Every essay — page 12
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors Mirrors that are not cameras Measuring from one picture Through water and glass The real instrument The second eye What a pair is for Many pictures at once The eye that moves Systems that kept the measure What each system gave up What a machine computes The rectangle behind the lens The second projection Drawn confidently
Many pictures at once
Raise the count and the answer changes character. A sequence determines its own camera track and the scene together — up to seven numbers that no quantity of pictures supplies, counted here in the Jacobian and then walked along to show that they are free rather than merely small.
A chain and an adjustment
Composing pairwise poses along a sequence is supposed to drift. Measured over five links it wanders instead — one chain ends closer to the truth than its own worst link — and the real cost of chaining turns out to be something else entirely.
Where the adjustment stops
Given exact marks the reprojection error falls to a hundredth of a billionth of a pixel, which is arithmetic. Given the same marks read to a whole pixel it falls to a third of a pixel and stays there, and a solver that reached zero on those would be fitting the rounding.
Another picture of the same sweep
Going from three views to seven across the same sixty degrees leaves the reconstruction exactly where it started, and at one point makes it worse. What a reconstruction is short of is angular spread, not photographs.
Far enough away, a pair is one eye
Hold the baseline and walk the scene away, and the parallax a single homography cannot explain falls as the distance to the power −0.968 — one over the distance, which says the ratio of baseline to depth is the whole of it. The recovered translation direction follows it down, from 3.3° at four metres to 74.5° at two hundred and fifty-six.
An uncertainty is quoted from something
One adjustment, one set of marks read to a pixel, and its uncertainty written four ways. Held at the first camera, the last camera is 105 mm from certain; held at nothing, every camera is within 4 to 6 mm. A ratio of two distances carries 0.1465 per cent in all four, to two parts in a billion.
A longer lens buys parallax and not the direction
Walk a pair away from a scene and the parallax a homography cannot absorb falls. Lengthen the lens to keep the scene filling the frame, and one power of that loss comes back, exactly. The recovered direction between the cameras does not come back with it. At the same parallax in pixels, a long lens is three times further from the answer, because what fixes a direction is the angle the marks span and no lens can widen that.
The eighth held number bends the scene
Four surveyed points, each 10 mm out in a different direction. Hold seven of their coordinates during an adjustment and the courtyard's shape moves by a trillionth of a millimetre; hold eight and it moves by 0.59 mm, because seven numbers choose a frame and the eighth makes a claim the pictures disagree with.
A narrow view keeps a second answer, inside out
Six pictures of a courtyard through a 3° field, every mark exact. Started from the scene turned inside out, the adjustment settles there — near points far, far points near — and misfits the marks by under a pixel. The misfit grows in proportion to the field and to the sweep of the cameras, and vanishes only where perspective does.
A scale chain leans rather than wanders
A camera driven 200 m along a street, each pair's scale handed to the next through the points they share. Across sixteen streets the scale does not spread the way a random walk would — 7.3 per cent after 199 links, where independent ties would give 108 — and its average leans by an amount the choice of average decides, from +28.6 per cent to −12.2.
Closing a loop mends its ends
Twenty-four cameras walked once round a ring of walls, every mark read to a pixel. Recognising the twenty-two points seen at both ends makes the last camera six and a half times better placed relative to the first — and the worst camera, across the ring, only a fifth better. A closure mends the ends of a walk and barely touches its far side.
The spread a point gets
A track of sixty degrees gives no point of the scene sixty degrees. The nearest receive 89 and the furthest 41, a factor of 2.2, and their errors run from 1.5 to 8.4 millimetres — following the angle at the point as its −1.68 power, with 94 per cent of the variation explained. The arc a track covers is one number for forty-four different situations and predicts none of them.
A loop's far side is a length
The worst camera on a closed loop of twenty-four is 411 mm from certain along the line from the start and 64 mm across it. A picture taken across the ring from the start tripod measures directions and moves the worst camera from 418 mm to 405. One distance measured across the ring to 10 mm moves it to 74, and a four-metre length measured on the start wall to 112. What the far side of a loop lacks is not a connection but a length.
A survey is trusted at its own accuracy, unless its error has a shape
Entered into an adjustment with a stated accuracy, a survey whose errors are random gives the smallest shape error when the stated accuracy is the true one — at 1, 5 and 20 mm alike. Stated twenty times too tight it can cost thirteen times the error; twenty times too loose, almost nothing. But twelve surveys all 10 mm out in fixed directions are best trusted anywhere from 0.3 mm to 10 mm, and the size of their error cannot say which.
A start needs the sign of its depths, not their size
Started part of the way from a courtyard toward its inside-out twin, a bundle adjustment returns to the truth from every start less than 42 per cent of the way — a start with a sixth of the true relief, the right way round — and falls into the twin from every start past 56. Between, neighbouring starts settle in different answers. The band sits in the same place at a 3° field, where the twin misfits by under a pixel, and at 25°, where it misfits by seven and a half.
