The thread: One machine used twice — page 5
A wedge moves the centre, not the lens
A wedge of glass in front of a lens deflects every ray by a little more the further off the axis it goes, which looks like the shape a radial distortion coefficient describes. Fitted together, the two are nearly independent — correlated at 0.16 at most — and a calibration that knows nothing of the glass does not invent a lens: it reports a coefficient of about 0.002, moves its principal point by 4.5 to 14 pixels, and leaves a swirling residual that no radial model takes. The blame goes to the camera's centre, and the residual says so from twenty-two degrees off the axis.
The rectangle behind the lensA dolly zoom off the axis keeps a line, not a plane
Step toward a subject along a track that is not quite the line of sight, and zoom to hold its size, and the plane that stood still in the classic shot stops standing still. The step now spreads from a point beside the centre while the zoom still shrinks toward the centre, and the two cancel only along one row of the picture, one depth per column. The subject itself slides by f·d·sin ψ over its distance — a pixel once the track is a quarter of a degree out — and turning to follow it holds the subject at the price of bending the rest of its plane, while shifting the frame instead holds the whole plane exactly.
The real instrumentA stereographic fisheye is a division model
The division model divides the picture radius by one plus a coefficient times its square. The fits that compared it with the polynomial were not of that model: they multiplied instead, and the model they measured has neither a fold nor a horizon. Fitted as it is written, the division model follows every fisheye law more closely than the polynomial at every field from forty degrees, and the stereographic law it follows exactly — the law is the model, with a coefficient of minus a quarter. Its horizon then turns out to sit beyond the lens's own ninety degrees, and pinning it there is a trade rather than a free constraint.
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.
The other systemsThe shadow rules that hold here
Drop the foot, run a line from the top at forty-five degrees, take the intersection: the drawing manual's shadow construction is exact in a parallel drawing, to arithmetic noise, at every point of the picture and with one set square. It is where the rule came from, and carrying it into a perspective picture is what broke it.
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.
Surfaces that are not flatThe camera that is a cylinder
A swing-lens camera turns its lens about its own entrance pupil and sweeps a slit across film bent into a circle concentric with it. Compute where the light lands, unroll the film, undo the pinhole's inversion, and the result is not similar to the cylindrical picture surface — it is the same map, to the arithmetic floor. What it pays instead is detail, and a shear on anything that moves.
The rectangle behind the lensA frame's shear knows travel only over depth
Read a frame row by row while the camera turns and travels, and every vertical post leans — the near ones more. The lean is the turn plus the travel over the post's depth, and that sum is all the frame holds: twice the travel past posts twice as far draws the same frame to eighteen decimal places. Two posts cannot separate turn from travel. A facade can, because a turn leans the edges of the frame more than its middle, but the two signals are 99.8 per cent alike, and reading them apart takes a pixel on every row.
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 real instrumentAn image circle pins a fisheye's horizon, if its field is known
A fisheye picture stops at a circle, and for a lens covering exactly 180° that circle is the ninety-degree radius a division model calls its horizon. Handed to a calibration as one more observation, read to two pixels, it takes the error at the edge of an equidistant lens from 9.4 pixels to 2.0 — at a cost of 0.3 pixels inside the field the targets covered. Read as ninety degrees on a lens that really covers 190°, it makes the edge five times worse than no circle at all.
Light and mirrorsThe ball stands at a focus
A ball on a floor casts an ellipse, and the point where the ball touches the floor is a focus of it — not near a focus, on one, to two parts in ten thousand million million. That is Dandelin's theorem arriving somewhere nobody puts it, and it is the only thing about the lamp that the shadow gives away.
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 rectangle behind the lensA rolling frame on a bend is right at its centre
Read a camera car's frame row by row as it rounds a bend and every still point moves — except the bend's centre, at every height, and the horizon row, at every depth. Each mark's displacement is its streak scaled by its row time, so the rolling frame is undistorted wherever the global one is sharp. And the usual repair makes it worse: a gyroscope that undoes the turn row by row puts 0.73 px back into the centre and is a net loss everywhere nearer than twice the bend's radius.
The second projectionKeystone correction spends the panel unevenly
Turn a projector fifteen degrees and keystone correction throws away 18 per cent of its panel — the number a specification quotes. It is the smaller half of the price. What remains is spread unevenly: the near edge of the corrected picture gets a panel pixel for every source pixel, the far edge three-quarters of one, so the far edge carries 58 per cent of the source's detail. A tipped projector keeps more than a turned one, and a lens shift that places the same picture loses nothing at all.
