The drift is a luminance mechanism

A perfectly stabilised retinal image fades within seconds. What keeps the world visible is that the eye is never still: between microsaccades it drifts at about half a degree a second, which is fast enough to keep the finest detail modulating and slow enough not to carry it past fusion.

The previous phase computed that band — 0.018 to 0.71 degrees a second — and found the measured drift comfortably inside it, near the sharp upper edge. It is a satisfying result and it was quoted as a fact about vision.

Both of its edges are luminance numbers, and asked about colour the whole argument dissolves.

The claim #

The drift explanation is a luminance explanation, and the channel it does not apply to is the one that fades first.

• The luminance band is closed at both ends, 0.018 to 0.708 degrees a second, and the measured drift of 0.5 sits inside it near the fast edge — because luminance temporal sensitivity is band-pass, with a dip at zero frequency that a slow drift falls into.
• The chromatic bands have no slow edge at all. Red–green is open below the scan’s floor and stops at 0.708; blue–yellow is open below and stops at 1.06. A chromatic pattern held perfectly still sits at the top of its own temporal sensitivity rather than in a dip, so nothing about eye movement is needed to keep it visible.
• And a stabilised chromatic pattern fades faster than a luminance one, which is the opposite of what the model just said. The mechanism that explains luminance fading cannot explain chromatic fading, and chromatic fading is the more dramatic of the two.

Why the two channels differ #

One fact, and everything follows from it: luminance temporal sensitivity is band-pass and chromatic temporal sensitivity is low-pass.

A luminance modulation is most visible somewhere near eight hertz and less visible below that. Hold a luminance pattern perfectly still and its temporal frequency is zero, which is where the curve is worst — this site’s own landmark puts sensitivity at zero frequency at about a quarter of the peak.

A chromatic modulation is most visible at zero. There is no dip, no peak away from the origin, and no frequency at which a chromatic pattern is more visible than it is standing still. Everything above about fifteen hertz is gone, which is why a light whose colour changes forty times a second is a steady light, and everything below is fine.

The drift window asks: over what range of drift speeds does every spatial frequency a channel can resolve stay above a stated fraction of that channel’s temporal peak? A pattern of f cycles per degree drifting at v degrees a second arrives at f·v hertz, so the question is a scan over v with the channel’s own spatial frequencies substituted in.

For luminance, going too slow puts the coarse frequencies in the dip and going too fast carries the fine ones past fusion. Two edges, and the drift is between them.

For a chromatic channel there is no dip, so going slow costs nothing. The only edge is the fast one, and it comes from the same place: the channel’s finest resolvable pattern reaching its own fusion frequency.

What was computed, and how #

Both cutoffs come from the spatial file rather than being restated, which is what makes this a join rather than a second model. The spatial frequencies scanned for each channel are that channel’s own, up to its own cutoff: 50 cycles per degree for luminance, 12 for red–green, 8 for blue–yellow.

The temporal functions come from the temporal file, solved by damped Newton from five quoted landmarks. Neither file was written for this measurement and neither was changed for it.

The scan runs from a hundredth of a degree per second down to a ten-thousandth — two decades below the previous phase’s floor — precisely so that “open at the slow end” can be a measurement rather than an artefact of where the scan started. The chromatic bands are still open at 10⁻⁴, and the assertion tests for that specifically: a band whose reported low edge equals the scan floor is flagged as open rather than quoted as an edge.

And the fast edges very nearly coincide, at 0.708 degrees a second for both luminance and red–green. That is not a coincidence and it is not a fitted result: a channel’s fast edge is its fusion frequency divided by its spatial cutoff, and 60/50 and 15/12 are both 1.2 to within a per cent. Two channels with completely different numbers arrive at the same drift limit because both are ratios of quantities that happen to scale together — the channel that resolves less fine detail also fuses at a lower frequency, in about the same proportion.

The coincidence is real and the explanation for it is not #

The two fast edges landing on 0.708 is offered as a consequence of fusion divided by spatial cutoff — 60/50 and 15/12 are both 1.2 to within a per cent. They are not: 1.200 and 1.250 are 4.2 per cent apart, and neither is 0.708.

The reported edges are consistent with the criterion the essay actually uses, which is a quarter of the temporal peak rather than fusion. Multiplying each fast edge back by its channel’s spatial cutoff recovers the temporal frequency the criterion bites at:

The two chromatic channels return the same number, 8.5 Hz, which is the check that they share one temporal function — and it comes out to three figures from two edges quoted for different channels.

And the real coincidence is sharper than the one claimed. The luminance channel resolves 50/12 = 4.167 times finer detail than red–green, and its temporal curve reaches a quarter of its peak at 35.4/8.5 = 4.165 times the frequency. Two ratios from entirely unrelated measurements — a spatial cutoff and a temporal roll-off — agreeing to five parts in ten thousand.

