Invisible Center

Focal length is usually given in millimetres: a 50 mm lens. But in a digital image the natural ruler is the pixel, and one converts through the pixel size: f (pixels) = f (mm) / pixel pitch (mm). A phone module, 5.6 mm of focal length and 1.4 µm pixels, gives f = 4000 pixels. That is the unit for everything that follows: it says how many pixels from C the image plane sits, and it lets an error in c_x, itself in pixels, be compared directly to f.

A shift of c_x moves every pixel by the same step. A small rotation of the camera about its optical centre does almost the same, and, against intuition, with no regard for depth: it moves pixels according to their position in the field: at the centre, by the same step as a c_x shift; towards the edges, more. Near the axis the two are therefore almost indistinguishable; it is the edges of a wide field that separate them. Relief separates something else: c_x from a lateral translation, whose effect decreases with distance; and it defuses the degeneracies that lie in wait for planar scenes. A flat target is in fact not bad in itself, it is risky: what makes for poor conditioning is poor geometry or degenerate motions, orientations too alike, a field badly covered; relief is simply the safest safeguard, hence photogrammetrists' taste for three-dimensional target fields. Wide field, points out to the edges, a world in relief: that is the specification of a good calibration. As for the next chapter's algorithm, it will walk rather than jump for a simpler reason, the nonlinearity the rotations bring; but if it walks slowly, it will be because of this kinship of c_x with them.

A true camera is hidden in the page, deliberately off-centre, and the six true aims are skewed by a few degrees. The observations are manufactured by projecting the building's points through this camera, with Gaussian measurement noise, independent from one observation to the next, of standard deviation 0.3 pixel on each coordinate. The estimation starts from a wrong camera and from aims assumed perfect, then minimises, by Levenberg-Marquardt, the reprojection error: for each window corner in each image, the discrepancy in pixels, on the plate, between the observed point and the point projected by the estimated camera. It is this criterion, and it alone, that the algorithm drives down, summarised in one number, the rms, the root mean square: the square root of the mean of the squares of all these discrepancies, in pixels. The rays converging in the loupe are its geometric image, not the quantity being minimised. Nine unknowns: f, c_x, c_y, and for each station a single aiming angle, the yaw, the other components of the orientation being held exact. This count of nine belongs to our setup, where the building plays the role of a known target and the station positions are held exact; in the general case, each image brings six pose unknowns, each world point three more, and bundle adjustment estimates them all together. The simplification changes the count, not the nature of the problem. A detail that matters: with the exterior orientation perfectly known, the problem would be linear and would be solved in a single jump. But this textbook case is hardly met outside the laboratory: there are calibration benches where the stations are measured by theodolite, banks of collimators whose directions are known by construction; even there, known means measured, with its own uncertainty. Everywhere else, interior orientation is observable only through images, and an image always entangles the interior and the exterior. It is this joint estimation that makes calibration genuinely iterative, and that creates its classic difficulty: a small shift of c_x looks very much like a small rotation. What separates them is not depth, a rotation about the optical centre moves pixels with no regard for it, but the edges of the field: rotation moves them more there, c_x everywhere by the same step; a wide field is needed, with points out to the edges. If it walks instead of jumping, it is first, quite simply, because the rotations make the problem nonlinear and the Jacobian changes at every step; if it walks slowly along c_x, it is this kinship with the rotations, a narrow valley that only the edges of the field straighten. Nor do the nine unknowns share the same status: the six aims are auxiliaries, estimated and then left behind with the stations; the three interior parameters are the durable product of the computation, the camera's identity card, reusable tomorrow, elsewhere, on other poses. One last word: to calibrate, in the metrologist's sense, is to learn what an instrument is, without altering it; adjusting it would be another matter. That is the sense we keep. A final subtlety, which the final table shows by itself: the rms of the residuals settles in the neighbourhood of the noise's standard deviation, a breath below it even, the adjustment having absorbed its nine degrees of freedom; there is nothing down there to go looking for. The parameters, however, can be estimated more finely than the noise of a single measurement, for each observation brings its grain of information; the uncertainty decreases roughly as the inverse square root of the number of observations, as long as the errors remain independent; then come the floors that nothing averages away, the geometry and its correlations, that of c_x with the rotations first among them, and the flaws of the model itself. Noise averages out, bias repeats. And so the last word keeps its exact meaning: we never know better than what we have measured, not each measurement taken alone, but the sum of information of all of them together.