The thread: Determined up to something — page 3
One picture and three people
A curved screen can be pre-warped for one seat, and the search over which seat to choose returns the middle one to a quarter of a per cent — there is nothing to be clever about. What the correction buys the sofa as a whole is six per cent, and the worst seat grows at fourteen pixels for every metre of audience, with no width at which it is zero except one person.
The second eyeAn ambiguity is not an uncertainty
Eight marks to forty cuts a solid scene's pose error from 19.8° to 0.7° and leaves a flat one at 48°. The two failures look identical from inside — a confident answer, a residual at the floor — and they respond to opposite remedies, so telling them apart is worth more than either measurement.
Mirrors that are not camerasTwo matches are enough
A general fundamental matrix has seven free numbers and needs eight correspondences. A mirror pair's has two, and two correspondences fix it — with a straightedge, on a print, by drawing the line from each mark to its reflection and marking where the two cross. Given the same sixteen marks read to four tenths of a pixel, the constrained fit lands 4.8 times closer to the truth than the eight-point algorithm.
What a pair is forRaise the gaze, and the line is gone
Turn two eyes 20° aside in the plane they share and the horopter keeps its vertical line — but the line stays in the median plane, 1.277 m ahead, not at the point being looked at. Raise the gaze as well and the rule by which each eye rolls decides the rest: Helmholtz's rule keeps a line; Listing's law and Fick's rule make the eyes' relative motion slide 3.93 mm and 7.09 mm along its axis, no point stays put, and the horopter becomes one curve.
The real instrumentThe lines that calibrate a lens
One straight edge through the centre of a picture says nothing about a lens's distortion, and one 180 px from the centre determines k₁ to 9.0 × 10⁻⁴ — the precision rises in proportion to the offset. But distance from the centre is not enough. Crowd three edges on one side and, the moment the distortion centre is also unknown, the coefficient is ten times worse, because a bend on one side looks like a moved centre; put one edge across the centre and it barely changes.
Many pictures at onceA 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.
Mirrors that are not camerasSquare to the camera is the worst mirror
A mirror pair's baseline runs along the mirror's normal, so a mirror facing the camera puts the second eye directly behind the first — the forward-motion arrangement, with the epipole in the middle of the frame and the rays to a mark crossing at 23°. Turning it forty-four degrees opens that to 65° and cuts the worst depth error threefold, and the number to watch is not the angle but where the reflected lens sits on the print.
Measuring from one pictureAn area, out of one photograph
A patch of ground comes back at 5.205 m² from one photograph, to 1.8 × 10⁻¹⁴, through a homography built from four marks and their four known positions. What is worth knowing is how it degrades — the patch's extent across the picture is read with an error growing as the depth, its extent into the picture with an error growing as the depth squared, and at twenty-eight metres the two are sixteen times apart — which is the depth divided by the camera's height.
What a pair is forA mismatch on its own line needs a third eye
Slide one mark of a correspondence 30 px along the epipolar line the other mark fixes, and every test two photographs can run stays at the arithmetic floor — epipolar distance 2.2e-14 px, the two rays meeting to 1.5e-15 m, reprojection 1.1e-13 px — while the point is reported half a metre too near. A third picture exposes it by 21.4 px from a third eye two metres off the first line of sight, and by exactly nothing from an eye on that line.
Many pictures at onceClosing 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 second eyeTwo marks off a known plane find the other eye
Map a courtyard's ground from one picture into the other, and every raised mark lands somewhere the map did not send it — displaced along a line through the image of the other camera, to a fifth of a billionth of a pixel. Two such marks put that image where it is, and with it the whole epipolar geometry.
The eye that movesA scroll can be asked its own radius
The two marks a bend leaves separate exactly. The along-roll scale alone fixes the angle in the disparity, so one point and a neighbour at its depth give back the radius and the depth in closed form — 200 m and 40 m returned to a part in 10⁹, with no search. The two answers are not equally held: a scale read one per cent too large under-reads the depth by one per cent and over-reads the radius by tan(φ − α)/α, which is 50 for a point ten metres from a five-hundred-metre bend. And a painter who evens the scale out by eye reports a gentler bend, never a bend that was never there.
