What a still eye stops seeing

Hold an image perfectly still on the retina and it disappears. That is Troxler’s observation from 1804, it is the reason a suction-cup contact lens with a tiny projector on it became a piece of laboratory equipment in the 1950s, and it is the standard explanation for why the eye never stops moving.

This site has carried both halves of the mechanism for two collections of essays without ever putting them in the same computation. The temporal filter knows about frequencies and has no state. The adaptation model knows about state and has no idea where on the retina anything is. Between them there is one number missing, and it is not a time constant.

The claim #

Fading is a spatial high-pass filter that switches itself on over a minute, and the frequency where it divides is nowhere near any limit of the eye.

• A gain that follows the mean of its own neighbourhood is a low-pass filter with a clock on it. Nothing needs to be added to the adaptation model except an extent, and the extent is one number.
• What is left after it settles is an outline. The middle of a stabilised disc goes to the colour of the ground — 0.2 per cent of the step it arrived with — while the border keeps 98 per cent of its gradient. A single gain over the whole field would take the two down together and could never produce that.
• A patch has to be about 1.8 degrees across before a still eye loses it, which is roughly the size of the fovea and roughly a thumbnail at arm’s length. Anything finer stays for as long as the eye holds still.
• And the drift defeats one adaptation pool and not the other. At the speed a real eye drifts, the one-second pool still follows a ten-degree pattern at 95 per cent while the sixty-second pool has lost it to 5. The eye is not defending itself against adaptation in general.

The one number #

The adaptation machinery here has two pools, a fast one at a second and a slow one at a minute, and a share between them. What it does not have is a size: the local pool is called local because it is not the whole field, and how much retina it covers has never been stated because nothing that used it needed to know.

Give it one and the arithmetic is immediate. If a pool tracks the image blurred by itself, then what it can cancel is what a blur of that size can see, and the transfer function of a Gaussian is a Gaussian. The contrast still driving a channel at spatial frequency f after t seconds is

$$r(f,t) = 1 - a(t)\,e^{-2\pi^{2}\sigma^{2}f^{2}}$$

with a(t) the fraction of the adaptation that has happened, taken from the clock the afterimage essays already use rather than restated. At t = 0 the residual is one everywhere and nothing has faded. As the pools fill, everything coarser than the pool is cancelled and everything finer is untouched.

Half a degree is the value used, and it is not a measurement anybody publishes directly. An afterimage has the size of the thing that made it and a soft edge about this wide; local adaptation is demonstrable between neighbouring degrees of field and not between the two eyes; a stabilised disc fills in from its border inward over seconds rather than jumping. Because it is soft, every conclusion below is re-run across a factor of four in it and the ones that do not survive are not here.

The threshold is one number in disguise #

The size threshold is quoted at 1.8 degrees for a pool half a degree across, and the sweep across a factor of four in that pool has a simpler answer than a range.

Across the whole permitted range the threshold is 0.987° at σ = 0.25, 1.302 at 0.35, 1.843 at 0.5, 2.432 at 0.7 and 3.688 at 1.0. Divided by σ those are 3.95, 3.72, 3.69, 3.47 and 3.69 — constant to within seven per cent across a fourfold change of the parameter.

A patch fades when it is about 3.7 pool-widths across, and that is the whole of it. Multiplied by the pool’s own half-power frequency instead of divided by its extent, the product runs 0.74, 0.70, 0.69, 0.65, 0.69 — the same statement in the other units, a dimensionless number near 0.69.

So the sweep is not really a robustness check. It is a demonstration that the unmeasured parameter sets only a scale: every conclusion here is about the ratio of a patch to the pool, so none of them can survive or fail as the pool moves, because they all move with it.

That makes the guess honest in a specific way. Half a degree is a guess; 3.7 pool-widths is not, and anybody who measures the pool by some other route gets the size threshold without measuring it.

How far the divide sits from any limit of the eye #

Nowhere near any limit is right, and the numbers are larger than the phrase implies.

The pool’s half-power frequency is 0.375 cycles per degree. The luminance channel’s acuity limit is 50, a ratio of 133. The red–green channel’s is 12, a ratio of 32; the blue–yellow channel’s is 8, a ratio of 21.

So the luminance case is not one order of magnitude but better than two — and the two chromatic channels, which are usually introduced as the coarse ones, are still twenty and thirty times finer than the pool that erases things.

