The parallax you cannot shoot away

The previous rung ends with a law and a question. The law is that a ray of a panorama shot about the wrong pivot passes that pivot by $e \sin\gamma$, where $\gamma$ is how far off its own frame’s axis the ray points. The question is whether shooting more frames makes the problem go away, since narrowing the strips narrows $\gamma$.

Half of it does. The other half is the subject of this essay and it is the half nobody says out loud.

γ has two angles in it #

A direction inside a frame has an azimuth $\alpha$ and an elevation $\beta$, and the angle off the axis is not their sum. It is the hypotenuse of a spherical right triangle:

$$\cos\gamma \;=\; \cos\alpha \, \cos\beta$$

Which is an ordinary identity and carries the whole finding. The photographer controls $\alpha$ — stitch $n$ frames round a turn and each contributes $|\alpha| \le \pi/n$, so more frames is less azimuth. The photographer does not control $\beta$. The frame is as tall as the frame is, and the top of it is at $\beta_{\max}$ whether there are six frames or six hundred.

So the two worst cases are

$$e \sin(\pi/n) \qquad\text{and}\qquad e \sin\beta_{\max}$$

and only the first has $n$ in it.

The floor is a property of the frame’s height #

Fix the pivot error and change nothing else. Shoot with a wide lens held upright and the frame is tall: sixty degrees of vertical field, so $\beta_{\max} = 30°$, and the floor is half the pivot error. Shoot with a long lens and the frame is short: twenty degrees, so $\beta_{\max} = 10°$, and the floor is a sixth of it.

That is the practical form. A photographer who cannot fit a rail to a tripod head — or who does not have one, or who is holding the camera — improves the picture more by turning the camera on its side and shooting narrow frames than by shooting more of them. The first changes $\beta_{\max}$; the second only changes $\alpha$.

It also inverts the usual advice, which is to shoot in portrait orientation to get more vertical coverage from fewer frames. That is right about coverage and wrong about parallax: a portrait frame is taller, so its floor is higher, and fewer frames means each contributes more azimuth as well. The two errors move the same way and the advice moves against both.

The crossover has no offset in it #

The two halves are equal when $\sin(\pi/n) = \sin\beta_{\max}$, so

$$n \;=\; \frac{\pi}{\beta_{\max}}$$

and $e$ has cancelled. The frame count past which every remaining scrap of parallax is vertical is a property of how tall the frames are and of nothing else — not of the mounting error, not of the lens, not of the scene.

At a thirty-eight degree vertical field that is nine point four seven, so ten frames. Past ten frames the photographer is buying nothing.

That is an unusually clean result for a practical question, and the reason it is clean is that the sine has already separated the variables. Both halves are $e$ times a sine; the ratio between them has no $e$; and the ratio is what decides which one to work on.

It is worth noticing what kind of statement that is. This site keeps finding quantities that survive a change of everything else — the cross-ratio under a projection, the axis scales under a parallel drawing, the seven directions no quantity of pictures fixes. A crossover with the offset cancelled out of it belongs on that list: it is a fact about the shape of the arrangement rather than about how badly the arrangement was made, so it holds for a tripod head with a rail nearly right and for one with no rail at all.

The vertical half is what a spherical camera is made of #

The most familiar instrument this applies to is not a tripod at all.

A consumer 360° camera has two fisheye lenses back to back, and the format it writes asserts, by being one rectangle, that its contents are the directions from one point. They are not: they are the directions from two pupils a few centimetres apart, and the stitch treats them as halves of one sphere.

Read against this essay, that camera is the extreme case of the horizontal half being bought off entirely — two frames, but each covering a full hemisphere, so there is no seam inside either. What is left is the join, and the join runs all the way round, top to bottom. Every angle on it is a $\gamma$ near ninety degrees, so the miss is nearly the whole separation of the two pupils, everywhere along it.

That is why the ghost on a spherical camera is a ring rather than a pair of lines, and why nothing about the file records it. The format has no place to write down a second centre.

The assertion that makes it a measurement #

“The vertical miss does not change with the frame count” is the kind of sentence that gets written down and never tested, and there are two ways to test it badly.

The first is to check that it changes little. That passes for a quantity that changes slowly, and it would pass for a wrong implementation in which $\beta_{\max}$ leaked a dependence on $n$ through the strip geometry.

The second is to check it at two counts. That passes for a quantity that happens to agree at those two.

assertOneHalfOfTheParallaxIsForSale requires the set of distinct values of the vertical miss, over eight frame counts from two to sixty-four, to have one element. Bit for bit, not to a tolerance. And it requires the crossover to land at the two frame counts either side of $\pi/\beta$ — one short of it the horizontal is still the larger, at it the horizontal is not — which is a statement about the closed form rather than about a list of counts somebody chose.

