The eye is never still

Every figure on this site assumes a stimulus that stays where it is put. So does every colorimetric standard, every appearance model, and every contrast sensitivity measurement quoted here.

None of them is a description of a real observation, because a real eye does not hold still. Between the saccades that move the gaze from place to place there is a continuous slow drift, of roughly half a degree a second, with small corrective jumps a few times a second. It is small enough that nobody notices and it is not a defect — hold an image perfectly still on the retina and it disappears.

The claim #

Fixational drift converts every spatial frequency into a temporal one, and the measured drift speed sits inside the narrow band in which none of them is lost.

The arithmetic is the one the stroboscopic argument uses, run backwards. A pattern of f cycles per degree, drifting at v degrees a second, modulates each receptor at f × v hertz.

Every frequency lands in the temporal passband. The finest detail the eye can resolve arrives at twenty-five hertz, which is well below the sixty-hertz fusion frequency; the coarsest lands at a quarter of a hertz, where the temporal function’s own low-frequency dip is the limit rather than fusion.

The band of drift speeds for which that holds is 0.018 to 0.71 degrees a second, and the measured drift is 0.5 — inside it, near the top.

Why there is a band at all #

Two constraints, from the two ends of the spatial range, pulling opposite ways.

Too fast and the fine detail goes. At four degrees a second, a fifty cycle-per-degree pattern arrives at two hundred hertz, which is three times past fusion. A drift that quick would blur away the eye’s own acuity, and the upper edge of the band — 0.71 degrees a second — is exactly where the finest resolvable pattern reaches the fusion frequency.

Too slow and the coarse detail sinks. The temporal sensitivity function is band-pass: its response at zero frequency is a quarter of its peak. At a hundredth of a degree a second every spatial frequency arrives near zero hertz, where the response is at that floor.

So there is a window, its upper edge is sharp and its lower edge is soft, and the measured drift sits just under the sharp edge — which is the arrangement that gets the most out of both ends without losing the top.

Nothing arranged that. The spatial cutoff is a landmark from grating experiments, the fusion frequency is a landmark from flicker experiments, and the drift speed is a landmark from eye-tracking. The three were measured by different people for different reasons, and multiplying two of them lands on the third.

The chromatic band, which the same two landmarks give #

The end of this essay files the chromatic version as unfinished, on the ground that both edges move in the same direction so the answer is not obvious. Both parts of that are right, and the answer needs no new machinery — only the two cutoffs the site already carries for each channel.

The upper edge is set by one relation: the finest resolvable pattern must not be carried past the frequency where the temporal response has fallen to a quarter of its peak. For luminance that is 50 cycles per degree against about 35 hertz, and dividing gives 0.70 degrees a second — the 0.71 quoted above, reconstructed, which is the check that the criterion is the one that was used.

Applying it to the other two channels, with the chromatic fusion frequency at fifteen hertz:

The chromatic bands are wider at the top, not narrower, and the luminance channel is the binding constraint for all three.

That is the non-obvious part and the arithmetic behind it is worth one line. The upper edge is the ratio of a channel’s temporal cutoff to its spatial one, so what matters is not how far each falls but whether they fall together. Going from luminance to red–green the spatial cutoff drops by a factor of 4.17 and the temporal one by 4.00 — so the ratio moves by 1.04, and the two chromatic channels end up with essentially the same drift ceiling as the luminance channel rather than one four times lower.

Red–green is four per cent behind luminance, which is inside the reported ranges of every landmark involved and is not a margin anybody should read as an ordering. The honest statement is that the luminance and red–green channels have the same drift ceiling, to the precision the landmarks support, and the blue–yellow channel has half again as much room.

Two things follow, and one of them is a genuine addition to the essay’s claim.

The coincidence is stronger than it looked. The measured drift of half a degree a second sits below the ceiling of all three channels at once, and the three ceilings are not spread over a factor of four — they are 0.70, 0.73 and 1.09. A drift speed chosen to suit the luminance channel suits the chromatic ones automatically, and the reason is a near-cancellation between two independently measured pairs of numbers.

