Square to the camera is the worst mirror

Everything in one shutter, two views and two matches are enough is about how well a mirror pair is determined. Neither says whether it is any good, and those are different questions: an arrangement can determine its answer perfectly and determine it terribly, which is the distinction an ambiguity is not an uncertainty is written to keep apart.

The mirror pair is well determined everywhere and very badly conditioned in one particular arrangement, and the arrangement is the obvious one — a mirror facing the camera.

The baseline has no choice about its direction #

A pair of photographs can be taken from anywhere. A mirror pair cannot: the second centre is the first reflected in the mirror, so the segment joining them is perpendicular to the mirror and that is the end of it. The photographer chooses how far from the glass to stand, which fixes the baseline’s length, and chooses how the mirror is hung, which fixes its direction. There is no third choice.

Set the mirror square to the camera and that direction is the direction the camera is looking. The second eye sits directly behind the first, which is a forward-motion pair: the arrangement a camera makes by walking toward what it is photographing rather than stepping sideways.

Forward motion is the worst-behaved geometry in the two-view subject, and the collection has already measured why in a different costume. A turn of the head is not a step sideways is about the opposite extreme — a baseline of zero, where nothing at all can be recovered — and the material between the two extremes is the same material: how much parallax an arrangement actually produces, against how much a count of its unknowns suggests.

The epipole comes to the middle #

A forward-motion pair puts its epipole where the camera is heading, which is near the middle of the frame. In a mirror pair the epipole is where the reflected lens appears, and a mirror square to the camera puts the reflection of the lens in the centre of the picture.

That matters because of the constraint from the previous rung. Every mark and its reflection are joined by a line through the epipole. When the epipole is in the middle of the frame, those lines radiate from the centre, and a mark near the centre lies close to its own reflection — so the two views of it are nearly the same view, and there is almost nothing to triangulate with.

The angle the rays cross at #

The quantity underneath all of this is the angle at which the two rays to a mark cross. The ray to the direct image is a ray of the real camera; the ray to the reflected image is that camera’s ray reflected in the plane, which is a ray from the reflected centre. They meet at the mark, and the angle they meet at is the parallax the arrangement supplies.

A displacement along either ray moves the intersection by an amount that grows as one over the sine of that angle, which is the law two rays that do not meet sets out for the general case. At sixty-five degrees a pixel of error is worth roughly a pixel’s worth of depth; at twenty-three degrees it is worth two and a half times as much, and at five degrees it would be worth eleven.

What the tilt is worth, in millimetres #

The honest way to price this is not to nudge one image coordinate and report the displacement. That is a measurement of the direction the nudge happened to be in, and on this site’s own room it gets the answer backwards: nudging the direct image sideways in a square-mirror arrangement moves the recovered point along the well-determined axis, and reports the degenerate case as the good one.

So all four image coordinates are perturbed in turn, the four displacement vectors are collected as the columns of a Jacobian, and what is quoted is the largest displacement any perturbation of a given size can produce — the largest singular value, which is the number a worst case actually needs.

Threefold, bought with nothing but where the mirror hangs. And the curve has a feature worth reading rather than smoothing: it rises slightly over the first few degrees before falling. That is not noise. Turning the mirror from square sweeps the epipole across the frame, and for the first few degrees it sweeps toward this particular mark before carrying on past it. A mark the epipole is passing over is the worst-placed mark there is.

The crossing angle accounts for three quarters of it #

Two numbers in the sections above invite being divided into each other, and the division is worth doing because it does not quite come out.

The crossing angle goes from 23° square to 65° turned, so the $1/\sin$ law predicts an improvement of $\sin 65° / \sin 23° = 2.32$. The measured worst-case displacement goes from 4.8 mm to 1.6 — a factor of 3.0. So the crossing angle explains about three quarters of the gain, in logarithms, and a quarter comes from somewhere else.

