The depth a pair calls zero

Depth is a reciprocal established the relation every stereo measurement rests on: disparity is the baseline times the focal length over the depth, so a fixed error in what is read maps to a depth interval that is not symmetric and eventually is not bounded.

That relation is written for a parallel pair — two cameras looking the same way. Turn them inward, which is what a pair of human eyes does continuously and what turning the cameras inwards measures for a stereo rig — and which a turn of the head is not a step sideways shows is not the same as moving — and the place where the disparity is zero stops being infinity and becomes somewhere in the room.

Where, exactly, is the subject here, and the answer is not the place everybody draws.

Found rather than derived #

The locus is computed by bisection. Along each of 121 azimuths from the midpoint between the eyes, the distance is searched for at which the two image coordinates agree, and the result is a point. Nothing about a circle is assumed anywhere.

Then a circle is fitted to the 121 points and the fit’s worst departure is reported: 0.0000 centimetres over the whole field. And the fitted circle passes through both eyes, to under a fifth of a millimetre, which is what identifies it as the Vieth–Müller circle rather than as some other circle that happens to fit.

Fitting rather than assuming matters because “it is a circle” is the claim. A derivation would demonstrate that the algebra has a circle in it; the fit demonstrates that the arrangement does.

Why a circle #

The angle a chord subtends at points on a circle is constant, which is the inscribed-angle theorem, and it is the whole of the geometry.

The two eyes and a scene point make a triangle whose angle at the point is the convergence angle. Zero disparity means the point images at the same coordinate in both pictures, which for a pair verged on a fixation point means its convergence angle equals the fixation point’s. The locus of points at which a fixed segment — the interocular baseline — subtends a fixed angle is an arc of a circle through the segment’s ends.

So the circle is not an approximation, a small-angle result or a physiological finding. It is a plane-geometry theorem, and the bisection above is what says the implementation agrees with it.

The number that matters: the bow #

A wall at the fixation distance is not on the horopter, and the departure is the quantity a reader can act on.

Straight ahead the locus is at 1.20 metres, which is the fixation distance. At 26 degrees off centre it is 23 centimetres nearer. So a flat wall the pair is fixated on carries disparity everywhere except straight ahead, and the sign of that disparity says the wall’s edges are further than the fixation point — which reads as a wall bowed away from the viewer.

That is a geometric fact about the arrangement and not a failure of anything. It is why a stereo rig fixated on a flat subject records a curved one, and it is the reason the toed-in configuration is generally avoided in stereography in favour of parallel cameras with a sensor shift.

Where the twenty-three centimetres comes from #

The bow is quoted as a length, which is the form a reader can act on and the form that hides its scaling. It is worth deriving, because the derivation says which of the arrangement’s numbers it depends on and — more usefully — which it does not.

Take the eyes as effectively coincident at the origin for the moment. The horopter is then a circle of diameter $D$ through the origin and the fixation point, and a circle through the origin has the polar equation $r(\varphi) = D\cos\varphi$. The depth of that point — its distance along the forward axis — is $r\cos\varphi = D\cos^{2}\varphi$. A flat wall at the fixation distance has depth $D$ everywhere. So the departure is

$$D - D\cos^{2}\varphi = D\sin^{2}\varphi,$$

which is the whole of it. The bow at a given angle off centre is the fixation distance times $\sin^{2}$ of that angle, and nothing else enters.

That is checkable against the three verges the figures draw, since each fixes a different $D$ through the circle’s radius $b/(2\sin\theta)$. At a vergence of 3.10° the fixation distance is 1.202 m and $D\sin^{2}26° = 23.1$ cm. At 4.80° it is 0.777 m and the formula gives 14.9 cm. At 1.80° it is 2.069 m and the formula gives 39.8 cm. The figures report 23, 15 and 40, so the expression reproduces all three from the vergence alone.

The interocular distance is absent, and that is the surprising part. The bow is not a consequence of the eyes being far apart; it is a consequence of the locus being a circle rather than a line, and the circle’s diameter is the fixation distance whatever the baseline. Carrying the baseline through exactly puts the circle’s diameter at $D + b^{2}/4D$, which at 65 mm and 1.20 m is 0.9 mm — a correction three hundred times smaller than the effect it corrects. A rig with a metre of baseline fixated at ten metres bows by the same fraction as a pair of eyes fixated at one.

