The thread: The round trip — page 4
A shadow decides which landscape it is
One sun over a several-station landscape is one sun: every band images the light's direction and its shadows' at the same page point, and each reports the altitude as 22.000000°. A shadow is still dislocated at the join, and the reading that could not be settled by any ground point is settled by one — because a ray crossing the seam has a riser to descend that the flat reading does not give it, and the two tips land 5.94 m and 11.7 px apart.
What survivesThree constructions, one map
A shadow cast on a floor, an anamorph painted on one, and a reflection in a mirror are usually treated as three different subjects, each with its own derivation and its own figure. Decomposed into their fixed points and lines, three of them are the same kind of map — a central collineation with a line of fixed points — and the fourth, a rectification, is not. That difference separates changing where a picture is seen from, from changing one picture into another.
The rectangle behind the lensA close picture carries its own distance
A 50 mm lens focused at one metre stands 2.632 mm beyond its focal length, and a picture records where the lens stood. Recover that from two vanishing points, put in the engraved focal length, and the focus distance comes back — to 0.7 per cent at 1 m, 3.7 at 5 m, and as an interval 28 per cent wide at 30 m. A single picture has given a scale. What it has given is where the lens was focused, and the subject can be anywhere in a sharp band two to fifteen times wider.
Many pictures at onceA 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.
Through water and glassThe residual does not warn
Fit a straight line to the stick a stereo pair puts back and the fit looks best exactly where the reconstruction is least supported: the residual is 0.535 mm at a level baseline, where the two rays meet perfectly, and 0.038 mm at sixty degrees of roll, inside the band where they miss by more than a pixel covers. Over the same sweep the fitted line is 285 to 584 mm short of the stick's metre — fourteen thousand times its own residual at worst.
Through water and glassA fit weighted by the miss trusts only the surface
Weight each point of a reconstructed underwater stick by how well its two rays meet, and the fit hands all but a ten-billionth of its trust to the one point where the stick enters the water — and reports a residual of nothing at every rolled baseline. Floored at the reading error, the weighting changes the answer by a few thousandths of a millimetre. And the miss itself, the one honest number, is exactly zero at a level baseline where the stick comes back 285 mm short.
Many pictures at onceThe 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.
What survivesTwo lines at infinity
A picture of a plane has two of them and they are not the same line. One is the horizon, where the plane's own infinity went; the other is where the picture's coordinates run out. The words ellipse and hyperbola are about the second, and every scrap of metric information is on the first — so a circle whose photograph is a hyperbola calibrates exactly as well as one whose photograph is an oval, to 7e-14 of a degree.
Where to standThe height a flat floor cannot give
The marks of a floor anamorph name the eye's position on the floor exactly and say nothing about how high it was — every candidate height explains them perfectly, to one part in a thousand trillion. That is a fact about planes rather than about anamorphs. Ripple the floor by six centimetres and the family collapses: the true height explains the marks exactly and the nearest wrong one, five centimetres away, leaves two millimetres on a design 1.8 metres wide.
Constructing a viewThree-point, laid out with a straightedge
Recovering a camera from a drawing is the familiar direction. The other direction — stand somewhere, measure a room, and lay the picture out — had never been taken in three-point, because the third axis needs a measuring point on a line nobody draws. With it, every corner lands where the camera puts it to three parts in ten million million of a pixel, and nothing anywhere is judged.
The eye that movesA scroll of a climbing road measures its grade
Every reading of the two-slit scroll has leaned on its two drawings of a point sharing a row, because the eye is at one height at both moments. On a road that climbs they do not — and what parts them is the height climbed between the two moments over the reach, which on a straight climb is 2·f·g·sin φ for every point at every depth and height. The scroll does not lose its rows to a hill. It gains a third mark, a gradient meter that a level bend cannot counterfeit.
Many pictures at onceA 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.
What survivesFive marks and the sixth
Five points determine a conic exactly — five coefficients up to scale, five equations, nothing left over — so a fit through five marks on a photograph is not a fit at all. The sixth mark, withheld, lands on the curve to 1.9e-13 px. And the moment a sixth mark is used, the arithmetic changes character completely: it becomes a least-squares problem, and the residual starts telling you something the five could never say.
Light and mirrorsA shadow across a second object
A straight edge held in front of a lamp defines one plane, and the shadow's boundary is wherever that plane meets something. That turns a picture of a shadow into a measurement: a camera ray and a known plane meet in one point, and the object the shadow is falling on comes back out.
Constructing a viewA drawing has three horizons
The triangle three vanishing points form is not a scaffold. Each of its sides is the vanishing line of one of the three coordinate planes — the floor's horizon and two walls' — and its orthocentre is the centre of the picture. So the horizon a perspective drawing is built on is one side of a triangle, and it is special only because the ground is where things stand.
The eye that movesAn eye that pitches with the road keeps its rows
An upright eye climbing a road parts each point's two drawings by the same few rows, and that offset reads the grade. Fix the eye to the vehicle instead, so it pitches with the road, and the offset vanishes exactly — for every point, at every depth and height. The grade has not gone. It has moved into the posts, which now lean by an amount that grows with their depth, and into one drawing, which can now read the grade on its own.
The real instrumentA 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.
What a machine computesOne homography makes a shadow map the eye's picture
A shadow map for a lamp facing the eye needed 706 thousand texels however its rows were re-spaced, nearly five times the one texel per pixel no map can beat. Warp the whole map by a projective transformation, not only its rows, and it needs 154 thousand — within five per cent of the bound — and every texel, carried into the eye's picture, lands at one pixel. The reason is exact: the eye's picture of a floor and the lamp's picture of the same floor are one homography apart.
The other systemsThe drawing that gives the solid back
One parallel view of a general point determines nothing: two equations, three unknowns, and the kernel is free. What closes it is not a second view but the correspondence — knowing which drawn edge runs along which world axis — and with it the whole solid comes back out of one drawing, exactly.
Systems that kept the measureNo solid casts an aspective figure
Fitting the best single rigid view to an aspective figure — head and legs in profile, eye and shoulders turned square — misses its own marks by 2.6% of the drawn height, and no yaw does better than 3.0% in a full sweep. A genuine single-view drawing of the same body fits to 7.6e-13 pixels, and the five rotations recovered from the marks alone match the convention's own list to 0.0e+0°.
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.
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 survivesThe polar with a straightedge
Two secants through a point cut a conic at four places; the complete quadrangle they make has two more diagonal points; the line through those is the polar. Not one length, angle or midpoint is used, so the whole construction survives the projection that made the picture — and three unrelated pairs of secants land on the same line to 4.3e-13, while moving the point moves it by fifteen orders of magnitude more.