Vergence moves the shells and does not respace them

Whole pixels cut space into shells finds that a stereo pair reading disparity to whole pixels does not measure distance on a scale at all: it chooses among a finite set of depths, 113 of them between half a metre and twelve, 6.7 centimetres apart at two metres and 1.39 metres apart at ten. A level floor comes back as thirty-five standing plates.

Everything in that measurement assumes the two eyes look the same way. Turn them inward onto a point and the shells stop being planes.

The shells become circles, and one of them is the horopter #

For a pair of eyes verged on a point, the quantity that decides disparity is the angle the baseline subtends at the point being looked at. Two points where that angle is equal have equal disparity; and the locus of points at which a fixed segment subtends a fixed angle is a circular arc through the segment’s ends, which is the inscribed-angle theorem and nothing more.

So every shell is an arc through both eyes, and the family is a nested set of them. Traced across ninety degrees of field and fitted, the constant-subtense locus is a circle to 3.6 × 10⁻¹⁴ metres — not a fit but arithmetic, since the geometry says so exactly.

The zero of that family is already an object here. The depth a pair calls zero finds it by bisection along 121 azimuths and reports it as a circle to four decimal places of a centimetre, lying 23 centimetres nearer than a flat wall at 26 degrees aside. What is new here is that the zero is not special: every whole-pixel disparity has its own arc, and the family fills the space in front of the eyes.

What a pinhole quantises is not that angle #

The theorem is about an angle, and a camera pair does not measure an angle. It measures a difference of two image coordinates, and an image coordinate is a focal length times a tangent.

So the loci a camera pair actually cuts are near-circles rather than circles. Fitted to the best circle through both eyes, the eight-, eighteen- and thirty-two pixel shells depart from it by up to 5.7 centimetres over ninety degrees of field — small against the shells’ own spacing at the middle of the field, which is 2.5 centimetres, but not small against nothing, and not at all small at the edge.

The departure has the same shape at every disparity, which is what says the cause is the projection rather than the depth. A wider lens makes it larger and a longer one smaller, because the tangent departs from the angle at a rate set by how far off-axis the point is in the picture rather than by where it is in the room.

Straight ahead, the old formula is exactly right #

A verged pair reduces to the rectified one in the middle of the field, and it is worth checking rather than assuming, because the two arrangements are described by different pictures.

Along the line of sight the shells of a pair verged at 1.2 metres are 22.3, 23.2, 24.1, 25.1, 26.2 and 27.4 millimetres apart across the six intervals nearest the fixation. The rectified pair’s own expression — depth squared times the step, over the focal length times the baseline — gives 24.6 millimetres at 1.2 metres, which is the middle of that run.

So nothing is lost in the middle. The verged family and the parallel family agree exactly where a reader would expect them to, and any difference has to be looked for off to the side.

Off to the side: the shells come nearer and get closer together #

Forty degrees aside, the arrangement looks quite different. The zero shell is 0.920 metres from the midpoint of the eyes where it is 1.200 straight ahead, and consecutive shells there are 10.4 to 11.8 millimetres apart rather than 22 to 27.

That is exactly the picture the circles predict: an arc through both eyes bends back toward them, so a ray leaving the midpoint at forty degrees crosses the family much closer in and crosses more of it per metre.

A reader could stop there and conclude that a verged pair resolves depth more finely off to the side. That conclusion is wrong, and finding out why is the rest of this essay.

The comparison that has to be made at the same place #

The two families put their shells in different places, so comparing the spacing “at forty degrees” compares two different ranges as well as two arrangements. The comparison that separates them fixes the range and the azimuth and asks how far apart the shells are there.

Done that way, the answer is flat. At 1.2 metres of range the spacing is 24.63 millimetres for the verged pair and 24.62 for the parallel one; at forty degrees aside and the same range, 18.85 against 18.86. Across the whole field the worst difference anywhere is 0.073 per cent.

So the shells are in quite different places and are spaced identically. Vergence moves the labels and does not change how finely the space is divided.

