Two mirrors are three cameras

One shutter, two views reads a photograph with a mirror in it as a stereo pair, and finds the pair unusually well behaved: its fundamental matrix is skew-symmetric, both its epipoles are the same point, and that point is where the camera would see its own lens. Two matches are enough then fits the whole geometry from two correspondences and a straightedge.

Hang a second mirror and the exposure holds three views rather than two. What is worth working out is whether the third one is worth anything.

Three appearances, three places #

A dressing table with two wings is the ordinary arrangement, and a camera in front of it photographs every object three times over: once directly, once in the left mirror, once in the right. The three appearances are three photographs in the sense that matters — each is a central projection of the scene from a definite point, and the three points are different.

The first is the lens. The second is the lens reflected in the left mirror, as far behind the glass as the camera stands in front. The third is the lens reflected in the right. Nothing about the exposure is unusual and nothing is composited; the three views are simply lying in one frame, in different parts of it.

With the mirrors a metre and a half away, the two mirror baselines come to 2.90 and 3.10 metres — twice each distance, which is the one length a single photograph cannot supply and a tape measure can. The third separation, between the two reflected eyes, is 2.26 metres, and it is the interesting one: there is no mirror between those two eyes, and neither of them is the real one.

The third pair is not a mirror pair #

The two pairs that contain the real view behave as expected. Their matrices are skew-symmetric to about a part in ten million million, which is arithmetic rather than a fit, and their epipoles are the two lens-reflections.

The third pair is different in kind. Going from the left mirror’s view to the right mirror’s is a reflection undone and another applied — two reflections, which compose into a rotation. A rotation is a proper motion, a genuine displacement of one camera to another, and the pair it produces is an ordinary stereo pair with nothing skew about it.

Measured on the same marks, the two mirror pairs score 1.1 × 10⁻¹² and 2.0 × 10⁻¹³ on the distance-from-skew that one shutter, two views uses, and the pair of reflections scores 2.000 — the far end of the same scale. Its own residual over those marks is 6.2 × 10⁻¹⁴ pixels, so it is an exact geometry of a different kind rather than a worse approximation of the same one.

The rotation is by twice the angle between the mirrors, which is two mirrors make one turn’s result arriving in a different costume. With the mirrors 44 degrees apart the turn is 88.0000 degrees, and the axis of the rotation is the corner where the two mirrors meet.

A line in the picture that is its own reflection #

A rotation moves nothing on its own axis. So a point on the corner where the mirrors meet is in the same place in both reflected views: it is its own correspondent, and it therefore lies on its own epipolar line.

That is a visible statement about the photograph. The corner is drawn in it — it is the seam between the two mirrors — and every point along that seam is a mark which the two reflected views agree about exactly. Measured along the part of it inside the frame, the worst such point misses its own epipolar line by 2.0 × 10⁻¹³ pixels. The same test on an ordinary mark, which is separated from its partner by a good part of the frame, misses by hundreds of pixels.

A reader with a photograph and a straightedge can use this. The seam is easy to find and hard to mistake, it is a line, and it is the one line of the picture the third geometry fixes without any correspondence being read at all.

Four epipoles, one line #

Three pairwise geometries have four epipoles between them — one for each mirror pair, since those two coincide, and two for the third pair, since it is an ordinary pair and its two do not.

All four lie on one line, to three ten-thousand-millionths of a pixel. The reason is short enough to be worth giving, because it turns the fact into something a reader can predict rather than something to be taken on trust. Each mirror’s normal is perpendicular to the corner the mirrors make, since the corner lies in both planes. The displacement between the two virtual eyes is perpendicular to it too, because a rotation moves nothing along its axis. So every direction the four epipoles are the vanishing points of lies in one plane — the plane square to the corner — and the vanishing points of one plane’s directions share that plane’s vanishing line.

The control is what makes it a claim rather than a coincidence: the vanishing point of the corner itself lies 3,534 pixels off the same line. Two points always lie on a line; four do not, and a fifth that should not is the evidence.

Eighteen numbers, or five #

Three general views of a scene are described by eighteen numbers. Two of the three pairwise geometries account for fourteen of them, and the remaining four are the reason a third photograph is worth taking: knowing how views one and two are related, and how views one and three are, leaves a four-parameter family of possible three-view arrangements.

The arrangement here is described by five. Two mirror planes are three numbers each, and an overall scale is invisible to every fundamental matrix, which leaves five. Read off the print they are the two lens-reflections — two numbers apiece — and one ratio.

