Model error — where it appears
Named by 50 essays across 12 fields — each of them below, with the objects they name alongside it.
The lens a pavement can hide
A photographed pavement reads as a correct drawing up to a radial coefficient of about four tenths — a lens strong enough to bow a straight edge across the page by nearly six pixels and to print as twenty per cent distortion at the frame's corner. The reason is that a pavement sits near the principal point, which is the one part of the frame a radial map barely touches.
A pane gives a product before it gives two numbers
A flat pane of glass displaces every point it is seen through, and the displacement at small angles is the thickness times one minus the reciprocal of the index. So the two numbers arrive multiplied together. Four panes from 6.5 to 13.2 millimetres thick, with indices from 1.35 to 2.1, agree to under two microns over an eight-degree fan and separate by more than a millimetre over sixty — and a fit over the narrow fan returns whichever pair it started near.
A 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.
A 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.
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.
A model that inverts has a horizon instead of a fold
The polynomial every calibration fits turns around at a finite radius and stops being a map from direction to picture. The division model, chosen because it inverts in closed form, never turns around — it rises for ever toward a horizon at one over the root of its own coefficient, so the whole hemisphere of directions lands inside a finite disc. Fitted to the four fisheye laws it follows every one of them more closely than the polynomial at every field from forty degrees to eighty — a hundred times more closely for the equidistant law at forty, and the stereographic law exactly.
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 rows count hands, not cameras
Rows drawn between two straight sides charge a camera nothing — every strip of a four-strip divergent picture reads back as a flat plane, whatever placed them, so a single viewpoint redraws the whole picture exactly. What they do fix is one number per strip, and that number survives the lens. Read the same drawing at focal lengths twenty to one apart and the lean runs from 36.4° to 86.0° while the habit stays at 1.000000000.
What survives being copied
A workshop copying a drawing from a drawing is a random walk — the spread across lineages grows as the square root of the generation, with a fitted exponent of 0.5001. A workshop copying the method is not, and its exponent is 0.012, which is no growth at all, and after forty generations two lineages started from different originals end up 5 × 10⁻¹⁵ apart. Copying the marks loses the picture; copying the recipe loses the original and keeps the recipe.
A vanishing line with a slope in it
Turn the plane about the view direction and no family of any surface's edges is level; each vanishing line acquires a slope, and a group must agree about two numbers rather than one. The count does not change character — a hand of four pixels costs the test 3.00 px at no slope and 3.27 at thirty-eight degrees of it. What the slope does expose is the redraw: holding each far edge at its drawn height charges 0.95 px to a picture one camera really took.
The floor a better camera cannot reach
Sweep the marking error from four pixels down to a hundredth and the measurement's error falls thirty-fold and then stops — at 6.0 per cent, which is exactly the six per cent the reference's assumed shape was wrong by. With the closure exact the same sweep keeps falling to 0.06 per cent. The crossing is at half a pixel, and it can be computed before the photograph is taken, which makes it a decision about equipment rather than a discovery about it.
The workshop that throws drawings away
Adding a rule that discards a drawing which looks wrong turns the mark copyist's spread from a growing random walk into a stationary process — the fitted exponent falls from 0.542 to 0.022, indistinguishable from the method copyist's 0.033 — and the two mechanisms then separate only by where they settle, 0.7002 against 0.8000, seven and a half spreads apart.
A tiring hand draws a different habit
A painter whose bays creep two per cent wider down a strip has drawn, to within a fiftieth of a pixel, what a steady painter with a different habit would draw — and on a weakly splayed floor the difference is the whole distance from a hand's even rows to a camera's log placement. One strip cannot tell fatigue from habit. A whole picture can, because a creep counterfeits a habit in proportion to the number of bays over the logarithm of the splay, and that is different on every strip.
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 tiring panel keeps its order, not its direction
Let a painter's creep grow from one strip to the next as the panel is worked, and the strips' habits carry the order they were drawn in — but only as a line, never as a direction: tiring from the floor to the book and steadying from the book to the floor put the same drift on every strip. Four strips find the order a quarter of the time against a twelfth by chance; six find it nineteen times in twenty. And the order costs the reading its refusal: once it is free, two steady hands fit one tiring hand nearly as well as a tiring hand does.
A 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.