Why it could not have been otherwise #

The result is a negative one, and a negative result is worth an argument rather than only a number.

Turning a camera about its own centre does not change which rays it collects. It changes which image coordinate each ray is given, and nothing else: the bundle of rays through the centre is the same bundle, relabelled. Disparity is a difference of two such labels, so vergence adds a quantity that depends on the two rotations and not on the point, which shifts every disparity by a constant.

A constant shift moves the shells — the zero goes wherever the shift sends it — and leaves the rate at which disparity changes with range untouched. The spacing between consecutive shells is one over that rate, so the spacing cannot change. The 0.073 per cent that does change is the whole of the second-order effect: a pinhole’s label is a tangent, and the tangent’s derivative depends on where in the field the ray sits, so a rotation moves a ray to a slightly different part of the tangent’s own curve.

There is a way to see that it must be so without any algebra, and it is the cleanest form of the argument. Cover one eye, and the pattern of rays that eye collects is fixed by where it is, not by where it points. Uncover it and turn it: the rays are the same rays. Whatever a pair of eyes can distinguish is a property of the two bundles and of how finely each ray can be identified within its own bundle, and a rotation changes neither. Vergence is a choice of coordinates on a measurement that has already been made.

That also says what would change it. A pair whose baseline is longer, or whose focal length is longer, or whose reading is finer, divides the space more finely; a pair that merely points somewhere else does not. Depth is a reciprocal is the same statement about a single pair, and the three quantities in it — focal length, baseline, reading precision — are exactly the three that vergence is not.

What vergence does buy, since it is not resolution #

None of this says vergence is useless, and saying what it is for is the honest end of a negative result.

The parallel pair has no zero-disparity shell at all: its disparity falls toward zero as the range grows and reaches it only at infinity. A verged pair brings that zero into the room, onto a circle 0.44 metres in radius through both eyes.

That matters for matching rather than for measuring. A matcher searches a range of disparities, and the range it must search is set by the spread of disparities in the scene; verging on the object of interest puts that object’s disparities near zero and shrinks the search. It matters for a viewer, too, since the region around zero disparity is the region a pair of eyes can fuse, and fixation places that region on whatever is being looked at.

Both coordinates agree on a circle and a line and raise the gaze, and the line is gone take that zero apart in the directions this essay does not — the vertical coordinate, and what happens when the gaze is raised as well as turned. The shells measured here are horizontal disparity alone, which is the quantity a rectified matcher reports and the only one a pair of horizontally-separated cameras is usually asked for.

How the arcs were found #

The shells are traced rather than plotted from a formula, and the method is worth a paragraph because it is what makes the departure from circles measurable at all.

Along each of forty-one directions from the midpoint of the eyes, the disparity is computed at a range and compared with the target; the range at which it crosses is found by bisection to two hundred halvings. Nothing about circles enters. The circle is then fitted afterwards, with its centre constrained to the perpendicular bisector of the baseline so that it passes through both eyes by construction — one free parameter, found by a ternary search on the worst residual.

That order matters. Fitting a general circle to a near-circular locus would absorb the departure into a moved centre and report a smaller residual about a circle that does not pass through the eyes, which is a different claim. Constraining the fit to the family the theorem is about is what makes the residual a test of the theorem rather than of circularity.

The same discipline runs through the rest of the subject. Two rays that do not meet measures a miss rather than assuming an intersection; a third ray is worth what its picture is worth measures two estimators against a truth rather than against each other. A quantity computed from the thing it is meant to test is not a measurement of it.

What the numbers are for a human pair #

The arrangement drawn is a human one, and stating it in those terms makes the sizes concrete.

Sixty-five millimetres between the eyes, fixation at 1.2 metres, and a resolution of one pixel at nine hundred pixels of focal length — which is about half an arcminute, and is in the right region for a person’s own stereoacuity. The shells straight ahead are then two and a half centimetres apart at the fixation distance, and the zero shell is a circle 0.44 metres in radius passing through both eyes and the point being looked at.

