The thread: Determined up to something — page 6
A strip's scatter points to the end drawn last
The size of a tiring painter's bays cannot say which edge of a strip they started from, because a creep read backwards is a creep. The scatter of the bays can: a hand that worsens as it goes leaves its largest slips on the rows it drew last. One six-bay strip names its edge three times in four when the slip triples, a four-bay strip carries no evidence at all, and a workshop that divided every strip from one edge is named nineteen times in twenty from six strips — with no creep in the picture.
What a pair is forA railing matched one along stands underground
Match every post of a railing to the next one along and two photographs build a perfect railing at 19.8 metres instead of 9.0 — straight, evenly spaced, every post on both its rays. Nothing near it can say so. What convicts it is what it stands on: its feet are 1.32 metres under the ground, and one ground mark read to a pixel shows every foot a whole period out from sixty metres away. Its length helps only through its ends, and once those leave the frame, counting matches prefers the wrong railing.
Many pictures at onceTwo 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.
Measuring from one pictureA wrong model shows at its own level
Give a single-view measurement the wrong length and the error stays in the length. Give it the wrong model — a slope where it assumed level ground, a horizon five pixels out, a focal length off the barrel — and the error lands on the level that model belongs to and on every level above it, never below. That is a diagnostic. Two equal things at different depths catch a wrong plane; two equal things at right angles catch a wrong shape; marks read to 0.4 px see a slope of 0.38°. A wrong length, alone, nothing in one picture can see.
Through water and glassA calibration through glass reports a prism
Give a calibration that knows nothing of a 2° wedge in front of the lens the two decentring terms every standard model carries, and it takes three-quarters of the swirl the wedge leaves, moves the principal point further rather than less, and reports a lens decentred by about 0.013 — a lens that does not exist. Give it thin-prism terms instead and it takes more — nine-tenths at twenty-five degrees off the axis, almost all of it nearer — and reports what the glass is: a prism. Either way the residual that announced the glass goes quiet, out to 41° and 46° off the axis instead of 22°.
The second eyeA turn moves the picture, not the blind point
A vehicle that yaws for a moment between frames points its camera somewhere new without taking it anywhere new, and that is exactly what a turn cannot help with. The mark it is driving toward stays blind at every turn; the disc around it moves by two hundredths of a pixel for a 3° yaw; with a sway, a turn changes a mark's depth cost by 0.2 per cent. Aiming the camera off the road moves the blind point across the picture — and off it past 30° — but the thing it blinds is still the thing the vehicle is heading for.
Through water and glassA water surface's height trades against its index
Three eyes that must find the water's surface as well as its index find the two along a valley: a surface assumed higher bends every ray sooner, and an index assumed weaker makes up for it. From one stick at a quarter pixel the index is known to ±0.14 and the surface to ±48 mm, and the fits do worse than that promises — they slide down the valley, and the metre-long stick comes back 881 mm. A second stick eight metres out narrows the valley to ±7.6 mm. Holding water's own index at 1.333 closes it: the surface to ±23 mm, the stick to 994.
The real instrumentA 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.
The second eyeSecond pictures reach a floor unless they go round
A point with little parallax can be read against several second pictures, and each should count in proportion to the square of its parallax. Doing that does less than expected, because every second picture shares the same first reading. In one direction, no number of them goes below one pixel over the longest parallax. Spread them round the point and a fit can find out where the first reading really was, and the floor goes away.
What each system gave upA stepped hand passes for a tiring one on a short strip
A painter who steps dividers from each row to the next leaves an error that accumulates — a walk from the edge begun at, with no fatigue in it. The fit of habit and creep takes about four-fifths of that walk before any reading of the scatter starts, because a walk is slow and so are the two numbers. What is left names the starting edge nearly as well as fatigue does, and on a six-bay strip it cannot say which of the two mechanisms left it: the verdict needs twenty-four bays on one strip, or sixteen six-bay strips by one hand.
Many pictures at onceA 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.
What a pair is forA facade matched a column out comes forward, not back
A railing matched one post along is rebuilt 2.2 times further away. A facade of windows 2.4 metres apart cannot be: a one-column shift that way needs a negative disparity, and the only mistake left in front of the cameras brings the facade to 6.3 metres from twenty, a third its size, hanging 1.1 metres off the ground with nothing under it to sink into. Its corners hold it only if the wall is assumed flat. Two corners then convict it at every period; one corner and a drainpipe need the pipe half a metre in for windows and four and a half for a fine repeat.
The second eyeSteering swings a camera too little to see ahead
A vehicle yaws about its rear axle, so a camera mounted ahead of the axle is swung sideways whenever the vehicle steers — a real baseline, where a turn alone gave the blind point ahead nothing. Half a degree of yaw swings a windscreen camera 1.3 centimetres and a bumper camera 3.1. Straight ahead at eight metres a pixel then costs 96 and 40 per cent of the depth. The cost falls as one over the arm times the yaw, and a usable 10 per cent needs 7.2 metre-degrees: two degrees of steering every frame on a bumper. Ordinary lane-keeping gives a fraction of that.