So the accounting is worth stating exactly. The two mirror pairs give the two epipoles, and each epipole is the vanishing point of that mirror’s normal, so the two mirror directions come free. What they do not give is either distance, and the third pair’s geometry depends on both. It depends on them only through their ratio, because scaling the whole room scales nothing a photograph can see — and the ratio is one number.

The corner supplies it. The corner lies in both mirrors, so the plane through the lens containing the corner’s image contains the corner, and that is one linear condition on the two distances. With the two mirrors at 1.45 and 1.55 metres the ratio comes back as 0.935483871 against a true 0.935483871.

What the rebuild is worth, and what it costs #

Rebuilt from those five numbers, and never shown the correspondences it is tested on, the third geometry puts every mark within 9.4 × 10⁻¹⁴ pixels of the line its partner fixes.

The control is the part that makes this a measurement. Assume instead that the two mirrors are equidistant — wrong by ten centimetres in a metre and a half, which is a mistake a photographer would make without noticing — and the same rebuild misses by 14.8 pixels. So the ratio is load-bearing, and the rebuild is capable of being wrong.

What this buys is not accuracy; a fit to the marks is exact too. It is independence. The correspondences between two reflected views are the hardest of the three sets to establish, because neither appearance is the direct one and both are laterally inverted with respect to the scene. A geometry rebuilt from two lens-reflections and a seam does not need them, and can be used to find them: every mark’s partner in the other mirror lies on a known line.

What the third baseline is worth #

Three baselines are only worth having if they are worth having separately, and the quantity that decides it is not the separation but the angle at which two rays cross at the mark.

At a mark 0.83 metres from the lens, the three pairs cross at 33.0, 46.6 and 57.7 degrees, and the widest of the three is the pair of reflections. Three tenths of a pixel of marking error moves the recovered point by 1.463 millimetres for the narrowest pair and 1.281 for the widest — a gain of twelve per cent, which is real and is not large.

It is not large because the three eyes are at similar distances from the scene and the mirrors are not far apart. The two virtual eyes are 2.26 metres from one another and both are about three metres from the marks, so the arrangement is a moderately wide rig rather than a dramatic one. Turning either mirror outward opens it, at the cost of the direct view losing the mirror from its frame, and that trade is the same composition problem square to the camera is the worst mirror sets out for a single mirror, with one more mirror in it.

What is not measured here is the three-ray answer. Three rays through one point are not two pairs of rays, they are an over-determined system, and how it should be weighted is a question about which ray’s picture is worth more — the subject of the midpoint is a choice of ruler, which finds that the obvious answer depends on what the world is measured in. Three rays that do not meet are a sharper case of two rays that do not meet, and the pairwise numbers above are a floor rather than what the arrangement can do.

Both reflected views are written backwards, and the pair is not #

A reflection reverses handedness, so each of the two mirror views shows the scene laterally inverted — text in them reads backwards, and a right hand appears as a left one.

The pair made of those two views is not inverted. Two reversals compose into no reversal, which is the same parity accounting the corner that answers every eye uses to decide what an arrangement of mirrors does: an odd number of reflections reverses handedness and an even number does not. So the third pair is an ordinary stereo pair of an ordinary scene, showing the scene the right way round twice, from two places neither of which is where the camera stood.

That matters for more than tidiness. A matcher looking for correspondences between the direct view and a mirror view has to cope with reversed appearance, which defeats most descriptors; between the two mirror views it does not. The pair which is hardest to think about is the easiest to match, and it is the one the rebuild above supplies the geometry for without needing any matches at all.

The one thing that has to come from outside the picture #

Everything above except the rebuild is a straightedge operation — mark each lens-reflection, draw the seam, and the four epipoles and the self-corresponding line follow.

The rebuild is not, and it is honest to say where it stops being one. Turning an epipole into a mirror’s normal direction needs the focal length: the epipole is a point on the picture, and the direction it stands for depends on how far the picture plane is from the lens. With the focal length known the two normals follow exactly, and with it unknown the mirror angle is not determined by the two epipoles at all — the same two points are consistent with any angle, for a suitable lens.

That is not a difficulty so much as a place where the arrangement joins the rest of the subject. A photograph of two mirrors contains a good deal of structure that a calibration can be read from, and recovering the camera sets out what a single view supplies when its scene has right angles in it. A room with two mirrors in it has more than most: three views of every mark, two of them related by a rotation of known axis.