Two consequences follow immediately. The shell spacing grows with the square of the range, so at three metres it is about fifteen centimetres and at ten about a metre and a half — and the range a pair cannot see past computes where that growth ends the instrument entirely, at fifty-eight and a half metres for a human pair. Nothing in this essay moves that limit either: it is the focal length times the baseline over the reading precision, and vergence is not in it.

And the whole family is small. The circle through both eyes and a point 1.2 metres away has a radius of 0.44 metres, so every shell in the picture is a modest arc rather than a nearly-straight line, and the bending is not a subtlety at the edge of the field but the dominant feature of the geometry.

Where this changes what a rig should do #

Two practical readings follow, and they point in opposite directions, which is why both are worth stating.

For a rig that measures, vergence is neutral and carries a cost. It buys no resolution, it complicates rectification — the standard pipeline warps a verged pair back to a parallel one before matching — and it makes the shells depend on where the cameras are pointed, which is a state that has to be known. A parallel pair is the simpler instrument and gives up nothing measurable.

For an observer, or for a rig whose problem is matching rather than measuring, it is worth a great deal. It puts the disparities of interest near zero, which is where the search is cheapest and where a human pair fuses. That the geometry of the shells changes shape at the same time is a consequence rather than a purpose.

The distinction is worth keeping because the two are easy to confuse, and the confusion has a direction: a rig builder who verges expecting finer depth has spent complexity on nothing, while an observer who does not verge cannot fuse.

What this does not settle #

Only the horizontal coordinate is measured. A verged pair produces vertical disparity too, and everywhere but on one line it is not zero. Nothing here asks how the vertical component would change the count of distinguishable places, which it must, since two coordinates carry more than one.

The eyes are verged symmetrically and in one plane. Asymmetric vergence, and a gaze raised out of the plane of the eyes, change the family’s shape and are the subject the horopter measurements above take up rather than this one.

And the reading error is a clean quantisation. A real matcher has sub-pixel interpolation and a correlated error, and whole pixels cut space into shells already separates the depths a reading can print from the depths it can tell apart — 449 against 149 at a quarter pixel. That separation applies here unchanged and has not been recomputed for the verged family, where the printed shells are arcs and the distinguishable bands are the regions between them.

Still open: what the second coordinate adds to the count #

The shells above are cut by one number, and a verged pair reports two.

A point off the plane of the eyes has a vertical disparity as well as a horizontal one, and the two are independent measurements of position. Quantising both cuts space into cells rather than shells — the intersections of two families of surfaces — and the count of distinguishable places is then a count of cells rather than of shells, which must be larger. How much larger is the open question, and the interesting part is that the two families are very differently shaped: the horizontal family is the near-circles measured here, and the vertical family is nearly flat, since vertical disparity varies slowly with range and quickly with height.

The measurement that settles it counts the cells inside a stated working volume for a verged pair, against the shells the horizontal coordinate alone provides, and asks where the gain is: whether the second coordinate adds most where the first is coarsest — far away, and off to the side — or whether the two families are nearly parallel over most of the volume and the extra coordinate buys little. If it is the first, a matcher that discards vertical disparity after rectification is throwing away precisely the part of the measurement it is shortest of.

The short version #

A verged pair’s shells of constant disparity are arcs through both eyes rather than planes of constant depth. The exact statement is about the angle the baseline subtends, whose loci are circles to 3.6 × 10⁻¹⁴ metres; the loci a camera pair actually quantises are near-circles, 5.7 centimetres off them across ninety degrees of field, because a pinhole measures a tangent.

Straight ahead the spacing matches the rectified pair’s expression exactly, 24.6 millimetres at the fixation distance. Off to the side the shells are nearer and closer together — 0.920 metres and 11 millimetres at forty degrees — but so are a parallel pair’s at the same place, to 0.073 per cent. Vergence moves the shells and does not respace them, so it buys matching and fusion rather than resolution.