A refocused panorama is six lenses

Focusing moves the pivot past its best place found that a fifty-millimetre lens focused at a metre carries its entrance pupil 2.63 millimetres forward of the camera body, and at half a metre 5.56 — more than the whole distance the pupil walks with field angle. A panorama head aligned at one focus and used at another pays for that along every seam. That essay treated the panorama as shot at one focus, and ended by asking about the case that is not: a scene deep enough that the photographer refocuses between frames, a table’s edge at half a metre in one direction and a wall at four in another.

Every frame then has its pupil in its own place, and a head that moves the camera back as the lens extends — a rail geared to the focus ring — would keep every one of them on the pivot. The question was whether that is worth building: whether it removes an error of the first order or only a correction to one.

It removes a first-order error, and indoors it is worth a factor of four or more. But the same measurement turns up a larger error, which no head repairs, in the way the frames are read.

Six frames, six pupils #

The room below is shot as six frames sixty degrees apart, through the same lens as before — a fifty-millimetre design with a strongly curved front element, whose pupil walks with field angle — and each frame is focused on what is in front of it: a table at half a metre, a chair at seventy centimetres, a doorway at a metre and a half, the far wall at four metres in two frames, and a shelf at 1.2 metres. Each seam is judged at the nearer of its two frames’ subjects, since that is where a wrong centre shows most.

On a head that does not move with the focus ring, the frame focused on the table has its pupil 5.56 millimetres forward of where the frame focused on the wall has its pupil, and the pivot can be in only one place. The best single place — found by trying every one — leaves 12.9 arcminutes along the worst seam, which at sixty pixels to the degree is thirteen pixels of misregistration on the table’s edge. A head that slides the camera back by each frame’s extension holds every pupil at the lens’s own best pivot, and leaves 2.82 arcminutes along the same seam.

That 2.82 is not zero, and it could not be. The entrance pupil walks with the angle established that the place a wide lens’s picture is a projection from is not a point: the chief rays near the axis cross it at one place and those at the edge of the field at another, 5.53 millimetres apart for this front element, and no pivot sits on both. The tracking head reaches that floor at every seam — 2.82 arcminutes at half a metre, 0.35 at four — and goes no further. Everything above the floor on the fixed head is what refocusing added.

Which seams carry the cost #

The six bars are not six samples of one error, and reading them seam by seam says where the cost of refocusing sits. On the fixed head the two worst seams, 12.9 and 11.4 arcminutes, are the two beside the table: the table’s frame is focused at half a metre and its neighbours at seventy centimetres and 1.2 metres, so the extensions either side of the two seams differ by 1.7 and 3.4 millimetres, and the seams’ subject is the nearest thing in the room. The seam between the two frames on the far wall, focused alike at four metres, carries 2.8 — and the tracking head carries 0.35 there. Two frames focused at the same distance have their pupils in the same place, so the fixed head’s error on that seam is not a difference of foci at all. It is the compromise pivot sitting 2.8 millimetres from where those two frames wanted it, because it was placed to serve the table.

That is the general shape. A fixed head’s error on a seam has two parts: how differently the two frames were focused, and how far the compromise pivot sits from the place the pair would have chosen for themselves. The first is zero where neighbours share a focus; the second is never zero anywhere except at the one seam the compromise happens to favour. The eye is a place, not a point described a wrong pivot as a single offset that the sky forgives and the foreground does not. A refocused panorama on a fixed head has a different offset at every frame, and the room’s near objects find every one of them.

The tracking head’s bars follow the room’s distances and nothing else — 2.82 arcminutes at half a metre, 0.94 at a metre and a half, 0.35 at four — because its one remaining offset, the walk, is the same for every frame. The pivot that is not the eye draws that offset for a rotating camera with no lens model in it, and a rig is right on one surface finds the depth at which such a rig’s parallax vanishes. With the frames refocused and the head tracking, every frame shares that surface again, which is what makes the six frames stitch as one instrument once more — the camera that is a cylinder, rebuilt from six lenses.