The cameras decide where a narrow view settles
Scatter a narrow-field bundle adjustment's starting depths and the answer it reaches stops following them: of eighty scattered starts whose depths had the right sign, twenty-six fell into the inside-out twin, and of a hundred with the wrong sign, forty-three came home. Split the start in two and the reason is plain. With the cameras where they are, the courtyard comes home from its own inside-out points; with the cameras on the twin's side, it falls in from the true ones.
A scale bar is worth its ends, not its tape
A length measured at the start of a walk round a ring pins the walk's scale — but only to the precision with which the pictures place the length's two ends, 3.4 mm here, whatever the tape says. Measured to a thirtieth of a millimetre or to three millimetres, a one-metre bar leaves the worst camera at the same 89 mm. So a short length is not a long one measured well: a bar under about 13 cm tells the walk less about its scale than the walk already knows. And on an open walk no length at the start does better than 214 mm, because what is left there is drift, not scale.
Split the track and the needles turn
A point triangulated from a track of cameras is not known equally well in every direction: its error is a needle, long along the direction its lines of sight barely constrain. That direction is not the mean line of sight but the lines of sight weighted by how near each camera is — and it can be turned. Two groups of cameras more than a right angle apart swing every needle across, and the same six pictures of a facade then measure its depth four times better.
Two short bars are not one long one
A long scale bar is awkward to carry, so the natural substitute is two short ones set far apart on the same wall. It does not work. Two 25 cm bars 3.5 metres apart leave the worst camera of a closed walk at 219 mm; one bar over the same span leaves it at 72. The gap between the bars is read by the pictures like any other distance, so it carries no scale — and the same holds for a long length measured as separate pieces. What does work is a tape read at every mark: pieces that share their ends add up to the whole, and pieces that do not, do not.
A length fixes the scale where it lies, and the worst camera nowhere
Move one measured metre round a ring of twenty-four cameras and the worst camera hardly notices: 199 to 225 millimetres on the open walk, 80 to 103 on the closed loop, wherever the bar lies. What moves is the scale. On the open walk a length is known to 0.41 per cent beside the bar and 1.58 per cent twenty walls away, and no longer bar at the start brings the far end below 1.53. A survey that must measure lengths everywhere on an open walk needs its references spread along it; one that closes its loop can put a single reference anywhere.
The bundle keeps the needles turned, if the wall has three rows
Six cameras in two groups at sixty degrees either side of a wall measured its flatness four and a half times better than six on a narrow arc, with their poses known. Found from the same pictures in one bundle adjustment, they still do: thirty-nine marks cost them three per cent. Ten marks in two rows cost them seventy-eight, and the wall's likeliest error becomes a twist — but a third row of marks takes most of that back, and fifteen marks in three rows beat fifty in two.
A wall's twist can be tested against the error the adjustment reports
A wall reconstructed from two groups of cameras either side of it is likeliest to come out twisted, and the adjustment that found it also reports how far to believe the twist. The report is honest — a flat wall's twist clears twice its stated error one time in twenty, at every layout — so the question is only how large a real twist must be to be seen. Ten marks in two rows state ±5.05 millimetres and see a five-millimetre twist seventeen times in a hundred; fifteen in three rows state ±2.31 and see it fifty-six. The test goes wrong only when its error is assumed rather than measured, and a mismatched mark big enough to fake a twist always shows in its own residual.
One square-on picture ties the two groups a third row only steadies
A wall surveyed from two groups of cameras either side of it is likeliest to come out twisted, and the twist is the two groups pitching in opposite senses. A third row of marks steadies it: the reported twist falls from ±5.05 to ±2.31 millimetres. One picture taken square to the wall, which belongs to neither group, ties them: ±1.25, from a third of the extra views, and a five-millimetre twist is then seen ninety-eight times in a hundred instead of fifty-six. The picture need not be square — twenty degrees off is nearly as good — and it can be taken from four times as far away and still beat the row.
An uncalibrated picture ties the two groups, if the height of its centre is known
One picture taken square to a wall ties a survey's two oblique groups and removes their twist, and the question was whether it still does when it comes from a phone whose lens the adjustment must find. Its focal length costs nothing; so does the column of its principal point. The height of its principal point costs everything: with it free, a picture ten degrees off square reports the twist to ±4.83 millimetres instead of ±1.30, because to a level picture its own centre's height and the wall's twist are one number, correlated 0.96. A tape on the wall does not help. A second picture through the same lens, or the same phone used for the whole survey, gives the tie back exactly; so does a centre known beforehand to ten pixels.
One camera known to be level ties the groups; a level phone picture does not read its centre
A phone picture held level puts its centre on its horizon, so it seemed that being level would give back the number an unknown lens takes away. Near square it barely does: the horizon is where the wall's rows of marks meet, thousands of pixels out of the frame, and even an exactly level picture 10° off square reads its centre to ±61 px. But being level does something the question did not ask of it. The wall's twist is the two survey groups pitching against each other, and one survey camera that knows its own pitch against the wall's verticals to a tenth of a degree holds its group still: the twist is reported to ±1.37 mm, as well as a calibrated square-on picture manages. Two such cameras, one in each group, also cancel the false twist a wall out of plumb would otherwise put in.