what the previous phase’s guess allowed. And here a product of two factors is out by eight orders of magnitude, because one of the factors was acting on the frequency axis and the other on the amplitude axis and multiplying them treated both as amplitudes.

The rule that would have caught all three is the same: before multiplying two corrections, ask what each of them is a correction to. A threshold elevation multiplies a response. A magnification moves a frequency. A detector change replaces the reading. Three different operations, and only the first composes the way arithmetic expects.

The surprising connection is with viewing distance. Doubling the viewing distance also halves the angular size of everything, which is a frequency-axis effect of exactly the same kind — so the finding here says that moving a page from thirty centimetres to sixty removes far more of a screen’s visibility than the corresponding change in angular subtense would suggest. The trade’s rule that a poster can be screened coarser than a book is usually justified by size; the arithmetic says it is justified by the slope of the sensitivity function, and that the margin is much larger than anybody claims.

Who found it, and when #

Cortical magnification was described by Daniel and Whitteridge in 1961: equal areas of visual cortex correspond to unequal areas of visual field, with the fovea enormously over-represented. Everything in this essay follows from that one anatomical fact, and the two consequences the essay is about — sensitivity and resolution — were understood as one consequence from the start.

The E2 formulation is Levi and Klein’s, from the 1980s: a linear scaling of the relevant spatial constant with eccentricity, parameterised by the eccentricity at which the constant doubles. It is quoted here per channel because the three channels’ constants have been measured separately and differ by more than a factor of three.

Halftone visibility has its own literature in the printing trade, largely empirical and largely about the loupe: rulings, angles and dot shapes chosen by looking closely at proofs. The two bodies of work do not cite each other, which is unsurprising — one is about the visual periphery and the other is about a printed sheet, and nobody had a reason to put them in the same calculation until somebody asked what a page looks like rather than what a dot looks like.

And the previous phase’s guess is the reason this essay exists. It said, in writing, that the two reductions should multiply, and it was wrong by eight orders of magnitude at the edge of a page. Recording that is more useful than quietly computing the right answer, because the guess was reasonable and the reason it failed is a rule rather than an accident.

What the pictures cannot show #

They cannot be looked at peripherally on purpose. A reader who tries to check this by looking away from the figure has moved their fixation, not their attention, and the screen in the figure is drawn at whatever ruling fits on a display — which is far coarser than the twenty-six cycles per degree the computation is about.

And the figures are drawn at the fovea, necessarily. Every curve, label and axis on this page is a foveal object. The essay’s whole subject is what happens to a pattern that is not being looked at, and the only honest way to show that is a graph of it — which is why this family has no picture of a screen fading and a lot of arithmetic.

What the three channels do with the same screen is the other half of the same object, and it is what says the effect belongs to luminance.

Where the ladder goes next #

The nearest unfinished piece is the chromatic case. The three channels have three E2 constants and three cutoffs, so a chromatic screen — which has no best angle in this model — should disappear from the periphery even faster than a luminance one. That is a one-line change and a real prediction.

The second is the scanpath. Everything here is an instant, and reading is a sequence of fixations. A model with a trajectory could say what fraction of a page’s area is ever inspected foveally in ordinary reading, which is the number that decides whether screen quality anywhere but the point of inspection has any consequence at all.

And the third is the moiré asymmetry, which this essay states and does not compute. A beat is at a low spatial frequency and therefore survives eccentricity where its parent screens do not; putting a number on the difference would explain, quantitatively, why the printing trade’s fear of moiré is out of all proportion to its fear of visible dots.