The thread: Fitted, not assumed — page 2
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.
Systems that kept the measureCounting the eyes needs the room
How many eyes made a picture is not a question the picture can be asked. Told what the room really measures, the rays refuse to meet and a second eye has been caught; told instead that the room is the one the picture is consistent with, the same rays meet exactly, at the first eye. The refusal is real and it belongs to the room.
Mirrors that are not camerasA fitted radius is wrong before it is uncertain
A sphere and a paraboloid of the same vertex radius agree to second order, so a fit over a small aperture cannot separate them. What it does instead is return a confident radius that is wrong by a stated percentage, with a residual far below any measurement floor — 0.03% of bias behind a residual of three ten-thousandths of a degree. The residual only clears a two-hundredth of a degree at six times the aperture, by which point the bias is thirty-six times larger.
The second eyeSeven marks, three answers
Seven correspondences leave a two-dimensional nullspace, and the requirement that a fundamental matrix be singular is a cubic in the mix — one or three real roots. Here it has three, and all three satisfy every one of the seven marks to 8.9 × 10⁻⁹ pixels. The eighth mark, withheld, separates them by more than an order of magnitude.
Many pictures at onceThe 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.
What a machine computesOne plane is nearly free
The near and far planes enter a depth buffer's precision through 1/near − 1/far, and one of those reciprocals is enormous. Pushing the far plane out by a factor of a thousand costs a tenth of a per cent; bringing the near plane in by the same factor costs a factor of a thousand — and an infinite far plane is the limit of the first rather than a separate case.
Through water and glassA ball of water has no eye either
A flat interface is not a projection through a centre and misses by ten millimetres. A sphere of water misses by more than that on a ball the size of a plum — 1.3 mm on a fifty-millimetre radius, and the axis crossings spread over 18.6 mm at seven tenths of the aperture. But at two per cent of the radius the same fit returns 24 nanometres, so a ball does have a centre — one at zero aperture and none by the time it is gathering any light.
What a pair is forBoth coordinates agree on a circle and a line
Two eyes fixating a point straight ahead see their horizontal image coordinates agree on a whole vertical cylinder over the Vieth–Müller circle, the same radius at every height to the last bit. Their vertical coordinates agree on almost none of it — 7.35 px apart at 26° aside and 30 cm up, 29.26 px when the fixation is brought to 60 cm. The points where both agree are the circle and one vertical line, and the line is the axis of the motion that carries one eye onto the other.
The real instrumentA tilted sensor is not a distortion
Tilt a sensor 3° out of square with its lens and every point of the picture moves — up to 7.5 px on the frame drawn here — yet every straight line stays straight to 10⁻¹³ px and the cross-ratio survives to 10⁻¹⁶. The picture is an ordinary pinhole picture whose principal point has moved 22.30 px. A calibration that frees its principal point absorbs it exactly; one that holds the principal point and reaches for tangential distortion terms leaves 1.87 px, and used as a correction it bends straight rows by 4 px.
What each system gave upA camera count needs a tolerance
Asked how few cameras could have drawn a splayed picture, a constructed stack answers four, three, two or one depending on how many pixels of redrawing a reader will allow — one camera at 17.6 px, two at 11.1, three at 4.6. The count is real: a picture built with two groups hidden in it gives back exactly those two, anywhere between nothing and 47.0 px. What it is not is a property of the picture alone, and the floor under it belongs to the hand — a one-camera drawing made by a hand that scatters its far corners by σ splits below about 0.57σ, where the 0.57 belongs to that hand and runs from 0.17 to 1.34 across a hundred and twenty of them.
Drawn confidentlySix tangents and the point nobody drew
Brianchon's theorem is a test a reader can run on a finished drawing with nothing but a straightedge — six tangents, three diagonals, and a question about whether they meet. Pointed at the drawing office's four-centre ellipse it rejects the curve by 1.7 per cent of the figure's own width, 546 times the instrument's own floor, with no true ellipse to compare against.
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.
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.
Through water and glassA point under water has two depths
The apparent depth of a submerged point is not one number even along one line of sight. The rays it sends to an eye pass through two focal lines, and at 60° from the vertical a point 1.50 m down has an image 0.740 m down and another 0.320 m down. Two eyes side by side read the first, a head moving up and down reads the second, and a pair of eyes tilted between them reads neither — their two rays miss each other by as much as 6.39 mm.
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.
Surfaces that are not flatUndoing a picture made on a curve
Three rounds of work here have measured what a curved receiving surface costs going forward — 5.67 mm on a dish, 9.07 mm on a ridge, 529.4 mm on a vault. None of them asked whether the design can be got back. It can, exactly, and the price is stated precisely: you have to supply the surface. Told the floor, the recovery returns a design to 1.1e-12 mm; told nothing, and fitting the four marks every rectification tool fits, it is 111 mm out.
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.
The real instrumentThe entrance pupil walks with the angle
The place a picture is a projection from is not a point in a wide-angle design. Chief rays traced through a strongly curved front element cross the axis 15.07 mm behind its front vertex when they are nearly on the axis, and 4.23 mm nearer the front at 80° of field. So no pivot makes a wide panorama seam clean: at one metre, pivoting at the paraxial pupil leaves 4.39 arcminutes of misregistration along a seam, and the best pivot still leaves 1.41.
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.
Surfaces that are not flatHow well the floor has to be known
“Supply the surface” is an instruction with no number in it, and an instruction with no number is a wish. Here is the number: the error in the recovered design is very nearly proportional to the error in the assumed curvature — the constant varies by 1.4% across a twentyfold range — so about nine parts in a thousand of the curvature buys one millimetre on a design 1800 mm wide, and 40% out costs 43.5 mm.
Systems that kept the measureSize that means rank
In a great many pictures the drawn height records importance rather than distance. That is a decision rather than a mistake, and it can be caught with a straightedge by carrying one figure's height across the room by the taught construction and see where it lands. A tenth of rank in the picture already misses the drawn head by 21 px, the miss is exactly linear in how much rank is there, and the whole test needs two references and no arithmetic.
Measuring from one pictureCounting is a measurement
A tiled floor gives its area with no reference length at all — count the tiles and multiply. The count is an integer, so it is exact wherever it can be made, which is a completely different error law from the rectifier's smooth decay. And the distance at which it fails is set by the tile's depth edge, which foreshortens as one over the depth squared, so 18.7 m for a 62 cm tile, where the across edge alone would have allowed 217.