Counted in octaves, the pool leaves 8.54 octaves of luminance detail untouched, 7.25 of red–green and 6.66 of blue–yellow. Whatever a still eye loses, it is not detail. It is the part of the image that has no detail in it at all.

Filling in, rather than dimming #

The word usually used for what a stabilised image does is fade, and the word is wrong in a way that decides which model can be right.

A disc whose border is stabilised does not get dimmer. Its colour spreads outward and the whole field takes on the surround. That is Krauskopf’s arrangement from 1963: stabilise only the edge of a red disc on a green ground, leave everything else free to move, and the red is replaced by green. The disc does not go grey and it does not go dark. It goes the colour of the ground.

A model with one gain over the whole field cannot produce that, because one gain scales the interior and the border by the same factor and the picture merely gets flatter. A gain with an extent produces it without being asked: the interior of the disc is a region the pool sees perfectly, so the gain there cancels it exactly, while the border is a step the pool cannot follow and survives almost intact.

The measurement of that, on a three-degree disc, is an interior at 0.2 per cent of its arriving step and a border at 98 per cent of its arriving gradient — a ratio of about four hundred. assertTheInteriorFillsAndTheBorderDoesNot requires both halves, because either one alone is satisfied by a model that is simply wrong.

What is lost is what is large #

The pool’s own cutoff is 0.37 cycles per degree, which is a feature about two and a half degrees across. Every channel the eye has resolves far past that: 50 cycles per degree for luminance, 12 and 8 for the two chromatic ones. The coarsest of those is twenty-one times finer than the pool and the finest a hundred and thirty times.

So a stabilised eye does not lose its picture. It loses the fill and keeps everything with any structure in it — which is why the phenomenon needed a suction cup and a projector to demonstrate rather than being obvious to anybody who stares at a wall.

The size at which the crossing falls is the useful form of the answer, because it is checkable against a thumbnail. It moves with contrast in the obvious direction and it does not move much: a patch at half the contrast fades a little sooner and a patch at twice the contrast a little later, and the crossing stays inside the same degree.

What the eye’s own motion is worth #

The drift is usually described as keeping the image moving so that nothing has time to fade. That is right and it is not specific enough, because a pool cannot be defeated by motion in general — it can only be defeated by motion fast enough against its own time constant.

A pattern moving at v degrees a second presents each pool with an input modulating at v·f hertz. A single-pole pool follows what is slow against 1/τ and averages away what is not. The two pools here are sixty seconds apart in time constant, so they are in two completely different regimes at any drift speed worth the name.

What that buys, at the frequencies the pool could otherwise cancel, is a factor of 27 more surviving contrast at a twentieth of a cycle per degree, 8 at a tenth, and 3 at a fifth. The drift is worth most exactly where the pool is most effective, which is the arrangement a designer would have chosen and is presumably not a coincidence.

It also completes an argument left open elsewhere on this site. The band of drift speeds that keeps every spatial frequency modulating is a luminance band, computed from a band-pass temporal function that a chromatic channel does not have. This is the other half: what the drift is doing is not keeping the signal inside a temporal passband but keeping the adaptation pools from tracking, and that half applies to every channel equally because the pool is not a channel.

Two absences, asserted #

The site’s habit is that a claim it cannot make is written down as an assertion that will break the build if it ever starts making it. Two apply here and both were expected to go the other way.

The model gives no order between the channels. Colour is reported to go first in a stabilised image, and it is not derived here. What survives a stabilisation is an edge; an edge is a high-frequency object; all three channels resolve high frequencies comfortably; and the residual left at a given frequency is the same number for each of them because the pool cancels what it can see and does not care what the contrast is along. Deriving an order would need thresholds in commensurable units across the three channels, which the spatial model says plainly it does not have — the chromatic thresholds are anchored at the luminance one as a deliberate simplification. assertNoChannelOrderIsPredicted requires the three residuals to stay identical.

And the periphery goes the wrong way. Troxler fading is stronger off axis; that is why the effect carries his name and not somebody else’s, because a peripheral blob is the easy demonstration. The obvious extension is to scale the pool with cortical magnification, which makes it larger off axis — and a larger pool cancels less of a fixed stimulus, not more. The model therefore predicts weaker fading where more is observed. That is a refutation of the obvious extension rather than a caveat, so assertTheTroxlerGradientIsNotExplainedHere holds it in place.

