Far enough away, a pair is one eye

A turn of the head is not a step sideways shrank the baseline to nothing and found that the algebra does not notice: a two-millimetre baseline recovers a courtyard to nine digits from exact marks, and what fails is the ratio of the baseline to the reading error.

This is the other end of the same ratio. Hold the baseline and walk the scene away.

The arrangement #

A pair thirty centimetres apart photographs a scene of fixed angular size with fixed relative relief, at distances from four metres to two hundred and fifty-six. The scene grows with the distance so that the picture looks the same at every step: same framing, same texture, same spread of marks across the frame. The only thing that changes is the ratio of the baseline to the depth.

That is deliberate. Holding the scene’s physical size instead would shrink it in the frame and confuse two effects — fewer pixels on the subject, and less parallax — and only the second is the subject here.

What falls, and how fast #

The quantity to watch is not the disparity of any one mark. It is how much of the disagreement between the two pictures a single homography cannot explain — because a distant scene is well described by a homography, and what a two-view solver has to work with is the residual that description leaves.

Measured across six doublings of the distance, that residual falls from 42.5 pixels to 0.76, and a straight line through the logarithms gives an exponent of −0.968. That is one over the distance to within three per cent, and it says what the arrangement’s only variable is: parallax is the baseline over the depth, and nothing else in the arrangement is changing.

An exponent of −2 would have meant something else was happening — a scene shrinking in the frame, say — and finding −1 is what confirms the sweep is measuring the ratio and not an artefact of how the scene was built.

What rises, and when #

The recovered translation direction is 3.3 degrees wrong at four metres, dips to a third of a degree at sixteen, and then climbs: 3.3 degrees at thirty-two, 25.8 at sixty-four, 63.5 at a hundred and twenty-eight, 74.5 at two hundred and fifty-six.

The climb starts where the parallax falls below about three pixels, which at three tenths of a pixel of marking error is ten times the reading error. That is the same crossing how flat is flat enough finds by flattening the scene instead of retreating from it, and finding it twice by two unrelated routes is what makes it a rule rather than a number.

Past that point the recovery is not merely imprecise. Seventy-four degrees out of a direction that only has ninety degrees of range either way is an answer with no information in it — the solver is returning whichever direction the noise preferred.

What the slider says about the ten-times rule #

The crossing above is read off one setting of the reading error, and the rule it states is a claim about all of them. The figure’s slider runs five, so the claim can be checked rather than carried, and checking it is worth the paragraph because it holds at three settings, holds approximately at a fourth, and has nothing to locate at the fifth.

At three tenths of a pixel the recovered direction passes ten degrees of error between 32 and 64 metres, and the parallax there is ten times the reading error. That is the rule exactly, and it is the setting the rule was read from.

At eight tenths the crossing moves in to between 16 and 32 metres, where the parallax is seven times the reading error. Close enough to be the same rule and far enough to say the coefficient is a rough one.

At a tenth and at a twentieth of a pixel there is no crossing inside the sweep at all. The pose error stays under 9° out to 256 metres, where the parallax is still eight and fifteen times the reading error respectively — above the threshold, and so consistent with the rule rather than a counterexample to it. Fine marks push the crossing past the far end of the arrangement, which is the whole reason a stellar parallax works.

And at two pixels the rule has nothing to say, because the recovery is already 23° wrong at four metres — where the parallax is twenty-one times the reading error and the rule predicts a good answer. The sweep never gets to a regime boundary, because it starts past one.

That last setting is the informative one and it fixes what kind of statement the rule is. It compares two resources and identifies which runs out first; it presupposes that neither has already run out. At two pixels of hand the marks are too coarse for the linear step to be well conditioned anywhere in the sweep, so the arrangement is mark-limited at every distance and there is no transition to find. A ratio of parallax to reading error is a necessary condition for a good recovery and not a sufficient one, and the coarsest setting of this figure’s own slider is where the difference shows.

