Geometry and Reconstruction from Symmetry

Abstract

In Chapter 6 we have illustrated how prior assumptions on the scene can be exploited to simplify, or in some case enable, the reconstruction of camera pose and calibration. For instance, the presence of parallel lines and right angles in the scene allows one to upgrade the projective reconstruction to affine and even Euclidean. In this chapter, we generalize these concepts to the case where the scene contains objects that are symmetric. While we will make this notion precise shortly, the intuitive terms of “regular structures,” (deterministic) “patterns,” “tiles,” etc. can all be understood in terms of symmetries (Figure 10.1).

So their (the five platonic solids) combinations with themselves and with each other give rise to endless complexities, which anyone who is to give a likely account of reality must survey.

— Plato, The Timaeus, fourth century B.C.