A strip keeps its ratio, not the end it began
A painter dividing a strip into bays tires as they go, and each bay comes out a little larger than the last. Divide the strip from its far edge instead of its near one and the tiring runs the other way — but the rows record none of it: a strip divided from the far edge by a tiring hand is, to the last digits, a strip divided from the near edge by a steadying one. A whole picture recovers every strip's direction anyway, because one hand shared one rate of tiring across strips of different splay.
An 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.
A strip's scatter points to the end drawn last
The size of a tiring painter's bays cannot say which edge of a strip they started from, because a creep read backwards is a creep. The scatter of the bays can: a hand that worsens as it goes leaves its largest slips on the rows it drew last. One six-bay strip names its edge three times in four when the slip triples, a four-bay strip carries no evidence at all, and a workshop that divided every strip from one edge is named nineteen times in twenty from six strips — with no creep in the picture.
A wrong model shows at its own level
Give a single-view measurement the wrong length and the error stays in the length. Give it the wrong model — a slope where it assumed level ground, a horizon five pixels out, a focal length off the barrel — and the error lands on the level that model belongs to and on every level above it, never below. That is a diagnostic. Two equal things at different depths catch a wrong plane; two equal things at right angles catch a wrong shape; marks read to 0.4 px see a slope of 0.38°. A wrong length, alone, nothing in one picture can see.
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°.
The residual has a shape
A flat-floor map mispredicts a shadow by millimetres on any floor that is not flat, and the number everybody quotes is the worst one. Tune a dish, a ridge and a step until all three mispredict by exactly 25.0 millimetres and the scalar can no longer tell them apart — by construction. The signed residual around the ring still can. The second harmonic of it reads 0.09%, 2.79% and 19.37%, a factor of two hundred across three floors the headline number calls identical.
A stepped hand passes for a tiring one on a short strip
A painter who steps dividers from each row to the next leaves an error that accumulates — a walk from the edge begun at, with no fatigue in it. The fit of habit and creep takes about four-fifths of that walk before any reading of the scatter starts, because a walk is slow and so are the two numbers. What is left names the starting edge nearly as well as fatigue does, and on a six-bay strip it cannot say which of the two mechanisms left it: the verdict needs twenty-four bays on one strip, or sixteen six-bay strips by one hand.
A plumb wall tells a slope from a horizon, faintly
A sloping ground and a misplaced horizon fail the ground's depth test alike. A wall standing plumb beside the ground separates them in kind: a slope leaves the wall's test at 1e-16, and a horizon moves it. But it moves it a tenth as far as it moves the ground's — 0.20 per cent for a horizon five pixels out, against 2.11 — so a pair of windows a metre wide cannot see the difference, and a whole facade six metres wide can tell a one-degree slope from its matching horizon 95 times in a hundred. A wall running straight away from the camera sees nothing at all.
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.
The 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.
A stepped hand's walk is in the creeps it was given
Fit each strip of a picture alone and a stepping hand's walk vanishes into a creep of that strip's own. Fit the picture once, with one habit and one creep for the hand, and the walks come back as a disagreement between strips that their small residuals cannot explain. Read against a line set once, eight six-bay strips name the hand ninety-five times in a hundred; the residuals of the same strips name it eighty-six, and needed sixteen strips for what the creeps do in eight.
A facade's corners see the horizon its widths could not
Read by its level widths, a six-metre facade saw a misplaced horizon only when it was 11.5 pixels out. Read by its two corners — the same wall's plumb edges — it sees 4.2, as well as the ground does, and four posts beside it bring that to 2.4. The corners' vanishing point is nearly six thousand pixels below the picture, but what the lines measure is its reciprocal, and in pixels of horizon their precision does not change from a camera pitched two degrees down to one pitched eight.
The 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.
One more habit takes two hands, and only part of a walk
A picture whose strips disagree was not painted by one steady hand, but the disagreement does not say whether a stepping hand or a second painter put it there. Refit the picture with a second habit and the answer is in how much of the disagreement that one number takes: nine-tenths and more for two hands, a share that shrinks with every strip added for a walk. Read against a line set on separate paintings, eight strips tell the two apart eighty-nine times in a hundred — and the split the second habit chooses is a map of who painted what, wrong mostly where an unknown starting edge lets one habit counterfeit the other.