The pivot’s share is first order #

Whether a tracking head removes a first-order error or a second-order one is a question about how the seam error grows as two neighbouring frames are focused further apart. The figure below isolates it: one seam at half a metre, the first frame focused on it, the second refocused anywhere from half a metre to infinity, and a fixed head aligned perfectly for the first frame.

The fixed head’s seam grows from 2.82 arcminutes, when both frames are focused at half a metre, to 17.13 with the second frame at a metre and 30.78 at infinity, where the two extensions differ by 5.56 millimetres. Against the difference of extensions it is nearly a straight line, about five arcminutes a millimetre at half a metre. That is what first order means here: the error is proportional to how differently the two frames are focused, with no threshold below which it vanishes and no cancellation that makes a small refocus free. The tracking head leaves 2.82 arcminutes wherever the second frame is focused.

The arithmetic the earlier essay gave explains the slope. A centre a distance $\delta$ from where it should be, on frames whose axes are $2h$ apart, misregisters a subject at distance $D$ by about $2\delta\sin h/D$. On a fixed head, two frames focused differently have centres whose offsets differ by the difference of their extensions, so $\delta$ is that difference. At half a metre and sixty degrees between frames, a millimetre of $\delta$ is two milliradians, 6.9 arcminutes; the measured slope, 5.03, is less because the walk with field angle already offsets the centres a little in the same direction and the two do not simply add.

The best a fixed head can do #

A fixed head is not helpless: its pivot can be put where it suits the room best, not where it suits one frame. The figure below slides it through every position.

The worst seam is a V with a rounded floor. Put the pivot where the far frames’ pupils are and the near frames misregister badly; put it where the near frames’ pupils have gone and the far frames do. The best compromise, 2.57 millimetres forward of the lens’s paraxial pupil at infinity, sits between the two and leaves 12.9 arcminutes. It is set by the two seams beside the table, whose frames are focused at half a metre and at 1.2 metres or seventy centimetres, and whose seam is also the nearest to the lens. No choice of fixed pivot gets below it, and the tracking head’s 2.82 is 4.6 times better.

That is the answer to whether a tracking pivot is worth building, for this room and this lens. It is — but only because the frames were refocused. A panorama shot at one focus throughout, on a fixed head aligned at that focus’s best pivot, is exactly the tracking head: every frame’s pupil is in the same place and the head has put the pivot there. The check on that is exact, to a billionth of an arcminute. What refocusing buys is sharpness, and the question a photographer is really choosing between is a sharp near edge with thirteen pixels of seam, a soft near edge with three, or a geared head with a sharp edge and three.

A refocused frame is a different lens #

The measurements so far assumed that the stitcher reads each frame at its own principal distance. A refocused frame’s picture is made further from the sensor than a frame focused at infinity: 55.56 millimetres at half a metre, 52.63 at a metre. Focusing is a zoom measured that as a narrowing of the angle of view, and it matters here much more than the pupil does.

A stitcher that links every frame to one lens reads all six with one principal distance — the fifty millimetres on the barrel. A frame focused at half a metre, read that way, has every direction off its axis turned by the wrong angle: a point thirty degrees off the axis, which the frame drew at $v\tan 30°$ with $v = 55.56$ millimetres, is read back at 32.7 degrees. Along the seam beside the table that is 270 arcminutes of misregistration, four and a half degrees, on a head whose centres are as good as this lens allows. The far wall’s seam, between two frames focused at four metres, still carries 37.

That is ninety times the largest error the pivot makes, and it has nothing to do with the pivot. A refocused panorama is not one camera turned six times; it is six cameras with six focal lengths, which happen to share a lens barrel. Read each frame with its own principal distance — the focal length times the subject distance over the subject distance less the focal length, or better, fitted from the overlap as a stitcher can when it is allowed to — and the whole of that error goes, leaving only the centres’.

So the order of business for a refocused panorama is not the one the question implied. The first thing to repair is the reading: unlink the frames’ focal lengths, or give the stitcher each frame’s focus. The pivot comes second, and matters only once the first is done.

Near rooms pay twice #

How much a tracking head is worth depends on the room’s size, and it depends on it more steeply than the pivot error alone would suggest.