What a machine computesA sky is carried as a direction
A triangle of sky has all three vertices at infinity, so the weight that lets a texture reach the horizon is zero everywhere and there is nothing to divide by. The attribute that belongs to such a triangle is the direction itself: carried over w like any other, it is every pixel's own ray to 5e-14 degrees. Carry the vertices' azimuth and elevation instead and a 60° triangle is 7.7° out, a 4096-texel sky needs triangles under ten degrees wide, and a triangle across the seam where azimuth wraps is painted with the opposite sky.
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.
What a machine computesGround and sky meet at the horizon without a crack
A ground of rate triangles and a sky of direction triangles share their vertices along the horizon, and rasterised as a graphics processor does it — positions snapped to a fraction of a pixel, every centre given to one triangle by the top-left rule — they lose no pixel and claim none twice, at any roll and any snapping. What the horizon does have is a sliver: a centre that falls within half a snapping step of it goes to whichever side the snapped edge puts it, and one that falls exactly on it, rolled one way, is given to the ground at a weight of zero. A ground stopped at a far plane leaves the crack the shared vertex never does: f·h/D rows.
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 rectangle behind the lensA rolling frame on a bend is a coarse curvature gauge
Read row by row on a bend, a frame leans every vertical by an amount proportional to one over its range less one over the bend's radius, so posts at several ranges lie on a straight line that crosses zero at the radius — exactly, to a thousandth of a metre. Off the axis the zero moves to R·cos of the bearing, the circle on which a camera on a bend is sharp. As a gauge it is coarse: one frame's eight posts read a 50 m bend as 45 m (35–74) with tops placed to half a pixel, and sixteen frames as 51 m (47–55). The slower the sensor reads, the better it measures the road.
Light and mirrorsThe floor is a choice of coordinates
Four rungs of this field have measured what a curved floor costs a shadow reading, in millimetres. It costs nothing. A shadow mark sits on the ray from the lamp through the occluder's edge, and where along that ray the floor caught it is a fact about the floor alone — so un-casting in rays returns the occluder to three parts in ten thousand of a millionth of a millimetre on a plane, a dish, a ridge and a step alike, while the same shadow read in the plan is wrong by up to 216 millimetres.
What a machine computesA ground and a sky share an elevation, not a distance
Read from its own attribute — the ground's rate and weight, the sky's direction — each side of a rasterised horizon reports the elevation of its pixels' rays to 2.2 millionths of a radian, in 32-bit arithmetic as in 64, so a haze painted by elevation crosses the edge without a seam. A fog by distance cannot, and should not: the ground a row below a walker's horizon is 1.2 kilometres away, and at thirty kilometres of visibility the fog steps by 0.85 in one row. A camera's pixel, averaging its area, records a step of 0.69.
The rectangle behind the lensA tracked post reads the bend, if the speed is known
A rolling camera aimed at a bend's centre passes across its roadside posts rather than towards them: in half a second a post twelve metres out moves from 12.0 to 12.8 metres away and its lean changes by five per cent. Followed through those sixteen frames it is one lean read sixteen times, and one lean is a ratio of the car's speed to the bend's radius. With the speed known it reads the radius as 49.9 metres (48.1–52.9), closer than eight posts in a single frame; with it unknown, it reads nothing. Aimed forty-five degrees ahead, the camera closes on the post, its lean doubles, and one post gives both.
The other systemsThe drawing and the development
A bent plate gets two flat pictures on the same sheet and each is exact in what the other loses. The parallel drawing keeps the generators at one scale and stretches the arc over a factor; the development keeps every length on the surface and keeps nothing of the shape in space. Neither is the plate and the pair of them is.
Light and mirrorsThe lamp comes out in rays and not in plan
One drawing of three posts and their shadows yields two points, and a curved floor treats them completely differently. The lines through each post's top and its shadow's tip meet at the lamp's image to a ten-thousandth of a pixel at every curvature, because a top and a tip are two points of one real ray. The lines through each foot and the same tips meet 113 pixels from the lamp's foot — and the lamp placed from an exact point and a wrong one lands 1.3 metres away.