Every essay — page 18
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors Mirrors that are not cameras Measuring from one picture Through water and glass The real instrument The second eye What a pair is for Many pictures at once The eye that moves Systems that kept the measure What each system gave up What a machine computes The rectangle behind the lens The second projection Drawn confidently
Through water and glass
A refracted picture is not a projection at all — its rays, continued into the water, miss each other by millimetres rather than meeting at a point. Everything the other fields rest on is measured here against the case where it fails, from the cross-ratio to Snell's window to a dome port that turns out to be an exact pinhole.
A straight stick in water is a kink and a curve
The bent stick is described as one kink at the surface. Traced point by point, the image two level eyes see leaves the surface 14.96° off a stick leaning 30° — the same whichever way it leans — and then keeps turning, by 1.11° when it leans away from the eye and 7.62° when it leans toward it. A photograph from the eye shows the kink and almost none of the curve, 1.63 px over 166 px, and when the stick leans straight toward or away from the eye it shows neither: the picture is one straight line.
A wedge of glass turns the camera behind it
A pane with parallel faces moves every point and no direction, so the camera recovered through a window is the camera that took the picture. Tilt one face 2° and every direction turns, by 1.04° on the axis and 1.67° forty degrees off it. The best rotation of the frame, 1.14°, still leaves 0.94 px, and no homography does much better, so the picture is no longer a projection from the camera's centre. The camera recovered from three vanishing points through the same glass is turned 3.37° — three times as far — because vanishing points lie where the glass bends most.
The stick a stereo pair puts back
Two eyes side by side reconstruct a submerged stick exactly as the sagittal image — kinked 14.96° — and two eyes one above the other exactly as the tangential one, kinked 9.59°. Roll the baseline between them and the two rays to a point miss each other by up to 3.13 millimetres, past the 2.85 a pixel covers at that range, and the reconstruction is a third stick that is neither — 551 millimetres of a one-metre stick, with its tip at 0.405 metres against a true 0.866.
The wedge recovered with the camera
Admit the glass into the model and the fit finally has something left over. A two-degree wedge is invisible to a reading confined inside eighteen degrees of the axis — the fit calls the whole displacement a rotation of the camera and is right to — and by fifty-five degrees it comes back to 0.07 of a degree. What the picture does not separate is the ordinary glasses — assuming an index of 1.50 for a true 1.52 costs one per cent in the angle and 0.07 pixels of residual.
The 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.
A 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.
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.
A third eye sees the water the pair cannot
A level stereo pair looking at a stick in water gets two rays that meet exactly and a stick 285 mm too short, with nothing to warn it. A third eye raised ten centimetres above the pair sees a disagreement of 2.6 pixels at the stick's tip. Asked to agree with the other two as if there were no water, it hides nearly all of that and makes the stick shorter still. Told only that there is a flat surface, the same three eyes find its refractive index and put the metre back.
A 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°.
A 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.
Turning 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.
A straight stick tests the index only across the sight
Told that each stick is straight, three eyes fitting a water surface's height and its index still slide down the valley between them — when the stick leans away from them. A wrong index carries such a stick into another straight stick, off its line by 0.016 mm and 116 mm short. Turned to lean across the line of sight, the same wrong index bows it by 2.2 mm, and the straight-line fit's slide falls from 0.058 to nothing. A tape on the stick, read only to ±50 mm, stops the slide outright.
A bowed stick is read as a wrong index
Holding a stick to a straight line cured the slide in a refraction fit — but only for a stick leaning across the sight, where a wrong index bows the read stick by two millimetres. A stick that is itself bowed by a millimetre in the same plane is, to that constraint, a straight stick seen through the wrong index: read without error, it moves the fitted index by 0.074 one way or 0.062 the other, and at a quarter pixel the constraint's bias overtakes the slide it was removing at about three-quarters of a millimetre of bow. Its misfit is a fiftieth of a pixel, so nothing in the residuals says so. A tape, which measures length, does not notice the bow at all.