Measuring from one pictureA plumb wall tells a slope from a horizon, faintly
A sloping ground and a misplaced horizon fail the ground's depth test alike. A wall standing plumb beside the ground separates them in kind: a slope leaves the wall's test at 1e-16, and a horizon moves it. But it moves it a tenth as far as it moves the ground's — 0.20 per cent for a horizon five pixels out, against 2.11 — so a pair of windows a metre wide cannot see the difference, and a whole facade six metres wide can tell a one-degree slope from its matching horizon 95 times in a hundred. A wall running straight away from the camera sees nothing at all.
Through water and glassTurning the glass tells the window from the lens
A calibration done through a two-degree wedge of glass reports a lens decentred by about 1e-2 that does not exist, and hides the glass inside it. Calibrate twice with the glass turned between, and the glass's part turns while the lens's stays: a half turn finds a real lens decentring of 2.2e-3 to 5.2e-4 with readings to 0.4 px, where one calibration was 1.0e-2 out. Three calibrations spread evenly round the turn do better, because the loop the glass traces is not a circle. What never separates is the radial coefficient, which has no direction for a turn to carry.
The eye that movesA vehicle's pitch lags the road by its wheelbase
A camera fixed to a vehicle does not pitch with the road under it; it pitches with the chord between its wheels. Over a step from level to six per cent, a two-slit scroll's rows — silent on any steady grade — depart by up to 2.18 rows over 6.5 metres of road for a 2.7-metre wheelbase, against 4.26 over 4.0 for a camera that pitched at the point. The excursion's width is the eye's own chord plus about the wheelbase, and one line of posts reads the wheelbase back to ±8 centimetres. A vertical curve does not silence the rows either; it shrinks them as one over its length.
Drawn confidentlyA steady creep draws the rule's bow
The arc reading was built to find the constant-ratio rule by the shape its pavements bow the diagonal. A correct pavement painted by a hand that lets each transversal gap grow 4.5 per cent on the last bows it the same way — to 0.31 pixels, a cosine of 0.9994 — on eight braccia, and 1.5 per cent is enough on twenty-two. The arc reading measures how much a pavement's tiles lengthen, not what lengthened them, and against its counterfeit it is a coin. The rule's own definition still tells them apart: its gaps' logarithms are straight, a correct pavement's bend. Read to half a pixel, that names the rule on eight braccia 94 times in a hundred.
The other systemsNo view draws a curved plate true
The auxiliary view is descriptive geometry's answer to a foreshortened plane — turn until the plane is parallel to the paper and it draws at true shape. A bent plate has no such direction and a dished one has none twice over: the best view of the first is out by 1 − cos w and the best of the second by 1 − cos²w, worse by exactly 1 + cos w, because its normals need two parameters rather than one.
Where to standThe eye that reaches the most
A higher eye buys the faces occlusion was hiding and loses design off the far end of the object, so "the best eye" is not a question with an answer until somebody says which of the two they are paying for. On three objects the answer is as high as possible; on a corridor with a doorway in it the two quantities cross and the best height is two and a bit metres.
Surfaces that are not flatThree conditions, and three prices
A matched picture needs the seat, the horizontal scale and the vertical law all at once. Each is broken alone here with the other two held, and all three turn out to be linear in the mismatch — the unforgiving case, with no margin at all. The vertical law costs a third of an arcminute on a television and eighty-four on a dome, because it is the difference between an angle and its tangent and that difference is cubic in the picture's vertical field.
What each system gave upA stepped hand's walk is in the creeps it was given
Fit each strip of a picture alone and a stepping hand's walk vanishes into a creep of that strip's own. Fit the picture once, with one habit and one creep for the hand, and the walks come back as a disagreement between strips that their small residuals cannot explain. Read against a line set once, eight six-bay strips name the hand ninety-five times in a hundred; the residuals of the same strips name it eighty-six, and needed sixteen strips for what the creeps do in eight.
What a pair is forA raised eye reads the storeys as the pair read the columns
Put a third camera 3.5 metres above the left one and its epipolar lines run up the facade: the storeys repeat along them as the columns repeated along the level pair's rows, the roofline and the foot carry depth where the corners did, and a raise shorter than a storey can only pull the windows nearer. The three pictures together refuse the level pair's one-column mistake at every raise except those where the storey over the raise equals the period over the baseline — 1.375 and 2.75 metres here — and even there the facade's lowest storey lands on the ground.
Many pictures at onceThe 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.
The eye that movesA scroll camera rings with its vehicle's suspension
A vehicle does not pitch with the chord between its wheels; its body follows that chord through springs, lags it, overshoots and swings back. Over a step to six per cent at ten metres a second, a body sprung at 1.3 hertz parts a scroll's rows by up to 2.6 pixels where a rigid vehicle parted them by 2.2, and leaves them parted eight metres past the step. No rigid vehicle can take the difference, and one line of posts read to a tenth of a pixel gives the suspension back: 1.302 ± 0.011 hertz, damped 0.303 ± 0.008.