Series

Bundle — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. view 1view 6the scene, in plan — recovered points and cameras over the true onesreprojection 0.332 px · track 2.0e-3264 observations, 168 parameters

    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.

    part 1 · manyviews
  2. -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

    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.

    part 2 · manyviews
  3. each point joined to its place in the inside-out answer — nearer the camera is uptoward the cameratrue 2.4e-13 px · inside out 0.946 px3° field, 60° of arc

    A narrow view keeps a second answer, inside out

    Six pictures of a courtyard through a 3° field, every mark exact. Started from the scene turned inside out, the adjustment settles there — near points far, far points near — and misfits the marks by under a pixel. The misfit grows in proportion to the field and to the sweep of the cameras, and vanishes only where perspective does.

    part 3 · manyviews
  4. 34384246505458626660° arc, 3° field60° arc, 25° field20° arc, 3° field20° arc, 10° fieldhalfway: the flattened scenelight: back to the scene · dark: into the twin · column: per cent of the way to the twinexact marks · one adjustment per cell68 starts

    A start needs the sign of its depths, not their size

    Started part of the way from a courtyard toward its inside-out twin, a bundle adjustment returns to the truth from every start less than 42 per cent of the way — a start with a sixth of the true relief, the right way round — and falls into the twin from every start past 56. Between, neighbouring starts settle in different answers. The band sits in the same place at a 3° field, where the twin misfits by under a pixel, and at 25°, where it misfits by seven and a half.

    part 4 · manyviews
  5. 020406080100points: per cent of the way to the twin'scameras 0%cameras 10%cameras 20%cameras 30%cameras 40%cameras 50%cameras 60%cameras 70%cameras 80%cameras 90%cameras 100%solid: back to the scene · empty: into the twin · diagonal: the blend that moves both together60° arc, 3° field, exact marks121 adjustments

    The cameras decide where a narrow view settles

    Scatter a narrow-field bundle adjustment's starting depths and the answer it reaches stops following them: of eighty scattered starts whose depths had the right sign, twenty-six fell into the inside-out twin, and of a hundred with the wrong sign, forty-three came home. Split the start in two and the reason is plain. With the cameras where they are, the courtyard comes home from its own inside-out points; with the cameras on the twin's side, it falls in from the true ones.

    part 5 · manyviews

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