Relative Pose Problems and their Branched Covers

Abstract main talk:   Methods for reconstructing 3D scenes from images often assume a pinhole camera model. I will recall one such method that estimates the relative orientation of two calibrated cameras from the input data of five pairs of points in the projective plane. In algebro-geometric terms, this method is inverting a branched cover of degree 20 over the space of input data. A classical construction known as the "twisted pair" implies that the Galois/monodromy group associated with this branched cover acts imprimitively on the 20 solutions. Based on numerical evidence (joint w/ Korotynskiy, Regan, Pajdla), we conjecture this group is even smaller than what mere imprimitivity would suggest. Shifting gears slightly, I will consider another reconstruction problem involving “radial cameras", which can be used model a wide variety of optical imaging devices (eg. fisheye lenses.) At first glance, we are faced with inverting a branched cover of degree 3584. Somewhat miraculously, this problem decomposes into many smaller problems, allowing us to devise a practical solution (joint work w/ Hrubý, Korotynskiy, Oeding, Pollefeys, Pajdla, Larsson.)