Sampling — where it appears
Named by 18 essays across 8 fields — each of them below, with the objects they name alongside it.
One depth per sample is not enough
A depth buffer keeps a single distance at each sample, so a post-process blur can only ask how far away the thing at this pixel is. Across an occluding edge that answer is two depths and an occlusion, and the gather it produces differs from the pupil's own integral by 70 per cent of full scale over a band eleven pixels wide.
Counting 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.
A fog band softens the horizon only by repainting the ground
Carry the sky's full haze a few rows down into a distance-fogged ground and the step at a walker's horizon is not removed but spread into a ramp, whose steepest row changes by about one and a half over the band's width of the whole rise. Two rows barely move a step of 0.63; softening it to the unaided horizon of an eye 150 metres up takes 32 rows at every visibility from one kilometre to thirty — and repaints all the ground beyond forty metres hazier than it is. The choice between an honest distance fog and a seamless horizon is real. What four rows of band do buy is the shimmer: the last row's saw-tooth falls from 0.061 a column to 0.012.
A rough road rings the suspension well enough to read, over four hundred metres
A body on springs responds to every few millimetres of unevenness in a road, and a scroll camera's rows respond to the body. With no ramp anywhere, 400 metres of posts along a good road give the body's frequency to within two hundredths of a hertz and its damping to within a tenth of itself — not from the highest point of the rows' spectrum, which is a mixture of the spring and the scroll's own answer to the road, but from its whole shape fitted against both. A step read the same spring from 40 metres. The road's own roughness needs ten times the length to come within twice the step's spread.
A climbing eye names the column mistake, if it knows how high it is
A level pair that matches a facade a column out rebuilds it nearer, and a third camera climbing above the left one watches the rebuilt windows slide down its picture at 98 pixels a metre for every column of the mistake. Read against the real windows, that slide is a ruler: a metre of climb from level names a one-, two-, three- or four-column mistake every time. What it costs is not frames but knowledge — the climb must begin before the rebuilt facade slides off the lattice, and the camera must know its own height to about a quarter storey over the mistake's slide, 34 centimetres for one column and 9 for four.
Four marks before anything is said
A reader fitting a correct perspective to a row of transversals has three numbers to choose, so three transversals fit whatever they are and the fourth is the first that can disagree. Below that count a pavement is unfalsifiable, and a great many painted pavements are below it.
A staircase winds the mistake round the lattice
A third camera climbing a stair along a facade sees a stereo pair's column mistake slide through its picture along the stair's own direction, wrapping at the columns across and the storeys up — a line wound round a torus. Such a line comes back onto a real window only when the stair's rise over its run is a ratio of whole storeys to whole columns; a degree off the diagonal, no two readings come within 3 px. The stair names the mistake from one frame, as a vertical climb does, but its readings sit twice as far apart in a frame, so a camera that knows where it is only to 10 cm still names it 95 times in 100 against 82, and a 30° stair keeps naming it a metre and a half along, where a vertical climb has lost half.
Jittered frames rebuild a camera's horizon, saw-tooth and all
A renderer that moves its sample a fraction of a pixel every frame and averages the frames turns each pixel's single sample into an average over the pixel's area — which is what a camera's pixel records. Along a hazy, rolled horizon that removes the quarter of the shimmer point sampling adds, 0.082 a column down to 0.061, and stops there: eight Halton frames are within a hundredth of the camera, and sixty-four are no better. The rest of the saw-tooth is the camera's own, and so is the horizon's step. Only a band of repainted ground goes below them, and jitter's own price is a flicker of 0.29 a frame until a history averages it away.
One pass reads the road's slope as well as its spring; a second speed only doubles the road
Two passes over one rough road at two speeds see the scroll's own answer to the road in the same place and the spring's swelling in two. That looks like the way to read the spring without being told the road's spectrum. It is not needed. A single 800-metre pass, left to choose the road's slope for itself, reads it to a tenth — 2.57 for a road of 2.5 — and the damping to 0.29 against 0.30, where the same pass told the survey standard reads 0.51. The second speed buys what doubling the road buys, and shares it out between the frequency and the damping.