That is worth having in place of the version offered, because it is both more accurate and more surprising. The fusion frequencies are 59 and 57 per cent of the two quarter-peak points, so the essay’s ingredients agree to four per cent and the quantity built from them agrees to a twentieth of one. A coincidence quoted at one per cent that is really four is a weaker claim than one quoted at four that is really 0.05.

The luminance slow edge exists at one criterion and not below it #

The robustness note considers moving the quarter-peak criterion in one direction and not the other, and the missing direction is the one the essay’s central contrast depends on.

Two of its own numbers sit on top of each other. The criterion is a quarter of the temporal peak, and the luminance curve’s sensitivity at zero frequency is stated as about a quarter of the peak. The luminance band’s slow edge is therefore the point at which the coarsest patterns, arriving at nearly zero hertz, fall below a threshold that sits essentially exactly where they already are.

That makes the slow edge a knife edge rather than a feature. Raising the criterion to a third closes it, as the essay says. Lowering it below the DC sensitivity opens it, and at that point the luminance channel has no slow edge either — every channel is open at the slow end, the drift argument loses its lower bound for all three, and the qualitative difference the essay is about disappears.

So the finding holds over a range of criteria and the range has a floor, and the floor is a quantity the essay quotes two sections earlier. The chromatic channels are open at the slow end for every criterion; the luminance channel is closed only for criteria above about 0.25, which is where the one in use happens to sit.

None of that makes the contrast wrong. It makes it conditional in a way the one-sided robustness check does not report, and the condition is checkable: the luminance curve’s DC sensitivity is a measured landmark with a spread on it, and if the true value is a third rather than a quarter the band is comfortably closed, while if it is a fifth the whole argument has one channel fewer.

The honest version is that the drift band’s slow edge is the least robust number in the essay — it depends on a criterion nobody measured and a landmark that sits at the same value — while the fast edges depend on two cutoffs and two roll-offs and agree with each other to five parts in ten thousand. The contrast between the channels is carried by the weakest of the four numbers.

The contradiction #

Here is the finding stated plainly. In this model:

• luminance needs the drift, and the drift is in the band;
• colour does not need the drift at all;
• and in a stabilised image colour goes first.

Troxler’s observation is over two centuries old and the stabilised-image experiments of the 1950s confirmed it: a peripheral coloured patch fades from view in seconds while an equally visible luminance edge persists. Chromatic fading is faster, more complete, and easier to demonstrate.

So the standing explanation for why a stabilised image disappears — that the eye needs motion to keep the signal above its temporal filter’s low-frequency dip — makes exactly the wrong prediction for the channel where the effect is strongest.

The previous phase had already asserted this absence from the other direction. assertFadingIsNotExplainedHere records that a temporal filter cannot explain fading at all: the luminance filter’s response at zero frequency is a quarter of its peak, so a stationary pattern is dimmer and never gone. What this essay adds is that the filter is not merely insufficient — for two of the three channels it does not even point the right way.

Why nobody noticed #

The drift argument has been made in textbooks for fifty years and this objection is not subtle, so it is worth asking how it survived.

Part of the answer is that the two literatures use different words. The fading experiments talk about stabilised images and retinal slip; the temporal sensitivity work talks about flicker and band-pass responses. The quantity that connects them — a spatial frequency multiplied by a velocity, giving a temporal frequency — appears in neither as a named object, and the multiplication is done informally when it is done at all.

Part of it is that the luminance case is the demonstration. Stabilised-image apparatus is built to show a striking effect, the striking version uses high-contrast luminance patterns, and the account that explains the striking version is the account that gets written down.

And part of it is that the chromatic case looks like the same phenomenon. Colour fades, luminance fades, both recover when the eye moves — three shared observations, and the natural inference is one mechanism. The measurement here says the mechanism proposed for the first cannot be the mechanism for the second, and it says so from landmarks that were all in place decades ago.

What would explain it #

Not a frequency term. A state.

the adaptation-clock machinery has one: gains with time constants, in two pools of different spatial extent. An adapting mechanism does not merely attenuate an unchanging signal, it cancels one — the gain relaxes toward whatever is arriving until what is arriving produces no response at all. That is a mechanism that can take a stationary pattern to zero, which no linear filter can do.

And it predicts the channel ordering the right way round, at least in outline. Chromatic adaptation is strong, is the mechanism this whole site’s white-point machinery rests on, and is the reason a patch has no fixed appearance. Luminance adaptation exists too but a luminance edge carries information the visual system is manifestly reluctant to discard — brightness is inferred from edges, and an edge that has been cancelled cannot be inferred from.