The somewhere else is a real feature of a mirror pair rather than slack in the measurement. The $1/\sin$ law is derived for two rays perturbed independently, and a mirror pair’s two rays are not independent: both are rays of one camera, so the four image coordinates being perturbed are four coordinates of a single photograph and the resulting displacements are correlated. The largest singular value of the Jacobian therefore picks up a contribution the two-ray argument has no term for, and turning the mirror changes that contribution as well as the crossing angle.

Which is the reason for taking a singular value rather than a nudge in the first place, and the reason this essay’s own warning about nudging one coordinate is not a fussy one. The 1.6 and the 4.8 are worst cases over every perturbation direction; the 2.32 is what a two-ray idealisation would give; and the gap between them is the arrangement’s own coupling, visible only because the two were computed separately.

The number to watch is on the print #

The mirror’s angle is not something a photographer measures. What is visible, in the viewfinder and in the finished photograph, is where the reflection of the camera falls in the frame — and that is the epipole, which is the quantity the conditioning actually depends on.

So the instruction is a compositional one. Arrange the shot so the camera’s own reflection is well away from the centre of the frame, and away from whatever is being measured. That is a rule that can be followed while looking through a viewfinder, and it is the same rule a surveyor follows when refusing to sight along a line: the evidence has to have a direction to it.

The other three degeneracies #

Turning the mirror fixes the arrangement’s worst case. Three other failures are not about the tilt at all, and each has a different signature.

A mark on the mirror plane is its own reflection. Dust on the glass, a scratch, the frame of the mirror itself: the direct image and the reflected image coincide, the joining line is undefined, and the correspondence carries no information. A fit handed several of these does not fail loudly; the rows of the design matrix are near zero and the answer is determined by whatever else is present, which is the quiet kind of failure. The remedy is to notice, and the way to notice is that the two marks are in the same place.

A scene confined to one plane parallel to the mirror gives every mark the same depth, so every joining line has the same length and the arrangement carries no depth information to recover. That is the planar degeneracy a flat scene fixes no second eye measures at length, and it arrives here unchanged.

A curved mirror has no reflected centre. There is no second camera, the joining lines do not concur, and the skew fit returns a matrix whose residual reports the curvature rather than the marks. That is the useful diagnostic in the whole family: the concurrence test is a flatness test.

Where the ill-conditioning actually lives #

It is worth being exact about which quantity is badly determined in the square arrangement, because “the reconstruction is poor” hides a structure that is easy to use once it is seen.

The Jacobian of the triangulation has three singular values, and in a forward-motion arrangement they are not merely small — they are unequal by a large factor. The largest belongs to a direction that runs roughly along the line of sight, and the other two belong to directions across it. So the recovered point is well placed on the page and badly placed in depth, which is the same anisotropy the metrology field finds when it unprojects a mark onto a ground plane and gets an ellipse rather than a disc.

That structure is usable. A measurement that only wants the shape of something in a plane facing the camera is barely affected by the square arrangement; a measurement that wants how far one object stands in front of another is affected by all of it. Quoting one error figure for the recovered point throws that distinction away, and the distinction is generally the thing a reader wanted.

Two mirrors, and why they do not simply add #

The obvious response to a badly placed mirror is a second mirror. It works, and it works less well than counting the images suggests.

Two mirrors produce a set of virtual cameras rather than one, and every one of them is a genuine extra view of the scene taken in the same exposure. But their centres are not scattered: they are the original centre reflected and re-reflected, so they all lie on one circle about the line where the mirrors meet, at the same distance from it. A set of viewpoints confined to a circle is a narrower set than the count suggests, and the directions it fails to constrain are the directions along that circle’s axis.

That is the same complaint as the one above, one level up: what an arrangement supplies is not a number of views but a spread of viewpoints, and the second is what the conditioning reads. The geometry of that circle is the subject of the next essay in this field, which is about what two and three mirrors generate rather than about what they measure.