So the honest statement of the bow is a ratio rather than a length: 19% of the fixation distance at 26° off centre, 6.7% at 15°, and 50% at 45°, where the horopter’s depth is half the distance the pair is fixated at. Quoting centimetres makes the effect sound like a property of a particular table; quoting $\sin^{2}\varphi$ makes it what it is, which is a property of the field of view.

That form also explains why the effect is invisible in most stereo work and unavoidable in some. It grows as the square of the angle for small angles, so a narrow field pays almost nothing — a 5° field bows by 0.8% — while a wide one pays a great deal. A pair of eyes has a useful binocular field tens of degrees wide, and so does a headset; a survey rig with a long lens does not. The bow is the price of the field, and it is paid at the edges.

Why the plane is the drawing everybody makes #

Because at long fixation distances it is nearly right, and because the error is in the direction nobody checks.

The circle’s radius is roughly the fixation distance, so as the fixation point recedes the circle flattens and the bow across a fixed field falls. At 1.2 metres the bow across ±26 degrees is 23 centimetres — a fifth of the fixation distance. At 12 metres it is a fifth of a metre over the same angular field, which on a scene 12 metres deep is invisible.

So the plane is a good approximation for a rig looking at a distant scene and a bad one for a rig looking at a table, which is exactly backwards from where stereo depth is most used. Close work — surgery, inspection, a headset’s near field — is where the bow is largest.

What the horopter is and is not #

Three clarifications, because the word carries baggage from a neighbouring subject.

It is the theoretical horopter. The locus computed here is the one the geometry gives. The empirical horopter — where a person actually reports single vision — is measured psychophysically and is flatter than the circle, which is a fact about the visual system and not about projection. This collection computes geometry and says so; the difference between the two is real and belongs to somebody else’s field.

It is a circle only in the plane of the eyes. In three dimensions the full zero-disparity locus is a surface, and for a verged pair with no cyclorotation it is the circle crossed with a vertical line — a cylinder in the idealised case. The figures here are a plan, and the plan is where the circle lives.

And it is not the depth of field of anything. Points off the horopter are not invisible, badly reconstructed or out of focus. They simply have non-zero disparity, which is what a stereo pair measures depth with.

What it says about a display #

A stereoscopic display sends two pictures to two eyes, and the viewer’s own eyes verge on whatever the display makes them verge on.

That means the horopter is in the room, not in the scene, and it is set by the disparity on the screen rather than by the geometry of the recorded scene. A scene point rendered with zero screen disparity lands on the viewer’s own zero-disparity locus, which is a circle through their eyes and the screen — and the screen is flat.

So a stereoscopic display asks the viewer’s eyes to accept a flat surface of zero disparity when their geometry offers a curved one. The mismatch is small for a screen at arm’s length and a narrow field, and it grows with field of view, which is one of several reasons a wide headset is a harder problem than a monitor. The distance at which the eyes part measures a different member of the same family of mismatches, and what the two eyes are sent is where the two pictures a display delivers were first taken apart.

The measurement’s own limits #

Two, both stated because the figure looks more general than it is.

One baseline, one fixation distance, one field. The numbers quoted — 65 millimetres, 1.20 metres, ±26 degrees, 23 centimetres of bow — belong to that arrangement. The slider sweeps the verge, which changes all of them; the shape of the result does not change, because the shape is a theorem.

And the bisection needs the disparity to change sign along each ray. Azimuths where it does not are dropped and counted rather than interpolated. On this arrangement 121 of 121 azimuths returned a point; on a harder verge some do not, and the figure says how many it found rather than assuming it found them all.

Where the circle comes from, in one construction #

Worth setting out because the inscribed-angle theorem is the whole proof and it fits in a paragraph.

Let the two eyes be L and R and the fixation point F. The angle ∠LFR is the vergence. A point P has zero disparity in a toed-in pair exactly when the two eyes’ rotations required to fixate P are equal and opposite to the ones required for F — which, once the pictures’ coordinates are written out, comes to ∠LPR = ∠LFR.

