Visual Plane
every picture states the point it is correct from
Theme

The thread: The round trip

Draw the picture from a known camera, forget the camera, recover it from the drawn edges alone, and compare. Agreement to fifteen digits is a statement about the geometry; anything less is a statement about the drawing.
horizon — the eye's own heightbase — 0 mhorizon crossing — 1.62 mtop — 3.40 m recoveredthe vertical vanishing point is 8586 px above this framerecovered 3.400 m · true 3.400 m1.8 cm per pixel of click error Measuring from one picture

A height, out of one photograph

Four points on a vertical, one cross-ratio, and the height of something nobody measured. The only metric input is the photographer's own eye height, because the horizon is at eye level and that is the one piece of perspective folklore that is exactly true.

 horizondepicted height ÷ side0.3278corner angles90.000° — forced by the methoddepicted side ratio0.7195lens this drawing implies60° acrossdrawn exactly as the method prescribesthe free step is where the far edges go Drawn confidently

The cube that is a box

The two-point cube every book teaches has a step it supplies no construction for. Place the two far edges symmetrically and the drawing depicts a square plan for free; place them eight points apart — a difference invisible on the page — and it depicts a box 1.4 times shallower than it is wide.

 epipoleepipoleleft pictureright pictureepipole from 44 correspondences vs the projected eye: 1.1e-9 px2.60 m between the eyes The second eye

The image of the other eye

Two photographs of one courtyard, and in each of them a point that is the other camera. It is computed from forty-four matched marks and nothing else, and it lands on the projection of the other eye to about a billionth of a pixel.

 view 1view 6the scene, in plan — recovered points and cameras over the true onesreprojection 0.332 px · track 2.0e-3264 observations, 168 parameters Many pictures at once

The track and the scene together

Six photographs go in and one hundred and sixty-eight numbers come out — every camera's position and orientation and every point's place in space, solved for at once. Nothing in the solve was ever told where a camera or a point was.

 the picturethe same plane, rectifieddoor — 0.5068 widths (true 0.5068)sill — 0.3026 widths (true 0.3026)diagonal — 1.1766 widths (true 1.1766)worst error 4e-16 relativethe probes were not used to build the map Measuring from one picture

Flattening a façade out of the photograph

Four corners of a rectangle whose proportions are known are enough to undo the projection of one plane. After that the plane can be measured with a ruler — lengths, angles, areas, all of it — in units of the rectangle's own width, and lengths the map was never given come back to fifteen digits.

 the water surfacethe pinholethe rays miss by 9.9 mmno single viewpoint — the rays miss by 9.9 mmdry control: 0e+0 m Through water and glass

A picture through water has no viewpoint

Continue the rays of a refracted picture into the water and fit them to a common point. They miss it by ten millimetres. The same fit with the water taken away misses by zero, which is what makes ten millimetres a measurement rather than a number.

 × 4.6e+6seven flatand the rest stiffsingular value ÷ the largest, log scale, smallest firstσ₈/σ₇ = 4.6e+6168 parameters · 528 residuals Many pictures at once

Seven numbers no picture can name

Shift a whole reconstruction by a metre and a half, turn it half a radian, scale it by 2.7, and every photograph of it stays where it was to a hundredth of a billionth of a pixel. Move one point by fifty millimetres and they move by two thirds of a pixel.

 near edge 155 pxfar edge 196 pxrecovered: ratio 1.000000, corners 1e-14° from squarecorrect from 12 cm, at 160 mm wideplane tilted -45° What each system gave up

An inverse perspective is a leaning plane

Ask a divergent construction what solid it depicts and it answers: a rectangle, four right angles, near edge equal to far. What the splay encodes is not the shape but the plane's tilt — and a real square on a plane leaning toward the camera really does photograph with its far edge wider.

 grey: the reflectiontwo routes, agreeing to 0e+0 px after one flip Light and mirrors

A mirror is a second camera

Reflect the scene and photograph it, or reflect the camera and photograph the scene. The two routes disagree by 315 px and agree to the last bit once one axis of the image is reversed — which is the whole of why a mirror is said to swap left and right, written down.

 fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples Surfaces that are not flat

Stereographic keeps every angle, and only stereographic

One surface in the family preserves shape exactly — every right angle stays a right angle and both its arms are magnified equally, to the last bit the arithmetic has. It also sends every circle in the world to a circle in the picture, which the site's existing conic fit can be pointed at and asked to confirm without being told what it is looking at.

 recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across What survives

Recovering the camera from the picture it drew

Draw a box from a known camera, forget the camera, and get it back from the twelve drawn edges alone. Agreement to one part in 10¹⁵ is a statement about the geometry, because the only thing that crossed between the two halves was a list of line segments.

 fitted k₁ = -0.280000true -0.280000, off by 5e-15 The real instrument

Fitting a lens from straightness alone

No calibration target, no known scene, no camera. Only the knowledge that some edges in the picture were straight — and the coefficient comes back to fifteen digits. Then it comes back with a companion, and the two are correlated at −0.997.

 no single viewpoint — the rays miss by 4.5 px, depth-dependentf recovered from it: 396.88 px Through water and glass

What survives a pane of glass

A slab of glass moves every point of a picture and moves no direction at all. So the camera recovered from a photograph taken through a display case is exactly the camera that took it — out of a picture in which nothing is where it was.