A leaning post is worth its lean, not its length
A facade's two plumb corners read a misplaced horizon at 4.2 pixels; a street's lamp posts and poles are more numerous and never quite plumb. Each lean is random from post to post, so it averages away with the count and puts no floor under the reading — but it is large beside a long line's reading error, so it sets the price of every post: four steel columns leaning a twentieth of a degree match the two corners, and it takes sixty-four timber poles at a quarter. Counted as the corners' equals, eight such poles make the corners' horizon 2.4 times worse; weighted for their lean, they leave it where it was.
A followed post's own lean is read as the bend
A rolling camera on a bend reads the bend from how its roadside posts lean, and a post that leans by itself adds its own tilt to every frame, unchanged. That tilt is indistinguishable from the bend's curvature: a tenth of a degree on a post twelve metres out turns a 50-metre bend into 58. No number of frames removes it, and fitting each post's own lean along with the bend loses the bend — a sideways camera's reading widens from ±7 metres to ±29. A followed post has to be plumb to six hundredths of a degree, finer than a well-set steel column; timber posts at a quarter of a degree help only in their dozens.
Groups in a row are one painter twice; groups in turn say little
A second habit splits a divergent picture into two groups, and nothing in the split says whether they are two painters or one painter before and after a change. The order of painting can, because a hand that changed once leaves its groups in a row and two hands taking turns leave them alternating. On eight strips read to a tenth of a pixel, 93 panels in 100 by one hand who changed habit are read with their groups in a row and none by two hands in turn. But the change and the fatigue the order is read from are both drifts in time, so a change at one moment costs the rate of tiring three-quarters of its precision again — and a panel read as alternating has acquitted nobody.
A photograph grades its posts across the sight, and not along it
Once two plumb corners have fixed a picture's vertical vanishing point, every other post in it can be read for its own lean: the angle between its line and the plumb line through its foot. Posts 2.5 metres tall at 7.5 metres are read to about a twelfth of a degree, enough to tell a quarter-degree lean from a twentieth ninety-one times in a hundred. But the reading sees only the lean across the line of sight. A post leaning towards the camera is caught five times in a hundred near the centre of the frame and a third of the time at its edges — and the posts read against one another lose the corners' share of the error, though not the street's own shared lean.
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.
A post read both ways sets its own lean aside
A rolling camera reads a bend from how its roadside posts lean, and a post's own tilt is read as bend: a quarter of a degree leaves a 50-metre bend read to ±12 metres. Followed again by a car coming the other way, the same post shows the same tilt and the opposite shutter lean. Half the sum of the two passes is the tilt and half the difference is the bend, and a timber pole read both ways gives the bend to ±1.7 metres — better than a plumb steel column read one way, at ±2.3.
Two painters on their own clocks leave no record of their turns
If each of two painters tires with their own work rather than with the day, the habits a panel's strips report record each painter's own sequence and nothing of how the two sequences were interleaved: painted in turn or one after the other, the panel is the same to the last digit. What does carry the difference is the shape of the fatigue — one trend through the panel, or two that start fresh — and that shape names one hand or two painters 59 times in 60 at a tenth of a pixel. But only once the comparison is calibrated: taken at face value, the model that searches seventeen times as many orders wins by searching, and calls two painters one hand four times in five at a quarter of a pixel.
A second photograph reads the lean the first cannot, if it is taken round the corner
One photograph with two plumb corners grades a street's posts for their lean across its line of sight and is blind to a lean towards or away from it. A second photograph, read the same way, sees each post's lean across its own sight — so the pair reads both components wherever their two sight lines through a post cross at a wide angle. At right angles they read the lean along the first sight to a tenth of a degree, and catch 65 in 100 of the posts the first picture missed, against 26. The picture the question proposed, from across the street, is nearly the worst place to stand: its sight lines run back along the first, it sees the same component reversed, and it catches 29.
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 body that bounces leaves a second swelling in the rows, and one pass reads both
A vehicle's body pitches and bounces on its springs at two frequencies, and a scroll camera's rows answer both: the pitch by turning the camera, the bounce by lifting it through the height climbed between the slits' two moments. One oscillator fitted to a body pitching at 1.4 Hz and bouncing at 1.0 lands between the two, at 1.22 Hz, with the camera over the body's centre, and on the pitch, at 1.41, with it 1.5 metres ahead — too lightly damped either way. Two modes fitted to the same single 800-metre pass read both, the pitch to ±0.02 Hz and the bounce to ±0.03, say plainly whether a bounce is there at all, and expose a bias the earlier readings carried even on a body that does not bounce: a camera ahead of the centre reads a 1.4 Hz pitch as 1.5 unless the fit knows its lever arm.