Two things follow for anybody using the rule. Check the near end first. If the closest, most favourable configuration in an arrangement already returns a poor pose, the ratio is answering the wrong question and the fix is better marks rather than a longer baseline. And read the far end’s value rather than its trend: the pose error saturates around 84°, which is what a direction carrying no information looks like, and a curve that has flattened there has stopped measuring the arrangement and started measuring the noise’s preferences.

Why a distant pair is a rotation #

The limiting statement is worth having exactly, because it explains why the failure has the character it does.

As the scene recedes, every point’s depth becomes large compared with the baseline, and the map between the two pictures approaches the homography induced by the plane at infinity — which is a pure rotation composed with the two calibrations. Two photographs of a very distant scene are related by a rotation, exactly as two photographs from one point are.

So a distant pair and a rotating pair are the same arrangement in the limit, which is why a turn of the head is not a step sideways and this essay have the same finding, and why the parallax a stitcher cannot remove treats the residual as the thing that identifies the arrangement. It is also why panorama stitching works on landscapes and fails on interiors: the assumption a stitcher makes is that the scene is at infinity, and a landscape is.

What this says about adding views #

The many-views field’s own finding is that angular spread rather than photograph count is what a reconstruction is short of, and this sweep says the same thing from the geometry’s side.

Another picture of the same sweep measured going from three views to seven across the same sixty degrees and found the reconstruction exactly where it started. The reason is visible here: filling in a sequence adds photographs whose baselines relative to the scene are the same ratio as the ones already there, and the ratio is what matters. Widening the sequence changes the ratio.

For a distant scene the ratio is small whatever is done with the count, so the remedy is a longer baseline — literally walking further, or an aerial survey’s flight line, or the Earth’s orbit for a parallax measurement of a star. In every case the fix is the same quantity: baseline over depth — and where the adjustment stops is where that quantity finally stops buying anything at all.

The rule of thumb, and why it is not one #

Practitioners quote base-to-height ratios: one to ten for aerial photogrammetry, one to five for close-range work, one to fifteen for reconnaissance. This sweep says where those come from and why they are not transferable.

The quantity that decides is the parallax in pixels, which is f·b/Z times the relative relief. A ratio of one to ten at f = 620 pixels gives sixty-two pixels of parallax for a scene with a hundred per cent relief and six for a scene with ten per cent. So the same ratio is comfortable on a hilly landscape and marginal on a flat field, and quoting a ratio without a focal length and a relief is quoting a third of the calculation — which is the same incompleteness a focal length is not an angle records for the other end of the chain.

The version that transfers is: at least ten times the marking error of out-of-plane parallax, measured as the residual of a fitted homography. It has units, it is computable from the photographs before any reconstruction is attempted, and it does not need to know what is being photographed.

Which kind of failure this is #

By the round’s own vocabulary, an uncertainty rather than an ambiguity — but one that approaches an ambiguity in the limit.

At any finite distance there is a unique correct answer and it is determined; the determination is simply weaker than the noise. More marks help, better marks help, a longer baseline helps a great deal. Nothing is exactly undetermined.

In the limit, though, the arrangement becomes exactly the rotational degeneracy, where the fundamental matrix does not exist and the homography is the whole relationship. So this sweep is the path between the two categories, walked, and it is why an ambiguity is not an uncertainty has to say that the categories are exact and the boundary between them is the instrument’s.

Why the parallax is measured after a homography #

A detail of method, because it decides what the falling curve means.

The obvious measurement is the disparity of a mark — how far it moves between the two pictures — and it is the wrong one. Most of that movement is the rotation between the two cameras, which carries no depth information at all, and it does not fall as the scene recedes.

What does fall is the part a single plane map cannot account for. Fit a homography to four marks, apply it to the rest, and measure the worst disagreement: that is the parallax the arrangement genuinely offers, and it is exactly the quantity a two-view solver has to distinguish from noise. A rotating camera gives zero; a distant scene gives a little; a near scene with relief gives a lot.