Crooked sticks cancel each other's bows slowly, and eight of them match a tape
A stick bowed by a millimetre fools a refraction fit that holds it to a straight line: the fit reads the bow as a wrong index. Several sticks each bowed in its own random direction pull the index both ways, and the pulls cancel — but not as one over the square root of the count, because the sticks do not pull equally: a near stick pulls two and a half times harder than a far one, and a stick leaning just off the line of sight tens of times harder than one leaning straight along it. Eight crooked sticks leave the index ±0.016 out where one leaves it ±0.037, and with a quarter pixel of reading error they read it to ±0.037 — exactly what one straight stick with its length taped does. The earlier guess was that matching a tape would take a large number.
A crooked stick hides in a quarter pixel, and its residual finds it only below a tenth
Eight sticks stand in still water, one of them bowed three millimetres and the rest one, and a refraction fit holds every stick to a straight line. Read exactly, the bad stick carries the largest share of the fit's residual in about seven sets of ten. Read to a quarter pixel, as a careful photograph is, it carries it in one set of five, barely more than chance: a millimetre of bow moves the index more than the reading error does and leaves its rays missing by a twelfth of a quarter pixel. The residual stays blind until the bad stick is bowed five or ten millimetres, and dropping the stick it ranks worst does nothing measurable for the index.
The second projection
A picture that has been made still has to be shown, and the display is a projection with a correct point of its own. Here the assumed figure width every earlier essay quoted is replaced by an actual chain — sensor, focal length, screen — and the answer is that almost nobody is standing where the picture says to.
The screen sets the distance
Every viewing distance quoted for a picture on a page is conditional on an assumed figure width. Replace the assumption with an actual chain — focal length, sensor width, display width — and the same 50 mm frame is correct from 9 cm on a phone, 83 cm on a monitor and 16.7 m in a cinema. Nobody is standing at any of them.
A wide field on a small screen
A picture rendered at a hundred degrees and shown on a screen that subtends forty-nine is being read from two and a half times its own station distance, so the depicted space is two and a half times too deep. The stretch at the edges everybody complains about is correct; the complaint is really that nobody is sitting where it would be invisible.
Two pictures on one screen
A stereoscopic display puts a point where two sightlines cross, so the depicted depth is b·D/(b−d) and the disparity that reaches infinity is exactly the separation of the reader's eyes — 63 mm, at any screen distance whatever. The depth budget is set by the width of a head and by nothing about the scene.
The render is distorted on purpose
A headset renders a bent picture so its lens can straighten it, which is a lens's distortion polynomial run backwards, and the one case in which distortion is introduced deliberately. The round trip closes to a thousandth of a millionth of a pixel, and the price is that one rendered pixel becomes 0.493 delivered pixels at the edge of the field and one at the centre.
A projector is a camera run backwards
Turn a projector fifteen degrees from square and it throws a trapezium; keystone correction cannot add light outside it, so it shrinks the picture until it fits and discards a sixth of the panel. And the instrument itself comes back out of the picture it threw — 2880 panel pixels recovered against 2880, by the function the wrong field wrote for hand-drawn cubes.
Turning the cameras inwards
A stereo pair made by rotating two cameras toward a common point puts the same world point at different heights in the two pictures — up to thirty pixels here, on a frame of four hundred. Two eyes level with each other see every point at the same height, so a pair with vertical difference is a pair of pictures of no scene at all.
The screen is a picture surface too
A curved television is one of the six named picture surfaces sitting in a living room, and from its own axis it delivers azimuth in proportion to the picture, exactly. What it is shown is a rectilinear picture from a sofa, and the difference is not a matter of degree — a flat screen from any seat shows a homography of the intended picture, so it is a correct picture of a transformed scene, and a curved one shows a map that is not a homography from any seat at all.
The screen that names the seat
A flat screen shows a homography of the intended picture from every seat in the room, and an observer's own framing is free to be a homography too — so a flat screen's picture is consistent with every seat there is. A curved one is not, and the seat comes back out of the picture in all three directions, in units of the screen's own radius.
The seats a screen will accept
Collect the seats whose picture is within a pixel of the one intended and the result is a solid — half a cubic centimetre in front of a curved desk monitor, a litre in front of a curved television. Ten times the tolerance is a thousand times the room, which is the pavement anamorph's own law arriving on an object that has nothing else in common with it.