The second disparity cuts cells

Whole pixels cut space into shells found that a stereo pair does not measure distance on a scale. Reading disparity to whole pixels, it chooses among 113 named depths between half a metre and twelve — shells 6.7 cm apart at two metres and 1.39 m apart at ten — and a level floor comes back as thirty-five standing plates. Vergence moves the shells and does not respace them then turned the eyes inward and found the shells relabelled rather than sharpened: the family becomes circles through both eyes, the twenty-pixel shell moves from 3.82 m to 0.74 at forty degrees aside, and consecutive shells stay the same distance apart to within 0.07 per cent.

Both of those count one number. A point that does not lie in the plane the two eyes share has two disparities — its images differ in row as well as in column — and the second is an independent reading of where it is. Quantising both cuts space into cells rather than shells, and the count of places a pair can name becomes a count of cells.

How many more that is, and where in a room the extra ones are, is what this measures. The answer to the second question is not the one the question was asked in expectation of.

Two families of slabs #

Reading disparity to a whole pixel divides space into slabs: everywhere the horizontal disparity rounds to the same integer is one slab, and a pair cannot tell two points inside it apart. Reading the vertical disparity to a whole pixel divides space into a second family of slabs, and a pair that reads both can tell two points apart unless they share a slab of each.

The picture is the answer to the first thing worth checking. Two readings only cut a space usefully if they disagree about direction: readings whose contours run nearly parallel leave cells that are long slivers, and a sliver is barely better than a slab. These two are nearly perpendicular. Over a room reaching 2.4 m across, 1.2 m tall and out to twelve metres, the angle between the two gradients runs from 72.5° to 90.0°, and it is exactly 90° straight ahead, where the horizontal reading responds to range and the vertical one to lateral position and neither to the other.

So the cells are compact rectangles rather than slivers, and the second reading is genuinely a second reading.

Forty-three times as many places #

With that settled the count is arithmetic. Sample a room of that size at 86,400 points, round both disparities at each of them, and count the distinct labels.

A pair verged at 1.2 m returns 179 distinct horizontal disparities and 7,663 distinct pairs, a gain of 42.8. That is a large number and it is worth saying what it is not: it is not forty-three times the resolution in depth. It is forty-three times the count of places the pair can put a name to, and most of the new names distinguish points at the same range from each other rather than points at different ranges.

The control is the striking half. A pair whose eyes look straight ahead returns 64 shells and 64 cells — the same number, exactly, not nearly. Two eyes that are not turned inward image every point on the same row in both eyes, so their vertical disparity is identically zero at every point of the room, and there is no second reading to have. The whole of the gain belongs to vergence.

That is a stronger statement about vergence than vergence moves the shells and does not respace them made, and it does not contradict it. Turning the eyes inward does not respace the shells, which is what that essay measured, and it creates a second family of them from nothing, which is what this one measures. A reason to verge exists; it is just not the reason usually given.

Both readings go coarse together #

The question that essay actually left was about where the gain is — whether the second coordinate adds most where the first is coarsest, far away and off to the side, in which case a matcher that discards it is throwing away exactly the part of the measurement it is shortest of.

They fall together. Straight ahead, a metre of motion buys 162 px of horizontal disparity at 0.6 m and 0.41 px at twelve; the vertical reading buys 34.0 px and 0.085. Both are one over the square of the range, so the ratio is 4.80 at 0.6 m, 4.80 at 1.2, 4.80 at 2, 4.80 at 4, 4.80 at 8 and 4.80 at 12 — a constant, to three figures, across a twentyfold change of range.

That settles it in the unwanted direction. The vertical reading is a fixed fraction as sharp as the horizontal one everywhere, so it is coarse exactly where the horizontal one is coarse. It is never a rescue; it is always the same auxiliary.

Off to the side the ratio is smaller — 4.80 straight ahead falls to about 2.0 by twenty degrees, because a point away from the median plane sees the two eyes at more different vertical angles — so the vertical reading does improve relative to the horizontal one across the field. That is a change of a factor of two, against a change of more than four hundred in the sharpness of either reading between 0.6 m and twelve.