Scale the room — every subject nearer or further by the same factor — and the tracking head’s worst seam falls in proportion to the distance, from 5.64 arcminutes when the table is a quarter of a metre away to 0.56 at two and a half metres. That is the $\delta/D$ law with $\delta$ fixed: the walk with field angle does not depend on the focus.

The fixed head’s seam falls about as the square, from 60.9 arcminutes to 0.8. Its $\delta$ is the difference of the frames’ extensions, and the extension is itself the focal length squared over the subject distance less the focal length — it falls as one over the distance too. So the fixed head’s error is one inverse distance for the geometry of the seam and another for how far the focus moved the pupil. At a quarter of a metre the tracking head is eleven times better; at two and a half metres, 1.3 times, which is to say not worth the gearing.

That is the practical rule. A geared head earns its complexity in small rooms, close to furniture, with a wide lens refocused on near subjects — exactly the interiors a panoramic photographer is most often asked to shoot. In a hall, a courtyard or a landscape it buys a fraction of an arcminute.

What the extension is, and what it is not #

The figure below is the quantity all of this is built on, carried over from the earlier essay: how far the lens stands beyond its focal length at each focus.

The extension is a closed form only for a lens that focuses by moving as one piece, so that its pupil travels with the glass. Most modern lenses focus internally, moving one group while the front element stays put, and their pupils move by an amount that is neither the extension nor zero and is not published. The tracking head described here, geared to move the camera back by the extension, is then the wrong gearing: it would need the lens’s own pupil-against-focus curve, which the hole a scene actually sees shows how to trace for a given design, and which a photographer can measure with two marks and a focus ring.

The principal distance has the same caveat and a better escape. An internally focused lens changes its focal length as it focuses — often shortening it, which is the reverse of unit focusing — and the principal distance a frame was made at is again not the barrel’s number. But the principal distance is readable from the picture: a close picture carries its own distance recovers it from vanishing points, and a stitcher recovers it from the overlap whenever it is allowed to treat each frame’s focal length as free. The pivot has no such escape. A wrong centre leaves parallax, and the parallax a rig cannot shoot away is what that essay’s title says it is.

What was assumed #

The lens focuses as one piece. Its pupil moves by exactly the extension and its principal distance is exactly the thin-lens image distance. An internally focusing lens breaks both, and changes the numbers without changing the order of the two errors: the reading’s error is set by how much the principal distance changes, the pivot’s by how much the pupil moves, and in most lenses the first is the larger.

Each seam is judged at the nearer subject. A real seam crosses whatever is there, and a seam over the table’s edge at half a metre and the wall at four behind it carries both errors at once. Taking the nearer is the worst case, and what a stitcher’s control points most often sit on.

The frames are sixty degrees apart. More overlap brings the frames’ axes closer, shrinks $\sin h$, and shrinks every pivot error with it; the reading’s error shrinks too, because the seam then sits nearer each frame’s axis. Tighter spacing is a remedy for both, and the only one that needs no equipment.

The subject is still. Refocusing takes time, and in that time a room with people in it changes, which is a larger seam than any of these.

Still open: whether one focus and a stopped-down lens beat six focuses #

The comparison this leaves undone is the one a photographer actually faces. Refocusing buys sharpness at the cost of a seam that needs a geared head and an unlinked stitcher. Not refocusing — one focus for all six frames, set so that the table’s edge and the wall both fall inside the depth of field — buys a perfect seam on a fixed head at the cost of stopping the lens down, which costs light, and past a point costs sharpness everywhere to diffraction.

The measurement that settles it takes the room above, a sensor with a stated pixel pitch, and the lens’s aperture as the free variable. For each aperture it finds the single focus that best covers half a metre to four metres, the blur circle that leaves at the table and at the wall, and the diffraction blur the aperture adds, and compares the worst blur with the thirteen-pixel seam of the refocused, fixed-head panorama and the three-pixel seam of the tracking head. If some aperture keeps every frame’s worst blur under three pixels, one focus and a fixed head beat any amount of refocusing; if none does, the geared head and the unlinked stitcher are the only way to a sharp, seamless room, and the question becomes how many pixels of blur a photographer will trade for how many of seam.