And the lower edge is the part this cannot settle. The luminance band has one because the temporal function is band-pass: its response at zero frequency is a quarter of its peak, so a stationary coarse pattern sinks. Whether the chromatic temporal functions dip in the same way is a fact about their shape that nothing quoted on this site establishes, and it decides whether the chromatic bands have a lower edge at all or run down to zero. A low-pass chromatic temporal response — which is the usual report — would mean a stabilised chromatic pattern is not attenuated by frequency at any speed, and that everything about chromatic fading is the gains rather than the filter. That is a sharper division of labour than the luminance case has, and it is one measurement away.

What drift cannot do, and asserted as such #

The obvious question is why a stabilised image fades, and this model does not answer it.

A temporal filter with a response of a quarter at zero frequency says a perfectly stationary pattern is dimmer and never gone. assertFadingIsNotExplainedHere requires it to keep saying so, in the shape the dither null established: if a revision ever starts predicting fading from the filter alone, the build stops and somebody reads why.

What is missing is not a frequency term but a state, and the site has one. A gain with a time constant follows what it has been receiving, so a receptor looking at an unchanging patch converges on it, its gain cancels it, and there is nothing left. That is the same mechanism that produces the afterimage, and it produces Troxler fading with no extra machinery: an afterimage is the residue after a stimulus is removed, and fading is the same process while the stimulus is still there.

So the two halves of this essay are two different files and neither does the other’s job. The filter says why drift is the right speed; the gains say why any drift is needed.

What this does to everything measured on this site #

The consequence worth stating is not about fading. It is that every threshold quoted here was measured on a drifting retina, because every threshold anybody has ever measured was.

A contrast sensitivity function is measured by showing an observer a grating and asking whether it is there. The observer’s eye drifts throughout, so the retinal stimulus is a grating and a temporal modulation, and the measured sensitivity is the product of the two functions rather than a property of the spatial one alone.

That is not a criticism — it is the condition every visual measurement is made in and therefore the condition the numbers are valid for. What it means is that the spatial and temporal sensitivity functions this site treats as separate are two projections of one surface, measured at whatever drift the observers happened to have.

A stabilised measurement would give different numbers, and the ones that exist do: contrast sensitivity measured on a stabilised retina falls at low spatial frequencies, which is exactly what this arithmetic predicts, because a stabilised low-frequency grating sits at zero hertz in the dip.

What it costs the figures on this page #

Every figure on this site is drawn for a stationary reader looking at a stationary page, and the drift argument says the assumption is wrong in a way that is small and not zero.

The spatial claims are safe. A contrast sensitivity function measured on a drifting retina and applied to a stimulus seen by a drifting retina is being used in the condition it was measured in, which is the only thing a landmark can be asked for.

The stated frequencies are not. Every figure here states a viewing geometry — a pixel pitch and a distance — because a spatial frequency needs one. The drift adds a second unknown the page has no access to either: a reader holding a phone in one hand has a retinal drift that includes the hand’s motion, which is very much larger than half a degree a second.

And the illusion figures are the ones that care. An asserted identity between two patches is a claim about two stimuli; whether a reader sees them as identical involves an adaptation state that moves while they look. A reader who stares at one of the two patches for twenty seconds is a different observer from one who glances, by an amount the gain essay puts at several units.

What was computed, and how #

Two landmarks and a multiplication. The drift speed is quoted at half a degree a second, with the range reported; the temporal and spatial functions are the site’s own, solved from their own landmarks. driftWindow multiplies and evaluates, and there is nothing else in it.

The band is scanned rather than solved, because the constraint at the slow end comes from the lowest spatial frequency and the one at the fast end from the highest, and there is no single expression whose root is the answer.

The floor is a choice and is stated. A quarter of the temporal peak is used, which is the value the low-frequency dip itself sits at — so the lower edge of the band is soft by construction and the upper edge is not. A stricter floor removes the band entirely; a looser one widens the lower edge and leaves the upper one where it is, because the upper edge is set by fusion rather than by the criterion.