The constancy is not a coincidence and it is the same fact depth is a reciprocal is about. Both disparities are differences between two projections of one point, both projections divide by the distance along the eye’s own sight line, and the two sight lines differ by an angle fixed once the eyes are aimed. Everything that varies with range in one varies with range in the other in the same way, and a ratio of two quantities that both go as one over the square of the range is a number. So the constant 4.80 is a property of this pair’s baseline, vergence and height off the plane, and not of anywhere in the room.

A cell is a tube, not a box #

Before the gain can be located it is worth saying exactly what a cell is, because “cell” suggests something it is not.

Two numbers cannot fix three, and a pair reading two disparities has two numbers. A cell is therefore the intersection of two slabs in a three-dimensional room, which is a tube — bounded across the two readings and unbounded along the direction in which neither changes.

At 1.2 m straight ahead the tube is 25 mm across the horizontal reading and 118 mm across the vertical; at six metres, 615 mm and 2.94 m. And the direction it runs in is not arbitrary: straight ahead it is within 4.6 degrees of vertical. Moving a point half a metre straight up changes the horizontal disparity not at all and the vertical disparity by 0.12 px, while moving it two centimetres sideways changes the vertical disparity by 0.061 px — the same amount, for a twenty-fifth of the distance.

So the second reading is not a reading of height, which is the natural guess from the words “vertical disparity”. It is a reading of lateral position, and the height of a point is exactly what neither disparity fixes. A pair reading both still knows where a point is only up to a vertical line.

Where the gain comes from, and it is not distance #

That says what the second reading measures. What remains is where in a room it measures anything at all, and that has a different answer again.

How sharply the vertical disparity responds to a sideways motion is proportional to how far the point stands out of the plane the eyes share — 0.292 px a metre at five centimetres above it, 1.358 at twenty-five, 5.417 at a metre, in exact proportion. So the second reading is worth nothing at all in that plane and more the further out of it a room goes. A room reaching ten centimetres either side gains 17.6 times; sixty centimetres, 42.8; a metre and a fifth, 55.0. A pair looking along a corridor it lies in the middle of gains the least it can, and one looking at a tall room gains most.

That is the sense in which the second reading is a height effect while being a lateral measurement. Height sets how sharp it is; what it then tells a reader is where the point stands across the field.

Split by range instead and the direction is the one nobody wanted. The near metre of the room gives a gain of 60.8, the middle band 19.8, and the far band from four metres to twelve gives 3.7. The second reading multiplies the count of nameable places most where the horizontal reading already names them 6.7 cm apart, and least where they are 1.39 m apart.

So the worry was right about the loss and wrong about its shape. A matcher that reduces a verged pair to one coordinate throws away a factor of forty-three in this room — a great deal — but it throws away almost none of what it is shortest of. What a reader short of far-field depth loses by discarding the vertical disparity is a factor of 3.7 on a scale where the shells are already a metre and a half apart.

What a matcher would have to do to collect it #

A gain of forty-three is only a gain if something can take it, and the second reading is small in exactly the way that makes it hard to take.

Straight ahead at two metres and twenty-five centimetres above the plane, the vertical disparity is 0.061 px. A metre of lateral motion changes it by 1.36 px. So the whole of the second reading, over a room, lives in the first pixel or two — and the count of 7,663 is made almost entirely of places far enough off the axis and far enough out of the plane for it to reach a whole pixel at all.

Two things follow for a reader deciding whether to collect it. A matcher reading disparity to whole pixels gets almost nothing from the second coordinate near the middle of its field, where its first coordinate is sharpest and where most of what it is looking at usually is. And a matcher reading to a quarter of a pixel gets the second coordinate over four times as much of the room, because the region where the vertical disparity exceeds the reading step grows as the step shrinks.

That makes the second coordinate a sub-pixel measurement in a way the first is not. Whole pixels cut space into shells found that refining the reading and refining the matcher are different purchases; here they are the same purchase, because the thing being refined and the thing being enabled are one quantity.