Measuring from one pictureAn outline reaches a waist and not a dish
In a cross-section nothing a silhouette does can enter a concavity. On a solid that stops being true, because a solid has lines a section does not contain. A spool's waist comes back in full from a ring of outlines, and a dish in the same object's lid never does. What decides it is whether a straight line can leave, and the answer is water.
What survivesThe two points a picture hides
The stratification rung says a photograph of a plane is projective, becomes affine when the vanishing line is known, and becomes metric with one more fact. The one more fact has a size and a shape: it is two points, complex conjugates on the vanishing line, and a rectification built from those two and nothing else returns every world angle to a tenth of a picodegree and every ratio of lengths to five parts in a quadrillion — and refuses to name a length.
The other systemsNothing moves when the object does
Slide a box 3.2 m across the world and its parallel drawing is the same drawing translated 131.5 px — every edge the same length to 4e-14 px. The perspective drawing of the same move changes its edge lengths by 87.2%. One family's pictures depend on where a thing is; the other's do not, and almost everything the two families disagree about follows from that.
The second eyeAn epipole in the picture leaves a blind disc
Step a camera half a metre straight forward and the image of the other eye sits in the middle of both pictures. Around it lies a disc where one pixel of reading costs a tenth of the depth or more — 20 px across a surface 2 m off, 247 px at 16 m — and at its centre no depth is recovered at any range.
What a pair is forA 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.
Many pictures at onceThe 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.
The eye that movesThe stations are also a staircase
A level eye draws a ground point on a row that depends on the eye's height and the ground's only through their difference. So a landscape drawn from three stations at 1.6, 4 and 11 m over flat ground is, to 6 × 10⁻¹⁴ px across ninety-nine samples, the same picture as one eye at 11 m over ground stepped 9.4 m and 7.0 m above the far plain. The station heights a reader recovers are recovered on an assumption the marks neither supply nor test — and what would decide between the two readings is a cliff 103% of a band's own height, which is exactly where the convention lays its mist.
Measuring from one pictureA groove is reached only along its own plane
A groove round a drum is the kind of hollow outlines can clear, and a level ring clears it completely. Raise the camera three degrees and more than half of it stays filled at any number of views. On this object the height of the camera decides the result, and the count of views buys almost nothing.
Light and mirrorsA light far enough away
The evidence in a photograph that its light is in the room rather than at infinity is one number — how far below the horizon the shadow lines put the light's foot. It falls off as one over the distance, from 211 px at 4 m to 10.8 px at 266 m, while the recovered height stays exact to 5e-13 of itself. What fails first is not the arithmetic; it is the evidence, and one pixel of error costs 0.21 mm of height at the near end and 0.07 m at the far one.
Where to standThe room the eye may stand in
An anamorph is correct from one point, and one point is not a thing a person can occupy. Fix a tolerance on the picture and the set of eye positions that meet it is a solid — for a design 1.8 m wide and a ten-millimetre tolerance it is 36 mm long, 14 mm across and 31 cubic centimetres altogether, a spindle pointing along the line of sight. Ten times the tolerance is a thousand times the room.
Measuring from one pictureAn angle on the ground
An angle needs no length at all — it is scale-free, so the one thing a single view cannot give is not the thing an angle wants. What it wants instead is the plane's shape, and the difference is measurable — assume the reference rectangle's aspect a quarter wrong and the angle moves 11.5°, a length running into the picture is out by exactly that quarter, and a length along the reference's own direction does not move at all.
What a pair is forVergence moves the shells and does not respace them
Turn two eyes inward and the depths a whole-pixel reading can report stop being planes and become a family of near-circles through both eyes — the twenty-pixel shell standing at 0.74 m forty degrees aside where a parallel pair puts it at 3.82. The spacing between consecutive shells is the same to 0.07 per cent across the whole field, so vergence relabels the rays and does not sharpen them, and the resolution argument for turning the eyes in does not exist.