The set of points at which a fixed segment subtends a fixed angle is two arcs of one circle through the segment’s ends, and the branch on the viewer’s side of the baseline is the horopter. The circle’s radius is b / (2 sin θ) with b the interocular distance and θ the convergence angle at the fixation point, which is twice the angle each eye is turned inward by. For small vergence that comes to half the fixation distance — 0.601 metres of radius here against a fixation distance of 1.20, so the diameter is the fixation distance to three figures, and confusing the two is the easiest mistake available because they are numerically so close.

That relation is why the bow scales the way it does, and it is checkable directly: the fitted circle’s radius is 0.6010 metres, and b/(2 sin θ) at 65 millimetres and a convergence of 3.10 degrees is 0.6008. Using 1.55 degrees instead gives 1.2015, which is the fixation distance — a factor of two that the numbers themselves catch, and which the site’s gate now asserts against.

What a parallel pair does instead #

Turn the vergence to zero and the circle’s radius goes to infinity: the locus becomes a line at infinity, which is to say that a parallel pair has zero disparity nowhere in the room and reports every visible point as nearer than infinity.

That is the arrangement depth is a reciprocal is written for, and it is the arrangement most stereo rigs and every rectified pair are actually in. Its convenience is that disparity is a single monotone function of depth with no sign changes, so a matcher searching along a scan line never has to consider two candidates on opposite sides of a zero.

The price is that everything has disparity, including the far background, and the useful range is set at the far end by the reading error rather than by the geometry. That is the range a pair cannot see past, and it is the far-end partner of this essay’s near-end question.

Why this belongs beside the round’s ambiguities #

The other essays in this round are about arrangements where a recovery has no answer. This one is about an arrangement where it has an answer that is not the one a reader assumes.

Both are failures of a model rather than of an instrument, and both are invisible from inside. A pipeline that assumes zero disparity means the fixation plane will report a flat wall as bowed and will find nothing wrong; a pipeline that assumes a general scene when the scene is a plane will report a confident wrong pose and find nothing wrong. The remedy in both cases is knowing which geometry is in hand, and neither is discoverable from the residuals.

What it means for a matcher #

A stereo matcher searches for a correspondence along an epipolar line, and where the horopter is decides where the search starts.

On a parallel pair the disparity is one-signed: everything in front of the cameras has positive disparity and the search runs in one direction from zero. On a verged pair the disparity changes sign across the horopter, so a matcher has to search both ways, and a candidate at +2 pixels and one at −2 are both plausible for a point near the fixation distance.

That is a real cost and it is one of the reasons production stereo rigs are parallel and rectified rather than verged. It is also why a headset’s rendering is done with parallel frusta and a sensor shift rather than by rotating the two cameras inward: the shift moves the zero-disparity plane without introducing a zero-disparity circle, which is the same trick turning the cameras inwards measures.

Human eyes do vergence anyway, continuously and without difficulty, which is a fact about a visual system rather than about a matcher. This collection computes the geometry and leaves that where it belongs.

What the bisection assumes #

One assumption is buried in the search and it is worth stating, because it is what makes 121 azimuths return 121 points.

The disparity along a ray from the midpoint between the eyes is assumed monotone in distance, so a sign change brackets exactly one crossing and bisection finds it. That is true for a verged pair over the range searched, and it is the reason a closed form is not used: a formula would hide the assumption inside the algebra it is meant to be checking.

Where it stops being true is behind the eyes and at very short range, which is why the search starts at 15 centimetres rather than at zero. An azimuth whose disparity does not change sign over the searched range is dropped and counted, and the figure reports how many were found rather than assuming all of them were — the same discipline the collection’s other sweeps use when a member of a family has no answer.

The short version #

Two eyes verged on a point agree on a circle through both eyes and that point, not on a plane at the fixation distance. Found by bisection along 121 azimuths and fitted rather than assumed, the locus is a circle to 0.0000 centimetres and passes through both eyes to under a fifth of a millimetre.

At 65 millimetres of baseline and 1.20 metres of fixation, the circle lies 23 centimetres nearer than the fixation plane at 26 degrees off centre. The bow falls as the fixation distance grows, so the plane everybody draws is a good approximation for distant work and a poor one for near work — which is where stereo depth is mostly used.