A number that is not the miss, and is the one to quote #

There is a temptation to report the floor as a single figure of merit and it should be resisted, for a reason the lens field has already met.

A miss in metres is a length in the room. Whether it matters depends on how far away the thing being drawn is, because what a reader sees is an angle: the miss divided by the distance, times the panorama’s own focal length. So the same floor is invisible on a landscape and gross on a table, and a specification that quotes the metres alone has stated a fact and answered no question.

The honest quantity is a pair — the floor, and the nearest distance at which it stays under a pixel. For twenty millimetres of floor on an eight-thousand-pixel panorama that distance is a little over a hundred and fifty metres, which is a way of saying that a badly-mounted panorama of anything indoors is a badly-mounted panorama.

What is left at the limit, and what it looks like #

Take the frame count to infinity and the horizontal miss goes to nothing. What is left is a picture whose horizon row is exact and whose top and bottom rows are drawn from somewhere else.

That is a strange object and it is worth saying what it is not. It is not a picture with a soft edge, and it is not a picture that is worse at the top. It is a picture whose rays are tangent to a sphere of radius $e\sin\beta_{\max}$ — the same tangent sphere the previous rung draws, with its equator collapsed to a point.

So the defect changes shape as the frame count rises. At two frames it is a horizontal doubling at the seam. At twenty it is a vertical disagreement between the top of one frame and the top of its neighbour, which reads as a slight shear of anything tall near a join rather than as a ghost.

Anybody who has stitched a room with a doorway in it has seen the second and not recognised it as the same defect, because it does not look like a doubling. It looks like the doorframe leaning.

The same shape of answer, twice on this site #

This is the second time this collection has found an error with two terms that behave differently, and the other one arrives from a completely different question.

A silhouette reconstruction has an error outside the convex hull that falls as one over the square of the view count, and an error inside a concavity that does not move at any count. Same structure: one term for sale, one not, and the useful engineering fact is which is which rather than how large either is.

The pattern is worth naming because it is the difference between a specification and a complaint. A quantity that falls with effort belongs in a shooting instruction; a quantity that does not belongs in the description of what the instrument is. Reporting them as one number — “the parallax is thirty-four millimetres” — hides which of the two a reader is looking at.

When an error has more than one term, measure whether each of them responds to the thing a user can change. A term that does is a setting; a term that does not is a property of the instrument, and calling both of them “the error” is what turns a design into a grievance.

What sets the floor in practice, and the number #

The floor is $e \sin\beta_{\max}$ and the two factors are independent, which means there are exactly two ways to lower it.

Move the pupil onto the pivot. That is the rail, and it is the only thing that removes the defect rather than reducing it — at $e = 0$ both halves are exactly zero, which the control checks at the arithmetic floor rather than at a tolerance.

Or shoot shorter frames. That is a longer lens and more of them, and it costs shooting time and stitching area rather than hardware.

There is no third way, and in particular there is no post-processing way. A warp applied to a delivered panorama is a map from picture coordinates to picture coordinates; the defect is that different depths want different maps; and a map that does not know the depth cannot be the right one for two depths at once. That is the same argument a rig’s stitching depth makes with a real baseline instead of a mounting error, and it has an exact optimum there because the baseline is large enough that choosing a depth is the honest thing to do.

Why this is not the rolling shutter’s problem #

It is worth separating this from a defect it resembles, because both are about a frame not being one thing.

A rolling shutter reads its rows one after another, so a frame taken while the camera moves is a stack of projections indexed by height — the rays leave from wherever the eye was at that instant. The miss there is the spread of the eye’s own track, and it is zero whenever the world and the camera are still.

The pivot error is not conditional on anything moving. Its miss is $e \sin\gamma$ whether the shot is a thousandth of a second or an hour, and it is there in a picture of a stone building on a windless day.

The two do compose, and unpleasantly: a rolling shutter on a badly-mounted camera has a miss that varies down the frame for one reason and across it for another. Neither figure on this site draws both at once, and the reason is that the sum would be a picture of two things nobody can separate by looking.

What this does not settle #

It does not say the floor matters. On a landscape at a hundred metres, twenty millimetres of miss is a fifth of a milliradian and is under the lens’s own errors; on a table two metres away it is not. The floor is a length in the room and whether it matters is a division the scene performs.

It does not say the frame count is free. Sixty frames is sixty exposures and sixty registrations, and the registration itself has errors that grow with the count.

And it does not say a stitched panorama should be avoided. It says the thing being bought when frames are added, and where the buying stops — which is a specification, and is what the previous rung’s law is for.