Frames read up and down part a post's lean in one drive
A rolling camera reads a bend from how its roadside posts lean, and a post's own tilt is read as bend unless a second car drives the road the other way. A sensor that reads alternate frames from the bottom row up reverses the shutter's lean without reversing the car. Thirty-two such frames of one drive read the 50-metre bend from a timber pole to ±1.8 metres, against ±1.7 for two passes of sixteen — and a post swaying in the wind, which costs two passes three metres, costs one drive a quarter of one.
How many lamps a drawing has
The shadow field recovers a lamp by intersecting drawn lines. Two lamps make that a partition rather than an intersection — and two centres fit any bundle better than one, on a one-lamp drawing as readily as on a two-lamp one, so a count is a decision that needs a noise level before it exists. A criterion built on a penalty instead of a noise level returns four.
A third photograph closes the hole the second leaves, if it brings a building
A second photograph of a row of posts reads every post's lean except the one whose two sight lines run nearest parallel, and that post sits at the end of the row away from the second camera. A third photograph from the other side puts its own worst post at the other end, so between them no post is left: sixty degrees round either side, the worst post is read to 0.10° where either picture alone left it at 0.23 or 0.20. But the third picture does not buy what was hoped of it. Without buildings of their own, three pictures catch fewer of the hidden leaning posts than two at a right angle with one, and the spare equation each post gains cannot read the further pictures' verticals: a lean the whole row shares, or one that grows along it, looks exactly like a vertical misread.
The road under a car ahead gives its growth a distance, while the road stays flat
One camera that knows its own height reads the row where the car in front meets the road, and that row is the camera's height seen from the car: a baseline of 1.3 metres standing on the road, read in every picture. With the car's growth it ranges a car 20 metres ahead to 0.80 per cent a pixel, six times better than a stereo pair's disparity, and reads the car's width to 0.74. A camera height a few centimetres out scales the distance and leaves the time untouched. A road that starts rising by one degree between the camera and the car moves the distance by up to 24 per cent, and neither horizon puts it right. The lane's own edges bend where the road does and read the rise back.
The rows read where the camera rides on the body, and a camera behind the centre hides the pitch
A scroll camera's rows over a rough road say how far ahead of the body's centre the camera stands, as well as how the body pitches and bounces. One 800-metre pass reads a camera 1.5 metres ahead to ±0.15 metres, costs the pitch's frequency nothing to do it, and removes the bias a fit that puts the camera at the centre carries — a 1.4 hertz pitch read as 1.54 from 2.5 metres ahead, and as 1.80 from a metre behind a body that does not bounce. Behind the centre the camera's rise works against its turn, and a metre back on a body that does not bounce it very nearly cancels the pitch where it rings.
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.
A turned part's outline reads its swelling from every side, and a straight taper hides it
A turned part that swells by 15% of its radius two fifths of the way along, drawn to half a pixel, gives up the swelling's height to ±0.0024 of its radius and its crest's place to ±0.003 of its length from one side-view drawing — and to about the same from every other view, square to the picture or lying across it. A swelling of one hundredth of the radius, under a pixel on the page, is read as 0.0098 ± 0.0022. Read as a straight taper, the earlier essays' reading, the same drawings put the swelling into the taper and read a 1° part as 2.2°.
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.
A floor cannot fake a second lamp
Cast the same two lamps onto four floors at four curvatures and ask how well one centre explains the drawing. Every one of the thirteen answers is 138.3277 pixels — the same to fifteen digits, because a floor decides where along a ray the shadow's tip landed, and a line through a point and another point that has slid along it is the same line.
A floor with a referent
Recover a focal length from two vanishing points and measure more points along each line. Through a pinhole the error falls from 0.34 per cent to 0.05 and the instrument finds no floor at all. Through a lens of k₁ = −0.05 it falls, turns, and rises to 0.70 per cent — because the noise the extra points removed had been partly masking the lens's bend. The floor is 0.72 per cent of the focal length, and doubling the distortion coefficient doubles it to 1.44. It is not noise and not conditioning; it is the model, priced.
Named alongside it
The objects these essays reach for when they reach for this one.
least squaresResidualIdentifiabilityVanishing pointdegrees of freedomConditioningRefractive indexCamera calibrationFalsifiabilityHorizonRadial distortionsingle-view metrology