Why this is not the same as choosing a stereo baseline #

A photographer setting up an ordinary stereo pair trades baseline against matching: a wide baseline gives better depth and makes correspondences harder to find, because the two views look less alike. The range a pair cannot see past prices one side of that and a wrong match is not a small error prices the other.

A mirror pair’s trade is different in an interesting way. Turning the mirror does not make the two views less alike — they are the same photograph, and a mark and its reflection are visible together, in one exposure, with the same lighting and the same exposure interval. What turning the mirror costs is field: a mirror at forty-four degrees reflects a narrower slice of the room, so there are fewer correspondences to be had even though each is better placed.

That is a more forgiving trade than the ordinary one, and it is the practical argument for the whole arrangement. The hard part of stereo is correspondence, and a mirror hands the correspondences over: two appearances of one object, in one frame, at one instant, joined by a line a ruler can draw.

A rule of thumb, and what it is worth #

A reader wanting one sentence to carry away can have this: a mirror pair behaves like an ordinary stereo pair whose baseline is twice the distance to the glass, reduced by the sine of the angle between the mirror’s normal and the line of sight to whatever is being measured.

The reduction is the whole of the essay. A mirror square to the camera and a mark straight ahead give a sine near zero, and a baseline multiplied by nothing is not a baseline. Turning the mirror, or measuring something off to the side, or both, brings the factor up.

Two cautions go with it. The rule is a first-order statement and it is stated on one mark rather than on a scene, so a photograph with marks spread across the frame has a different factor for each of them — the ones near the reflected lens are badly served while the ones at the edge are well served, in the same exposure. And the rule says nothing about how many correspondences exist, which the mirror’s field of view decides and which turning the mirror makes worse. The arrangement to aim for is the one that trades a little field for a lot of angle, and there is no formula for where that balance sits, because it depends on what the photograph is of.

The one case where square is right #

The essay’s title is a claim about triangulation, and there is a measurement a mirror pair makes for which square is the best arrangement rather than the worst. Saying so is not a hedge; it is what separates a claim about an arrangement from a claim about mirrors.

A mirror square to the camera puts the epipole at the principal point, and the epipole is the vanishing point of the mirror’s normal. So a square mirror is a mirror whose normal points at the camera, and the position of the epipole in the frame is a direct reading of the mirror’s orientation. Anyone wanting to align a mirror — to set it perpendicular to an optical axis, which is a common enough problem on a bench — has exactly this instrument: turn the mirror until its own reflection of the lens sits at the principal point, and it is square to within whatever the reading precision allows.

The two uses want opposite arrangements for the same reason. Triangulation wants the epipole far from the marks, because that is where the parallax is. Alignment wants the epipole at a known place, because that is where the reading is. One geometry, two jobs, and the quantity both of them turn on is where the reflected lens falls.

What is measured here #

One mark, triangulated across a mirror in one room, with the mirror turned through forty-four degrees.

The worst displacement three tenths of a pixel of marking error can produce falls from 4.8 millimetres to 1.6, a factor of three. The angle at which the two rays cross rises from 23° to 65°. The epipole moves from a hundred and seventeen pixels off the centre of the frame to four hundred and ninety-one. And the error rises before it falls over the first eight degrees, because the epipole passes over the mark on its way out of the picture — which is what says the governing quantity is the mark’s distance from the epipole rather than the mirror’s angle.

The short version #

A mirror pair’s baseline runs along the mirror’s normal, so hanging the mirror decides the geometry and nothing else does. Square to the camera is the forward-motion case: the reflected lens sits in the middle of the frame, the joining lines radiate from there, and the marks nearest the middle are the least separated from their own reflections.

Turning the mirror carries the epipole out of the picture and opens the crossing angle from twenty-three degrees to sixty-five, cutting the worst-case depth error threefold. The quantity that decides it is where the camera’s own reflection falls on the print, which is a thing to compose rather than a thing to compute.