 0.5120.1110100how finely each point is read off the picture (px, log scale)worst epipolar error against the exact geometry (px, log scale)raw pixelscentred and scaled1.1× apart at 0.25 px, 29.9× at 4 pxspread 102 against 2.5e+5 The second eye

Eight points and the basis they are read in

The linear system that recovers a fundamental matrix is written in whatever coordinates the marks were read in, and pixel coordinates are a bad choice. Centring and scaling them first is worth nothing at a quarter-pixel reading and a factor of thirty at four.

 0.30.50.7125×10⁻⁴0.0010.0020.0050.010.02how finely each point is read (px, log scale)worst camera-centre error, as a fraction of the track's mean radius (log)chained pairsadjusted togetheradjustment is 4.0–9.2× better6 views · identical observations Many pictures at once

A chain and an adjustment

Composing pairwise poses along a sequence is supposed to drift. Measured over five links it wanders instead — one chain ends closer to the truth than its own worst link — and the real cost of chaining turns out to be something else entirely.

 the picturethe same plane, rectifiedacross — 0.6000 widths (true 0.6000)along — 0.7667 widths (true 0.7667)diagonal — 1.2023 widths (true 1.2023)worst error 1e-15 relativethe probes were not used to build the map Measuring from one picture

The plan hidden in the photograph

Rectifying the ground is the same operation as rectifying a wall, and it turns a photograph into a site plan. The horizon is not an input to it and comes out as a consequence — the plan's points at infinity land on it, at first order, which is the check that the plan is a plan and not a plausible warp.

 points in front of both camerasR₁, t0 of 44180° from the truthR₁, t reversed0 of 44180° from the truthR₂, t44 of 44chosenR₂, t reversed0 of 44180° from the truththe chosen pose is the true one to 0.0e+0°cheirality, not algebra, picks it The second eye

Four cameras fit, and one of them can see

The essential matrix does not determine a camera pair. It determines four, all of which reproject every correspondence exactly, and the thing that picks one is not more algebra — it is the assumption that the photographer could see what was photographed.

 -10-50012345iterationreprojection error (px, log scale)exact marksread to 1 px4.67 px → 0.3324 px in 5 iterationsexact marks reach 1.3e-11 px Many pictures at once

Where the adjustment stops

Given exact marks the reprojection error falls to a hundredth of a billionth of a pixel, which is arithmetic. Given the same marks read to a whole pixel it falls to a third of a pixel and stays there, and a solver that reached zero on those would be fitting the rounding.

 one pixel is an areacentrescornersprincipal point moves0.707 pxfocal length changes by4.5e-13 pxan edge-versus-centre viewport1.303 pxa half-pixel convention is a principal-point error; an off-by-one viewport is a focal-length error8 vertices, all shifted by the same 0.7071 pxspread across marks 0.0e+0 px What a machine computes

A pixel is not a point

Where the sample sits inside a pixel is a convention, and getting it wrong shifts every mark by half a pixel in each axis. What that costs can be measured by recovering the camera from the picture — the answer is a principal point exactly 0.707 px from the truth with the focal length untouched, and the other half-pixel mistake does precisely the reverse.

 05101550100150200distance from the camera (metres)error in the recovered height, per pixel of click error (%)6 m — 0.55%25 m — 2.27%60 m — 5.43%120 m — 10.84%one pixel, on a 690 px picturelinear in distance Measuring from one picture

How wrong a measurement from one picture can be

The formula divides by a difference of two nearly equal numbers when the object is tall, which looked like the instability and is not. Measured, a taller object is recovered better and a more distant one worse — half a per cent per pixel at six metres and eleven per cent at a hundred and twenty. The argument that was wrong is as much the finding as the curve that replaced it.

 an anchorcorrect from 19 cm, at 160 mm wideflat 1e-13 px · rolled 30.5 px What survives

A projection of a projection

Photograph a photograph and the composite map is a homography — four marks determine it and every other lands where they say, to 1e-13 px. Roll the print and the same four mispredict the rest by 30.5 px, which is why straightening a page of a thick book never quite works.

 pixels 2.00 : 1recovered focal length219.9 pxthe camera's actual one436.4 pxwhat the recovery returnsspread across three estimates 0.0e+0worst bundle residual 9.1e-13 pxboth are what a wrong picture would trippixel aspect 2.00 unmodelledfocal length 49.6% short, every diagnostic clean The rectangle behind the lens

The pixel that is not square

A camera with two focal lengths is a real thing — anamorphic cinema optics, non-square photosites, a stretched video format. Hand the round trip of camera recovery a picture from one and it returns a focal length 49.6% out, a principal point far from the truth, three independent estimates agreeing to 1e-16, and bundle residuals at the noise floor. Every alarm the site has stays silent.

 15° of yaw, 6° of pitch, 1.50 throw ratio81.3% of the panel reaches the corrected rectangleouter: the thrown quadrilateral · inner: what correction can keep18.7% of the panel discarded The second projection

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.

 what was drawnthe solid it depicts — reading 1 of 2xyzlooked at 45.00° off the normalcube edge 1.0000 of the drawn unitresidual 5e-16 The other systems

Any three lines you draw are a cube

Four earlier essays said that cavalier projection is not the projection of anything — because its axis scales sum to three where every orthographic projection sums to two. That is true of orthographic projection and false of projection. Three lines from a point, drawn by hand, are a picture of an actual cube seen from an actual direction, and the cube and the direction come out of the drawing in closed form.

All themes