Covariance — where it appears
Named by 20 essays across 4 fields — each of them below, with the objects they name alongside it.
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.
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.
A third ray is worth what its picture is worth
Three eyes on one point, two at seven metres and one walked back to seventy. The point nearest all three rays in metres is 132 millimetres from the truth and the point of least reprojection error is 34 — the same 34 the near pair gives alone — and the first is pulled 12 millimetres along the line to the distant eye. And arrangement beats count outright — two rays spread over fifty-five degrees beat eight rays inside four, by a factor of 4.4.
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 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.
A tilted target pays for its tilt in perspective
A calibration target of printed circles held square to the lens fixes a lens's first radial coefficient to 4.77 thousandths from ninety-six marks. Tilt it thirty degrees, fit the tilt along with everything else, and the same ninety-six marks fix it to 2.46 — nearly twice as well. Squash the circles by the same angle without perspective and nothing is gained. What pays is the near half of each circle being drawn larger than the far half, which spreads a circle's marks across a band of distances from the lens's centre.
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 closer pose sharpens one coefficient and not the pair
Photograph a calibration target tilted four ways at several distances, and the two radial coefficients of the lens stay locked together near a correlation of −0.97. A target further off adds nothing at all. Bringing it closer sharpens the first coefficient by about a half, then stops, and pushes the pair closer together. What moves the correlation furthest is being told each pose's distance.
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.
A stereo pair and a lane wander add as two readings of one depth
A pair of cameras 0.3 metres apart across a car ranges the road twenty metres ahead to 15.8 per cent a pixel from one moment's two pictures — and to 4.6 per cent from the last second's sixty, because every picture is another reading. Add the car's ordinary drift across its lane and the second's pictures give 4.5: exactly what the pair's look and a single camera's wander would give as two independent readings. The two baselines neither waste each other nor multiply, while the pair's is the larger; only a narrow pair on a car that drifts half a metre does better than the rule, by a quarter.
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.
A car ahead costs a pair its speed, and a car that may drift costs a single camera everything
A stereo pair's last second of pictures ranged the still road twenty metres ahead to 4.5 per cent a pixel. A car twenty metres ahead is not still, and each old picture saw it somewhere else. Read together with its speed, a car keeping pace is ranged to 4.5 per cent all the same — it stood twenty metres from every picture, and reading its speed spends exactly what that was worth. A car that may also drift across the lane costs the pair a fifth more and takes a single wandering camera from 38 per cent to 185. A braking car caps the useful look at about a second.
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.
A growing car times the gap without its width
A pair of cameras ranges the car in front from its disparity, and reads how soon the gap will close only as well as it reads the car's speed: to 39 per cent a pixel from a second's pictures, for a car 20 metres ahead closing at 5 metres a second. One camera watching the same car grow reads the same four seconds to 8.8 per cent, without knowing how wide the car is. The car's own width is a baseline six times the pair's. The distance needs the width, and the pair, reading both, measures the width as well.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
bundle adjustmentgauge freedomTriangulationControl pointDepth uncertaintyleast squaresBaselineConditioningdegrees of freedomDriftLoop closurecamera track