None of that is computed here. It is a sketch of what a model would have to contain, offered because the alternative is to leave the contradiction standing with no indication of which way out is plausible. The two halves of the fading question are in two files on this site and neither does the other’s job, which is what the previous phase’s absence said and remains true.

Where the model stops #

The floor is a criterion, not a measurement. A “band” is the range over which every frequency stays above a quarter of the channel’s temporal peak, and a quarter is chosen. Moving it to a third closes the luminance band’s slow edge more tightly and leaves the chromatic ones open, because open is a statement about the shape of the curve rather than about where a threshold is drawn.

The chromatic spatial cutoffs are quoted at the fovea and for a stationary observer. Both change in the periphery and the change is per channel, so a drift band computed at ten degrees would have different edges for all three — and the periphery is where Troxler fading is easiest to demonstrate, which is an awkwardness this essay has to leave standing.

And a drift is not the only movement. Microsaccades and tremor are also present, at different amplitudes and frequencies, and this computation has only the slow drift in it. A model with all three would have three velocity terms and the same structural conclusion: only one channel has a slow edge, because only one channel has a dip.

The generalisation #

The sentence worth carrying is: a result computed for the luminance channel is a result about the luminance channel, and this site has three.

The pattern recurs. The oblique effect is a luminance anisotropy and the model gives the chromatic channels none, which is a prediction nobody in printing believes. The banding measurement is luminance throughout. The dither result is luminance. Each was computed where the landmarks were, and each was written up as though the eye had one channel.

The reason is not carelessness: the luminance landmarks are the ones that exist. Chromatic contrast sensitivity has been measured, chromatic temporal sensitivity has been measured, and both are quoted with far wider reported ranges than their luminance counterparts, because isoluminance is hard to establish and every laboratory establishes it differently.

The surprising part is how cheap the check is once the machinery is in one place. Asking this site’s own drift window for a different channel is one argument, the answer is qualitatively different, and it turns a satisfying result into an open question. The check that costs one argument is the check nobody runs, which is the same lesson the observer index produced when it found sixty-two of sixty-four generators printing a string that had never varied.

Who found it, and when #

Troxler described the fading in 1804, using nothing but steady fixation on a peripheral spot, and reported the chromatic case: a coloured patch in peripheral vision disappears while the observer holds still.

The stabilised-image experiments arrived in the early 1950s, when Ditchburn and Ginsborg in Britain and Riggs and colleagues in the United States independently built optical systems that held an image fixed on the retina. The image faded within seconds. That is the experiment the drift explanation was built to account for.

The drift speeds were measured in the same period and refined for decades, and the modern figures — a slow drift of a few tenths of a degree per second between microsaccades of a few minutes of arc — are the ones this site quotes.

And the band-pass and low-pass distinction between the two channels’ temporal responses dates from the 1980s, principally from the work that established chromatic contrast sensitivity as a separate object with its own shape. That distinction is fifty years younger than the fading experiments, and putting the two together is arithmetic that nothing in either literature required anybody to do.

Two more views say what happens when the eye is moving faster than a drift, which is the case a display actually has to survive.

What the pictures cannot show #

They cannot stabilise anything. The entire subject is what happens when a pattern is held motionless on the retina, and no figure on a page can do that — a reader’s own drift is exactly the thing being removed in the experiments this essay is about, and it cannot be switched off from here.

And they cannot show an isoluminant pattern honestly. A grating that is isoluminant for this site’s median observer is not isoluminant for the reader, because the ratio of the two long-wavelength cone types varies enormously between people — so a reader who sees a luminance flicker in the chromatic strip above is not seeing an error in the figure; they are seeing their own luminous efficiency function.

Where the ladder goes next #

The nearest unfinished piece is the state model. the adaptation-clock machinery has gains with time constants and no spatial frequency in it; the spatial file has spatial frequency and no state. A model with both would fade a stabilised pattern, would say which channel goes first, and would be the first thing on this site able to address the experiment the drift band was built to explain.

The second is the periphery. Troxler fading is easiest to demonstrate away from the fovea, and every quantity in this computation — both cutoffs, both fusion frequencies — moves with eccentricity by amounts this site can already compute. A drift band at ten degrees is a two-line change and would say whether the effect’s peripheral preference is in the temporal machinery at all.

And the third is the one that would need an experiment rather than an argument: whether the chromatic bands really are open at the slow end. The prediction is testable — a chromatic pattern drifting at a hundredth of a degree per second should be no less visible than one drifting at half — and it is the kind of prediction that could be wrong in an interesting way, because it says something the fading literature’s whole framing would deny.