What this counts and what it does not #

A tube is still not a point. Everything above counts labels, and a label names a tube a metre long rather than a place. Depth is a reciprocal established the shape of the first coordinate’s coarseness and the range a pair cannot see past put a limit on it; this adds a second coordinate and does not add a third. What would is the image row itself, which the next section says is not counted here.

It counts disparity labels, not everything a pair can see. A matcher returns a pair of disparities, and the count above is a count of distinct pairs. A reader who also uses the row a point is drawn on — which constrains its elevation directly — has more than this, and a rectified pair whose vertical disparity is zero by construction still knows a point’s height that way. The comparison is between what the disparities distinguish, which is the quantity the two earlier readings counted.

The room is a choice and the number depends on it. Forty-three is the gain in a room 2.4 m across, 1.2 m tall and 0.5 to 12 m deep, at a baseline of 65 mm and a focal length of 900 px. A taller room gains more and a flatter one less, which is the point of the height sweep; but no single number here is a property of the pair alone.

Whole pixels are the crudest reading there is. Whole pixels cut space into shells found that a quarter-pixel reading with a quarter-pixel of matcher error prints 449 depths where whole pixels print 113, and the same refinement applies to both coordinates here. A finer reading and a second coordinate are different purchases and nothing above says which is the better one.

And the eyes are pinholes that yaw and do not pitch. Real eyes obey a torsion rule as they move, and raise the gaze and the line is gone is where the consequences of that live. A pair whose eyes carried torsion would have a vertical disparity in the plane as well, and the clean zero the control rests on would not be clean.

The second reading, priced #

A point off the plane two eyes share has a second disparity, and quantising it cuts a second family of slabs across the first. The two families are nearly perpendicular — 72.5° to 90.0° across a room — so the cells they cut are compact and the second reading is genuinely independent of the first.

Counted, a verged pair returns 179 distinct horizontal disparities in a stated room and 7,663 distinct pairs of them: a gain of 42.8. A pair looking straight ahead returns 64 and 64, the same number exactly, because its vertical disparity is identically zero. The second coordinate belongs to vergence and to nothing else.

A cell is a tube rather than a box, because two numbers cannot fix three: 25 mm across the horizontal reading at 1.2 m and 118 across the vertical, running within 4.6 degrees of vertical. What the second reading measures is lateral position, not height; height is what sets its sharpness and is also the one thing it leaves free.

And it is not where it was wanted. Both readings fall as the square of the range and their ratio is 4.80 at every range in the room, so the vertical reading is coarse wherever the horizontal one is. The gain is 60.8 in the near metre and 3.7 beyond four metres, and it comes from height rather than from distance: 17.6 in a room ten centimetres deep either side of the plane, 55.0 in one reaching a metre and a fifth.

Still open: what a rectified pair spends when it flattens the rows #

A rectified pair has no vertical disparity by construction — that is what rectification is for, and rectification is a family, not an operation found the family of ways to achieve it, every member putting all matches on common rows and disagreeing only about what it does to the pixels.

That raises a question the count above cannot answer, because it compares a verged pair with a parallel one rather than a verged pair with its own rectification. Rectifying is a warp of the two images and therefore a bijection: no ray is lost and nothing about the scene becomes unknowable. What changes is where the pixel grid sits relative to the disparities, and the count of distinguishable labels is a property of the grid rather than of the rays.

The measurement that settles it takes one verged pair, rectifies it by each member of that family in turn, and counts the cells in the same room after each — against the 7,663 the unrectified pair gives and the 179 its horizontal coordinate alone gives. If the rectified counts land near the first, then rectification costs nothing and the second coordinate is a bookkeeping artefact of an unrectified frame. If they land near the second, then rectification really does spend most of a pair’s resolving power for the convenience of a scanline, and the family’s members differ in how much of it they spend — which would make the choice among them a resolution decision rather than the pixel-stretching one it is currently made on.