Rotation Averaging and Optimization on Manifolds

I will discuss averaging on manifolds, mainly the manifold SO3 of 3D rotations. A number of theoretical results will be given, in particular conditions for convexity of distance measures on SO3, and basins of convergence of averaging. Different averaging problems, such as single rotation and multiplerelative rotation averaging will be discussed along with their applications in different Computer Vision problems, such as structure from motion and hand-eye coordination. Recent work in L1 averaging on SO3 will be presented, based on the classical Weiszfeld algorithm (1937), which gives a solution to the so-called Fermat or Fermat-Weber problem, concerning L1-averaging in R^n. Using this algorithm, we can compute the orientation of all 595 cameras in the Notredame data set with accuracy of 1 degree, in about 3 minutes. Extension of the Weiszfeld algorithm to averaging on the essential manifold will be discussed, including a new (I think more natural) metric on this manifold.Key is the computation of the geodesics on this manifold, and hence the exponential and logarithm maps