Where the count of images comes in #

An arrangement of two mirrors makes more images than two. Two mirrors make one turn counts the whole orbit — nine images at 36 degrees, and more at angles which do not divide a half turn — and each of those images is another view of the scene from another place.

That does not give an arbitrarily large rig, for two reasons and both of them are about the room rather than about the arithmetic. The deeper images are the ones an eye may not reach at all, which is what two mirrors show fewer images than they make measures; and the ones it does reach are dimmer and smaller, since each bounce costs reflectance and adds path length. A photograph of a two-mirror corner typically contains three usable views and a fourth that is faint.

The accounting extends, though, and cleanly. Every image in the orbit is the lens reflected by the same word that makes the image, so the whole rig’s geometry is still five numbers however many views are visible. A twelve-view arrangement with five parameters is not something any arrangement of separate cameras offers.

The arrangement where the third pair is a mirror pair after all #

Every claim above needs the two mirrors to meet. Setting them parallel is not a small perturbation of that and the accounting changes in a way worth following, because it is the one arrangement in which the third geometry costs nothing at all.

Two reflections in parallel planes compose into a translation, not a rotation — twice the distance between the planes, along their common normal. So the two virtual eyes are displaced along the normal, and a pair of views related by a pure translation has a skew-symmetric fundamental matrix for the same algebraic reason a mirror pair does. Measured on two parallel mirrors 1.45 and 1.55 metres in front of the camera, the third pair’s matrix comes back skew to 6.0 × 10⁻¹¹, its baseline is 0.200 metres — twice the ten-centimetre difference — and the two lens-reflections fall on the same point of the print, because the two normals are the same direction.

So at exactly parallel the arrangement has one epipole rather than four, all three matrices are skew, and the third geometry is fixed by that single point: no ratio is needed, because there is no corner to read it from and no rotation for it to parametrise.

The approach to that limit is quick rather than gradual, which is the part worth knowing for a real arrangement. At two degrees between the mirrors the third matrix is still 1.999 of the way to symmetric; at four tenths of a degree it is 1.839; and only at zero does it fall to nothing. A pair of mirrors nearly but not quite parallel therefore behaves like a rotation pair with a very short baseline — 0.226 metres at two degrees — which is the badly-conditioned case rather than the degenerate one, and the two are worth telling apart.

What this does not settle #

The reconstruction has not been done, only its geometry. Three views with known relative pose and a known calibration determine the scene up to the same one scale a single mirror pair leaves, and a tape measure across either mirror removes it; what has not been measured here is whether the third pair’s baseline is worth anything for precision, which depends on the crossing angles rather than on the separations, and square to the camera is the worst mirror is the reason to expect the answer to be composed rather than computed.

Nothing here treats the mirrors as having thickness. A real domestic mirror is silvered on the back of four millimetres of glass, so each reflection is displaced by a refraction through a slab on the way in and another on the way out, and what survives a pane of glass is the measurement of what that displacement does and does not preserve. The effect is small, it is not zero, and it is largest for the doubly-reflected views, which pass through four slabs.

And the corner has been taken as visible and sharp. A mirror pair joined by a frame has no visible seam, and the ratio has to come from somewhere else — a second correspondence in the third pair would do, which is one number for one number, but it is exactly the measurement the rebuild was meant to avoid needing.

Still open: what a third mirror is worth #

Two mirrors give three views for five numbers. Three mirrors give more, and the accounting does not obviously extend, because the three pairwise corners are not independent: three planes meet in three lines which are themselves concurrent, so a third mirror adds three numbers to the description and considerably more than three views.

The question with a measurement in it asks where that stops paying. Adding a plane adds three numbers and multiplies the reachable orbit; counting the views a photograph of a three-mirror arrangement actually contains, against the eight numbers describing it, gives a ratio that cannot rise for ever — the images get dimmer, and the deeper ones are reached from narrower ranges of eye position. Finding the arrangement at which the ratio is largest, and whether it is the right-angled corner or something shallower, would say what shape a mirror rig should be when the point of it is to photograph one object from as many places as one shutter allows.

The short version #

A photograph with two mirrors in it contains three views of the scene, taken from the lens and from its reflections in each mirror, at separations of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew to a part in ten million million; the third is a rotation by twice the angle between the mirrors, about the corner where they meet, and is not skew at all.

The third geometry is not new evidence. Two lens-reflections and the image of the corner — five numbers — rebuild it to 9.4 × 10⁻¹⁴ pixels, against the 14.8 pixels the same rebuild misses by if the two mirrors are wrongly assumed equidistant. A general three-view arrangement takes eighteen.