The second one is worth more than the first. It says the pool is not simply a receptive field, because a receptive field is exactly the thing that scales with eccentricity. Whatever sets the extent of a local adaptation gain, it is not the same thing that sets the size of the units underneath it.

What was computed, and how #

Nothing in the file is fitted. The two time constants come from the adaptation model, the three acuity limits and the threshold anchor come from the spatial model, the drift speed is quoted with the other velocities, and the pool’s extent is the single number added.

The residual is the expression above. Visibility is that residual times the presented contrast, divided by the threshold at that frequency in that channel — so a value above one is a pattern still there. Fade times are bracketed and bisected rather than solved, because the two-exponential clock has no closed-form inverse and the bracket is unambiguous: the residual is monotone in time, so a pattern still visible when the pools have finished is a pattern that stays.

The disc is convolved directly with the Gaussian rather than transformed, because a profile with edges wrapped circularly would fold the far side of the field onto the near one — the same reason the profile filter mirrors before transforming.

And assertTheConclusionSurvivesThePool re-runs the whole of it across a factor of four in the extent, from a quarter of a degree to a full one. The cutoff moves with the parameter, as it must; every conclusion drawn from it does not.

Where it stops #

The model is a gain and a blur, and there is a great deal it is not.

It has no receptive fields, so it says nothing about the shape of what fills in — a stabilised square does not become a stabilised circle here, because there is no mechanism that could make it one. It has no filling-in dynamic: the interior arrives at the ground because the pool cancels it, not because anything propagates inward from the border, so the model predicts the endpoint of Krauskopf’s demonstration and not the several seconds it visibly takes.

It has no eccentricity, as asserted. It has no second eye. And it treats the pool as linear, which cannot be right at large contrasts — a gain that cancels the mean of a black-and-white field is being asked to do something a multiplication cannot do.

What it does have is the property the site asks of a model: it makes a prediction that could be wrong in a stated way. A patch smaller than about two degrees, stabilised, should not fade. A patch of the same size at the same contrast, four times larger, should be gone in seconds. That is not a matter of degree, and an apparatus that can stabilise an image can settle it.

Who found it, and when #

Troxler reported the effect in 1804 with no apparatus at all: fixate steadily on a mark and the objects in the periphery grow indistinct. That is the version everybody can reproduce, and it is also the version this model does not explain, because it happens off axis and the eccentricity term goes the wrong way.

The controlled version arrived a century and a half later and needed engineering. In 1952 Ditchburn and Ginsborg, and independently Riggs, Ratliff, Cornsweet and Cornsweet, built optical arrangements that cancelled the eye’s own motion — a contact lens carrying a small mirror or a whole projector, so that the image moved with the eye instead of across it. What they found was not a gradual dimming but a disappearance in a few seconds, with regeneration when anything at all was allowed to move.

Krauskopf’s 1963 experiment is the one that decides between models, and it is the one usually left out of the summary. He stabilised only the border of a disc and left the rest of the field free. The interior did not fade to grey; it took the colour of the surround, and the observer reported a uniform field of the ground’s colour where a disc had been. That is a statement about what the visual system does with a region whose edge has stopped saying anything, and it is the reason a fading model has to be spatial.

The interval between the observation and the apparatus is the useful part of the history. For a hundred and fifty years the phenomenon was a curiosity about staring, because the only version available was the peripheral one; once it could be produced on demand it became an argument about what the visual system is doing when nothing changes, and the answer turned out to be cancelling whatever it can predict. That is the same thing an adaptation gain does, and this file is one way of saying that the two are not two mechanisms.

Where the ladder goes next #

The pool has an extent and nothing else. Two obvious arguments follow from giving it a second property.

Giving it a shape — an ellipse rather than a circle — makes the fading anisotropic, which is testable against the oblique effect that the two-dimensional spatial model already computes. And giving it a boundary rather than a Gaussian falloff is the difference between a blur and a region, which is what would be needed before anything here could say what a filled-in shape looks like rather than only how much of it is left.

The more interesting one is upstream. Every clock on this site so far — the two neural pools, the pigment’s two minutes — belongs to the observer. A room has a fourth, and it does not.