It is also the quantity that appears in the flatness sweep, and using the same instrument in both is what lets the two crossings be compared. A rule derived from two different measurements would have been two rules.

The sweep at the near end #

The left-hand end of the plot is worth reading too, and it is not simply “more parallax is better”.

At four metres the pose error is 3.3 degrees, which is worse than the third of a degree at sixteen. That is not noise: at a base-to-depth ratio of one to thirteen the scene subtends a wide angle, the two cameras see it from genuinely different directions, and marks near the edge of the frame move a long way — which makes the correspondence problem harder and the linearisation less good, not the geometry worse.

So the curve has an optimum rather than a monotone trend, and the optimum is where the parallax is comfortably above the noise and the two views are still similar enough to match reliably. That is a trade every practitioner knows and it is the reason base-to-height ratios have a recommended value rather than a recommended minimum.

The sweep here does not measure the matching difficulty — the correspondences are given — so the near end’s rise is the geometry’s own and the real one would be steeper.

What a surveyor already knows #

None of this would surprise anybody who has flown an aerial survey, and the point of measuring it is not to surprise them.

It is that the same quantity governs a stereo camera on a desk, a headset’s inside-out tracking, a phone’s two lenses, a self-driving car’s stereo rig and a spacecraft’s approach imagery — the one thing a single view cannot give being what all of them are buying — and in each of those the numbers are different and the rule is the same. Writing it as a residual in pixels rather than as a base-to-height ratio is what makes it portable between them.

And it explains a failure that looks mysterious from inside: a reconstruction that works beautifully in a room and produces nonsense outdoors has not encountered a bug. It has run out of the only resource the arrangement supplies.

The astronomical version, for scale #

The same ratio governs the oldest use of a two-view reconstruction there is, and putting the numbers beside each other says how extreme the arrangement has to be.

Stellar parallax uses the Earth’s orbit as a baseline — three hundred million kilometres — against distances of tens of trillions. The ratio is about one in a hundred thousand, which by the standards of this sweep is hopeless, and it works because the marking error is a few thousandths of an arcsecond rather than a few tenths of a pixel. The rule is the same one: parallax against reading error, and the reading error is what a century of instrument-building bought.

At the other extreme, a phone’s two lenses are a centimetre apart looking at a face half a metre away — a ratio of one in fifty, better than anything in this sweep — and the reconstruction is limited by matching rather than by geometry.

Both are the same inequality with the two sides moved by ten orders of magnitude, which is worth knowing because it says which side to work on. A stereo rig that is short of parallax cannot be rescued by a better detector; one that is short of correspondence cannot be rescued by a longer baseline.

Why the homography is fitted on four marks and not on all of them #

A methodological choice worth stating, because a least-squares homography over every mark would give a smaller residual and a worse instrument.

Fitting on all the marks lets the fit absorb some of the parallax — the very thing being measured — so the residual reported would be the parallax the arrangement offers minus whatever a plane could soak up, which shrinks as the number of marks grows. That would make the measurement depend on the count.

Fitting on four and testing on the rest is the withheld-evidence discipline the collection uses everywhere, and here it has a second virtue: four marks determine a homography exactly, so the fit has no freedom to absorb anything and the residual on the remaining marks is the whole of the disagreement.

The price is that the number depends slightly on which four, and the four here are taken evenly spread through the list rather than chosen. A reader wanting a robust version would take the median over several choices; the exponent it fits, which is the essay’s claim, does not move.

The short version #

Hold the baseline and walk the scene away. The parallax a single homography cannot explain falls from 42.5 pixels to 0.76 across six doublings, at an exponent of −0.968 — one over the distance, confirming that the baseline-to-depth ratio is the whole variable. The recovered translation direction is a third of a degree wrong at sixteen metres and seventy-four degrees wrong at two hundred and fifty-six.

The crossing is at about ten times the marking error of parallax, which is the same crossing the flatness sweep finds by a different route. Past it, a pair of photographs is a rotation, which is why panorama stitching works on landscapes; the remedy is a longer baseline and nothing else.