A floor is read along curves
Whatever a shadow says about the floor it landed on, it says only where the shadow is — and a shadow is a curve while a floor is a surface. Shadow curve length grows exactly linearly in the number of lamps, by a fitted exponent of 0.999, and the fraction of floor within two centimetres of one grows more slowly at 0.94, because the curves begin to overlap. At thirty-two lamps, seventy-one per cent of a nine square metre patch has still never had a shadow on it.
A rolling horizon outruns the history that was meant to quiet it
Jittered frames averaged in a history draw a still hazy horizon as a camera does, four times nearer than one centred sample. Roll the camera and the history has to follow. Left in place, it is worse than no anti-aliasing at all from a degree a second. Reprojected exactly through the roll, it still reads last frame's picture between pixels, and an interpolated step is a softened step, softened again every frame: past one and a half degrees a second the history sits further from the camera than a single centred sample, and at sixteen it paints the first row of ground as sky. No blend fixes both its error and its pops. Four rows of band move with the horizon popping a fortieth as much — at the price of a quarter of the fog's range in repainted ground.
A clamp keeps the sky clean and leaves the rolling horizon smeared
Clamping a jittered history to the range of each pixel's neighbours is the renderer's standard guard against a moving edge smearing, and on a rolling horizon it does half the job. Two rows from the horizon, where the neighbours agree, it pulls the smear back at once: a sky row the history had darkened to 0.88 is restored to 0.99. On the rows the horizon crosses, the nine neighbours hold sky and ground together, the window runs from 0.12 to 1.00, and the smear passes through it untouched. The crossing with a plain centred sample moves from 1.5° a second of roll to 1.8; at 16° a second the history is 0.070 from a camera instead of 0.093, still nearly three times a centred sample's 0.025. And a clip tight enough to reach the smear brings the jitter back: on a still camera it pops four times as hard.
An error with two terms
Two results from machineries with nothing in common have now found the same shape. A panorama's parallax separates into a term that halves every time the frame count doubles and a term with no frame count in it at all; a silhouette's error into an excess that falls as one over the square of the view count and the area of a concavity that is the same number at four views and at a hundred and twenty-eight. Fitting both terms turns the distinction into a measurement, and pointed at seven of this collection's own laws it reads every one of them the way its own essay does.
The distance at which two lamps part
Two lamps five centimetres apart are one lamp, and the drawing is right to say so. The separation at which they become two is proportional to how carelessly the picture was clicked — 3.6 centimetres at half a pixel, 7.2 at one, 28 at four — with no floor anywhere, so nothing but care stands between a reader and any separation at all.
What a null result is worth in decades
The first draft of this expected a short sweep to invent a floor, on the reasoning that least squares always spends a free parameter. It does not — on exact data the fitted floor of a floor-free law comes back at three parts in a quadrillion. The failure is the other one and it is worse because it looks like a result. Over a third of a decade at one per cent noise, floors of a fifth of the first sample are still consistent with the data, and the fit reports none while telling the truth.
Two distances to infinity
A parallel projection is the limit of a perspective one, and the limit arrives twice. Which faces get painted settles within three object radii, because a face is either round the back or it is not; where each mark lands falls like one over the distance and is still out at thirty-four. Far enough away has two answers an order of magnitude apart.
The bias out of reach
A single-view height recovered from the cross-ratio has an error with two terms. The spread of an average of m readings falls as one over the root of m, by a fitted exponent of −0.52 over three decades. The bias does not fall at all — and it is eight microns, against a single-shot spread of six millimetres, so the two cross at 535,000 measurements. There is a floor here and no experiment anybody will run can see it.
The arrangement the count cannot see
Five posts laid out five different ways give the same leverage to a sixth and the same separation limit to a quarter, and the sixth arrangement — posts strung out along their own shadows, built to be exactly degenerate — is no worse than the rest. The degeneracy belongs to the family the count does not use, and its conditioning there is exactly zero.
Named alongside it
The objects these essays reach for when they reach for this one.
instrument limitHorizonConditioningIdentifiabilityPower lawAsymptoticsClip spaceDemonstrationHomogeneous coordinatesleast squaresProjective mapResidual