And the absence is asserted rather than described. assertFadingIsNotExplainedHere requires the stabilised response to stay above five per cent of the peak, which it does at 0.25, so a future revision cannot quietly start agreeing with a textbook it has not earned.

Where the model stops #

Drift is not the only motion. Microsaccades — small jumps of a few tens of minutes of arc, one to three times a second — carry the low spatial frequencies that drift leaves in the dip, and a tremor at eighty hertz and tiny amplitude sits above both. This model has one velocity.

And drift is not uniform. Real fixational drift is a slow random walk rather than a constant velocity, so the temporal frequency at a given spatial frequency is a distribution with a mean rather than a number. That has a consequence the constant-velocity model cannot state: a random walk has power at every temporal frequency at once, so some of every spatial frequency lands in the passband whatever the mean speed, which makes the band’s edges softer than they look here.

There is no direction. Drift has one, patterns have one, and a pattern’s orientation matters — a grating drifting along its own bars produces no temporal modulation at all, and the arithmetic here assumes the drift is across them.

And the chromatic channels are absent. The chromatic temporal cutoff is fifteen hertz against sixty, so the drift band for a chromatic pattern is four times narrower and the measured drift is comfortably inside it. Working that out would need the chromatic spatial cutoffs too, and both exist.

The generalisation #

The sentence worth carrying: a static stimulus is not a physiological condition, and every measurement in this subject was made on a moving one.

Colorimetry’s whole apparatus assumes a stimulus that is what it is. That assumption is safe, because the thing it stands on — a match, an integral, three numbers — has no time in it. The moment a threshold enters, the assumption stops being safe, because a threshold is a measurement and every measurement was made on a drifting retina.

The surprising connection is with the stroboscopic effect. That essay found a lamp flickering at a kilohertz visible during a saccade, because motion turns a temporal pattern into a spatial one. This is the same conversion in the same direction with the roles of the two functions exchanged: there a known temporal modulation was made spatial and judged by the spatial function; here a known spatial pattern is made temporal and judged by the temporal one. One relation, f = v × spatial frequency, and two arguments that look unrelated.

And a third use of it is available and not taken. The same relation says what velocity of a moving object keeps its detail in the passband, which is the smooth-pursuit case and is why a tracked object is sharper than an untracked one at the same speed.

Who found it, and when #

Troxler described the fading of a stabilised peripheral stimulus in 1804, without stabilising anything — steady fixation is enough in the periphery.

The stabilisation experiments are Riggs, Ratliff, Ditchburn and Fender’s, in the early 1950s, using a mirror on a contact lens to hold an image fixed on the retina. The image disappears in a few seconds and reappears when the stabilisation is broken, which is the demonstration this essay’s second half is about.

The drift measurements are of the same era and later, and the speed has been refined but not moved: fractions of a degree a second, with microsaccades of a few tens of minutes of arc superimposed.

And the argument that drift is functional rather than a failure of muscular control has been contested for seventy years and is now largely settled in favour of function — with the observation that drift statistics are tuned to whiten the spatial statistics of natural images, which is a stronger version of the coincidence this essay measures.

What the pictures cannot show #

They cannot stabilise anything. Every claim here is about what happens when a retinal image does or does not move, and a page has no access to the reader’s eye. The fading result in particular needs an apparatus with a mirror on a contact lens.

And the coincidence is the hardest thing to draw. What is being claimed is that three numbers measured for unrelated reasons multiply to a fourth, and the only picture of that is a curve with its ends marked — which shows the result and not the surprise.

Where the ladder goes next #

The chromatic band’s upper edge is computed above and the answer is that there is nothing to worry about there. What is left is its lower edge, which needs the shape of the chromatic temporal response rather than its cutoff — a landmark this site does not carry, and the one that decides whether a stabilised chromatic pattern fades for the same reason a luminance one does or for an entirely different one.

The second is the join to the gains. This essay says the filter explains why drift is the right speed and the gains explain why any is needed, and nothing computes the two together: a gain following a drifting stimulus is a state driven by a modulated input, which would give the fading rate as a function of the drift rather than as an absence.